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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon th e quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 INVESTIGATION OF DURATION OF EARTHQUAKE RELATED STRONG GROUND M OTION Volume I by Novikova Elena Igorevna A D issertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree D O CTO R OF PHILOSOPHY (Civil Engineering) May 1995 Copyright Elena I. Novikova UMI Number: 9617129 UMI Microform 9617129 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 UNIVERSITY OF SOUTHERN CALIFORNIA T H E GRA DUATE SCHOO L UNIVERSITY PARK LOS ANGELES, CA LIFO RN IA 90007 This dissertation, w ritten by under the direction of h.Q.%.... D issertation Com m ittee, and approved b y all its members, has been presented to and accepted b y The G raduate School, in partial fulfillm ent of re quirem ents fo r the degree of M o v i . KOyA E .£ & t1 * Jr.. D O C T O R OF PH IL O SO PH Y D ean o f G raduate S tu d ie s D a te y.K rh/..^s:) .-x 0 DISSERTATION COMMITTEE ........ ............ * A . k ii To my father, who once gave me for a present one of his astrophysical books with a handwritten note: “To my daughter, wishing her an incredible joy of always loving to work hard” ACKNOW LEDGM ENTS I am indebted to my advisor, Prof. M.D. Trifunac, for his guidance and help which I enjoyed during my work on this D issertation. It was a pure luck to get an advisor w ith so broad scientific knowledge and so deep understanding of civil and earthquake engineering, who, on top of everything, is such an interesting person. I am very grateful to the other members of my Guidance and Defence Com m ittees, especially to Prof. V.W. Lee, whose knowledge and cheerful support helped me a lot. I will always be grateful to my first scientific advisors, D r. T.G. R autian and Dr. V.I. K halturin. In a rem ote village in Pam ir m ountains, on a seismic station, they taught me seismology, and still continue to support me. I can not forget my Alma M ater—Moscow Institute of Physics and Technol ogy, which provided me with solid education in physics and m athem atics. Later on this helped me to learn seismology and civil engineering. I do not have enough words to describe my gratitude to my family. To me, my farther was always an example of a hard working scientist, and it was him who gave me the love and appreciation for the scientists and the scientific research. My loving m other supported me during the years of hard work on the D issertation like only m others can do . She also helped me to typeset the text of my D issertation to fit the USC form at requirem ents. The support and understanding of my husband maid it possible for me to devote so much tim e to th e work on the D issertation. He was both m other and father for our daughter iv Irina. I hope, when the times comes, to be able to help my daughter as mjr m other helped me. V TABLE OF CONTENTS Page No. D edication...................................................................................................................... ii A cknow ledgm ents........................................................................................................ iii List of figures.................................................................................................................. viii List of ta b le s ................................................................................................................. xxiv A b s tra c t......................................................................................................................... xxvi 1. In tro d u c tio n .................................................................................................................... 1 1.1 The duration of strong ground motion in earthquake resistant design and in seismological stu d ie s............................................................... 1 1.2 T he definition of duration: a historical review .......................................... 5 1.3 O rganization of this s tu d y ............................................................................... 12 2 . Strong m otion d ata set and calculation of d u ratio n .......................................... 16 2.1 General considerations................................................................................... 16 2.2 Description of the d ata s e t ........................................................................... 17 2.3 C alculation of frequency dependent d u ratio n .......................................... 29 2.4 Q uality of th e duration d a ta ........................... 38 3. Some physical considerations on param eters involved in the regression an a ly sis............................................................................................................................ 47 3.1 M ag n itu d e............................................................................................................ 48 3.2 Epicentral distan ce............................................................................................ 61 3.3 Modified Mercalli in ten sity ............................................................................. 72 3.4 Vertical and horizontal dimensions of the sedim entary b asin................ 96 3.5 Geological and local soil site conditions.......................................................156 4. Scaling the duration of strong earthquake ground m otion in term s of the earthquake m agnitude and other p aram eters.......................................................160 4.1 G eneral considerations and definitions.........................................................160 4.2 Model dur = dur ( M ,M 2, A )........................................................................163 4.3 Model dur = d u r(M ,M 2, A ,h , R , h R ,R 2,h 2,<p)....................................180 4.4 M odels dur — du r(M ,M 2, A ,R , R 2 ,<p) and dur = d u r(M ,M 2, A ,h , h 2) .................................................................. 200 4.5 Models dur = dur[M ,M 2, A, s) and dur = d u r[M ,A ,s,s i,) ........... 207 5. Scaling the duration of strong ground motion in term s of Modified M ercalli Intensity and other p aram eters...............................................................224 5.1 Models dur = d u r { I i y i M , A ! and dur — dur[lMM)..........224 5.2 Models dur = dur(/M M , A ', /m m A ', h, R, AR, R 2, h2, < p) and dur = dur(/M M > h , R ,h R ,R 2,h 2,<p)................................................. 239 5.3 Models du r = dur(lMM, A ', Im m &',R, R 2,<p), dur = dur(lM M ,R ,R 2,<p), dur — d u r ( l M M , A ' , h , h 2) and dur = dur[lMM, h, h2) ............................................................................255 5.4 Models du r = dur (/mm> A ', 7m m A ', s), dur = dur(lMM,s )> dur = dur(/M M , A ', ZmmA'jSjSx,) and dur = dur(lMM,s,si,) ... 265 6 . Frequency dependent duration of strong ground m otion, duration of the ru p tu re at the source and duration and the num ber of the separate strong m otion pulses: their dependence on m agnitude.................................................. 287 6.1 D uration of strong ground motion as a function of m agnitude for different frequencies........................................................................................... 287 6.2 D uration of high frequency radiation from the source and the corresponding source dim ensions...................................................................299 6.3 D uration and the num ber of the separate pulses contributing to strong ground m otion as a function of m agnitude...................................305 6.4 T he num ber of the high frequency patches and the “asperity size” . 324 7. T he propagation effects: a closer look................................................ 331 7.1 Dependence of the dispersion on the propagation path ch aracteristics......................................................................................................331 7.2 Num ber of strong m otion tim e intervals and their duration as a function of epicentral distance........................................................................335 7.3 G eom etry of a sedim entary basin and separate pulses of strong ground m o tio n ..................................................................................................... 349 7.4 Soil and geological site conditions and separate pulses of strong ground m o tio n ..................................................................................................... 354 8 . Scattering and attenuation effects and the nature of strong ground m o tio n ...............................................................................................................................357 8.1 The distribution of the strong motion pulses along the record and their relative “strength” ................................................................................... 357 8.2 Scattered energy in the strong motion record—the “strong m otion coda” .......................................................................................................................390 9. C onclusions.........................................................................445 R eferences....................................................................................................................... 452 A ppendix A. Filters used in the com putation of duration and the lim itations they im pose..........................................................................465 A ppendix B. Accepted and rejected data: making the decision.........................471 A ppendix C. The solution of linear least square problem by the singular value decomposition and other related questions......................... 483 A ppendix D. Coupling of th e effects of the earthquake m agnitude, M , and the epicentral distance, A , on the duration of strong ground m o tio n ......................................................................................................... 487 A ppendix E. D uration of strong ground motion and the hypocentral d e p th ............................................................................................................494 LIST OF FIGURES Page No. 1.1 Records w ith different durations of strong m otion in high- and in low-frequency p arts of the sp ectru m .............................................................. 9 1.2 Exam ples of cases when the definition of strong m otion p a rt of accelerogram as one continuous tim e interval is not ap p ro p ria te 11 1.3 The flow-chart of the organization of this s tu d y ...................................... 14 2.1 D istribution of 3-component records with respect to the m agnitude of the earthquakes......................................................................... 19 2.2 D istribution of 3-component records w ith respect to epicentral d ista n c e .................................................................................................................. 20 2.3 D istribution of 3-component records with respect to the depth of the hypocenter.........................................................................................................21 2.4 D istribution of 3-component records w ith respect to the depth of sedim ents at the recording site, for different geological and local soil co n d itio n s.............................................................................................................. 23 2.5 D istribution of records w ith respect to geological and local soil site co n d itio n s.............................................................................................................. 24 2.6 D istribution of 3-component records w ith respect to the Modified Mercalli In ten sity ................................................................................................. 26 2.7 T he observed intensity d ata and the estim ate of the Modified M ercalli Intensity by Eq. (2 .1 )....................................................................... 28 2.8 A m plitude characteristics of the filters used in the band-pass filtering of the acceleration d a ta ........................................................................31 2.9a Original acceleration record and its 12 band-pass filtered channels .. 32 2.9b Original velocity record and its 12 band-pass filtered ch an n e ls 33 2.9c Original displacement record and its 12 band-pass filtered channels . 34 2.10 T he definition of d u ratio n ................................................................................... 36 2.11 Number com ponents available at each frequency band ........................ 40 2.12 Correlations between duration determ ined from acceleration, velocity and displacement d a ta ......................................................................... 43 2.13 Change of the position of the central frequency in a narrow band signal due to in teg ratio n ................................................................................... 45 3.1 M agnitude versus fault length and fault w id th ...........................................52 3.2a Channels # 1 and # 2 : observed duration is plotted versus m agnitude of earthquake................................................................................... 55 3.2b Channels # 3 and # 4 : observed duration is plotted versus m agnitude of earthquake................................................................................... 56 3.2c Channels # 5 and # 6 : observed duration is plotted versus m agnitude of earthquake................................................................................... 57 3.2d Channels # 7 and # 8: observed duration is plotted versus m agnitude of earthquake................................................................................... 58 3.2e Channels # 9 and # 10: observed duration is plotted versus m agnitude of earthquake................................................................................... 59 3.2f Channels # 1 1 and # 12: observed duration is plotted versus m agnitude of earthquake..................................................................................... 60 3.3 Phase velocities of Love waves are plotted versus frequency............... 62 3.4a Channels # 1 and # 2 : observed duration is plotted versus epicentral d istan ce.............................................................................................. 66 3.4b Channels # 3 and # 4 : observed duration is plotted versus epicentral d istan ce.............................................................................................. 67 3.4c Channels # 5 and # 6 : observed duration is plotted versus epicentral distan ce.............................................................................................. 68 3.4d Channels # 7 and # 8 : observed duration is plotted versus epicentral d istan ce.............................................................................................. 69 3.4e Channels # 9 and #10: observed duration is plotted versus epicentral distan ce.............................................................................................. 70 3.4f Channels # 1 1 and # 12: observed duration is plotted versus epicentral d istan ce.............................................................................................. 71 3.5a Channels # 1 and # 2 : observed duration is plotted versus Modified M ercalli In ten sity ................................................................................................ 75 3.5b Channels # 3 and # 4 : observed duration is plotted versus Modified M ercalli In ten sity ................................................................................................ 76 X 3.5c Channels # 5 and # 6 : observed duration is plotted versus Modified M ercalli Inten sity ................................................................................................ 77 3.5d Channels # 7 and $ 8: observed duration is plotted versus Modified M ercalli Inten sity................................................................................................ 78 3.5e Channels # 9 and # 10: observed duration is plotted versus Modified Mercalli Intensity............................................................................... 79 3.5f Channels # 1 1 and #12: observed duration is plotted versus Modified Mercalli Intensity................................................................................. 80 3.6a Channels # 1: observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity................................................84 3.6b Channels # 2 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity............................................... 85 3.6c Channels # 3 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity............................................... 86 3.6d Channels # 4 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity............................................... 87 3.6e Channels # 5 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity............................................... 88 3.6f Channels # 6 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity...............................................89 3.6g Channels # 7 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity............................................... 90 3.6h Channels # 8 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity...............................................91 3.61 Channels # 9 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity...............................................92 3.6j Channels # 10: observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity...............................................93 3.6k Channels # 1 1 : observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity...............................................94 3.61 Channels # 12: observed duration as a function of the hypocentral distance and the Modified Mercalli Intensity...............................................95 3.7 An idealized scheme for the definition of param eters R and < p 98 3.8a An example of the determ ination of the param eters for horizontal reflections, p and R. Livermore earthquake, January 26, 1980............103 3.8b An exam ple of the determ ination of the param eters for horizontal reflections, p and R. M am m oth aftershock, May 25, 1980....................104 3.8c An exam ple of th e determ ination of the param eters for horizontal reflections, p and R. M am m oth aftershock, May 27, 1980....................105 3.9 D istribution of 3-component records w ith respect to R for different geological and local soil conditions.................................................................107 3.10 D istribution of 3-component records w ith respect to p ........................ 108 3.11a Channels # 1 and # 2 : observed duration is plotted versus R 109 3.11b Channels # 3 and # 4 : observed duration is plotted versus R 110 3.11c Channels # 5 and # 6 : observed duration is plotted versus R I l l 3.l i d Channels # 7 and # 8 : observed duration is plotted versus R 112 3.l i e Channels # 9 and # 10: observed duration is plotted versus R 113 3.I l f Channels # 1 1 and # 1 2 : observed duration is plotted versus R 114 3.12a Channels # 1 and # 2 : observed duration is plotted versus p ............... 115 3.12b Channels # 3 and # 4 : observed duration is plotted versus p ................116 3.12c Channels # 5 and # 6 : observed duration is plotted versus p ............... 117 3.12d Channels # 7 and # 8 : observed duration is plotted versus p ............... 118 3.12e Channels # 9 and # 10: observed duration is plotted versus p 119 3.12f Channels # 11 and # 12: observed duration is plotted versus p 120 3.13a Channels # 1 and # 2 : observed duration is plotted versus h ................123 3.13b Channels # 3 and # 4 : observed duration is plotted versus h ................124 3.13c Channels # 5 and # 6 : observed duration is plotted versus h ................125 3.13d Channels # 7 and # 8 : observed duration is plotted versus h ................126 3.13e Channels # 9 and #10: observed duration is plotted versus h ..............127 3.13f Channels # 11 and # 12: observed duration is plotted versus h ........... 128 3.14 Reflection of a wave from the boundary of a sedim entary b a sin 130 3.15al Horizontal com ponent, channel # 1: observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 132 3.15a2 Vertical com ponent, channel # 1 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 133 3.15bl Horizontal com ponent, channel # 2 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 134 3.15b2 Vertical com ponent, channel # 2 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin............... 135 3.15cl Horizontal com ponent, channel # 3 : observed duration as a function of horizontal and vertical dimensions of a sedim entary basin............... 136 3.15c2 Vertical com ponent, channel # 3 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 137 3.15dl Horizontal com ponent, channel $ 4 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin............... 138 3.15d2 Vertical com ponent, channel # 4 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 139 3.15el Horizontal com ponent, channel # 5 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin............... 140 3.15e2 Vertical component, channel # 5 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 141 3.15fl Horizontal com ponent, channel jf 6 : observed duration as a function of horizontal and vertical dimensions of a sedim entary basin............... 142 3.15f2 Vertical component, channel # 6 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 143 3.15gl Horizontal com ponent, channel # 7 : observed duration as a function of horizontal and vertical dimensions of a sedim entary basin............... 144 3.15g2 Vertical com ponent, channel # 7 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 145 3.15hl Horizontal com ponent, channel # 8 : observed duration as a function of horizontal and vertical dimensions of a sedim entary basin............... 146 3.15h2 Vertical com ponent, channel # 8 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 147 3.1511 Horizontal com ponent, channel # 9 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ......148 3.1512 Vertical component, channel # 9 : observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ......149 3.15jl Horizontal com ponent, channel #10: observed duration as a function of horizontal and vertical dimensions of a sedim entary b a sin .........................................................................................................................150 3.15j2 Vertical component, channel #10: observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ......151 3.15kl Horizontal com ponent, channel #11: observed duration as a function of horizontal and vertical dimensions of a sedim entary b a sin .........................................................................................................................152 3.15k2 Vertical component, channel #11: observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 153 3.1511 Horizontal com ponent, channel #12: observed duration as a function of horizontal and vertical dimensions of a sedim entary b a sin .........................................................................................................................154 3.1512 Vertical component, channel #12: observed duration as a function of horizontal and vertical dimensions of a sedim entary b asin ............... 155 4.1 The coefficients a ,( /) in Eq. (4.9)................................................................. 167 4.2a Channels # 4 and # 5 : isolines of the duration as a function of m agnitude and epicentral distance (from Eq. (4.9)).................................173 4.2b Channels # 6 and # 7 : isolines of the duration as a function of m agnitude and epicentral distance (from Eq. (4.9)).................................174 4.2c Channels # 8 and # 9 : isolines of the duration as a function of m agnitude and epicentral distance (from Eq. (4.9)).................................175 4.2d Channels # 1 0 and #11: isolines of the duration as a function of m agnitude and epicentral distance (from Eq. (4.9)).................................176 4.2e Channel # 12: isolines of the duration as a function of m agnitude and epicentral distance (from Eq. (4.9)).................................177 4.3 The residuals of Eq. (4.9) and the model prediction are plotted versus channel num ber....................................................................................... 179 4.4a T he coefficients a ,( /) , i = 1 -j- 4, in Eq. (4 .1 2 )..........................................184 xiv 4.4b The coefficients «,■(/), t = 5 -f-10, in Eq. (4.12), horizontal co m p o n en t..............................................................................................................185 4.4c T he coefficients a ,( /) , * = 5 -r 10, in Eq. (4.12) vertical com ponent. 186 4.5a Channel # 5 : isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (4 .1 2 )) .188 4.5b Channel # 6 : isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (4 .1 2 )) .189 4.5c Channel # 7 : isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (4 .1 2 )) .190 4.5d C hannel # 8: isolines of the additional (relative to th e basem ent rock sites) duration as a function of R and h (from Eq. (4 .1 2 )) .191 4.5e Channel # 9 : isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (4 .1 2 )) .192 4.5f Channel # 10: isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (4 .1 2 )) .193 4.6 The residuals of Eq. (4.12) and the model prediction are plotted versus channel num ber....................................................................................... 199 4.7a The coefficients a t ( /) , t = 1 4, in Eq. (4.14)......................................... 203 4.7b The coefficients ae(f), o.s(f) and a io (/) in Eq. (4 .1 4 ).......................... 204 4.8 A dditional (relative to the basement rock sites) duration as a function of R (from Eq. (4.14))...................................................................... 206 4.9a The coefficients a t ( /) , t = 1 -r 4, in Eq. (4.15)........................................209 4.9b The coefficients a^{f) and ag(f ) in Eq. (4.15)..........................................210 4.10 A dditional (relative to the basement rock sites) duration as a function of n (from Eq. (4.15))........................................................................211 4.11a T he coefficients a t ( /) , * = 1 — 4, in Eq. (4.16)..........................................214 4.11b T he coefficients a i3( /) and a\4(f) in Eq. (4.16)......................................215 4.12a The coefficients a i ( / ) , 02(f) and 0.4(f) in Eq. (4 .1 7 )............................220 4.12b T he coefficients a i5(f), a u ( / ) and 012(f) in Eq. (4 .1 7 )...................... 221 5.1 T he coefficients a{(f) in Eq. (5.1).................................................................228 XV 5.2a Channels # 2 and # 3 : isolines of the duration as a function of intensity and hypocentral distance (from Eq. (5 .1 )).................................229 5.2b Channels # 4 and # 5 : isolines of the duration as a function of intensity and hypocentral distance (from Eq. (5 .1 )).................................230 5.2c Channels # 6 and #7: isolines of the duration as a function of intensity and hypocentral distance (from Eq. (5 .1 )).................................231 5.2d Channels # 8 and # 9 : isolines of the duration as a function of intensity and hypocentral distance (from Eq. (5 .1 ))................................232 5.2e Channels #10 and #11: isolines of the duration as a function of intensity and hypocentral distance (from Eq. (5 .1 ))................................233 5.2f Channel #12: isolines of the duration as a function of intensity and hypocentral distance (from Eq. (5 .1 ))................................234 5.3 The coefficients a ,( /) in Eq. (5.2).................................................................238 5.4a The coefficients ai(f), ox9( /) , a4(f) and a 2o (/) in Eq. (5.3)............... 242 5.4b T he coefficients a ,( /) , i = 5 -r 10, in Eq. (5.3), horizontal co m p o n en t............................................................................................................. 243 5.4c T he coefficients a ,( /) , i = 5 -f-10, in Eq. (5.3), vertical com ponent . 244 5.5a Channel #5: isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (5 .3 ))............245 5.5b Channel #6: isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (5 .3 ))............246 5.5c Channel #7: isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (5 .3 ))............247 5.5d Channel #8: isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (5 .3 ))............248 5.5e Channel #9: isolines of the additional (relative to the basem ent rock sites) duration as a function of R and h (from Eq. (5 .3 ))............249 5.6a T he coefficients o i( /) and a\g(f) in Eq. (5 .5 )...........................................252 5.6b T he coefficients a ,( /) , t = 5 4- 10, in Eq. (5.5), horizontal com p o n en t............................................................................................................. 253 5.6c T he coefficients a ,( /) , i = 5 -j- 10, in Eq. (5.5), vertical com ponent . 254 xv i 5.7a T he coefficients a i ( / ) , aig(f), 04(f) and 020(f) in Eq. (5.6)...............258 5.7b The coefficients a6(f), a8( f ) and aio(f) in Eq. (5 .6 )........................... 259 5.8 Additional (relative to the basem ent rock sites) duration as a function of R (from Eq. (5 .6 )).........................................................................260 5.9a The coefficients a i( f) and 019(f) in Eq. (5 .7 ).........................................263 5.9b The coefficients 05(f), a8(f) and Oio(f) in Eq. (5 .7 )........................... 264 5.10a The coefficients u i( / ) , a ig (/), 04(f) and 020(f) in Eq. (5.8)...............267 5.10b The coefficients 05(f) and ag(f) in Eq. (5 .8 )...........................................268 5.11 Additional (relative to the basement rock sites) duration as a function of h (from Eq. (5.8))..........................................................................269 5.12a The coefficients « i( /) and aig(f) in Eq. (5 .9 ).........................................271 5.12b The coefficients 05(f) and ag(f) in Eq. (5 .9 )...........................................272 5.13a The coefficients ai(f), 019(f), 0.4(f) and 020(f) in Eq. (5.10).............275 5.13b The coefficients 013(f) and o i4(/) in Eq. (5.10).....................................276 5.14 The coefficients a ,( /) in Eq. (5.11)..............................................................279 5.15a The coefficients ai(f), aig(f), 04(f) and 020(f) in Eq. (5.12).............282 5.15b The coefficients 015(f), a n ( /) and 012(f) in Eq. (5 .1 2 )..................... 283 5.16 The coefficients Oi(f) in Eq. (5.13)..............................................................285 6.1 “A ctual” duration of the rupture process in the source and the “observed” duration of strong m otion as functions of m agnitude . . . 289 6.2 D uration of the source rupture at different frequencies..........................290 6.3 T he coefficient 02(f) in Eq. (6 .4 )...................................................................293 6.4 Frequency dependent coefficients a( f) and (3(f) of the Eq. (6 .5 )___297 6.5 An estim ate of the length of the source area, responsible for the release of high-frequency ra d ia tio n ................................................................ 304 6.6 Comparison of the dependence of the total duration and the duration of one pulse, on m a g n itu d e.............................................................308 6.7a Channels # 1 to # 3 : observed duration of separate pulses and the observed total duration of strong ground m otion are plotted versus m agnitude...............................................................................................................311 6.7b Channels # 4 to # 6 : observed duration of separate pulses and the observed total duration of strong ground m otion are plotted versus m agnitude...............................................................................................................312 6.7c Channels # 7 to # 9 : observed duration of separate pulses and the observed total duration of strong ground motion are plotted versus m agnitude...............................................................................................................313 6.7d Channels # 10 to # 12: observed duration of separate pulses and the observed total duration of strong ground motion are plotted versus m agnitude...............................................................................................................314 6 .8a Channels # 1 to # 3 : observed number of pulses of strong m otion is plotted versus m agnitude..................................................................................315 6 .8b Channels # 4 to # 6 : observed number of pulses of strong m otion is plotted versus m agnitude..................................................................................316 6 .8c Channels # 7 to # 9 : observed number of pulses of strong m otion is plotted versus m agnitude..................................................................................317 6 .8d Channels # 1 0 to #12: observed number of pulses of strong m otion is plotted versus m agnitude..............................................................................318 6.9 T he coefficients scaling duration of one strong m otion pulse and num ber of strong motion pulses with respect to m agnitude (Eq. (6 .1 8 ))........................................................................................................... 322 6.10 Estim ates of the asperity size and of the length of the area involved in slipping due to the rupture of one asperity, versus m ag n itu d e 328 7.1 The coefficient a ^ f ) in Eq. (4.9) for three cases: all available d ata included, “soft” propagation paths only and “hard” propagation paths o nly...............................................................................................................334 7.2 “G radual” separation of different modes of surface waves, as those travel through layered m edium ........................................................................336 7.3 A schematic representation of the range of the duration of one strong motion pulse as a function of epicentral distance for a narrow frequency band channel...................................................................... 338 7.4a Channels # 1 to # 3 : observed duration of separate pulses of strong m otion is plotted versus epicentral d istan ce.................................................339 7.4b Channels # 4 to # 6 : observed duration of separate pulses of strong m otion is plotted versus epicentral d istan ce............................................. 340 7.4c Channels # 7 to # 9 : observed duration of separate pulses of strong m otion is plotted versus epicentral d istan ce............................................. 341 7.4d Channels # 1 0 and # 12: observed duration of separate pulses of strong m otion is plotted versus epicentral d ista n c e................................. 342 7.5a Channels # 1 to # 3 : observed number of pulses is plotted versus epicentral d ista n c e . ......................................................................................... 343 7.5b Channels # 4 to # 6 : observed number of pulses is plotted versus epicentral distance............................................................................................... 344 7.5c Channels # 7 to # 9 : observed number of pulses is plotted versus epicentral distance............................................................................................... 345 7.5d Channels # 1 0 to # 1 2 : observed number of pulses is plotted versus epicentral distance............................................................................................... 346 7.6 T he coefficients scaling duration of one strong m otion pulse and num ber of strong m otion pulses w ith respect to epicentral distance (E q .(6.18))............................................................................................................. 348 7.7 The coefficients scaling duration of one strong m otion pulse and num ber of strong m otion pulses with respect to R and h (Eqs. (7.2 )-(7 .5 ))................................................................................................. 351 7.8 Two mechanisms for increasing the num ber of pulses, “dispersion” and “reflection” m echanism s............................................................................353 7.9 T he coefficients scaling duration of one strong m otion pulse and num ber of strong m otion pulses with respect to geological and local soil conditions (Eqs. (7.6 )-(7 .9 )).................................................................... 355 8 .1a The correlation between the time of the S-wave arrival and the distance to the source, with all data points included...............................358 8.1b The correlation between the time of the 5 -wave arrival and the distance to the source, for different geological and local soil site conditions............................................................................................................... 359 8 .1c The correlation between the time of the 5-wave arrival and the distance to the source, for different magnitudes and depths of sedim ents under the recording s ite ................................................................ 360 8.2a Channel # 1 : the duration of a strong motion pulse versus the tim e when it was observed..........................................................................................363 8 .2b Channel # 2 : the duration of a strong motion pulse versus the tim e when it was observed....................................................................................... 364 8.2c Channel # 3 : the duration of a strong motion pulse versus th e tim e when it was observed....................................................................................... 365 8 .2d Channel # 4 : the duration of a strong motion pulse versus the tim e w hen it was observed........................................................................................... 366 8.2e Channel # 5 : the duration of a strong motion pulse versus the tim e when it was observed........................................................................................... 367 8.2f Channel # 6 : the duration of a strong motion pulse versus the tim e when it was observed........................................................................................... 368 8.2g Channel # 7 : the duration of a strong motion pulse versus th e tim e when it was observed............................................................................................369 8.2h Channel # 8 : the duration of a strong motion pulse versus the tim e when it was observed........................................................................................... 370 8.21 Channel #=9: the duration of a strong motion pulse versus the tim e when it was observed............................................................................................371 8.2j Channel # 10: the duration of a strong m otion pulse versus the tim e when it was observed.........................................................................372 8.2k Channel # 11: the duration of a strong motion pulse versus the tim e when it was observed.........................................................................373 8.21 Channel #12: the duration of a strong motion pulse versus the tim e when it was observed.........................................................................374 8.3a Channel # 1 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 376 8.3b Channel # 2 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 377 8.3c Channel # 3 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 378 8.3d Channel # 4 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 379 8.3e Channel # 5 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 380 XX 8.3f Channel # 6 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 381 8.3g Channel # 7 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 382 8.3h Channel # 8: the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 383 8.31 Channel # 9 : the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 384 8.3j Channel #10: the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 385 8.3k Channel #11: the relative weakness of the strong m otion tim e interval versus the tim e when it was observed........................................... 386 8.31 Channel #12: the relative weakness of the strong m otion tim e interval versus the time when it was observed........................................... 387 8.4 Overall trends of the distribution of the “relative weakness” of a strong m otion pulse w ith respect to epicentral distance and earthquake m agnitude........................................................................................388 8.5 T he frequency dependent coefficients, scaling the relative weakness of a strong m otion pulse w ith respect to M and A (Eq. (8 .2) ) ........... 389 8 .6 a C onstruction of an envelope of the scattered energy (Borrego M ountain earthquake, April 8, 1968, channel # 3 ) ................................... 396 8 .6b C onstruction of an envelope of the scattered energy (Borrego M ountain earthquake, April 8 , 1968, channel # 5 ) ................................... 397 8 .6c C onstruction of an envelope of the scattered energy (Borrego M ountain earthquake, April 8, 1968, channel # 7 ) ................................... 398 8 .6 d C onstruction of an envelope of the scattered energy (Borrego M ountain earthquake, April 8 , 1968, channel # 9 ) ................................... 399 8.7a Channel # 1 : the normalized envelopes of scattered w aves...................402 8.7b Channel # 2 : the normalized envelopes of scattered w aves................... 403 8.7c Channel # 3 : the normalized envelopes of scattered w aves...................404 8.7d Channel # 4 : the normalized envelopes of scattered w aves...................405 8.7e Channel # 5 : the normalized envelopes of scattered w aves................... 406 8.7f Channel # 6 : the normalized envelopes of scattered .w aves..........407 8.7g Channel # 7 : the normalized envelopes of scattered w aves.......... 408 8.7h Channel # 8 : the normalized envelopes of scattered w aves...........409 8.71 Channel # 9 : the normalized envelopes of scattered .w aves......... 410 8.7j Channel # 1 0 : the normalized envelopes of scattered w aves........411 8.7k Channel # 1 1 : the normalized envelopes of scattered w aves.........412 8.71 Channel # 1 2 : the normalized envelopes of scattered w aves........413 8.8 T he averaged normalized envelopes of the incoherent scattered m o tio n ..................................................................................................................... 417 8.9a Channels # 4 to # 6 : the shape of the strong m otion coda envelope, as formed by the shifted averaged normalized envelopes........................420 8.9b Channels # 7 to # 9 : the shape of the strong m otion coda envelope, as formed lay the shifted averaged normalized envelopes........................421 8.9c Channels # 1 0 to #12: the shape of the strong m otion coda envelope, as form ed by the shifted averaged norm alized envelopes .. 422 8.10 O ur estim ates of Q - 1 , compared w ith other m easurem ents of Q ~ 1 factor in C alifornia..............................................................................................424 8.11a Channel # 4 : comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................427 8.11b Channel # 5 : comparison of the envelopes of the “strong m otion coda” for sm all and large volume of sedim entary d e p o sits....................428 8.11c Channel # 6 : comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................429 8 .l i d Channel # 7 : comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................430 8 .l i e Channel # 8 : comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................431 8 .I l f Channel # 9 : comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................432 8.11g Channel # 10: comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................433 8.11h Channel # 11: comparison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................434 8 .H i Channel # 1 2 : com parison of the envelopes of the “strong m otion coda” for small and large volume of sedim entary d e p o sits....................435 8.12 An apparent quality factor can be assigned to a record obtained a t a point located on the surface of a layer-on-top-a-halfspace............439 8.13 A pparent Q ~l as a function of the angle of accidence O q and frequency f ............................................................................................................ 443 9.1a The-flow chart for selecting the proper scaling model in term s of the earthquake m a g n itu d e................................................................................ 448 9.1b The-flow chart for selecting the proper scaling model in term s of the Modified Mercalli In te n sity .......................................................................449 A .l The band-pass Ormsby filter............................................................................ 467 B .la Original acceleration record and its 12 band-pass filtered channels; the right-m ost column gives the value of the “index of acceptance” . 472 B .lb Original velocity record and its 12 band-pass filtered channels; th e right-m ost column gives the value of th e “index of acceptance” . 473 B .lc Original displacement record and its 12 band-pass filtered channels; the right-m ost column gives the value of the “index of acceptance” . 474 B.2 D istribution of cases accepted for the regression analysis: “perfect” d ata points and “acceptable” d ata p o in ts..................................................479 B.3 D istribution of cases rejected from the analysis due to low signal to noise ra tio ............................................................................................................... 480 B.4 D istribution of cases w ith duration equal to or longer than the length of the re c o rd .............................................................................................481 D .l T he coupling of the influence of m agnitude and epicentral distance on the d u ratio n ....................................................................................489 D.2 D uration as function of m agnitude and distance (from Eq. (D .l) ) .. .490 D .3a D uration as function of epicentral distance for various ranges of m agnitude (from Eq. ( D .l ) ) .............................................................................491 D.3b D uration as function of m agnitude for various ranges of epicentral distances (from Eq. (D .l)).............................................................492 E .l T he coefficients a ,( /) in Eq. ( E . l ) ................................................................496 E.2 T he num ber of strong motion pulses as a function of hypocentral depth in Eq. (E.2) ................................................................................................499 E.3 T he m agnitude as a function of the hypocentral d e p th ......................... 500 E.4 T he coefficients fl2( /) , an(f) and a 18(/) in Eq. (E .3 )...........................501 xxiv LIST OF TABLES Page No. 3.1 C haracteristic param eters the of horizontal reflection determ ined from Fig. 3.6............................................................................................................... 106 4.1 Source, propagation p ath, site param eters and the assigned com ponent num bers..................................................................................................161 4.2 Results of the regression analysis of Eq. (4-9). 4.3 Results of the regression analysis of Eq. (4.12) 4.4 Results of the regression analysis of Eq. (4.14) 4.5 Results of the regression analysis of Eq. (4.15) 4.6 Results of the regression analysis of Eq. (4.16) 4.7 Results of the regression analysis of Eq. (4.17) 5.1 Results of the regression analysis of Eq. (5.1). 5.2 Results of the regression analysis of Eq. (5.2). 5.3 Results of the regression analysis of Eq. (5.3). 5.4 Results of the regression analysis of Eq. (5.5). 5.5 Results of the regression analysis of Eq. (5.6). 5.6 Results of the regression analysis of Eq. (5.7). 5.7 Results of the regression analysis of Eq. (5.8). 5.8 Results of the regression analysis of Eq. (5.9). 5.9 Results of the regression analysis of Eq. (5.10) 5.10 Results of the regression analysis of Eq. (5.11) 5.11 Results of the regression analysis of Eq. (5.12) 5.12 Results of the regression analysis of Eq. (5.13) 6.1 N um ber of d ata points available for the analys .166 .183 .202 .208 .213 .219 .227 .237 .241 .251 .257 ,262 ,266 ,270 .274 .278 .281 .284 band for different m agnitudes.............................................................................. 294 XXV 6.2 Results of the regression analysis of Eq. (6 .5 )...................................... 296 6.3 Results of the regression analysis of Eq. (6.17)....................................320 6.4 Results of the regression analysis of Eq. (6.18)....................................321 8.1 P aram eters chosen for calculations of average normalized envelopes . . . 415 A .l T he properties of filters used in two-step process of calculating frequency dependent d u ra tio n ..............................................................................468 E .l R esults of the regression analysis of Eq. ( E . l ) ............................................... 495 xxv i ABSTRA C T In this work, the physical bases and empirical equations for modelling the d uration of strong earthquake ground m otion in term s of earthquake m agnitude, distance to the source, Modified Mercalli Intensity and the geological and local soil site conditions are investigated. At 12 narrow frequency bands, the du ratio n of strong m otion in a function f(t), where f(t) is acceleration, velocity or displacem ent, is defined as the sum of tim e intervals during which the in tegral f* f 2(r)d r gains a 90% portion of its final value. We present a variety of th e regression models of so defined duration which differ from the models in the literature by considering new model param eters and by the larger and more com plete database used in the regression analysis. T he model appropriate for each particular case can be chosen among those presented. The choice depends on th e availability of information about the source, the propagation p ath , and the site conditions. Special attention is being paid to physical interpretation of the regression results. T he regression coefficients of the models presented are found to be in good agreem ent w ith the known mechanisms of the earthquake source rupture and w ith the physics of the wave propagation. A special chapter is devoted to w hat can be described as weaker portions of the strong m otion records, and possibility of estim ating quality factor Q from those portions is being discussed. 1 1. INTRODUCTION 1.1. The Duration of Strong Ground M otion in Earthquake Resistant Design and in Seismological Studies The duration of strong earthquake ground motion is one of the im portant functions th a t characterizes the ground shaking caused by earthquakes. It is closely related to the am ount of energy “forced” into a stru ctu re and to the rate of this energy input. The significance of the duration of excitation becomes particularly im por ta n t in the case of nonlinear yielding structures, as the num ber of cycles dur ing vibration is the direct function of the duration. The influence of duration of earthquake-like excitation on the response of simple yielding structures was studied by Husid (1967). Several later studies were carried out which considered th e influence of duration of strong ground motion of a particular earthquake on some engineering structures. Villarverde (1989) investigated the prevalent fac tors th a t contributed to the collapse of the upper floors of a large num ber of buildings during the M ichoacan Earthquake of Septem ber 19, 1985. He found th a t this collapse m ight have occurred because the com bination of large accel eration and large duration of shaking induced the form ation of plastic hinges at the columns of some of the upper stories and because these plastic hinges, in tu rn , induced the lateral instability of those stories and of the stories above them . The work of Anderson and Bertero (1991) shows th a t a large accelera tion level may not be necessary to drive a structure into a nonlinear response. They found th a t the response of an instrum ented six-story steel building to the 2 W hittier Narrows Earthquake excitation was, most probably, completely linear, disregarding relatively high peak acceleration—about 0.22 g at the base. The linear n atu re of the response may probably be explained by very short duration of strong ground m otion—only about 5 sec. These authors also estim ated (by com puter modelling) the response of the same structure to several other strong m otion records, among them the Bucharest (1977) record w ith peak acceleration of 0 .2 g and duration of about 10 seconds and the Mexico (1985) record w ith peak acceleration of 0.17 g and with strong motion duration about 30 seconds. B oth of these excitations produced nonlinear response of th e studied model of the six-story building. In the case of the Bucharest record, it might be attrib u ted to large am ount of energy, "forced” to the structure and partially to the longer duration. The M exican record probably causes strong nonlinear response prim arily due to the long strong ground motion duration. Jeong and Iwan (1988) presented a theoretical analysis of the effects of du ration on the dam age of structures subjected to earthquakes. They found the expected dam age to depend on both the ductility of the structure and duration of the excitation. The influence of the strong m otion duration on the seismic per form ance of structures was also emphasized by Uang and Bertero (1988), Park et al. (1985) and by Zembaty (1987). U dwadia and Trifunac (1974), Amini and Trifunac (1984, 1985) and G up ta and Trifunac (1987, 1988a,b, 1990a,b,c, 1991, 1992) studied the probability rep resentation of the peaks of structural response to earthquake excitation. They developed the m ethod which allows one to predict the probability of exceedance of any given displacement level any given number of times at any floor of a 3 m ultistory building exited by an earthquake. This probability, am ong other param eters, depends directly on the duration of strong ground m otion. This probabilistic response spectrum represents the “state-of-the-art” in earthquake engineering research. It was shown th a t the duration of strong ground m otion influences also the standard definition of the response spectrum (Zembaty, 1988). It is therefore apparent th a t the duration of strong ground shaking m ust be con sidered am ong the param eters for the earthquake resistant design (Kennedy and Short, 1985; Peng et al., 1989). D uration is used as a param eter in models describing the space-time varia tions of earthquake ground motion (Vanmarcke, 1985), and in generation of the artificial accelerograms (Trifunac, 1971b; Wong and Trifunac, 1978, 1979; Lee and Trifunac, 1985c, 1987b). The knowledge of the duration of strong ground motion is also necessary for prediction of the seismic performance of soils at the sites where the liquefaction is possible (B artlett and Youd, 1990). Thus, the im portance of the duration of strong shaking as one of the char acteristics of an earthquake excitation of structures and soils cannot be over estim ated. The knowledge (or estimate) of the duration of future earthquake shaking, together w ith the knowledge (or estimate) of the intensity of such shak ing are the essential tools in the hands of a design engineer, and are needed for the design of future or evaluation of the existing structures. In this work, we develop some simple regression equations which allow one to predict the du ration of strong ground motion at a site as a function of earthquake and site param eters. For this purpose, we use the uniformly processed strong m otion ac celeration records (Lee and Trifunac, 1987a). Essentially, all the d ata included in 4 this d a ta bank where recorded in one region— California. O ther useful inform a tion about this region was already available in the literature (see, for exam ple, the m ap showing distribution and configuration of basem ent rocks in Califor nia by M.S. Sm ith, 1964). This allowed us to consider some factors, influencing strong m otion, which would not be possible to account for in other p arts of the world. T he duration of strong ground motion is a meaningful function not only from the pure engineering point of view. It is closely related to the earthquake source param eters, and, as we will show in this work, can be used as a tool in the study of an earthquake source. An interesting example, although not directly related to the content of this work, which shows the possible practical use of the strong dependence of the duration of strong ground motion on the source param eters, is th e study of Izutani and Hirasawa (1987). They developed a simple m ethod for rapid determ ination of fault param eters of large shallow earthquakes from the azim uthal dependence in duration of strong ground m otion. This m ethod may be applicable to near-field tsunam i warning, provided th a t a sufficient num ber of digital accelerographs are installed with a good azim uthal coverage. T he duration of strong earthquake ground motion also depends on the prop erties of the m edia along the propagation path. Once the relationships among some of the involved param eters are understood, those can be used in future investigation of, for example, dispersion phenomena. We will consider also the dependence of duration of strong shaking on the M odified M ercalli Intensity. If the dependence of the duration of strong ground m otion on earthquake m agnitude and propagation path param eters can give some estim ates of the duration of some future earthquake shaking and improve our 5 knowledge about earthquake source and wave propagation phenom ena, then the dependence of strong m otion duration on Modified Mercalli intensity can provide further inform ation on the advantages and drawbacks of this scale and help in possible improvements of this qualitative measure of the earthquake “stren g th .” All of the above examples call for careful investigation of the duration of earthquake related strong ground shaking and for the development of em pirical scaling equations for use in earthquake engineering. 1.2. The Definition of Duration: a H istorical Review T he first step in any study of a physical phenomenon is to adopt some def inition of the quantity which is studied. The duration of strong ground m otion is not easy to define. F irst studies of the dependence of duration on m agni tude (Housner, 1965) and on epicentral distance and m agnitude (Esteva and Rosenblueth, 1964) did not produce quantitative definitions of duration. Page et al. (1975) define duration as the time interval between the first and the last tim e when acceleration exceeds the level of 0.05 g (“bracketed” duration). Husid et al. (1969) define duration as the time interval during which 95% of the total en- ergy (eventually observed at the site) is coming to the recording station. Trifunac and B rady (1975b) define the duration of the excitation function f{t), which can be acceleration, velocity or displacement, as the time during which 90% of the integral f*° / 2(r)d r is achieved (to is the length of the digitized record). Bolt (1973) suggested th a t the duration of strong ground motion should be considered separately in several narrow frequency bands, as it is physically frequency depen dent quantity (due to the complexity of the earthquake rupture and frequency 6 dependent attenuation). Trifunac and Westermo (1976a,b, 1977, 1978, 1982) and W estermo and Trifunac (1978, 1979) developed frequency dependent definition of duration, based on the earlier work of Trifunac and Brady (1975b). Kawashima and Aizawa (1989) studied bracketed and introduced norm alized duration, which they defined as elapsed tim e between the first and the last acceleration excursions greater th a n \i times the peak acceleration (0 < // < 1). M cCann and Shah (1979) based their definition of duration on the tim e dependent root-m ean square acceleration, arms(t): aT ms{t) = ^ J a2(r)d(r)J , (1.1) w here t is the tim e along the record and a(t) is the acceleration. T he derivative of arsm (t) identifies the tim e after which a rma(t) is always decreasing, and this tim e is used as the upper cut-off time of the strong m otion portion. The lower cut-off tim e can be obtained by applying the above procedure to th e record with the reversed time. Vanmarcke and Lai (1980) introduced a definition of duration of strong ground shaking using an idealization of an earthquake as a segment of a limited duration of a random process w ith constant spectral density function. Assuming th a t the process is stationary Gaussian and using the results available for the peak statistics of such processes, they proposed th at , ( 2-dur\ ( “ T - ) , —-■ W p - ^ d u r> 1 .3 6 T o , (l2) V a max . 2I0/a 2 max dur < 1.36T0, where dur is the duration of strong motion, a max is the m axim um recorded acceleration, To is predom inant period of the earthquake m otion, and dur/To A 2 -fo dur - r — 5— m ax 7 is the num ber of cycles during the time interval dur. Jo here is the integral of the acceleration squared taken through the whole record of the length to- If we assume th a t the tim e to is long enough, so th a t the acceleration a(<) is practically zero for t > to, then the integral Jo can be related to the total power of the Fourier spectrum of acceleration, F(tjj), through Parseval’ s theorem : r oo i r oo Jo — I a2(r)dT = — I F 2(u)du, (1.3) J o * Jo and Jo is called in this case the Arias intensity (Arias, 1970). The peak factor r — Omax/flrms in Eq. (1.2), and a rma is the root-m ean square am plitude. The definition Eq. (1.2) relates param eters dur, Jo, a max, a rms and lb ) which are all im portant for engineering practice. M ohraz and Peng (1989) introduced the structural frequency and dam ping in the definition of duration and used a low-pass filter for com puting the duration. Theofanopulos and W atabe (1989) compared various definitions of duration and came up w ith their own which is closely related to the p art of the record th a t contains significant portion of the seismic energy. Thus, the definition of duration, as used by different investigators, pro gressed w ith tim e from simple “bracketed” definition towards frequency depen dent, structural response oriented functionals, w ith th e seismic energy considered as the main tool in the definition of duration. Many definitions utilize the integral of the type f* f 2{r)dr, where f( t) is acceleration, velocity or displacem ent. These integrals have specific physical meaning. For example, v 2(r)dT is proportional to the total energy, transm itted by the seismic waves past the recording point. Integral f g a 2(r)dr is proportional to the work done by all the forces acting on a single-degree of freedom viciously damped oscillator, exited by the accelera tion a(t). The sam e integral is closely related to Arias intensity, Eq. (1.3), and, 8 w hen divided by t, gives a 2m s ( t ) , Eq. (1.1). The prediction of the response, /( t ) , of a multi-degree of freedom structure can be done in term s of th e num ber of peaks of / ( t ) during the entire history of excitation (which depends on duration of strong m otion), the w idth of the power spectrum of f [ t ) and the value of (l/£ o )-( fo° f 2(T)dr^ (Udwadia and Trifunac, 1974; Amini and Trifunac 1984, 1985; G upta and Trifunac, 1987, 1988a,b, 1990a,b,c, 1991a,b). A lthough the examples, mentioned above, may seem to be com pletely unre lated, they have one common feature— an integral of the form f * f 2 ( r ) d r . The natu re of the growth of these integrals / g a 2(r)d r, f* v 2(t)<It and f * d 2{T)dr w ith tim e t is very similar (Trifunac and Brady, 1975b): they increase rapidly at first and then tend asym ptotically toward their final values. The portions of the record where f* f 2 ( r ) d r has its fastest growth can be related to the definition of th e strong m otion p art of exitation. Such definition of the strong m otion and of its duration can then be linked to various physical phenom ena, description of which involves integrals of this type. Hence, following the works by Trifunac and W estermo, we will accept the definition of duration of a function of m otion /(f), where /(f) is acceleration, velocity or displacement, as the sum of tim e intervals during which the integral f 2 ( r ) d T gains a significant portion of its final value. In the m ajority of Chapters of this study this “portion” was taken to be 90%. Some empirical models were tested with this portion being equal to 75% and 95%, and no significant differences in the major overall trends were found. O ur definition of strong m otion duration can be made frequency dependent by considering the record filtered through several relatively narrow band-pass channels, and calculating the duration separately in each one of them . Fig. 1.1 illustrates the im portance of taking into account the frequency dependent nature -1 0 0 - 5 0 50 100 10 15 20 25 30 35 40 45 50 55 60 0 5 C M -1 0 0 - 5 0 0 50 cd i o o 10 15 20 25 30 35 40 45 50 55 60 0 5 C D O 3 - 1 0 0 - 5 0 50 100 10 15 20 25 30 35 40 45 50 55 60 0 5 Time (sec) F ig . 1 .1 This figure dem onstrates the necessity for the frequency dependent n atu re of the definition of the strong motion duration. Several records w ith clearly different durations of strong motion in high- and in low-frequency p arts of the spectrum are shown: a) Im preial Valley E arthquake, October 15, 1979, m agnitude M = 6 .6 , epi central distance A = 27.8 km, vertical component; b) San Fernando Earthquake, February 9, 1971, M — 6.4, A = 42.0 km, N28E com ponent; c) N orthw est California Earthquake, O ctober 7, 1951, M = 5.8, A = 56.3 km, N46W com ponent. 10 of the of strong ground motion duration. Several records w ith substantially dif ferent duration of high- and of low-frequency strong m otion segments are shown. A positive feature of the definition of Trifunac and W estermo is th a t, unlike some other physically related definitions (McCann and Shah, 1979; Vanmarcke and Lai, 1980), it allows one to consider the strong motion p a rt as being com posed of several separate strong motion portions, and the beginnings and the ends of all of these portions (pulses) can be specified. Fig. 1.2 presents several exam ples of the records which have strong motion p art consisting of several sep arate pulses (which probably can be explained by the complexity of the source ru p tu re a n d /o r by the complexity of the wave propagation: dispersion and re flections from well-defined boundaries in the E arth ’s crust). It can be seen from Fig. 1.2 th a t the definition of the duration of strong motion as one continuous tim e interval is not meaningful for some records. We will show how the inform ation about the location in th e record and the duration of each separate strong motion pulse can be used to study the source of the earthquake and the wave propagation phenomena. Note also, th a t this additional inform ation about the “structure” of strong m otion can be used in further development of the definition of strong m otion. Thus, considering the energy dissipated by the structure in the time “gap” between two strong m otion pulses and the root-m ean square am plitude of the exitation function in this gap and in the pulses, one can determine w hether the structure is going the “see” both strong ground motion pulses as one continuous strong excitation. This depends on the stru ctu re’s param eters, such as natural frequencies and dam ping. Hence, a new definition of duration could be formulated and called “the duration of strong response” (although this is not the purpose of the present work). A - 1 0 0 - 5 0 50 100 10 15 20 25 30 35 40 45 50 55 60 0 5 CM o O O - 3 0 0 - 2 0 0 -1 0 0 0 100 200 300 0 C O - 3 0 0 -2 0 0 O -1 0 0 3 100 300 10 15 25 30 35 40 45 50 55 60 0) c ) J L J I I I 10 15 20 25 30 35 40 45 50 55 60 Time (sec) F ig . 1.2 Complex structure of the strong motion portion of the acceleration record is shown by several examples: a) Oroville aftershock # 1 1 , August 16, 1975, m agnitude M = 4.1, epicentral distance A = 4.9 km; b) M organ Hill Earthquake, April 24, 1984, M = 6.2, A = 4.4 km, S60W component; c) Imperial Valley Earthquake, May 18, 1940, M = 6.7, A = 9.3 km, S00E com ponent. In these cases, the definition of strong motion part of accelerogram as one continuous tim e interval is not appropriate. 12 design engineer might be interested more in this “duration of strong response” (which would depend on damping) more than in the duration of the strong m otion exitation. Various definitions of duration belong to two m ajor groups: “relative” and “absolute” definition. In th e “absolute” definition, like th e one of Page et al. (1975), the inform ation about the absolute level of acceleration is preserved. T his inform ation is not included into “relative,” or “norm alized,” definition of d uration of strong m otion, such as in the works of Husid et al. (1969), Trifunac and B rady (1975b) and M cCann and Shah (1979). The “relative” definition assumes th a t EVERY record, obtained by a strong m otion accelerom eter (trig gered by an earthquake) has a strong motion part, regardless of the absolute level of the acceleration recorded. Hence, the m otion, defined as “strong” is strong relative to some average am plitude, observed during the tim e of recording We will use this “relative”type of definition because the dependence of the average am plitude of motion (which is directly related to the Fourier spectral am plitude at a given frequency) on earthquake size and on regional param eters is already known (Trifunac 1989a,b,c). The knowledge of the frequency dependent duration of strong ground motion in this “relative” sense combined w ith the inform ation about the Fourier spectral am plitudes at all frequencies can give a fairly com plete description of an earthquake related strong motion excitation. 1.3. Organization of This Study T he m ajor purpose of this study is to develop simple regression equations which allow one to predict the duration of strong earthquake ground m otion in 13 term s of th e source, site and transm ission path param eters. These equations will be derived using the currently available uniformly processed strong m otion d a ta base (Trifunac and Lee, 1987a) and will be applicable to the W estern United States and in particular to California. The good quality of this d ata allows us to look into some effects, th a t were not studied before. We also will discuss the physical reasons for choosing the particular term s in the regression equations and will interpret the results we obtain (such as the dependence of the regression coefficients on frequency). Some results, dealing w ith the attenuation coefficient of th e m edium , Q ~ l , will be presented, although those are not directly related to the duration of strong m otion, but represent a useful side product of th e present effort. T he overall organization of this report is schematically presented in Fig. 1.3. T he first C hapter consists of introduction and is not included in the flow chart of Fig. 1.3. T he following two Chapters provide the necessary background for the understanding of our scaling equations. Chapter 2 gives the general description of th e d a ta set and shows the procedure used in identification of strong m otion portions of each record. Some technical details are presented in Appendices A and B. C hapter 3 deals with some physical considerations regarding the param eters involved in regression analysis. Each term that appears later in scaling equations, is defined and described there. Several models of duration of strong ground motion when it is scaled in term s of earthquake m agnitude, source-to-station distance, geom etry of the sedim en tary basin and geological and local soil site conditions are presented in C ahpter 4. T he m athem atical details of the method used in the regression analysis are given in Appendix C. C hapter 4 also includes prelim inary discussion of the re- Cap ter 3 Physical reasons for including particular terms into the regression equations Magnitude Distance to the source Modified Mercalli Intensity Dimensions of the sedimentary basin Geological and local soil site conditions Chapter 4 Models which include the magnitude of the earthquake Chapter 5 Models which include the Modified Mercalli Intensity Appendix D Appendix gj Chapter 7 Propagation effects Chapter 6 Duration of strong ground motion and the physical process at the source I-\ S ' Appendix C Chapter 2 Description of the data set and of the procedure for calculating the duration ( Appendix A Appendix B Y Chapter 8 Scattering and attenuation effects as those can be seen using current definition of the duration of strong ground motion F ig . 1.3 The flow-chart of the organization of this study. 15 suits (the dependence of the regression coefficients on frequency). Several models of secondary im portance, which include earthquake m agnitude, can be found in Appendices D and E. C hapter 5 gives the models of duration built on the assum ption th a t he M od ified M ercalli Intensity at the site (instead of the m agnitude of the earthquake) is available. The next two Chapters include the discussion of the physics of the source (C hapter 6) and of the wave propagation (Chapter 7) in connection w ith the results th a t can be obtained using the present definition of duration. Some of the models, developed in C hapter 4, are used here, and some other models, describing the overall duration of strong motion and the duration and the num ber (in one record) of separate strong motion pulses, are introduced. C hapter 8 concludes the main body of the report w ith some considerations on attenuation and scattering of strong earthquake waves. A m ethod of con structing the envelope of “strong motion coda” (which is different from coda definition usually used in seismology) is shown. Coda attenuation factor Q ~l is obtained from the late p art of “strong motion coda” and the discussion of possible mechanisms th a t alter the shape of the early p art of it is presented. 16 2. STRONG MOTION DATA SET A N D CALCULATION OF DURATIO N 2.1. General Considerations After choosing the definition of the duration of strong ground m otion, one can proceed w ith fulfilling the task of predicting the duration of strong shaking at a p articular site caused by a given earthquake. Such predictions can be m ade based on the previously recorded d ata and the assum ption th a t the future m otions will obey the statistical trends and correlations established by the previously observed sam ple data set. In our case these are the correlations relating the d uration of strong ground motion with the param eters of the earthquake, of th e propagation path and of the recording site. However, one should be careful w ith using these correlations, and apply them only in those areas where the d a ta have been recorded. Regression equations derived from the d ata obtained in California, may easily fail in the Eastern United States, in Indonesia or in P am ir, for example. Different geological structures and prevailing mechanism of the m otion of plates, along with other regionally specific differences, may influence the nature of the earthquake source, resulting in different correlations of the fault length and magnitude, and may alter the prevailing depth of the seismogenic zone. Different geological history, reflected in the structure of the upper crust, will influence the conditions of wave propagation, causing changes in the dispersion and attenuation laws. All this can drive the predictions, based on the d a ta from a geologically different region, far from the local reality. This leads to the conclusion th at every region has to be studied separately to avoid 17 possible biases. Perhaps, the future research will allow us to understand the nature of earthquake occurrence better and the physics of wave propagation, thus enabling us to link our knowledge of statistical properties of, say, duration of strong ground motion in one region with the predictions of the duration in another region. For the tim e being, however, all correlations, derived hereafter, are fully applicable only in the region where the d ata set was recorded. O ur d ata set covers only restricted range of earthquake m agnitudes, distances to the epicenter and other param eters. We assume th a t only predictions coming from interpolation, not extrapolation, are accurate enough. Thus, in application of our models the m agnitude of the earthquake (or the intensity of shaking), the epicentral distance and all other param eters of the earthquake and of the recording site should be w ithin the range of param eters of the sample d a ta set which served as a base for deriving the correlation formulae. 2.2. Description of the D ata Set T he d ata set used in this study is fully described in one of the EQINFOS reports (Lee and Trifunac, 1987a). So far, this is the most com plete uniform ly processed d ata set, consisting of 494 three component “free field” records ob tained in California since 1933. Each component of every record was digitized and filtered to be noise-free inside a frequency band which depends on the quality of the record, bu t is not wider than [0.05 25] Hz. M ethods used in digitization and processing of these records are described by Trifunac and Lee (1979a) and Lee and Trifunac (1984, 1990). We will outline here only those features of this d ata base which are im portant for the understanding and for using our results. 18 Each record consists of two horizontal and of one vertical com ponent of accel eration, velocity and displacement. Due to malfunctioning of some transducers few com ponents were lost. As a result, the d ata base has 486 vertical and 984 horizontal com ponents of acceleration, velocity and displacem ent, generated by 106 earthquakes and recorded at 283 different sites. The previous studies (Tri funac and W estermo 1976a,b, 1977, 1978, 1982; W estermo and Trifunac 1978, 1979), based on the same definition of duration as the present one, used only about one th ird of the data set available now. T he following characteristics of the records were considered in this study: m agnitude of an earthquake, epicentral distance, hypocentral depth, Modified M ercalli Intensity at the site and the recording site conditions. Locations of th e epicenters and the m agnitudes of earthquakes are available for the m ajority of EQ IN FO S records. This inform ation was adopted from various published reports, catalogs and papers. Inform ation on recording site conditions (depth of sedim ents and geological and soil site classification) has been collected earlier by Trifunac from various published maps and consulting reports. He also used N uclear Regulatory Commission, U.S.G.S., C.D.M.G. and University of Southern California reports and other publications. New param eters characterizing the records introduced in this study and not presented earlier, will be described in C hapter 3. The histograms in Figs. 2.1, 2.2 and 2.3 display the relative distribution of the records w ith respect to m agnitude, epicentral distance and hypocentral depth. We recognize th a t different m agnitude scales have contributed to those re ported for different earthquakes in this data base. Some of these scales describe the energy of the source through a “body wave window” (like m agnitude mb) Number o f records 19 2 0 0 180 - 160 - 140 - 120 - 100 - 80 - 60 - 40 - 20 - ^ y / / / a 3.0 3.5 4.0 4.5 5.0 5.5 6.0 V / / A Z Z Z f T A V ZZ f7 A 6.5 7.0 7.5 Magnitude F ig . 2.1 D istribution of 3-component records (adopted for the analysis) w ith respect to the m agnitude of the earthquakes. Number o f records 20 140 -i 120 - 100 - 80 - 60 - 40 - 20 - 80 100 120 140 160 180 60 20 40 0 Epicentral distance (km) F ig . 2.2 D istribution of 3-component records (adopted for the analysis) w ith respect to epicentral distance. Number o f records 21 140 i 120 - 100 - 80 - 60 - 40 - 20 - 10 12.5 15 17.5 20 22.5 25 7.5 2.5 5 Hypocentral depth (km) F ig . 2.3 D istribution of 3-component records (adopted for the analysis) w ith respect to the depth of the hypocenter. 2 2 while some are related to the energy of long period surface waves (like Ms)- However, the majority of the earthquakes were characterized by their local m ag nitude M l - One can notice th a t a substantial num ber of records were obtained from earthquake w ith m agnitude around M = 6.5. About half of these records were generated by the San Fernando earthquake in 1971, which had the m agnitude M = 6.4. Excluding this event, the d ata base has relatively uniform coverage of m agnitudes from M = 4.0 to M — 6.5 w ith ju st a few records available for M « 3 and M > 7 (Trifunac, 1991a). Epicentral distances are uniformly represented in the range A < 50 km w ith the num ber of records available progressively diminishing beyond A = 60 km. The determ ination of the hypocentral depth is generally not reliable. It can be obtained accurately only if the earthquake happens to be recorded by a dense array, which obviously is not always the case. Besides, all earthquakes recorded in 1930’s through 1950’s were reported to have the focal depth of 16 km. This results from the old practice of assigning the location of hypocenters to some selected “standard” depths. The histogram s in Figs. 2.4 and 2.5 show the relative distribution of records w ith respect to the geological and soil conditions at the recording site (for the records where this inform ation is available). Three param eters are considered here w ith two of them representing “geological scale.” These “large scale” pa ram eters are: h—the depth of sediments at the recording site and the simplified geological classification in term s of param eter s. Sites located on sedim ents are m arked by s = 0 , sites located on geological (basement) rock are labeled by s = 2, and s = 1 stands for the interm ediate sites. A description of the site in 100- c e h 3 80 > H O o c u J h 60 tt-t o S h c u 40 ,n Lh I I Y // ///A s=2 (G eological ro ck ) (SSI! s = l (In te rm e d ia te ) Im n\ s=0 (S ed im en ts) m m M L r i i a M L 2 3 a ) I I rrrr 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 100 W / / / A s,=0 ("R ock") b) C O T3 U O O C U J h « 4 - i O 80 60 > h c u 40 0 20 0 ISS3 s. =1 (S tiff soil) i/////l s .^ 2 (Deep soil) _ K 3 _ m w M iT l % 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Depth of sedim ents (km) F ig . 2.4 D istribution of 3-component records (adopted for the analysis) with respect to the depth of sediments at the recording site, h : a) for different geological conditions (were s and h are available); b) for different soil conditions (where sl and h are available). a) b ) 140- 120- CO o 100- O ( U S-c 80 - « t - i o u 60 - V S 40 - P z 20 - 0 - Soil conditions: mz z m sL=0 ("Rock") C 5 Z 5 H sL = l (Stiff soil) m il s l =2 (Deep soil) s=0 s= l s=2 Geological conditions Geological conditions: Y // ///A s=2 (Geological rock) K S s= l (Interm ediate) H U B s=0 (Sediments) o 100 g j 60 Soil conditions F ig . 2.5 Relationship of geological and soil site conditions (where s and sl are available): a) distribution of records (adopted for the analysis) w ith respect to geological conditions for different soil conditions; b) distribution of records (adopted for the analysis) w ith respect to soil conditions for different geological conditions. to 25 term s of s can be used when precise inform ation on the depth of sedim ents is not available (Trifunac and Brady, 1975a). The th ird param eter involved in the description of the recording site con ditions is the soil classification factor sl (as used by Seed et al., 1976). sl is assigned to have value equal to 2 for deep soil sites (soil layer deeper th an 100 m ), sl — 1 for stiff soil sites (soil layer 15-70 m deep), and both sl = 2 and sl = 1 having shear wave velocity less than 800 m /sec. If shear wave velocity in the soil m aterial exceeds 800 m /sec, the site is classified as “rock,” sl = 0 . Seed et al. (1976) also considered soft to m edium clays, which are not common in the W estern United States and, therefore, are not included in this analysis. P aram eter sl describes the recording site conditions on the “local scale” and gives additional information once “geological scale” is described by the depth of sedim ents or by the param eter s. Notice (see Fig. 2.5) th a t s and s l , though correlated, do complement each other: once one of the param eters is fixed, an other one can take practically any value from its allowable range. For example (Fig. 2.5b), sites w ith sl = 0 ( “rock”) can be located over either basem ent rock (s = 2), interm ediate (s = l) or sedim entary (s = 0) geologic conditions. Notice also th a t available data are not uniformly distributed between site classifications s = 0 , 1 and 2 and sl = 2, 1 and 0 . Fig. 2.6 shows the distribution of records in the d ata base w ith respect to th e Modified Mercalli Intensity at the recording site, I m m - This param eter was not available for the two thirds of records in our d ata base. The missing values were estim ated by using the equation, proposed by Lee and Trifunac (1985a). I m m = 1.5 -M + 1.12 - 0.856-ln A - 0.015-A - 0.26-s, (2.1a) Number o f records 26 200 - r e c o r d e d 175 - Y /////A e s t i m a t e d 150 - 125 - 100 - 75 - 50 - 25 - VI IV V VIII IX X II VII Modified Mercalli Intensity F ig . 2 .6 D istribution of 3-component records (adopted for the analysis) w ith respect to the Modified Mercalli Intensity. 27 where A is “representative distance” A = \ / a 2 + H 2 + L2. (2.16) A designates the epicentral distance, H is the hypocentral depth, L stands for the source dim ension, and s is the geological classification param eter. The estim ate of the source dimension was obtained from L = i ( M ) { l - e x p ( l ^ ) } , < 2^ where L(M ) is an empirically determined linear function of m agnitude, M , such th a t for M = 3 L(M) — 0.2 km, (2.Id) for M = 6.5 L(M) = 17.5 km. All length param eters of the source are measured in km. T he Eq. (2 .1) fits the d ata for short and interm ediate epicentral distances (say, up to 160 km ), and underestim ates Im m f°r l°ng epicentral distance. The Eq. (2.1) was obtained by the regression analysis on the same d ata set, as used in this work. The quality of fit is shown on Fig. 2.7. The observed intensities are plotted versus representative distance A. Each data point is shown as a shaded circle, w ith the diam eter of the circle proportional to the m agnitude of the earthquake. The estim ate of Im m with the help of Eq. (2.1) is presented by a family of solid lines, each line corresponds to a certain m agnitude. T he range of m agnitudes from M = 3 to M = 8 , with increments of 0.5 is shown, and the beginning of each line is marked by the circle in a square box, w ith the diam eter of the circle proportional to the magnitude it represents. The estim ate is shown for the case of H = 5 km and the recording station located on the basem ent rock, (s = 0 ). XII-i XI- • ^ x - m < d IX- + -> d M VIII- • rH VII- c tf o V I- u C D S V- T i IV- C D III - 'd o TT- I - o - Diameter of the circle is proportional to the m agnitude - observed data points — M=8 estimated: 100 120 140 160 180 200 220 Representative distance, A (km) F ig . 2.7 The observed data (shaded circles) and the estim ate of the Modified Mercalli Intensity by Eq. (2 .1) for the hypocentral depth H = 5 km and s = 2 (the basement rock site). Each line corresponds to one magnitude and is m arked by a shaded circle of an appropriate size, enclosed by a square to distinguish it from the observed data (M = 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5 and 8). to 29 Recalling the fact th at ju st a few records of our d a ta base were obtained at epicentral distances greater than 160 km, and th a t many of those were not considered in the regression models of the duration of the earthquake related strong ground motion due to their low signal to noise ratio (see Appendix B), we assume th a t Eq. (2.1) can be used to estim ate the missing d a ta on Modified M ercalli Intensity. This estim ate, together w ith the originally observed I m m , can be used further as one homogeneous d ata set in the studies of the relations between the duration of strong ground m otion and of the Modified M ercalli Intensity. 2.3. Calculation of Frequency Dependent Duration As it was noted long ago and explicitly shown in the previous studies (Tri funac and W estermo 1976a,b 1977, 1978, 1982; W estermo and Trifunac 1978, 1979), duration of strong ground shaking depends on the frequency of motion. One of the possible ways to account for this is to study the correlations of dura tion w ith source and site param eters in several frequency bands separately. This m ethod was used in all previous works of Trifunac and Westermo. They used 6 frequency bands w ith central frequencies varying from 0.22 to 18 Hz. In the present study, 12 frequency bands, later called channels, were used. Frequencies from 0.075 Hz to 21 Hz were covered by these channels. Signal, corresponding to each of the channels was obtained by band-pass filtering the original records from EQINFOS files by using the Ormsby filters. These filters, having reasonable am plitude characteristics, are known to posses very im portant property of preserving the original phase. The necessity of this property for the 30 filters used in earthquake engineering d ata processing was emphasized by Lee and Trifunac (1984). The com puter program for band-pass filtering was kindly w ritten by Professor V.W . Lee. The actual am plitude characteristics of the filters he used are displayed in Fig. 2.8. Precise values of roll-off and cut-off frequencies of Orm sby filters used are listed in Table A .l in Appendix A. Band-pass filtering was applied to all 1470 components (486 vertical and 984 horizontal) of acceleration a(i), velocity v(t) and displacem ent d(t). Fig. 2.9a- c show the examples of such filtering for one of the records. Original record is shown on the top and its 12 corresponding band-pass filtered channels are located below it. Each band-passed channel is labeled by its roll-off and cut-off frequencies and by its num ber (starting from the longest period band). Em pty box corresponds to the channels th a t appear to be outside the initial noise-free frequency band adopted for this record during initial d ata processing (Trifunac and Lee, 1979). Roll-off and cut-off frequencies are shown for each channel. Percentage sign (%) next to the channel # 3 characteristics denotes th a t the lower cut-off of the original frequency band of the record falls w ithin the channel band under consideration. Numerical values of the peak accelerations (velocities and displacem ents), in each channel, are also listed. Asterisks on a(t), v(t) and d(t) show the tim e when these peak values were observed. The right-hand side of the picture shows the graphs of f* f 2{T)dr as functions of tim e for each channel (f ( t ) is a(t), v(t) or d(t). The meaning of the dashed lines and num bers th a t appear on the right-hand side of Fig. 2.9a-c will be explained later. As in all previous studies by Trifunac and Westermo, we define duration of strong m otion as the sum of the time intervals over which the significant contribution to the total integral f*° f 2(r)dr is made. Here to is the length of Amplification i : J t f x U V $ i § i < S M \ |} ? ? ? f e r : : - i£ i£ g i $ % & & Frequency (Hz) F ig . 2.8 Amplitude characteristics of the filters used in the band-pass filtering of the acceleration data. Black dots show the location of the central frequencies of the channels. 3 2 SAN FERNANDO EARTHQUAKE FEB 9 , 1 9 7 1 - 0 6 0 0 P ST WHEELER RIDGE, CAL. COMP N 90E .15-.2,25.0-27.0 hz - 0.2 0.026 G 444 I 18.2s - 0.2 .05- 07, 08-. 10 hz (Outside passband) .0 8 -1 0 ,1 5 -. 17 hz (Outside passband) - 0.1 # 2 % .15-.17,.27-.30 hz - 0.1 0.001G '2.5 I 17.7s - # 3 .27-.30,.45-.50 hz - 0.1 0.001G '1 2 .6 I 24.1s - .45-.50,.80-.90 hz 2 -o.i 0.003 G '21.3 I 25.8s - # 5 .80-.90,1.30-1.50 hz 0.002 G - 0.1 '13.8 I 22.2s - o ^ -0 .1 - 1.3-1.5,1.9-2.2 hz 0.004 G '32.1 I 17.1s - 1.9-2.2,2.8-3.5 hz 0.007 G - 0.1 '40.0 _ 15.6s — # 8 2.8-3.5,5.0-6.0 hz 0.015G - 0.1 109 I 11.0 s - # 9 5.0-6.0,8.75-10.25 hz 0.010G - 0.1 "97.6 _ 10.2s - 8.75-10.25,16.0-18.0 hz 0.006G - 0.1 "28.9 I 9.9 s - #11 16.0-18.0,25.0-27.0 hz 0.002G - 0.1 '1.6 I 9.9 s - #12 30 0 10 20 40 0 20 40 TIME - SECONDS F ig . 2 .9 a Original acceleration record (top) and its 12 band-pass filtered chan nels. VELOCITY - C M /SE C 33 SAN FERNANDO EARTHQUAKE FEB 9 , 1 9 7 1 - 0 6 0 0 PST WHEELER RIDGE, CAL. COMP N 90E V15-.2.25.Q-27.0 hz 1 .4 cm /s 25.8s -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0^ 0.5 .05-.07,.08-.10 hz (Outside passband) # 1 .08-.10,.15-.17 hz (Outside passband) # 2 % .15-.17,.27-.30 hz 0.34 cm /s #3 20.3 s .2 22.2s 45-.50..80-.90 hz 80-.90.1.30-1.50 hz cm /s 24.1s 1.3-1.5,1.9-2.2 hz 0.32 cm /s l 33 l / s # 7 18.1s 1.9-2.2,2.8-3.5 hz . - 0.39 cm /s 16.1s 2.8-3.5,5.0-6.0 hz 0.63 cm /s # 9 0.2 11.0s 5.0-6.0,8.75-10.25 hz 0.22 cm /s l _L 23 i / s #10 l 0.05 10.4s 8 75-10.25,16.0-18.0 hz 0.061 ________ cm /s #11 I __________ I __________ I . . .. 0.004 12.3s 16.0-18.0,25.0-27.0 hz _L 0.032 cm /s #12 "0.000 23.3s 0 10 20 30 40 TIME - SECONDS 20 4 0 F ig . 2 .9 b Original velocity record (top) and its 12 band-pass filtered channels. SAN FERNANDO EARTHQUAKE FEB 9 , 1 9 7 L - 0 6 0 0 PST WHEELER RIDGE, CAL. COMP N 90E .15-.2,25.0-27.0 hz 0.52 cm -0.5 ‘1.0 I 24.3 s - 0.5 - 0.2 .05-.07,.08-.10 hz (Outside passband) 0.2 - 0.2 .08-.10,.15-.17 hz (Outside passband) 0.2 - 0.2 % /Y 15-. 17,.27-.30 hz 0.24 cm # 3 '0.4 I 16.8s - 0.2 - 0.2 27jr.3K.45r.50 hz 0.25 cm ro.4 I : 22.1s - 0.2 - 0.2 .45-.50,.80-.90 hz 0.17 cm # 5 70.10 I 1 27.5 s - 1 0.2 & - 0.2 w A 2 0 g 0.2 2i -0.2 Q - . co 0 Q 0.2 -0 .2 .80-.90,1.30-1.50 hz 0.056 cm '0.008 I 22.7s - 1.3-1.5,1.9-2.2 hz 0.035 cm #7 0.004 I 21.6s - 1.9-2 2,2.8-3.5 hz 0.022 cm 0.001 _ 17.0s - 2.8-3.5,5.0-6.0 hz 0.025 cm " 0.001 I 26.0s - 0.2 - 0.2 5.0-6.0,8.75-10.25 hz 0.006 cm #10 '0.000 I 22.0 s - 0.2 - 0.2 8.75-10.25,16.0-18.0 hz 0 003 * - cm '0.000 I 25.1 s - 0.2 - 0.2 16.0-18.0,25.0-27.0 hz 0.004 cm #12 " 0.000 I 25.1s - 0.2. 20 40 0 20 0 10 30 40 TIME - SECONDS F ig . 2.9c Original displacement record (top) and its 12 band-pass filtered chan nels. 3 5 strong m otion record, and f(t) is one of the functions a(t), v(t) or d(t). The “significance” of the contribution can be defined in term s of the portion p, of /o° f 2(T)d-T. Eight different values of p were considered in this study: 99%, 97.5%, 95%, 90%, 85%, 80%, 75%, 70%, with p = 90% chosen as the “basic” value adopted in the definition. The procedure for calculating duration in each channel is sum m arized on Fig. 2.10. Band-passed f(t) is shown on the top of the figure. The result of integration f* / 2(r)d(r) is shown in the center together w ith its sm oothed ver sion. B ottom graph displays the derivative of the sm oothed version of f* f 2(r)dr and eight different levels of this derivative, corresponding to different p in the definition of duration: dur = £ ( * » - 2 ) - * S 1 } ) » (2- 2«) *=1 where and 11 -2^ are the beginning and the end times of m tim e intervals during which the portion p of the total integral I = f*° f 2 ( r ) d T is achieved: rt0 rn m H-I = n- f 2(r)dT = J 2 f 2{r)dr = (2.26) Jo i= i Jt\l) 7^i Here to stands for the length of strong motion record, and /,• is the gain of the integral of f 2 ( t ) during the tim e interval These tim e intervals in Eq. (2.2a) are chosen so th at their sum gives the shortest possible tim e during which n portion of the total integral I can be achieved. M athem atically this corresponds to the requirem ent th a t the derivative of the integral of f 2 ( t ) is bigger th an some threshold level p^. We, however, consider not the derivative of the integral f* f 2 ( r ) d T itself, but the derivative of its sm oothed version: dt r t 1 m / f 2(r)dT > Pn if and only if te £,-2^ • (2.2c) V O J smoothed ,_ j 36 10 - 5 stro n g m o tio n -1 0 300 250 200 150 100 E l i = 0 .9 1 i=l 1 C M 50 30 T h r e s h o l d le v e ls : 2 0 = 0.99; 0.975; 0.95; 0.85; 0.80; 0.75; 0.70 10 o 10 20 30 Time (sec) 40 50 60 F ig . 2 .1 0 The definition of duration illustrated for the east acceleration com ponent of the M organ Hill earthquake, band-pass filtered by channel # 4 filters (central frequency 0.37 Hz): a) tim e history /(f) with strong motion intervals (shaded); b) f Q / 2(r)d r and its smoothed version; c) the derivative of the smoothed integral of f 2(t) and its threshold levels (see Eq. (2.2)). Time intervals giving contribution to duration w ith n — 0.9 are highlighted. 37 T he reasons for sm oothing the integral will be explained shortly. Eq. (2 .2) gives the com plete definition of duration. M ajor p art of this study considers fj, = 0.9 only. From this m om ent on, we will call the duration obtained by Eq. (2.2) as “observed” duration and will designate it as dur. The same notation will be used to designate the estim ate of the duration of strong m otion obtained by using any of our m odels, described later. The precise m eaning of dur will be obvious from the context. Smoothing of the integral of f 2(t) was introduced into the definition of the duration of strong ground m otion for two reasons, different for long and for short period channels. In long period channels, a substantial visible periodic com ponent is generated during narrow band filtering (this can be seen in Fig. 2.10). Frequency of this oscillations is equal to the doubled central frequency of the channel. T here is no good physical reason to consider a physical processes w ith characteristic tim e of the order of or shorter than the central period of the channel. Considering only one frequency, we notice th a t an engineering structure will not “feel” the duration of the excitation if this duration is shorter than, or equal to its fundam ental period (which usually dominates in the response). Such excitation will be “felt” by th e stru ctu re as a 5-pulse, which has short duration, bu t finite energy. So, as we are interested in the DURATION of strong ground m otion, we have to consider tim e intervals longer than one period of oscillations, th a t is, longer th a n the central period of the channel. This causes the cut-off frequency of the sm oothing filter to be of the order of the central frequency of channel. At low frequency channels, we decided to apply the filter which sm ooths the input function over two periods of motion. 3 8 Short period bands require more severe criteria on the corner frequency of th e sm oothing filter. Keeping this corner frequency of the order of the central frequency of the channel, one will end up having too many tim e intervals, con trib u tin g to the duration. These tim e intervals of strong m otion will be very short and w ith short gaps of “silence” in between them . T he engineering struc tures have small dam ping (not more than about 5%). T he sam e is true for soils (dam ping up to about 10%). It follows then, th a t the structure or soil will n o t “see” the tim e gap between the pulses in the excitation if those gaps are shorter th an the characteristic relaxation time T •£, where T is the fundam ental period of the structure and £ is the dam ping ratio. We conclude th a t for the high frequency channels, the cut-off frequencies of sm oothing filters should be substantially lower than the central frequency of the channel. The exact values chosen for corner frequencies of sm oothing filters in each channel are listed in Table A .l in Appendix A. This Appendix also presents further discussion on the filters used in this study and on the lim itations those impose. 2.4 Quality of the Duration D ata We consider again Fig. 2.9a-c. The right-hand side of these figures shows Jo f 2(T)dT- Dashed lines display the starting and finishing points of a single the tim e interval during which 90% of / 0 < o / 2(r)d r was achieved, where t0 stands for th e length of the record. This tim e interval corresponds to a simplified version of the definition of duration Eq. (2.2), where m is equal to 1, is the tim e were 5% of fo° f 2(r)dr is achieved and corresponds to 95% of j J 0 / 2(r)d r. This definition of duration was used in the early work by Trifunac and Brady (1975b). 39 We use it here ju st for estim ation of the quality of d ata in each channel for the record considered for the purpose of making decision of including or not including this particular channel of the record in the regression analysis. Thus, if this estim ate of duration (its value in seconds is printed for each channel on the right hand side of Fig. 2.9) is com parable w ith the length of the record, and f* / 2(r)d r continues to rise after t ^ \ one can expect th a t / 2(r)d r is not reaching its final value at t = to. Simply, the record is too short and the duration of strong ground m otion for this record cannot be estim ated for the channel under consideration. In th e case of Fig. 2.9a this happens for channels # 4 and #=5. A nother reason to disregard a channel is too low values of f*° f 2(T)dr for this channel as com pared w ith th e overall integral for original (non-filtered) record. T he num erical values of these integrals (in some arbitrary units) can be located at the right hand side of Fig. 2.9a-c above the estim ate of duration. Low values of f*° f 2(r)d r in the channel simply show th a t there is almost no signal present in this frequency band and th a t this channel can (and should) be disregarded. In the case of Fig. 2.9a this happens for channels # 3 and # 12. Considerations like this were applied to com ponents of acceleration, veloc ity and displacement (1470 plots each), and only “clean” d ata were used in all subsequent analyses. The complete description of this “cleaning-up” procedure can be found in Appendix B. As a result of this procedure only limited num ber of cases in each frequency band were considered to be qualified for the regres sion analysis. Fig. 2.11 show the num ber of “good” acceleration, velocity and displacem ent band-pass filtered records available for each channel, separately for horizontal and vertical components. As we will see later, the channels can be assum ed to represent narrow frequency bands and, therefore, can be referred to Number o f cases Number o f cases 40 BOO 700- 600- 500 400- 300 2 0 0 - 100-1 0 Horizontal components I / . / / / / I acceleration K S velocity m zzZA displacem ent I m k m k p i I I i i / 4 ll P i I m m 1 p /p m I 5 fi 1 I P P P P > /> A .075 .12 .21 .37 .63 1.1 1.7 2.5 4.2 7.2 13 21 400-) 350 300 250- 2 0 0 - 150- 1 0 0 - 50 0 Vertical components \!lUl\ acceleration K S velocity y ///ma displacem ent v 4 m 1 m m N m i / 4 / 4 m p P I 1 1 f i t 0 I I f a II k I P h i m Is I p P i 'P ? m Y H .075 .12 .21 .37 .63 1.1 1.7 2.5 4.2 7.2 13 21 Frequency (Hz) F ig - _ 2 .1 1 N um ber of adopted components of band-pass filtered acceleration, velocity and displacement, available at each frequency band (channel). Channels are identified by their central frequency. 41 by their central frequencies, as it is done in Fig. 2 .11. (We note here th a t we use th e sam e letter to designate the frequency of motion f and th e tim e history f(t). We hope th a t in each case the meaning of the notation can be understood from the context). As it was noted earlier (Trifunac and Brady, 1975b) for the original (not narrow band-pass filtered) record the behavior of a2(r)dr, f * v 2(T)dr and f t d2{r)dT is qualitatively similar. As a rule, these integrals first grow rapidly, som etim es having several noticeable time intervals of fast growth. The rapid grow th corresponds to the intervals of strong motion and is associated w ith the greatest portion of seismic energy coming to the site. The subsequent interval of tim e during which these integrals gradually approach their final lim its results from th e late arrivals of scattered, diffracted and other “coda” waves. However, if a wide frequency band record is considered, many quantitative differences in the behavior of f* a 2(r)d r, f * u2(r)d r and f* d2(r)dr can be found. On the other hand, for a very narrow frequency band [/i; / 2], where ( / 2 - f i ) / y / / 1/2 < 1» the functions a(<), v(t ) and d{t) differ in scale only. In the limit, where f \ — ■ ► / 2 = / , displacem ent can be represented as d(t) = sin(27r/<), and acceleration and ve locity as a(t) = (27rf)v(t) = (2nf ) 2d{t). Therefore, in the lim it for very narrow channels, integrals a 2(r)d r, f * v 2(r)dr, d2(r)d r differ by a constant only. In the case of the channels used in this work, ( / 2 — / 1) / \ / / i / 2 « 2 -r 3 (see Table A .I in A ppendix A). Detailed study of many records shows th a t these frequency bands can be considered to be narrow enough: the shapes of inte grals f * a 2(r)dr, f * v 2(r)dr and f* d 2{r)dT are practically undistinguished by eye (com pare the right-hand sides in the plots in Fig. 2.9a, 2.9b and 2.9c). 42 Due to the sim ilar shape of the integrals involved in the definition of duration Eq. (2.2), the duration of strong motion for a particular com ponent in a given channel comes out practically the same for acceleration, velocity and displace m ent. Exam ples of the correlations between duration of strong ground shaking determ ined from acceleration (dura), velocity (durv) and displacem ent (durd) of th e recorded m otion are shown on Fig. 2.12 (for several frequency bands). High positive correlation of dura, durv and durd allows one to tre a t all d ata for ac celeration, velocity and displacement as one more or less uniform set. This was done earlier by Trifunac and Westermo to increase the num ber of cases available for th e analysis. Here we make several remarks to support the usefulness of this approach. Firstly, as the w idth of the channels is not infinitesimal one still can see some scattering in the correlation graphs (Fig. 2.12). Therefore, considering acceler ation, velocity and displacement together, we do not ju st simply triplicate the data. One way to look at it is th a t we rather first introduce some “random noise” to the prim ary data set (acceleration) to get secondary data set (velocity), and then repeat operation to get displacement d ata from velocity data. Notice here th a t this “random noise” cannot be treated as completely random . Frequency bands used are not infinitely narrow, so the shapes of f* a2(r)dr, f* v2(r)dr and f* d2(r)dT do not coincide completely. Therefore, some additional inform ation is introduced to the d ata and consideration of dura, durv and durd together is not equivalent to simple repetition of the information. Second rem ark we make deals w ith the change of spectrum of a record during integration and w ith the increase of duration at each channel associated w ith it. Suppose we consider an acceleration record band pass filtered in a frequency Duration (sec Channel 3, f 0=0.21Hz 50 40 30 10 dur 50 40 30 20 dur D Channel 7, f 0 = 1.7Hz Channel 10, f0 =7.2Hz 40 30 20 10 dur 40 30 20 10 dur 20 30 40 20 10 dur 20 dur 10 20 30 F ig . 2 .1 2 Correlations between duration determ ined from acceleration (dura), velocity (durv) and displacement (durd) for several channels. Horizontal and vertical components are shown combined. Solid straight lines correspond to durd = durv or durv — dura. 44 band [/j; ^ * 2]. Let us assume th a t this filtered record has a flat Fourier spectrum Fa(f) inside [fc f 2\: Fa(f) = I COIlStan* / ■ < / < / » . (2 3 ) 1 0 otherwise (see Fig. 2.13a). An estim ate, of the central frequency of the channel / o ° \ using acceleration is: loS io (/o a)) = 0.5{log10 h + log10 f 2} , or f^ a) = \ / f i f 2. Fourier spectrum of the velocity (Fig. 2.13b) is simply w = t r and the new shape of Fourier spectrum gives the new estim ate for central fre quency of the channel, / qVK As one can see, characteristic frequency of the channel appears to be shifted towards low frequencies: < /o ° \ Further integration during calculation of displacement shifts f 0 even more. Same con siderations are applicable if Fa{f) does not have so simple form as in Eq. (2.3). Therefore, on the average, each channel can be characterized by slightly different central frequencies for acceleration, velocity and displacement and ft,d) < d " ' < /„(0)- It was shown in earlier works by Trifunac and Westermo (1976a,b, 1977, 1978, 1982) and W estermo and Trifunac (1978, 1979) th a t duration of strong ground m otion increases as frequency decreases. Therefore, in each particular channel, on average, dura < durv < durd. log [Fa(f)] a ) 45 fi f{f> log [f] 4 , log [Fv (f)] f i fl)v) log [f] F ig . 2.1 3 Change of the position of the central frequency in a narrow band [/ii fi] signal due to integration. a) Fourier spectrum of the acceleration, w ith the central frequency /o°^; b) Fourier spectrum of the velocity, w ith the central frequency . 46 T his trend can be noticed for the data displayed in Fig. 2.12. However, the relative increase in duration (durv — dura)/dura <C 1 and, as we will see later, is m uch sm aller than the standard deviation of dura, durv and durj from any of the regression models discussed later, and, therefore, will not disturb our analysis. This allows us to consider acceleration, velocity and displacem ent records as one uniform set of data. 4 7 3. SOM E PH YSICAL CONSIDERATIONS ON PA RAM ETERS INVOLVED IN THE REGRESSION ANALYSIS Properties of the acceleration, velocity and displacement recorded at a p ar ticular site from a given earthquake depend on three basic factors: source, prop agation p a th and site-specific features at the recording station. Each of these factors can be characterized by many param eters. However, only a few of them can be found readily available for m ost records, Thus, only a few can be in cluded in the regression analysis. The presence of some “detailed” param eters (like am ount of slip in the source, for example) in the final equation should be of little use because it is very hard to predict the value of such param eters for fu tu re earthquakes in the present state-of-the-art in seismology. Therefore, we will restrict ourselves only to consideration of simplest characteristics of the source, of the p ath propagation and of the local site features. Physical basis for including various param eters in regression analysis of the duration of earthquake related strong ground shaking is considered in this C hap ter. Several conventional param eters such as m agnitude, epicentral distance, Modified M ercalli Intensity, depth of sediments at the recording site (or simpli fied geological classification in term s of variable s) were used in earlier works (Trifunac and Brady, 1975b; Trifunac and Westermo, 1976a, 1977, 1978, 1982; W estermo and Trifunac 1978, 1979) and are only briefly discusses here. The new param eters which are introduced for the first time in this study will be defined and described in more detail. We designate the observed and the predicted duration of strong ground mo tion as dur. W hen we want to show explicitly w hat param eters of the model 4 8 are, we list those in brackets. For example, dur = dur (M , M 2, A) should be read as “the total duration of strong ground m otion as a function of m agnitude, m agnitude squared and epicentral distance” (free constant is always included in the form ula by default). Occasionally, we will refer to the specific functional de pendence of the duration on one, two, or several param eters, which cannot serve as a com plete model of duration, bu t can be treated as one of the contributing term s in the additive formula of the complete model. In this case, the underlined dur will be used. For exam ple, dur(M) stands for the functional form th a t only shows th e dependence of duration on the m agnitude of the earthquake. Similarly, dur(s.sr) will designate th a t we wish to concentrate only on th e dependence of duration on site param eters s and s l . 3.1 M agnitude Let us consider the duration of strong ground motion as the sum of three portions: dur T o ■ + ■ Tj\ + Tr e g jo n ( ancj si t e ) ( ^ ‘ -Q where dur is the total duration of acceleration, velocity or displacem ent, r0 stands for duration of the rupture process in the source, t a shows the increase in du ration due to dispersion effects and T reg;on (ancj a;te) describes the effects of some specific regional or local features. T he duration of an earthquake source T o may be m ost directly related to some fault dimension. Although not every earthquake can be described by a fault rupture initializing at one end of the fault and propagating tow ards another 49 one, the first estim ate of tq could be the length of the source L divided by the average velocity of dislocation v, which is of the order of shear wave velocity (3a (Brune, 1970). More generally, r0 = A — , A = 0.5 — 1, (3.2) v where, for unilateral faulting A « 1, and for bilateral A « 0.5. From the definition of seismic m om ent, Mo — JiAu, where /I is the average shear m odulus in the source region and u is the am ount of slip averaged over the area of the fault plane A, and the relationship for the work done during dislocation, oAu, where o is the m ean operating stress across the fault plane, the energy E released during ru p tu re process can be obtained: „ gMq E = -----------------------------------------------------3.3 A i Seismic efficiency factor r] < 1 (Wyss and Brune, 1968) relates the radiated seismic energy E a and energy available from rupture: E a = rjE. (3.4) G utenberg-Richter (1956) energy-m agnitude formula In E a — a + bM (3.5) and H ousner’s (1955) equation E = C-exp (3M) allow one to relate the seismic moment of a shock w ith its m agnitude. Many correlation equations “seismic mom ent - m agnitude” for several m agnitude scales and different regions can be found in literature, e.g.: 50 log10Mo = 1.7Ml + 15.1 3 < M l < 6 (Wyss and Brune, 1968), log10 Mo = M s + 19.2 5 < M s < 7 ( Brune and King, 1967), log10Mo = 1.5M l + 16.0 3 < M l < 7 (Thatcher and Hanks, 1973). A new m om ent m agnitude scale was introduced (Hanks and K anam ori, 1979), in which M is defined by 2 M = - log10 M 0 - 10.7. u Thus, the m agnitude M and log10M 0 can be related linearly. However, we need to relate seismic moment Mo and the fault length L to use B rune’s formula (Eq. (3.2)). Various geometries of the fault can be assumed: circular, rectangular or any other. This gives for the source area A = k ti1 1 < q < 2, (3.6) where k is some constant. A simplified observation th a t the average slip in big earthquake is proportional to the rupture length (Housner, 1955; Scholtz, 1982), gives u = k'L, (3.7) where k' is some constant. Combining Eqs. (3.2)-(3.7), the expression for the source duration can be w ritten as T° = ue x p ) ■ • e x p ( 4 t " ) ' (3 - 8 ) or, renam ing the coefficients, To = aexp(pM). (3.9) 51 T he applicability of Eq. (3.9) w ith constant a and /? throughout the wide m agni tude range depends on w hether earthquakes can be considered self sim ilar. For this to hold, constant apparent stress, t]o, and similar source geom etry (at least) are required. M any investigators support the assum ption of constant apparent stress, which they related to the constant stress drop by assum ing th a t th e stress drop is com plete (Kanamori and Anderson, 1975; Geller, 1976). O thers show evidence of great variations of apparent stress rjo observed in different regions (R autian and K halturin, 1978; R a u tia n e t al., 1981). Trifunac (1993) emphasizes th a t the question of self sim ilarity should be addressed for each region separately. He concludes th a t apparent stress rjcr is relatively constant for strike slip events in California (slowly increasing from 10 to 20 bars w ith change in m agnitude from M = 3 to M — 8), but the geometry of the source cannot be considered sim ilar for events of different magnitude. W hen the earthquake is sm all, it is reasonable to assume th a t both dimensions of the source, the length L and the w idth W, grow w ith m agnitude. Once the w idth W reaches the size of the seis- mogenic zone, the increase of the area of the fault A is only possible through the increase in source length L. The last statem ent is true for San A ndreas strike- slip type earthquakes. The results of Chinnery (1969) suggest the growth of W versus m agnitude up to W « 5 -r 20 km and M « 5 — 6 and no increase for larger earthquakes (Fig. 3.1). It follows from here th at, for sm all earthquakes, L ~ W and 7 « 2 in Eqs. (3.6) and (3.8). For large earthquakes, the w idth of the source is “fixed” and A ~ L in Eq. (3.6), i.e., 7 = 1 . Trifunac (1993) also questions the linear relationship between the slip u and the source dimensions (Eq. (3.7)). His models, based on linking strong m otion d ata on Fourier spectral am plitude w ith various estim ates of the source param eters, indicate th a t u may 8 7 - 5 - 4- 3 - 2 - Fault width, W Chinnery (1969) •A y S 3 0 4 * n x * 9 s ° * * o * ■ ♦ y 4 # A G- S4 •'6 * ,4 ^ o o 4- S 3 o o 4 £ 3 F ault len g th , L Chinnery (1969) “ a Gibowicz (1973) * Thatcher (1972) ★ Hanks and W yss (1972) o W yss and Molnar (1972) -------- Wideman and Major (1967) -------- Ozawa (1970) Takemoto and Takada (1970) Trifunac (1972a), San Fernando, P waves Trifunac (1972a), San Fernando , S waves o Trifunac (1972b), Imperial Valley T 0 T 3 l o g i o L, l o g 10 W ( k m ) F ig . 3.1 M agnitude versus log10 L and log10 W . Fault length (L) grows w ith m agnitude in an exponential manner, while fault w idth (W) grows at a slower rate and appears to be bounded by the w idth of the seismogenic zone (up to 20 km in California). C n 5 3 not be related linearly neither to L, nor to W. He also models the duration of the source radiation during big earthquake (L » W) as L W , To~ v + W.' ( ’ where v « 0.6/?B . Comparison of Eqs. (3.10) and (3.2) indicates th a t modeling duration of the source rupture by Eq. (3.2) in the wide range of m agnitudes may require the dependence of the coefficient A on magnitude. We conclude from the above discussion th a t variations in all param eters, and also some inperfections of the relations (3.2)-(3.7) will cause some “scattering” around the average value of source duration, which could otherwise be reasonably represented by Eq. (3.9). We will study the details of this later (see C hapter 6 ). The adequacy of Eq. (3.9) can be indirectly verified, by checking the corre sponding equation for the source length: L = a! exp(/?M ), (3-H ) where a ' — av/X. Static strain observation give, for example: M l = 3.25 + 1.71og10 L (W ideman and M ajor, 1967); M l = 3.2 + 2.25 log10 L (Ozawa, 1970) M l = 2.6 + 2.21 log10 L (Takemoto and Takada, 1970). T he above relationships, and selected data determ ined by direct m easurem ents, using the length of aftershock zones, or B rune’s spectral theory (Chinnery, 1969; Gibowicz, 1973; T hatcher, 1972; Hanks and Wyss, 1972; Wyss and M olnar, 1972; Trifunac, 1972a,b) are shown in Fig. 3.1. The scatter of the d ata points on “m ag nitude versus log10 L” is noticeable, but the general linear relationship between M and log10 L could be acceptable. Therefore, given th at v is approxim ately 5 4 constant and th a t the change of A w ith M is less than th e scatter of d a ta on Fig. 3.1 (Trifunac, 1993), we may assume th at To = a exp(0M) gives reasonable representation of the physical relationship m agnitude-source duration. M ore over, we are going to treat a and /?, as well as To itself, as functions of frequency, allowing additional flexibility in the description of the source. We discuss now the com putational aspects of im plem enting Eq. (3.9). Direct attem p t of fitting To = dur(M) = aexp(/?M ) to d ata results in nonlinear regres sion analysis. The disadvantage of this procedure comes from the poor stability of nonlinear fitting, especially for noisy data. However, attem p ts of com plet ing such analysis for some frequency bands were made, and some results are discussed in C hapter 6 . For more general case (which includes all frequencies), the linearization of Eq. (3.9) was considered. The fact th a t the Taylor series exp(x) = converges everywhere on real axis (except ±oo) allows one to use simple quadratic approxim ation: M f ) ~ « i (/) + «2[f)’M + a 3( /) - M 2, (3.12) where a, ( /) , i = 1 t 3, are frequency dependent coefficients to be defined from regression analysis. This is the final functional form adopted in this study for the description of the duration of the rupture process at the source as function of m agnitude. According to Eq. (3.1), r0 = dur(M), as defined by Eq. (3.12), is one of th e term s th a t contribute to the total duration. Previous studies by Trifunac and W estermo did not include the quadratic term o3 -M 2 because of insufficient d a ta and the absence of the “cleaning up” procedure (see A ppendix B). Fig. 3.2 shows the dependence of the total duration of strong ground motion (as determ ined from horizontal and vertical band-pass filtered com ponents of 60 50 40 30 O < D 20 tn ^ 10 5 5 0 °3 O ■ p h 60 0 3 50 40 P Q 30 2 0 10 0 f0 =.075 Hz f0 - . 1 2 Hz I i I : i ; • M a g n i t u d e F ig . 3 .2 a Channels # 1 and #2: observed duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus m agnitude of earthquake, for horizontal and vertical com ponents com bined. W ith the increase of m agnitude, the rupture tim e grows, causing an increase in duration. However, the possibility to detect this growth depends on the frequency of m otion. M a g n i t u d e F ig . 3 .2 b Channels # 3 and # 4 : observed duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacement, is plotted versus m agnitude of earthquake, for horizontal and vertical com ponents combined. W ith the increase of m agnitude, the rupture tim e grows, causing an increase in duration. However, the possibility to detect this growth depends on the frequency of motion. 60 50 40 ^ 30 O < D 20 in ^ 10 C °3 o ■ p h 60 H -> c tf 50 * h 40 30 20 10 0 f0 =-63 Hz : i • t : ■ • ■w i « i ; : l S ! 1 1 • i i» ; i * . i i ii i I : I I • a L f0 =1.1 Hz •! i l i i i '■ I i ! i • i Mjlii li l I I ' I 5 6 M a g n i t u d e 57 8 F ig . 3 .2 c Channels # 5 and # 6 : observed duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus m agnitude of earthquake, for horizontal and vertical com ponents com bined. W ith th e increase of m agnitude, the rupture tim e grows, causing an increase in duration. However, the possibility to detect this growth depends on th e frequency of m otion. 60 - 50 - f0 =1.7 Hz 58 40 30 O 20 w ^ 10 _ o. I •! : 1 . i I i | I ■ !l : i i 11 ! i ! • ■ ■ ■ 40 30 20 10 M a n i t u d e F ig . 3 .2 d Channels # 7 and # 8 : observed duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus m agnitude of earthquake, for horizontal and vertical com ponents combined. W ith the increase of m agnitude, the rupture time grows, causing an increase in duration. However, the possibility to detect this growth depends on the frequency of m otion. 59 60 50 40 30 0) 20 10 o 40 30 20 10 M a n i t u d e F ig . 3 .2 e Channels # 9 and # 10: observed duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacement, is plotted versus m agnitude of earthquake, for horizontal and vertical com ponents combined. W ith the increase of m agnitude, the rupture time grows, causing an increase in duration. However, the possibility to detect this growth depends on the frequency of motion. 60 60 f0 =13 Hz 50 40 30 10 O ■pH 60 40 30 20 10 n i t u d e M a F ig . 3 .2 f Channels # 11 and # 12: observed duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus m agnitude of earthquake, for horizontal and vertical com ponents com bined. W ith th e increase of m agnitude, the rupture tim e grows, causing an increase in duration. However, the possibility to detect this growth depends on the frequency of m otion. 61 acceleration, velocity and displacement) on the m agnitude of an earthquake. This d a ta indicate th a t the dependence becomes “stronger” when the frequency of m otion increases. We will discuss the reasons which cause this behavior in C hapters 4 and 6 . 3.2 Epicentral Distance We consider next the second term in Eq. (3.1) which describes the influence of epicentral distance on duration of strong ground m otion. Significant portion of strong earthquake ground motion comes from surface waves (Trifunac, 1971a) at least for low and interm ediate frequencies. T he surface waves are known to be subject to dispersion. To describe this phenom enon we approxim ate the propagation p ath between the source and the station by a layered m edium . The classical frequency equation for Love waves in a soft layer above elastic half space (Love, 1911) is: tan wh' / /?2 /?' V 1 ~ c2 y '\j c2/( /? ') 2 - 1 ’ where h' the thickness of the layer, c is the horizontal phase velocity of the propagating mode shape (which depends on frequency w and the num ber of the m ode), /?' and /? designate the shear wave velocities in the layer and in the halfspace respectively and y' and y stand for their rigidities. Fig. 3.3 describes Eq. (3.13) graphically. Only a finite number of modes is allowed for a given frequency and this num ber increases with frequency ( / = w/27r). T he curve, corresponding to each mode, starts at the level c — j3 (at some frequency) and gradually approaches c = as frequency goes to infinity. This results in the u -p • M o o < d > w < d A P " H cti -p 3 o N ■ i — i P O K m a x m m v Frequency f F ig . 3.3 Phase velocities c of the modes of Love waves, propagating in a “soft” layer which rests on top of a “hard” halfspace, are plotted versus frequency (solid thin lines). j3' and /? represent shear wave velocities in the soft layer and in the halfspace (solid thick straight lines). Dashed lines (effective maximum cmax and effective minimum cm- m) bound the shaded region of effectively seen phase velocities and show the effective concentration of mode velocities near /?; at high frequencies. 0 5 to 6 3 effective concentration of phase velocities near c — f}'. Resulting “effective” m axim um cmax and “effective” minimum cm;n phase velocities available at each frequency are designated by dashed curves in Fig. 3.3. The effectively “seen” range of c is illustrated by the shaded region. T he general features of the solution for the simple “one layer over a halfspace” model are preserved in multilayered models (Trifunac, 1971b, Lee and Trifunac, 1987b). Surface (Love and Rayleigh) waves, propagating through these more realistic models still possess the effective limits cm- , n and cmax on phase veloc ities of propagation. The behavior of these limits as functions of frequency is qualitatively the same as in Fig. 3.3. The contribution to the total duration th a t results from dispersion and the fact th a t different types of elastic waves propagate w ith different velocities, i.e. ta term in Eq. (3.1), is clearly a linear function of source-to-station distance A. At least two distances, hypocentral and epicentral, can be used here. To simplify the analysis and because hypocentral distance is not known for many records due to the lack of information about the focal depths, we will use the epicentral distance for the models involving m agnitude of an event. Considering dispersion and multimode propagation of surface waves, the epicentral (instead of hypocentral) distance being the main param eter characterizing the path of the wave is also appropriate physically. On the other hand, m ajority of earthquakes considered in this study are shallow strike-slip type California earthquakes, and, thus, hypocentral and epicentral distances do not differ too much. Nevertheless, the effect of the focal depth param eter can be seen, and is discussed in Appendix E. 64 In the case of the models involving Modified Mercalli Intensity, the hypocen tra l distance A ' will be used. This is the case because about 2/3 of the M M I d a ta used to construct the models of duration were not observed at a site, but estim ated using the Eq. (2.1) which involves a “representative distance,” defined by Eqs. (2 .1b )-( 2 .1d). This distance cannot be used in the regression analysis of duration models, involving MMI (because the earthquake m agnitude is assumed to be not known), and so the hypocentral distance is utilized to approxim ate the representative distance. We will further discuss the question concerning the models which involve MMI in the next session. The dispersion term t& = dur(A) in Eq. (3.1) can be w ritten as t a ( / ) = o4(/)-A , (3.14) where a4f) = c - 77) “ T^TTY (315) ° m i n \J ) c m ax\J ) T he direct calculation of cmm(/) and cmax(/) represents a problem by itself, so we will determ ine < z 4(/) from regression analysis. However, a simple estim ate can be made: 1 1 / 1 1 C m in ^ m a x ^ m i n ^ m a x w here umin and umax are the smallest and the largest wave velocities of the layered m edia in the region. W ith vmax = 6 km /sec and vm;n = 2.5 km /sec, a 4 < 0.23 sec/km . A nother pair vmax = 4 km /sec and um;n = 2 km /sec gives a 4 < 0.25 sec/km . As we will see later, these values of a 4 are in perfect agreem ent w ith our regression analysis (the value 0.22 sec/km serves at an upper limit of a 4( /) in th e m ajority of models considered). 6 5 For very high frequencies (/> 1 0 Hz) the main contribution to strong m o tion comes from the body waves (see C hapter 8). In this case the increase of the duration of the strong ground motion with the increase of epicentral (or hypocentral) distance might be partially attributed to the “forward scattering” on the random velocity structure of the lithosphere. The m echanism of this scattering was described by Sato (1982a) in his study of the am plitude atten u a tion of impulse waves in random media. Later he specifically studied the effect of the broadening of the seismogram envelopes in the random ly inhomogeneous lithosphere (Sato, 1989), and found th at this broadening (i.e. increase of strong m otion duration) increases w ith distance traveled and can be explained by the diffraction effects. W hen a wave with length A propagates through a m edia w ith random ly distributed fluctuations of velocity v, the “longwave com ponent” of these fluctuations (changes of v over the length much larger than A) causes the shift in the tim e of the wave’s arrival while the “shortwave com ponent” results in the random scattering and the decrease of the am plitude of the wave. A longer propagation tim e causes more “forward scattering” to occur and bigger variety of arrival times to become possible. As a result the effective duration of every pulse increases w ith distance, and the duration of strong ground m otion grows w ith A. The sim plest way to account for this growth is to consider the linear dependence (3.14), where, however, the coefficient ( 1 4 (f) should have the physical m eaning, different from the one assumed by Eq.(3.15). The observed d ata are shown in Fig. 3.4, which presents the dependence of the duration of strong ground motion on the epicentral distance. Horizontal and vertical com ponents of the band-pass filtered acceleration velocity and displace m ent are shown together. The absence of the growth of duration as the function 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 3 .4 a Channels # 1 and # 2 : observed duration as determ ined from the band pass filtered acceleration, velocity and displacement, is plotted versus epicentral distance, for horizontal and vertical components combined. T he increase of duration w ith distance is believed to occur due to dispersion (low frequencies) and scattering (high frequencies). The absence of the growth at large distances may be partially attributed to the weakening of the signal and late triggering of the recording device. 60 f0 =.21 Hz 50 40 30 C D 20 10 0 20 40 60 80 100 120 140 160 180 r-l 60 ctf 50 U 40 30 20 10 0 s ' V, _L fn =.37 Hz _L _L 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 3 .4 b Channels # 3 and # 4 : observed duration as determ ined from the band pass filtered acceleration, velocity and displacement, is plotted versus epicentral distance, for horizontal and vertical components combined. T he increase of duration w ith distance is believed to occur due to dispersion (low frequencies) and scattering (high frequencies). The absence of th e growth at large distances may be partially attributed to the weakening of the signal and late triggering of the recording device. 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 3.4 c Channels # 5 and # 6 : observed duration as determ ined from the band pass filtered acceleration, velocity and displacement, is plotted versus epicentral distance, for horizontal and vertical components combined. T he increase of duration w ith distance is believed to occur due to dispersion (low frequencies) and scattering (high frequencies). The absence of the growth at large distances may be partially attributed to the weakening of the signal and late triggering of the recording device. 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 3 .4 d Channels # 7 and # 8 : observed duration as determ ined from the band pass filtered acceleration, velocity and displacement, is plotted versus epicentral distance, for horizontal and vertical components combined. T he increase of duration w ith distance is believed to occur due to dispersion (low frequencies) and scattering (high frequencies). The absence of the grow th at large distances may be partially attributed to the weakening of the signal and late triggering of the recording device. 60 - 50 - 70 f0 =4.2 Hz 40 o •i - h 60 a so 40 30 f0 =7.2 Hz Epicentral distance (km) F ig . 3 .4 e Channels # 9 and # 10: observed duration as determ ined from the band-pass filtered acceleration, velocity and displacement, is plotted versus epi central distance, for horizontal and vertical components combined. T he increase of duration w ith distance is believed to occur due to dispersion (low frequencies) and scattering (high frequencies). The absence of the growth a t large distances may be partially attributed to the weakening of the signal and late triggering of the recording device. 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 3 .4 f Channels # 1 1 and # 12: observed duration as determ ined from the band-pass filtered acceleration, velocity and displacement, is plotted versus epi central distance, for horizontal and vertical components combined. T he increase of duration w ith distance is believed to occur due to dispersion (low frequencies) and scattering (high frequencies). The absence of the growth a t large distances may be partially attributed to the weakening of the signal and late triggering of the recording device. 7 2 of epicentral distance A, at large A, could be attributed to the weakening of the signal at a large A, which may lead to the late triggering of the recording in strum ent, and, therefore, to the underestim ate of the duration of strong ground m otion. Ju st for the completeness of the discussion dealing w ith the dependence of the duration on m agnitude and on distance to the source, we m ention here two m ore term s which describe the dependence of duration on hypocentral depth H: a 1 7 {f)-H + a 1 8 {f)-HM. (3.16) T he reasons for choosing this particular form of the dependence dur(H) (or dur(H , M)) and the results of the regression analysis of the models which include these term s, are discussed in Appendix E. 3.3 Modified Mercalli Intensity Instrum ental d ata provides valuable m aterial for investigation of the strong ground m otion. However, it takes time to accumulate the sufficient am ount of d a ta to initiate any research which deals with analysis of any sufficiently general statistical trends. As a substitute for the instrum ental data, a qualitative description of the earthquake in term s of Modified Mercalli Intensity (MMI) scale (W ood and N eum ann, 1931) or its equivalents can be used. Modified M ercalli Intensity rating of strong ground m otion is based on the subjective assessment of the level of shaking and on the resulting dam age as experienced by witnesses and as estim ated by experts who visit and analyzed the d a ta in the area after the earthquake. 7 3 Historically, the M M I scale has evolved dealing w ith older structures. The dam age of the m odern tall structures (with low natural frequency), long span bridges or large dam s is hard to describe by the MMI scale because such struc tures are not even mentioned in the description of the scale. Thus, the cor relations of frequency dependent duration of strong ground shaking w ith the M odified M ercalli Intensity at the site, Im m , should be very useful not only for duration studies, b u t also because of the possibility to obtain the new informa tion about the characteristics of the MMI scale at the low-frequency end. The trends of the frequency dependent duration with respect to the Modified Mercalli Intensity can also serve as a basis for the correlation of this scale w ith various instrum ental scales. Trifunac and Westermo (1976b, 1977) and Westermo and Trifunac (1979) presented the equations relating the frequency dependent duration w ith I m m and the recording site conditions (geological param eter s or the depth of sedim ents). In this work, we will also include other param eters (discussed later), and will consider two groups of equations. Both groups treat the first the two term s, To and t a , in Eq. (3.1) together because the source and the propagation effects cannot be uncoupled when the shaking at a site is measured by the M M I scale. One group will deal w ith the simplest description and will include r0 + t& = dur(lMM). i.e. scaling in term s of Im m only. The other group, more descriptive, will consider tq + t^ = dur{ I \ * A), th at is, it the distance to the source will be included as a param eter. We consider this last representation here, although we assume th a t no instrum ental inform ation on the earthquake is available and the precise position of the source is not known. There are at least two reasons for th a t. First, the consideration of tq + t a = dur(lMM. A) can provide some useful 7 4 inform ation on the nature of the MMI scale. Second, the position of the epicenter can be approxim ately located (even if there is no instrum ental d ata available) by creating the m ap of Modified Mercalli Intensities observed at various sites and finding th e point where I m m reaches its maximum. If we do not want to include distance in the regression model, then the sum of the first two term s from Eq. (3.1) has the simplest form To{f) + T A ( / ) = aig(f)-lM M - (3-17) T he coefficient aig(f) is of “contradictive” nature. On one hand, the duration of strong ground m otion should be longer when I m m increases due to the assumed increase in the m agnitude of the event. On the other hand, the duration is thought to be shorter when I m m increases due to the decrease in the epicentral distance. As a result, different trends prevail at different frequency bands (see C hapter 5). This can be noticed in Fig. 3.5, which shows the observed duration as function of Modified Mercalli Intensity for all 12 channels. At low frequencies, duration of strong motion tends to decrease when intensity increases. A t high frequencies, duration grows w ith an increase of the intensity level. We next proceed to the discussion of the first group of regression equations, which considers To + r a as function of both Modified M ercalli intensity and the distance to the earthquake. To be consistent w ith the estim ate Eq. (2.1) which serves as a source of 2/3 of the I m m data, we would like to consider the representative distance A in our equations. The definition of this distance, A = \/A * + H 2 + L 2, (2.16) involves, besides the epicentral distance A and the hypocentral depth H , the “size” of the fault L, as it is “felt” at the site. This “effective” size was obtained II III IV V VI VII VIII IX X Modified Mercalli Intensity F ig . 3 .5 a Channels # 1 and # 2 : duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus Modified Mercalli Intensity. Vertical and horizontal com ponents are shown com bined for each channel. For the clarity of the figure, a stripe of finite w idth on the horizontal axis is assigned to each intensity level I m m , and the abscissa of each d a ta point, corresponding to this Im m , is chosen arbitrarily w ithin this stripe. A t low frequencies, duration tends to decrease when intensity increases. At high frequencies, duration grows with the increase of the intensity level. II III IV V VI VII VIII IX X Modified Mercalli Intensity F ig . 3 .5 b Channels # 3 and # 4 : duration of strong ground m otion as determ ined from th e band-pass filtered acceleration, velocity and displacem ent, is plotted versus Modified M ercalli Intensity. Vertical and horizontal com ponents are shown com bined for each channel. For the clarity of the figure, a stripe of finite w idth on th e horizontal axis is assigned to each intensity level I m m , and the abscissa of each d a ta point, corresponding to this I m m , is chosen arbitrarily w ithin this stripe. A t low frequencies, duration tends to decrease when intensity increases. At high frequencies, duration grows w ith the increase of the intensity level. II III IV V VI VII VIII IX X Modified Mercalli Intensity F ig . 3.5 c Channels # 5 and # 6 : duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus Modified Mercalli Intensity. Vertical and horizontal com ponents are shown com bined for each channel. For the clarity of the figure, a stripe of finite w idth on th e horizontal axis is assigned to each intensity level Im m > and the abscissa of each d ata point, corresponding to this I m m > is chosen arbitrarily w ithin this stripe. A t low frequencies, duration tends to decrease when intensity increases. At high frequencies, duration grows with the increase of the intensity level. II III IV V VI VII VIII IX X Modified Mercalli Intensity F ig . 3 .5 d Channels # 7 and # 8 : duration of strong ground m otion as determ ined from th e band-pass filtered acceleration, velocity and displacem ent, is plotted versus Modified M ercalli Intensity. Vertical and horizontal com ponents are shown com bined for each channel. For the clarity of the figure, a stripe of finite w idth on th e horizontal axis is assigned to each intensity level I m m , and the abscissa of each d a ta point, corresponding to this I m m , is chosen arbitrarily w ithin this stripe. A t low frequencies, duration tends to decrease when intensity increases. At high frequencies, duration grows w ith the increase of the intensity level. II III IV V VI VII VIII IX X Modified Mercalli Intensity F ig . 3 .5 e Channels # 9 and #10: duration of strong ground m otion as de term ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus Modified Mercalli Intensity. Vertical and horizontal com ponents are shown combined for each channel. For the clarity of the figure, a stripe of finite w idth on the horizontal axis is assigned to each intensity level I m m , and the abscissa of each d ata point, corresponding to this I m m , is chosen arbitrarily w ithin this stripe. A t low frequencies, duration tends to decrease when intensity increases. At high frequencies, duration grows with the increase of the intensity level. 80 60 f0 =13 Hz 50 40 30 < D 20 III VII VIII o 60 50 40 Q 30 20 10 III VIII VII Modified Mercalli Intensity F ig . 3 .5 f Channels # 11 and # 12: duration of strong ground m otion as de term ined from the band-pass filtered acceleration, velocity and displacement, is plotted versus Modified Mercalli Intensity. Vertical and horizontal com ponents are shown combined for each channel. For the clarity of the figure, a stripe of finite w idth on the horizontal axis is assigned to each intensity level I m m , and the abscissa of each d ata point, corresponding to this I m m , is chosen arbitrarily w ithin this stripe. At low frequencies, duration tends to decrease when intensity increases. At high frequencies, duration grows with the increase of the intensity level. 81 by m ultiplying the linear approxim ation of the fault dimension, L (M ): for M ~ 3 L(M) = 0.2 km, for M = 6.5 L(M) = 17.5 km. by a function: (2 .Id) (2 .1c) T he logic th a t leads to Eq. (2.1c) is as follows (Trifunac and Lee, 1985, 1990). of the fault, the whole fault will be “felt” at this site. It may be questionable, however, th a t the sam e assumptions can be true for small distances from the fault. This is related to the fact th a t the fault size is not present in the expression of Fourier am plitudes of the near field motion (Brune, 1970; Trifunac, 1973). So, in Eq. (2.1), the “effective” size of the fault, L, is obtained from the estim ate of the fault length L(M) m ultiplied by a factor. This factor is equal to zero when the epicentral distance is zero. At A = L(M), the factor is equal to 0.9 which m eans th a t 90% of the fault size is “felt.” When the epicentral distance grows, the value of this factor gradually approaches 1. To take advantage of the representative distance (Eq. (2.1b-d)), some esti m ate of the source dimension L(M) should be available. We are do not wish to use the m agnitude of the earthquake in the development of the models of strong ground m otion duration in term s of the Modified Mercalli Intensity. The use of m agnitude would contradict the assum ption that no instrum ental d a ta are avail able. This restriction prevents us from taking advantage of the representative distance A. The best we can do it to consider the hypocentral distance It is reasonable to assume th a t if the site is “far enough” relative to the size A ' = \ / A 2 + H 2 8 2 as an approxim ation of A. The biggest discrepancies will occur at the distances com parable w ith the size of the fault and slightly further. Of cause, the use of the hypocentral distance requires the knowledge of the hypocentral depth H, which is not available if there were no instrum ental records of the earthquake. However, in the regions with limited seismozenic zone (like San Andreas fault system ), the prevailing hypocentral depth can be estim ated w ithout much error. This value can serve later as an approxim ation of the hypocentral depth for all earthquakes from this region. We assumed H — 5 km for all sources w ith unknow n hypocentral depth. This particular value comes from the distribution of records produced by the sources with known H (Fig. 2.3). A great num ber of these records were obtained during several big earthquakes w ith relatively deep hypocenters: San Fernando, 1971, H — 13 km; Borrego m nt., 1968, H — 9.2 km; Im perial Valley, 1979, H = 12 km and Coalinga, 1983, H = 9 km. The rest of th e records have the distribution with the average H m uch less th an it is seen from Fig. 2.3. We also w anted to pick such a value for the depths of the sources w ith unknown coordinates of hypocenters, which would be easy to remember. We now tu rn to the discussion of the particular functional form of the term s which should be included in the equations scaling the duration of strong ground m otion in term s of Modified Merclli Intensity at a site and the distance to the source. T he estim ate of /m m through the m agnitude of the earthquake and the representative distance A , Eq. (2 .1), and already established functional form of dur{M) = To, Eq. (3.12), suggest that the dependence of the duration of strong earthquake ground motion on Modified Mercalli Intensity and on distance to the source should be coupled. Thus, the simplest function w ith suae properties 8 3 accounts for the first two term s in Eq. (3.1) is To ( /) + TA ( f ) = aig(f)-IMM + a4(/)-A ' + (3.18) T he d a ta (pertinent to the functional form of Eq. (3.18)), are displayed in Fig. 3.6. For each channel separately, the observed duration of strong ground m otion is shown as a function of two param eters: the Modified M ercalli Intensity and th e hypocentral distance. D ata for the horizontal and vertical com ponents are displayed together. The duration is shown averaged over ranges of A ' and I m m given by J < A ' < AJ+1, A; = /-20, / = 0 -r 8 , [ { I m m ) = (I m m ) h > { I m m ) k = k + 2 , k — 0 + 8 . T he intervals of A{ and (Tmm)a: are indicated by the dashed mesh, w ith the density of shade inside every “box” proportional to the averaged duration. As it can be seen from the data displayed, the behavior of duration as a function of I m m vaties for different hypocentral distances A '. To conclude this section we will emphasize once more th a t we will use two representations of To + rA : the first group of the regression equations relating the duration of strong ground motion and the Modified M ercalli Intensity will use To + 7"a = dur[Ihfhf. A ') as defined by Eq. (3.18). The second group will use To + t a = dur{lMM) as defined by Eq. (3.17). As the first group of th e equations directly includes the dependence on distance, these models are expected to have sm aller residuals, bu t also to be more region dependent. The latter follows from the fact th a t different regions have different attenuation laws. All components f0 =.075 Hz 0 20 40 60 80 100 120 140 160 VIII VIII + -> ■ pH C O S o -p • — i VII VII III 0 20 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : 26 32 38 45 51 57 F ig . 3 .6 a Channels # 1: observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site I m m ■ The duration is shown averaged in the various ranges of A ' and I m m , specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of A 7 and I m m ■ Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m i s different for different A '. All components f0 =.12 Hz 0 20 40 60 80 100 120 140 160 ^ VIII f - g vi * V > — I Iff VIII VII 44 21 III 0 20 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : It: 16 24 32 40 48 56 F ig . 3 .6 b Channels # 2: observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site I m m • The duration is shown averaged in the various ranges of A 1 and ImMi specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of A ' and I m m • Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m is different for different A '. All components f0 =.21 Hz 20 40 60 80 100 120 140 160 21 VIII 320 177. VII - X i _____ ------1 152 124 III 0 20 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : Tl 13 22 30 39 47 56 F ig . 3.6c Channels # 3 : observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site I m m ■ The duration is shown averaged in the various ranges of A ' and I m m » specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the d ata points available for the particular range of A' and Im m • Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m is different for different A '. All components f0 =.37 Hz 0 20 40 60 80 100 120 140 160 VIII 70 20 194 VII 81 43 17 III 120 140 160 60 80 100 0 20 40 Hypocentral distance (km) A v . d u r a t i o n s e c : F ig . 3 .6 d Channels # 4 : observed duration of strong ground motion as a function of the hypocentral distance, A , and the Modified Mercalli Intensity at the recording site I m m • The duration is shown averaged in the various ranges of A and I m m 5 specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed boxes” give the number of the data points available for the particular range of A ' and I m m - Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of Im m is different for different fit t 13 20 27 35 42 49 JU 9I9JJip JOJ JlI9 J9 JJip SI W W j JO UOIJOUTIJ B SB UOtJBITip JO JO IA B qsq 9 q j ‘sp U B q j(0 U 9 n b 9 IJ 9UIOS UI ‘jBq^. U99S SI •I9qj9§OJ UM OqS 9IB SJU9UOduiOD {B01JI9A pUB JBJUOZIJOg •nvij PU B /V J° 82ubi xepoijiBd gqj ioj 9|qB|iBAB sjmod BjBp gqj jo jgqm nu gqj 9aiS uS 9Xo’ q„ pgqsBp ui sjgqranu gqj^ •gpBqs 9SU9p 9IOUI b OJ spuods9UOD uoijBinp ibSuoj 9qx 'M S9U I paqsBp gqj £q pggpgds ‘W Vj pu’ -g y jo sgSuBj snoiJBA gqj ui pgSBiOAB UMoqs si uoijBinp aqx a^is 8uipi099i oqj jb Xjisuojuj i[[B3J9jaj pggipop^ gqj puB ‘ y 99UBjsip |BJjU9oodXq gqj jo u o ip u n j b sb uoi'jom punoj§ B uoip jo uoi'jBinp p 9AJ9sqo :c^i spuuBq^) 9g*g "Sijj op ps i z iz si 8 :oas .:zz:t:::z: •::::::^ : • ! • ! • ! u o i^ vu n p 'ciy (ni5[) oon^^sip 091 OPT 021 001 09 09 OP 02 0 III ES IIA I-- BOS:; IIA OS 0 09T 02T 0 0 T 09 09 OP 0 2 OPT Z H 89'= °J s^uauoduioo jjV All components f0 = l .l Hz 20 40 60 80 100 120 140 160 4 -rrn \ nj;\ 1111111111 i m i it t t'i n 111111111111111111 n ri m 1 1 pi 1 1 n 111 [ti i i i i i i ± ranxLrji t I'i.g 4L'UJJ,£Lu 212 i s m 1 1 1 1 1 1 1 1 i i r 1 1^ * 11 ri -t i n i i i ........ I ...................I I 1 I I I I I II 1 I I I M I M I I I I I I III I I I I I 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : 11 17 22 28 33 F ig . 3 .6 f Channels # 6 : observed duration of strong ground m otion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site Im m • The duration is shown averaged in the various ranges of A ' and I m m > specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of A ' and I m m ■ Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m i s different for different A '. All components f0 =1.7 Hz 0 20 40 60 80 100 120 140 160 VIII VIII ■ r* v ii :213 VII 400 67 72 34 III III 0 20 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n ' • ' • ’ * :m : l j — !— i— I..* ..' • c I ............................. L.— ......~...........a x L i a i sec: 6 11 16 21 26 31 F ig . 3.6 g Channels # 7 : observed duration of strong ground motion as a function of th e hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site I m m - The duration is shown averaged in the various ranges of A ' and I m m > specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the num ber of the data points available for the particular range of A ' and I m m - Horizontal and vertical components are shown together. It is seen th at, in some frequency bands, the behavior of duration as a function of Im m is different for different A '. All components f0 =2.5 Hz o 20 40 60 80 100 120 140 £ m 0) -p a X IX VIII 3T i ir r r i txf'i 111111111 ii m m i ii 11 m n T iT | r m i m i p i i in 1111111 m "i i rpi i i t i i i i j- _ 3 • .'9 -v — . -• : 105 - - - - - - - - - - - - - - - -4 ._ _ _ _ ------ I ---------------- i----------------1 ----------------4--- 160 X 3.6 I I : • :p :E i4 i i VII =• 293 * •!: : :453::::!: : :193 m : :! 18 ; o I 17 VI =. 600 . : 429 : 4 0 6 :: :-\ 83 7 —s - - t - -r -|~ • * : r ■ T - - 4 ' - V =• 243 • ' • :103 •: ::! 100 9 — j „ . , ----- 21 = 27 i • I S) ' k r H F i I I 11 i I I I I I i i i I i i i i I I I I I i I I I I I I I I I ii IX VIII VII VI V IV III II 160 Hypocentral distance (km) A v . d u r a t i o n s e c : 15 2 0 26 31 F ig . 3 .6 h Channels # 8: observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site Imm- The duration is shown averaged in the various ranges of A ' and Im m > specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the num ber of the data points available for the particular range of A 1 and Im m- Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m is different for different A '. All components f0 = 4.2 Hz 20 40 60 80 100 120 140 160 90: 24 VIII -Trr r? 260 322 VII 594 325 204 49 284 63 III 0 2 0 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : . ' . 1 L . ..._L T tE 4- . £ : r 8 13 17 2 2 27 F ig . 3.6i Channels # 9 : observed duration of strong ground m otion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site Im m ■ The duration is shown averaged in the various ranges of A ' and Im m , specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of A ' and I m m ■ Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m is different for different A;. All components f0 = 7.2 Hz 20 40 60 80 100 120 140 160 ^ V I I I £ VII U i g w t v ► — I 2:4 VIII _fj [" i l i i ; i £ i l l : ^ i . 246 289 VII 579 286 141 41 316 III III 20 0 60 40 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : 1 2 17 2 2 27 F ig . 3.6j Channels #10: observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site Im m- The duration is shown averaged in the various ranges of A ' and ImMi specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the paxticulax range of A ' and I m m - Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m i s different for different A '. All components fo=13 Hz 20 40 60 80 100 120 140 160 S ' ™ * 5 5 vn g n * v i — i 59: : 11 VIII i .1 ri a i 170 73 VII •422 160 46 23Q 32 III III 0 20 40 80 60 10 0 1 20 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : E l 1 0 14 18 2 2 F ig . 3.6k Channels #11: observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site Imm- The duration is shown averaged in the various ranges of A ' and I m m j specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of A7 and I m m - Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m is different for different A '. All components fo =21 Hz 20 40 60 80 100 120 140 160 37 VIII .91 VII 71 115 III 0 20 40 60 80 100 120 140 160 Hypocentral distance (km) A v . d u r a t i o n s e c : 1 j 1 0 1 2 F ig . 3.61 Channels #12: observed duration of strong ground motion as a function of the hypocentral distance, A ', and the Modified Mercalli Intensity at the recording site I m m • The duration is shown averaged in the various ranges of A ' and I m m , specified by the dashed mesh. The longer duration corresponds to a more dense shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of A ' and I m m - Horizontal and vertical components are shown together. It is seen th a t, in some frequency bands, the behavior of duration as a function of I m m i s different for different A '. 96 3.4 Vertical and Horizontal Characteristic Dim ensions of the Sedimentary Basin D eviations from the uniform layered earth model may occur anywhere along the p ath of the waves propagating from the fault to the recording site. For the San Fernando, 1971, earthquake, which generated large percentage of records in our d a ta base, substantial deviations from multilayered model dom inate in the whole region, which covers the location of the fault and of the recording strong m otion stations. Typically, these deviations occur due to the appearance of the basem ent rock on the surface of the earth, so th a t several upper kilometers of the e a rth ’s crust can be looked at as a collection of irregular sedim entary basins separated by arbitrary shaped basement rock barriers, (Sm ith, 1964). These basem ent rock barriers often can be recognized at the surface as m ountains, thus coupling geological and topographical irregularities. Influence of such structures on the propagating waves was shown by studying idealized geometries (Todor- ovska and Lee, 1990, 1991), by generalization of previously obtained solutions to arb itrary shapes (Moeen-Vaziri and Trifunac, 1988a,b) and by interpretation of the recorded data (Vidale and Helmberger, 1988). San Fernando records were studied in the latter work. It was shown th at surface waves coming from the San Fernando valley (San Fernando earthquake, 1971) were “stopped” by Santa M onica m ountains. However, the direct S-waves incident at the edge of the Los Angeles basin on the far side of these mountains resulted in the secondary gen eration of the strong surface waves th a t traveled through the Los Angeles basin. E shraghi and Dravinski (1989a,b) presented tim e domain solutions along with th e frequency domain solutions for antiplane and inplane waves. They showed 97 th a t th e process of generation of strong surface waves at the edges of th e allu vial valley and propagation of those waves inside the valley, resulted in the main contribution to the ground motion. All theoretical solutions sited above show th e presence of standing waves inside the valleys and fast spatial changing in the am plification-deam plification factor. T he ability of the basem ent rocks, surrounding a sedim entary valley, to reflect waves back into the valley, and the effect of conversion of the body waves into the surface waves at the boundaries of a sedim entary basin, allows us to assum e th a t some param eters describing the characteristic horizontal dim ension of these sedim entary basins should play some role in the description of the strong ground m otion duration. We describe the prolongation of duration of strong m otion at th e stations which are situated on sedim entary deposits (due to the processes m entioned above) with two param eters. One of them is the horizontal distance R from the station to the nearest basement rock appearing on the surface and capable of producing the reflections (see Sm ith’ s m ap (1964)). We wish to choose the second param eter in such a way th at it can describe the significance (or the “efficiency” ) of these reflections. The angle < p with which the reflecting surface of the rocks can be seen from the station seems to be a reasonable m easure of the potential “efficiency” of secondary (reflected) waves. Only a p a rt of the boundary between the rock and the sediments which receives the direct waves from th e source is assumed to be capable of generating the reflected waves and was considered in determ ination of R and < p from the Sim ith’s (1964) m ap (see Fig. 3.7). T here are several distances from the rock to the station: the shortest distance, the distance to the middle of reflecting surface, etc. O ur R m eans such 9 8 th i s r o c k o u t c r o p is i n th e s h a d o w z o n e source station th e s e r o c k o u t c r o p s '> .1 / a r e to o f a r F ig . 3 .7 An idealized scheme for the definition of the generalized param eters R and ip. T he numbers below the scheme show the notation used. “1” represents the basem ent rocks, appearing on the earth ’s surface. “2” shows the angle, subtended a t the source by th e rocks from which reflection occurs, w ith the solid lines representing direct waves, em itted by the source towards the rocks. “3” shows the symbol for the angle, subtended at the recording station by th e surface of the rocks from which reflection occurs, with the dashed lines representing the waves reflected by the rock tow ards the station. For the t— rock, r, is th e distance from the source to the rock, i 2t- is the distance from the rock to the station, and < P i is the angle subtended at the station by the rock. 99 a distance from the rock to the station, which, when added to the corresponding source-to-the-rock distance, gives the shortest possible p ath source-rock-station. It can happen th a t several rocks (or clusters of rocks), capable of producing reflections, surround the station. Consider the hypothetical geometry, shown in Fig. 3.7. Two rocks are located in the vicinity of the source and the station. T he direct waves from the source can be reflected in the direction of the station, causing the prolongation of duration there. We designate the distance from the source to th e t— rock as r,-, and the distance from the t— rock to the station as R{. T he angles subtended a t the recording station by the surfaces of the t— rocks are < £ > ,- . We wish to combine now the individual param eters 72, and into some “overall” R and < p, which would characterize all rocks as a group. First, we consider the “energy equation,” which relates the energy of the waves reflected by individual rocks, towards the station. Taking the density of energy em itted by the source in frequency / as 1, we have for the density of energy arriving to the i— rock 1 / — 2ttfr j \ r ? eXp\ Qv ) ' where Q is the attenuation factor, n = 1 (for surface waves or n = 2 (for body waves), and v stands for the velocity of the waves. The energy reflected from the i— rock in the direction of the station is assumed to be proportional to the length of th e reflecting surface of this rock, Ri<Pi. The energy, which comes to the station after being reflected by the i— rock is then proportional to 1 ( - 2 * f R i \ „ 1 ( ~ 2 i r / r , \ R ?e x p \ e x p ( ~ ^ r j • 1 0 0 T he “overall” R and < p should represent some “fictitious” single rock which alone can reproduce the same effect as the combination of several real rocks. The energy, reflected by this “fictitious” rock alone should be the sam e as all the energy, reflected by the group of individual rocks: exp +r)} ■ w h = £ > p { - I t * * +ri>} < 3-1 9> T he above equation, however, is too complex to be considered at the present stage of this analysis. It has too many simplistic assum ptions and accounts for the effects which will further be disturbed and overshadowed by other effects which occur during wave propagation through the real earth. Thus, we disregard geom etrical spreading, because the size of the source and the length of the reflect ing surfaces of the rocks can be assumed to be large com pared to the epicentral distance and to the distances J fE , and r{. We also disregard the attenuation factor a t the present stage. We, however, recognize, th a t attenuation of the secondary pulses (reflected from rocks) may play a significant role in the form ation of the strong m otion parts of the record. Thus, we account for attenuation later, by assum ing a special shape of the functional dependence of duration on R, dur(R). To be specific, dur(R). which is the possible prolongation of duration due to reflection of waves from the nearby rocks in the direction of the station, will be accounted for only if R is not too large, i.e., the reflected pulses do not attenuate too much. After all simplifications the “energy equation” (3.10) is R<p = J ^ R i< p i , (3.20) i and states th a t the reflecting surface of the “fictitious” rock should be equal to th e sum of the reflecting surfaces of the individual rocks. 1 0 1 We next consider the “delay equation.” The pulse, reflected by a rock in the direction of the station, spends more time in the m edium (it travels the distance Ri + rt), and, consequently, arrives at the station later, th an th e direct pulses. T he “fictitious” rock has to be positioned at such a distance R from the station, th a t the delay of the pulses, coming from it, represent some “proper” com bination of the delays of pulses, coming from the individual rocks. The tim e, when the pulse from the “fictitious” rock should be expected at the recording station, (R + r)/v, can be approxim ated as a weighted sum of the tim es of arrivals of the individual pulses, (Ri -f- r,)/u . The weight of each individual pulse can be assumed to be proportional to the portion of energy it delivers to the station when compared to the total energy brought to this station by all reflected pulses: R + r = E Ri + rt (3.21) As for “energy equation” (3.19), we disregard here the geometrical spreading and attenuation. Then the Eq. (3.21) becomes R + r - E Ri + rt- Riipi v £ y RjV>j ’ i.e. the individual times of arrivals are weighted now according to the length of the reflecting surface of the rocks, Ri<Pi. The velocity can be factored ou t from the above equation if the properties of the region are assumed to be approxi m ately homogeneous. Next, we assume th a t the distance from the source to the “fictitious” rock, r, and all the distances from the source to the individual rocks are com parable, i.e. r « r, for all i. This is based on the fact th a t the linear 1 0 2 dimensions of the source and of the rocks are of the order of r and (in strong m otion seismology). Now, recalling the identity E Ri<Pi . r E ^ ' we get the simplified “delay equation” : < 3- 2 2 ) We note, however, th a t the Eqs. (3.20) and (3.22) were not used directly, when the actual m easurem ent of R and < p were carried out. Instead, we used the physical intuition and common sense, which, however were based on and guided by the above considerations. Some real examples of the determ ination of the param eters R and < p are shown in Fig. 3.8 and in Table 3.1. It can be noticed th a t in m any cases, displayed in Fig. 3.8, considerable uncertainty is involved in the m easurm ent of R and < p . For exam ple, which rocks should be considered as potential reflectors? This depends on the distances to different rocks, their location relative to the source, to the station and to each other, their horizontal dimensions, m agnitude of the earthquake, depth of the hypocenter the rate of attenuation of strong m otion w ith distance (Trifunac and Lee, 1985, 1990) and . . . the intuition of the investigator. The histogram s in Figs. 3.9 and 3.10 show the distribution of the 3- com ponent records w ith respect to the param eters R and < p . Figs. 3.11 and 3.12 present the dependence of the observed strong ground m otion duration (as defined from the band-pass filtered acceleration, velocity and displacem ent, hor izontal and vertical components) on R and < p . As we will see later, it is possible to extract some meaningful correlations from such noisy data. 122° 1 O' 1 22 121° 50' 121° 40' 121° 30f 103 38 - 38 122 ’ bays b asem en t rock on the surface * , 10 km . ) | scale source stations, th eir num bers and angles c p F ig . 3 .8 a An example of the determ ination of the param eters for horizontal reflections: the angle £> , subtended at the recording station by the surface of the rocks from which reflections occur, and the characteristic horizontal distance to these rocks, R. The chosen values of R and < p are listed in Table 3.1. D istribution and configuration of basement rocks are taken from S m ith’s m ap (1964). Livermore earthquake, January 26, 1980. 104 119 118°50' 118° 4 0 f 118° 3 0 f - 3 7 °5 0 37° 30f - 119 b asem en t rock on th e surface 10 km i , ^ scale source stations, th eir num bers and angles 9? F ig . 3 .8 b An example of the determ ination of the param eters for horizontal reflections: the angle < p, subtended at the recording station by the surface of the rocks from which reflections occur, and the characteristic horizontal distance to these rocks, R. The chosen values of R and ip are listed in Table 3.1. D istribution and configuration of basement rocks are taken from Sm ith’s m ap (1964). M am m oth aftershock, May 25, 1980. 105 119° 118° 50f U 8 ° 4 0 ' 11 8 °3 0 ' b a s e m e n t ro c k on th e s u rfa c e 10 k m | sc a le 9+6 so u rc e s ta tio n s , th e ir n u m b e rs a n d a n g le s (fi F ig . 3.8 c An exam ple of the determ ination of the param eters for horizontal reflections: the angle < p, subtended at the recording station by the surface of the rocks from which reflections occur, and the characteristic horizontal distance to these rocks, R. The chosen values of R and < p are listed in Table 3.1. D istribution and configuration of basem ent rocks are taken from Sm ith’s m ap (1964). M am m oth aftershock, May 27, 1980. 106 T a b le 3.1 Characteristic param eters of horizontal reflection determ ined from Fig. 3.6. a) Livermore earthquake, CA (Jan 26, 1980). S ta tio n R 9 1 2 0 k m 7 0 ° 2 3 0 k m 4 0 ° 3 2 5 k m 7 0 ° 4 4 5 k m 4 0 ° 5 4 5 k m 3 0 ° b) M am m oth aftershock CA (May 25, 1980). S ta tio n R 9 1 2 0 k m 5 0 ° 2 15 k m 9 0 ° c) M am m oth aftershock, CA (May 27, 1980). S ta tio n R 9 1 8 k m 8 0 ° 2 4 k m 9 0 ° 3 3 k m 1 2 0 ° 4 2 k m 1 9 0 ° 5 4 k m 2 0 0 ° Z Z Z Z Z Z A s=2 (G eological ro ck ) S S I s= 1 (In te rm e d ia te ) m m s=0 (S ed im en ts) « 50 7 7 7 7 7 7 / 107 a ) 10 20 30 40 50 60 E T T Z 3 s L=0 ("R ock") S S 3 s L = l (S tiff soil) 1 / / / / / 1 sl =2 (Deep soil) P 50 b) F ig . 3 .9 D istribution of 3-com pone^ records (adopted for the analysis) w ith respect to the characteristic horizontal dimension R — distance from the station to the rocks, which can reflect towards the station on the waves arriving from the source: a) for different geological conditions (where R and s, the param eter of simpli fied geological classification, are available); b) for different local soil conditions (where R and s^, the param eter of the soil classification, are available). Number o f records 108 140 - 120 - 100 - 0 0 - 40 - 2 0 - V / / (/ A Y / / j ( / A V / / / / A Y 7 7 7 7 7 A vttjtti____ 100 140 180 220 260 300 340 20 60 \P (degrees) F ig . 3 .1 0 D istribution of 3-component records (adopted for the analysis) w ith respect to the characteristic angle of effective reflections, (p. 109 O Q ) x n 60 50 40 30 20 10 f0 =.075 Hz ° ° 10 2 0 30 40 50 60 70 80 O 12 Hz 30 20 10 2 0 10 30 40 50 60 70 80 R ( k m ) F ig . 3 .1 1 a Channels # 1 and # 2 : ibserved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus horizontal characteristic dimension of a sedim entary basin, R. Vertical and horizontal com ponents are shown combined for each channel. 1 1 0 o CD CO 60 50 40 30 20 10 0 f0 = .21 Hz i.* :• • i. ■ • • u;i:i ■ i • •: i I I 10 2 0 30 40 50 60 70 80 O 37 Hz fH P 40 30 2 0 10 10 2 0 30 50 60 70 80 40 R ( k m ) F ig . 3 .1 1 b Channels # 3 and # 4 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus horizontal characteristic dimension of a sedim entary basin, R. V ertical and horizontal components are shown combined for each channel. I l l o < D 73 60 50 40 30 20 10 0 f0 =.63 Hz !.'■ I I • j! o . I • • . i : i . h i : ! l 2 - • ;n 10 20 30 40 50 60 70 80 O Hz aS J - H 2 40 Q 30 20 10 70 80 50 60 40 20 30 10 R ( k m ) F ig . 3 .1 1 c Channels # 5 and # 6 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus horizontal characteristic dimension of a sedim entary basin, R. Vertical and horizontal components are shown combined for each channel. 1 1 2 O < D W 60 50 40 30 20 10 :i f0 = 1.7 Hz • • i - i 1 i 1 1 . • 1 I i i. is 1 10 • I . • • I 2 0 30 40 50 60 70 80 O 2.5 Hz cfi 40 3 Q 30 2 0 10 2 0 10 30 40 50 60 70 80 R ( k m ) F ig . 3 .l i d Channels # 7 and # 8 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus horizontal characteristic dimension of a sedim entary basin, R. Vertical and horizontal com ponents are shown combined for each channel. 113 60 50 40 30 f0 =4.2 Hz o 2 0 10 2 0 30 40 50 60 70 80 O u 0 4 0 Q 30 20 10 2 0 30 40 50 60 70 0 0 R ( k m ) F ig . 3 .l i e Channels # 9 and # 10: obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus horizontal characteristic dimension of a sedim entary basin, R. Vertical and horizontal components are shown combined for each channel. 114 O < D C O 60 50 40 30 2 0 10 f0 =13 Hz iil'i !M i-1 I i ! cl °° 10 20 30 40 50 60 70 80 O f0 =21 Hz cti 1 -i 3 4 0 30 20 20 30 10 40 50 60 70 80 R ( k m ) F ig . 3 .I l f Channels # 11 and #12: obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displace m ent, is plotted versus horizontal characteristic dimension of a sedim entary basin, R. Vertical and horizontal components are shown combined for each channel. 115 O C D W 60 50 40 30 20 10 a ° ° f0 =.075 Hz 45 90 135 180 225 270 315 O 12 Hz 03 50 40 30 20 10 270 45 90 180 225 315 135 (p (degrees) F ig . 3 .1 2 a Channels # 1 and # 2 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus characteristic angle of reflections from rocks, p . Vertical and horizontal com ponents are shown combined for each channel. 0 45 90 135 180 225 270 315 cp (degrees) F ig . 3 .1 2 b Channels # 3 and # 4 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus characteristic angle of reflections from rocks, < p. Vertical and horizontal com ponents are shown combined for each channel. 117 60 .63 Hz 50 40 30 O 20 cn 10 315 180 225 270 90 135 45 G o cd 50 40 0 Q 30 20 10 315 225 270 135 180 90 45 (p (degrees) F ig . 3 .12c Channels # 5 and # 6 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus characteristic angle of reflections from rocks, (p. Vertical and horizontal com ponents are shown combined for each channel. 118 60 1.7 Hz 50 40 30 O 20 CD M 10 180 315 45 90 135 225 270 O ■ h 60 c d 40 30 20 10 225 315 90 135 180 270 45 (p (degrees) F ig . 3 .1 2 d Channels # 7 and # 8 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus characteristic angle of reflections from rocks, < p. Vertical and horizontal com ponents are shown combined for each channel. 119 60 f0 = 4 .2 Hz 50 40 30 O 20 <V 10 90 45 180 225 270 135 315 a o 40 n 30 20 10 90 180 270 45 135 225 315 cp (degrees) F ig . 3 .1 2 e Channels # 9 and # 10: obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus characteristic angle of reflections from rocks, < p. Vertical and horizontal com ponents are shown combined for each channel. 1 2 0 60 f0 =13 Hz 50 40 30 o 20 C O 10 45 90 135 180 225 270 315 f0 = 2 1 Hz C ti 50 53 40 Q 30 20 10 45 90 135 180 225 270 315 cp (degrees) F ig . 3 .1 2 f Channels # 1 1 and if 12: obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displace m ent, is plotted versus characteristic angle of reflections from rocks, tp. Vertical and horizontal com ponents are shown combined for each channel. 121 Next we consider the tim e history of strong ground m otion as it is seen at the recording station, located in a sedim entary basin and surrounded by rocks. D irect P , S and surface waves arrive to the station first, and those generate strong ground m otion of some duration. Later, some reflected, scattered and otherw ise delayed waves arrive. Many of those do not have am plitudes th a t are large enough (compared to the direct wave am plitudes) and, thus, they do not cause significant increase in initial duration of the strong m otion. However, some of those “late” waves, being reflected by well-defined and distinct inclined boundary between sedim ents and basem ent rock, significantly differ from other scattered waves. These “special” reflected waves cause additional pulses which will contribute to the duration of strong motion energy observed at the station. The above simple physical considerations allow us to search for the func tional form of the dependence dur = dur(R . (g). The angle 100° may probably be attributed to the uneven distribution of the d ata points am ong vari ous m agnitudes, epicentral distances and the geometries of the basin. In our d ata base, the stations located in small valleys, i.e. those having 100°, happen to record small earthquakes w ith short duration of the source function. Dependence on R is more complex. Suppose first th a t R is small. T hen the tim e intervals which correspond to initial pulse (of duration duri) and reflected pulse (of duration dur2) will be observed at the station alm ost sim ultaneously w ithout producing significant increase in duration of strong motion. Next, we increase R. The time delay between the two pulses of energy causes separation of the corresponding intervals of strong motion on the accelerogram and the total duration is longer now and equal to duri + dur2 . Further increase of R causes the 122 increase of tim e which reflected surface waves spend travelling through dispersive medium. This causes an increase of dur2 and results in further prolongation of the total duration dury + dur2 - Next we make R very big. Second pulse, the one th a t was generated by reflection from the rem ote rock, experienced strong attenuation and is now so weak th a t it hardly can be noticed in the background of the scattered waves. The further the rock is, the weaker the pulse becomes. Therefore, two ranges of horizontal characteristic dimension (distance to the reflecting rock) exist: small R, where duration of strong m otion grows w ith the increasing R , and large R, where the effect is the opposite. The simplest way to describe such a dependence on R is through a parabolic function, dur(R) = const 1 + const2 -R + constz-R 2 , (3.23) where const{, i = 1,2,3 are constants (different at different frequency bands) and we expect constz < 0 . We next tu rn to the depth of sedim entary deposits at the recording site, h. T he dependence of the duration of strong ground m otion, determ ined from our d ata base, on the depth of sediments under the station, h, is shown in Fig. 3.13. It has been recognized th a t param eter h plays an im portant role in scaling various characteristics of strong earthquake ground m otion (Trifunac and Lee, 1978, 1979b, 1985, 1990; Trifunac, 1980; Lee, 1991). Some studies on the influence of the depth of sediments on the duration of strong m otion were also perform ed (W estermo and Trifunac, 1978, 1979; Trifunac and W estermo, 1982). In the last group of works, the linear dependence dur(h) = const4 + const z-h, where const4 and constz are frequency dependent regression coefficients, was 123 O < L > C O 60 50 40 30 20 10 f0 =.075 Hz 0 0 6 o f0 = .12 Hz • 40 2 30 20 10 h k F ig . 3 .1 3 a Channels # 1 and # 2 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus the depth of sedim entary deposits under the recording station, h. Vertical and horizontal com ponents are shown combined for each channel. 124 60 fn = .21 Hz 50 40 30 O 20 0) 10 o 0 =.37 Hz cti -I Q 30 20 h ( k m ) F ig . 3 .1 3 b Channels # 3 and # 4 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus the depth of sedim entary deposits under th e recording station, h. Vertical and horizontal com ponents are shown combined for each channel. 125 60 50 40 30 O 20 C O 10 o f0 =1.1 Hz c d s 4 0 30 20 10 h k F ig . 3.13c Channels # 5 and # 6 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus the depth of sedim entary deposits under the recording station, h. Vertical and horizontal com ponents are shown combined for each channel. F ig . 3 .1 3 d Channels # 7 and # 8 : obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacement, is plotted versus the depth of sedim entary deposits under the recording station, h. Vertical and horizontal com ponents are shown combined for each channel. 1 2 7 O < D C O 60 50 40 30 20 10 0 f0 =4.2 Hz 5! 4. li i> • a • t I I a : I : • • • •• . s' iii • i t- • i*. ■ . I # !!h ii-S itt 'f ' *: ■ • i ' : h " ; 0 o 1 . 2 H z a J h S 3 4 0 30 20 10 h F ig . 3 .1 3 e Channels # 9 and # 10: obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displacem ent, is plotted versus the depth of sedim entary deposits under th e recording station, h. Vertical and horizontal components are shown combined for each channel. 1 2 8 60 f0 =13 Hz 50 40 30 O 20 C O 10 o f0 =21 Hz c O 2 40 Q 30 20 10 h ( k m ) F ig . 3 .1 3 f Channels # 1 1 and #12: obsreved duration of strong ground m otion as determ ined from the band-pass filtered acceleration, velocity and displace m ent, is plotted versus the depth of sedim entary deposits under th e recording statio n , h. Vertical and horizontal components are shown combined for each channel. 129 considered. In the present study, we assume the parabolic dependence dur(h) — const4 + consts-h + const e-h2 , (3-24) where consti, i = 4 ,5 ,6 , are the frequency dependent coefficients. The reason for this is th a t the dependence dur — dur(h) should be sim ilar to dur = dur(R) w ith the difference in scale due to the fact th a t h describes the dim ension, per pendicular to R. Consider now the wave travelling from the valley tow ards the edge of sedi m entary deposits. One way to interpret the process of propagation comes from geom etrical optics and Snell’ s Law. This interpretation is illustrated in Fig. 3.14. As it can be seen, the wave cannot penetrate all the way to the edge of the basin. We introduce here the “effective” horizontal distance R eff- the distance from the station to the region of reflection. Strictly speaking, it is the param eter 72eff which should be considered in Eq. (3.23) instead of R. R eff, however is hard to determ ine. Thus, we have to use only R and h in our equations and account for the existence of the “effective” horizontal distance i2eff in some way. The logic we use is as follows. By fixing the horizontal dimension R and by changing the depth of sediments h, one can change R efi. This means th a t the effects, pro duced by the param eters R and h are related to each other. T he sim plest way to account for this coupling is to combine Eq. (3.23) and Eq. (2.24) by adding one more, coupling, term : dur(R. h ) = consti + const^-R + ■ const^-R 2 (3.25) + const4 + const^-h + conste-h2 + consty-Rh. The coupling may be a way to account for the fact th a t _ R eff (not R) determ ines the characteristic horizontal dimension of the reflections. C onstant term consti + < 0 r R -3> W I W 7 ' ROCK SEDIMENTS R e f f -£ > F ig . 3.14 Reflection of a wave from the boundary of a sedim entary basin. The station is indicated by the black triangle, and the direction of wave propagation is represented by the arrows, h is the depth of sediments under the station, R is the distance to the reflecting rock as it is seen on the earth ’s surface, and R ef / is the effective distance from the station to the region were the reflection actually occurs. 130 131 const4 in dur(R ,h) should be equal to zero, because no prolongation of duration (of the type discussed above) is possible when R = h — 0 (which sim ply means th a t station is on rock). Thus, renaming coefficients in Eq. (3.25), recalling contribution from the angle of reflection < p and considering the influence of R, h and < p on duration as a frequency dependent phenom enon, the final form of dur(R,h,<p) can be obtained: dur(R,h,(p) = rregioD(f) = a 5(f)-h + a6 (t)-R + a7 (f)-hR+ (3.26) + a 8 {f)'R 2 + a ^ iD 'h 2 + aio(f)-<p. Here a ,( /) , t = 5 10, are frequency dependent coefficients to be determ ined from regression analysis. Eq. (3.26) could be interpreted to represent the last term , TVegion) in Eq. (3.1). Note th at the sum of the first five term s in Eq. (3.26) corresponds to the complete decomposition of dur(R , h) as a function of two variables into Taylor series up to the second order (assuming constant term is zero). The duration of strong ground motion, as defined by Eq. (2 .2) and calcu lated for the band-pass filtered acceleration, velocity and displacem ent, is shown in Fig. 3.15 as a function of two variables: horizontal, R, and vertical, h, charac teristic dimensions of a sedim entary basin. Horizontal and vertical com ponents are presented separately for each channel. The duration is shown averaged over certain ranges of R and h ( R i < R < R l+1, At = M 0, / = 0 + 7, 1 (3-27) t ^ ifc ^ ^ ^ j lifc -— A, h — 0 ~ 7 * 6 , which are indicated by the dashed mesh. The density of shade in each “box,” defined by Eq. (3.27), is proportional to the average duration, the scale used is shown below each graph. The scale was chosen to be the sam e for the horizontal Horisontal component f0=.075 Hz 0 10 20 30 40 50 60 70 80 7 6 5 4 3 2 1 0 0 0 10 20 30 40 50 60 70 80 R (km) A v . d u r a t i o n s e c : 26 31 36 41 46 51 F ig . 3 .1 5 a l Horizontal component, channel # 1: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the num ber of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 4 5 km ). Vertical component f0=.075 Hz 0 10 20 30 40 50 60 70 80 7 6 5 4 ;q;‘ w i : 3 2 1 0 0 20 30 40 50 60 70 80 R (km) A v . d u r a t i o n s e c : 2 6 3 1 3 6 4 1 4 6 5 1 F ig . 3.15a2 Vertical component, channel # 1 : obsreved duration of strong ground motion as a function of horizontal, R , and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 - T - 5 km). Horisontal component f0=.12 Hz A o 10 20 30 40 50 60 70 80 7 6 5 15 4 33 3 23 2 1 0 0 10 20 30 40 50 60 70 80 R ( k m ) A v . d u r a t i o n s e c : F ig . 3 .1 5 b l Horizontal component, channel # 2 : obsreved duration of strong ground motion as a function of horizontal, R , and vertical, /i, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For moderate frequencies the longer duration can be seen for the interm ediate values of R (25 -f- 55 km) and h (2-=-5 km). Tip I± ± t 16 23 29 35 42 48 Vertical component f0=.12 Hz 0 10 20 30 40 50 60 70 80 7 6 5 4 3 2 1 0 0 20 30 40 50 60 70 80 R (km) A v . d u r a t i o n s e c : 16 23 29 35 42 48 F ig . 3.15b2 Vertical component, channel # 2 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the num ber of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -f 55 km) and h (2 -j- 5 km). Horisontal component f0=.21 Hz 10 2 0 30 40 50 60 70 80 7 6 5 34 5;3 4 1(59- 3 72 2 1 87 23 0 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n sec: F ig . 3 .1 5 c l Horizontal component, channel # 3 : obsreved duration of strong ground motion as a function of horizontal, H, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -j- 55 km) and h (2 -f- 5 km). ♦ ♦ .I ' r :l. T -~ . -I- 1 *r J- ______L:_: :_• • — — .. i 11 Ml \ . 16 21 26 32 37 42 Vertical component f0=.21 Hz 0 10 20 30 40 50 60 70 80 7 6 5 X 8- 4 3 2 1 0 0 10 20 30 40 50 60 70 80 R (km) Av. d u r a tio n sec: F ig . 3.15c2 Vertical component, channel # 3 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the num ber of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for b o th types of components. For moderate frequencies the longer duration can be seen for the interm ediate values of R (25 -f 55 km) and h {2 4-5 km). IX i: 16 2 1 26 32 37 42 Horisontal component f0=.37 Hz 10 2 0 30 40 50 60 70 80 7 12 6 5 55- 49- 3. 4 4 109* 24 3 ^ — V 36 69 2 1 21.1 58' 20' 27 0 0 20 10 30 40 50 60 70 80 R (km) Av. d u r a tio n s e c : 16 21 26 32 37 42 F ig . 3 .1 5 d l Horizontal component, channel #4 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for th e particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -r 55 km) and h (2 -r 5 km). Vertical component f0=.37 Hz 0 10 20 30 40 50 60 70 80 7 6 5 4 54 3 2 1 90 0 0 20 10 30 40 50 60 70 80 R (km) Av. d u r a tio n s e c : 16 21 26 32 37 42 F ig . 3.1 5 d 2 Vertical component, channel # 4 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 4-5 km ). Horisontal component f0=.63 Hz 0 10 20 30 40 50 60 70 80 7 6 5 2.4 80 3.9 4 3 2 1 315 30’ 44 24 0 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n se c : F ig . 3 .1 5 e l Horizontal component, channel # 5 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the num ber of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 -j- 5 km). 13 19 25 30 36 42 Vertical component f0=.63 Hz 0 10 20 30 40 50 60 70 80 7 6 5 16- 4 3 35 2 1 |L35 41' 1 : 2 0 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n sec: F ig . 3.15e2 Vertical component, channel #5 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 4-5 km). 13 19 25 30 36 42 Horisontal component f0= l .l Hz 0 10 20 30 40 50 60 70 80 7 6 5 28 105- 66 4 2-7 U 1 3 36 88 2 1 162' 32 407 30' 50 26 0 0 2 0 30 50 40 60 70 80 R (km) Av. d u r a tio n sec: 10 14 18 23 27 31 F ig . 3 .1 5 fl Horizontal component, channel # 6 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedimentary basin. Duration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for b oth types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4 55 km) and h {2 4-5 km). Vertical component f0 =1.1 Hz w 43 10 20 30 40 50 60 70 80 7 6 5 26- 4 fcEE 3 2 27 1 162 63 15 0 ^ 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n ; • ; • ; • Sec: 10 14 18 23 27 31 F ig . 3.15f2 Vertical component, channel # 6 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 4- 5 km). Horisontal component f0 =1.7 Hz 2 0 30 40 50 60 70 80 7 6 5 130- 4 12 3 42 14 2 24 14 l - L ' d 36 1 5Q5 181 55 27 0 2 0 30 40 50 60 70 80 R (km) Av . d u r a tio n I I J . I - 1 t-l i j ... 1 1 s e c : 8 11 14 17 20 23 F ig . 3 .1 5 g l Horizontal component, channel # 7 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, /i, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For moderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2-j-5 km). Vertical component f0 =1.7 Hz 10 2 0 30 40 50 60 70 80 A 7 6 5 50 4 3 42 x J ri l 2 30 1 21:6 7:2 0 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n se c : F ig . 3.15g2 Vertical component, channel # 7 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -j- 55 km) and h (2 -r 5 km). M l 8 11 14 17 20 23 Horisontal component f0=2.5 Hz 0 10 20 30 40 50 60 70 80 7 6 5 129- 4 3 2 24 1 580 190' 33' 35 0 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n sec: F ig . 3 .1 5 h l Horizontal component, channel # 8: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the num ber of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -r 55 km) and h (2 -i- 5 km). 1 1 ■ J i- -- 1 r - 7 9 1 1 1 4 16 18 Vertical component f0=2.5 Hz 10 20 30 40 50 60 70 3t 1 1 n 1 1 ii 111111 11TT11I I111I I11 j:l;l:l;t|ll:l;| 111111 111 11 1 1 11 m i i m JiJJlLLL 249- 4171711 i 79 i-'l i-i i h 1111 Ii ..... 1111 i n ii i i l l ii. ii i ii 11) l-l-l-l-IN u l m LiMLMJ 40 R (km) Av. d u r a t i o n sec: i I . - - - 11 14 16 18 F ig . 3.1 5 h 2 Vertical component, channel # 8: obsreved duration of strong ground m otion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. Duration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -r 55 km) and h (2 -r 5 km). Horisontal component f0=4.2 Hz A 10 20 30 40 50 60 70 80 7 6 5 110- 28- 48 4 3 ;32 2 5ff 1 '457 176' 32' 2:4 37 2. 8 0 0 20 30 40 50 60 70 80 R (km) A v . d u r a t i o n s e c : 7 9 1 1 13 15 17 F ig . 3 .1 5 il Horizontal component, channel # 9 : obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for b oth types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2-r5 km). Vertical component f0=4.2 Hz 10 20 30 40 50 60 70 80 A 7 6 5 50 4 3 2 1 20 r 68' 0 0 10 20 30 40 50 60 70 80 R (km) Av. d u r a tio n sec: 7 9 1 1 1 3 15 17 F ig . 3.15i2 Vertical component, channel # 9 : obsreved duration of strong ground motion as a function of horizontal, R , and vertical, h, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For moderate frequencies the longer duration can be seen for the interm ediate values of R (25 -f 55 km) and h (2 -r 5 km). Horisontal component f0=7.2 Hz w £ 10 'ITT'I I (Ti 20 n 1 1 n 111 30 40 50 60 70 ._ _ _ _. _ I S p r l l i *4 * • } ' -26- * -| * 110- S :; : 30 =. * 10 1 0 8 20 : 46;: 15 . 3 70 4 1 lo ll rT 43 66 62 r * ’1 1 I - f - 1 $ $ 1 1 - 80 I I | I I I I I I I n I I I I I I l lT T 'l |:n.i:i | f:I :|: I I I I I I ' m'l I l'l I I I I I I H 7 6 5 \ m m 3 2 389 1 1 H f i t iT i 163 LI,1U-I 11 26 III |:ii l:!:l i-i-h n L I I I I I I I F 40 50 R (km) Av. d u r a tio n sec: 11 13 15 F ig . 3 .1 5 jl Horizontal component, channel #10: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -F 55 km) and h (2 -F 5 km). Vertical component f0 =7.2 Hz 7 6 ^ 5 S 4 A A 3 2 1 0 10 20 30 40 50 60 70 16 '■ 5 196 JJJLLLt I I I'l l 1 I I I | I I I I I I I I I I I I I iTrrriTTi i it r rn ry : 4 ; TT 80 B 4 I I •6 i l l • 7 67 I I 1 1 l‘l tl.l) 17 1 1: fill! 56 46 iM I& I 4 ■ S : 1 19 1 = 38 29 WmMrnm l : 1 1 i : j ; I j 1 13 :i l i t i:i: 2 [ 1 1 ; L :ri' 1 1 :1 4 R :!'I 10 A v . d u r a t i o n s e c : 20 30 40 50 60 70 I I I I I I I IB 80 R (km) 2 11 13 15 F ig . 3.15j2 Vertical component, channel #10: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the num ber of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for b oth types of components. For moderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 -7 - 5 km). 151 Horisontal component f0 =13 Hz 0 10 20 30 40 50 60 70 80 HI m I III III I I III N 1 1 1 I I I III I I II I III I IT] I I I " I I 1 1 !TTi I nil'I'! iT j I I I T I I I I I I I I I I I I I I f c j iiiiiiiiliiiiiiiiiiii! M : 20 ::4: : 2 3 i i _ . . . ----- ---- 14 4 11 75 4 M fci III 34 4 :xe r b L ;: 2 2: —T j - V-- 20 36 28 3 m m m 232 • m i n t ill 10 100 •: Lit.fh- 14 •5 • ’ ! U I * 1- . 1 0 . JJ.IU J.I 111i- n - m i 3 -i.l; 1 1 1 1 1 1 1 ih i i i i i i 20 30 40 50 R (km) 60 70 I S 80 Av. d u r a tio n sec: F ig . 3 .1 5 k l Horizontal component, channel #11: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 4- 55 km) and h (2 4-5 km). Vertical component fo-13 Hz 10 20 30 40 50 60 70 80 7 6 5 54 4 3 ' 3 2 1 46 0 0 10 20 30 40 50 60 70 80 R (km) Av. d u r a tio n sec: 3 4 6 8 9 1 1 F ig . 3.15k2 Vertical component, channel #11: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -r 55 km) and h (2 4- 5 km ). Horisontal component fo=21 Hz 0 10 20 30 40 50 60 70 80 7 6 5 4 3 2 1 ’ 114 4: l 0 0 20 30 40 50 60 70 80 R (km) Av. d u r a tio n se c : F ig . 3.1511 Horizontal component, channel #12: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, h, characteristic dimensions of a sedim entary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the d ata points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -r 55 km) and h (2 -r 5 km). Vertical component fo-21 Hz 0 10 20 30 40 50 60 70 80 7 6 5 3;1 4 3 2 1 2 : 2 0 s i 0 20 30 40 50 60 70 80 R ( k m ) Av. d u r a tio n sec: F ig . 3.1512 Vertical component, channel #12: obsreved duration of strong ground motion as a function of horizontal, R, and vertical, /i, characteristic dimensions of a sedimentary basin. D uration is shown averaged in the various ranges of R and h, specified by the dashed mesh. The longer duration corresponds to a darker shade. The numbers in dashed “boxes” give the number of the data points available for the particular range of R and h. Horizontal and vertical components are shown separately for each channel. The scale of shades is the same for both types of components. For m oderate frequencies the longer duration can be seen for the interm ediate values of R (25 -r 55 km) and h (2 -r 5 km). 156 and for the vertical com ponents of each channel, bu t changing from one frequency band to another. The num bers in “boxes” indicate the am ount of d a ta points available for the particular range of R and h. It can be seen from the figure, th a t the long durations tend to occur for the interm ediate value of the horizontal, R, and the vertical, h, characteristic dimensions of a sedim entary basin. 3.5. Geological and Local Soil Site Conditions T he recognition and the understanding of the local site effects on strong ground m otion am plitudes increased significantly during recent years (A nderson, 1991). M ajor advances have been achieved since the tim e when K anai (1949, 1951) studied the local soil site effects by using m icrotrem or m easurem ents, and w hen G utenberg (1957) considered the effects of geological site conditions. It is now recognized th a t local soil and geological conditions should be considered si m ultaneously in various scaling equations (Trifunac, 1987, 1989a,b,c, 1991b; Lee 1990, 1991, 1992). In the resent works, sited above, the local soil effects where m odeled by the param eter sl • Geological conditions were described through the depth of sedim ents h or by the simplified geological classification in term s of the param eter s. Both s and sl are defined as in C hapter 2 of the present study. T he geological param eter s was already considered in the regression m odels of th e frequency dependent duration (Trifunac and W estermo, 1976a,b, 1977,1978). Theofanopulos and W atabe (1989) studied the influence of local soil conditions. T heir definition of duration of strong m otion was not frequency dependent. More over, they analyzed the d ata from several geologically different regions (Japan, 1 5 7 U nited States, Mexico and Greece) all as one d ata set, and did not correct their results for this fact. We do not think th a t such an approach is satisfactory. Given th a t Fourier and Pseudo-Relative-Velocity spectra of strong ground m otion depend on s (or h) and sl simultaneously (Trifunac, 1987, 1989a,b,c, 1991b; Lee 1990, 1991, 1992), it would be logical to assum e th a t so does the duration of strong motion. This question was not studied previously and will be addressed in this work. The nature of the dependence of dur = dur(h) was already discussed. If h is not available, the use of site classification in term s of s is possible. However, pa ram eters s and s l , being qualitative variables (each taking on only three discrete values), need to be considered in a different way than conventional quantitative variables (M ontgomery and Peck, 1982). The term , accounting for the local soil effects, can be w ritten in the form: In this representation, coefficient an describes the change in duration of strong 0). Coefficient a \ 2 accounts for deep soil sites, com paring them w ith stations, would indicate the same in the case of stiff soil sites. By com paring a n and dur(sL) = a n - S ^ + a 1 2 - S ^ \ (3.28) where 1, if sL = 1, 0, if sl ^ 1, (3.29) 1, if sL = 2, 0, if sl ^ 2. m otion in the case of stiff soil conditions as compared with th e “rock” sites («£ = located on “rock.” This m eans th at positive a \ 2 would show th a t duration of strong m otion is longer at deep soil sites, than on “rock” sites. Positive a n 1 5 8 &12 conclusions about the degree of these prolongations can be m ade. Thus, ®12 > ®u would show th a t the duration of strong m otion is, on the average, longer at deep soil sites, than at stiff soil sites, for example. Equations analogous to Eqs. (3.28)-(3.29) were used by Trifunac and Lee in the studies m entioned above for scaling Fourier and Pseudo-Relative-Velocity Spectra am plitudes of strong m otion. It would be logical to consider the influence of geological conditions in a sim ilar m anner: dur(s) — ais-S^1) + a\4 -S^°\ (3.30) where s W = • f 1. if s = 1, I 0, if s ^ 1 , s (0) = • f 1. if 5 = 0, I 0, if 5 ^ 0 . (3.31) Here a 13 shows the change in duration for the interm ediate (s = 1) sites com pared w ith the basem ent rock sites (s = 2). Coefficient a i 4 displays th e difference between sedim entary sites (s = 0) and the basement rock. The reason for con sidering instead of S ^ (as it was done in Eq. (3.29) in the case of local soil conditions) is th a t we wish to treat all site effects taking rock sites (basem ent rock for geological classification and local “rock”soil for soil classification) as a reference. U nfortunately, the im plem entation of Eqs. (3.28)-(3.31) causes instability in th e regression solution. This is related to the highly uneven distribution of records am ong rock and sedim entary sites (see Fig. 2.5a)). Nevertheless, an a tte m p t to use Eqs. (3.28)-(3.3l) in regression was m ade and some results are describe later. In the case where the use of Eqs. (3.28)-(3.31) becomes impossible, 159 the simplified (but rough) version of dur(s) and dur(sr.) can be used: dur(s) = o 15-(2 - s ), (3.32) dur(sL) = a 1 6-sL. (3.33) T he term rregjon (and site) can be constructed now by using Eq. (3 .2 8 )-(3 .3 3 ) and recalling th a t all regression coefficients a,- should be treated as frequency dependent quantities. 160 4. SCALING THE DURATIO N OF STRONG EARTH Q UAK E G RO UND M OTION IN TERMS OF THE EARTHQUAKE M A G NITUDE A N D OTHER PA R AM ETER S 4.1 General Considerations and Definitions Several different equations modeling the duration of strong ground m otion will be discussed in this C hapter. All of them are of the form dur(f) = ^ 2 aik{f)Pik, (4-1) k where ptfc is the tjjr com ponent of the param eter vector p and a tfc( /) stands for the com ponent of the frequency dependent vector of coefficients. The correspondence of the various components of the vector p to the param eters of the source, propagation path and site is shown in the Table 4.1. Table 4.1 was com plied using Eqs. (3.12), (3.14), (3.16)-(3.18), (3.26) and (3.28)-(3.33), which specify how the various factors contributing to duration are accounted for. Sum in Eq. (4.1) is taken through only some of the term s a tp,. The specific set of diPi th a t makes up the sum is different for every model. Once the set of a ,p t - (Table 4.1) is chosen, the model is defined, and Eq. (4.1) is solved for unknown coefficient a,ik(f), separately in each frequency band. The solution is perform ed using singular value decomposition (SVD) technique (see, for exam ple, Press et al., 1986) which is briefly described in Appendix C. SVD provides a powerful control on the error of com putation allowing to spot those coefficients a,-f c in the Eq. (4.1) which cannot be obtained with reasonable accuracy. Then assum ing unreliable coefficients to be zero, the improvement of the overall accuracy of the solution can be achieved (see Appendix C for details). T ab le 4.1 Source, propagation path, site param eters and the assigned component numbers (see Eq. (4.1)). P r o p a g a t i o n p a t h S i t e c o n d i t o n s S o u r c e (and site c o n d itio n s ) S o i l G e o l o g y i 1 2 3 17 18 19 20 4 5 6 7 8 9 10 11 12 16 13 14 15 Pi 1 M M2 H HM 1mm I mmA' A,A' h R hR R2 h2 9 SL(1) s l (2) S L s(i) S(°) s a; ai a2 33 an ai8 aj9 320 34 35 36 37 38 39 310 311 312 316 313 314 315 G e o 1 o g y Here: M—earthquake magnitude, H—hypocentral depth (km), A—epicentral distance (km), A '—hypocentral distance (km), h—depth of sediments at the recording site (km), R—characteristic horizontal dimension (distance to the reflecting rock), (km), — angle subtended at the station by the reflecting rock (degrees), s—geological site conditions, sl —local soil site conditions, and S ^ —see Eq. (3.28), 5 ^ and —see Eq. (3.26). 161 162 We next state some definitions. We call “d ata points” the collection of com ponents of acceleration, velocity and displacement (horizontal, vertical or both combined) considered in the analysis. Each d ata point has several character istics, am ong those are: the observed duration, the properties of the causative earthquake, of the propagation path and of the recording station, and the “theo retical” (or predicted from a model) duration. For the j — d ata point considered, dur} ( f ) designates the value of duration of strong ground m otion, obtained ac cording to the definition Eq. (2.2), and du r“ odel(/) stands for the “theoretical” value of duration, predicted by the model Eq. (4.1) after all coefficients a ,f c ( /) were obtained by regression analysis. The residual of the j — d ata point, £j{f), is the difference between the actual value of duration and the value predicted by the model: ei(f) = durj(f) - durylodel(/). (4.2a) £j{f) gives the absolute error of the prediction. The relative error, £yel( /) , can be defined as , durAf) — dur^°del(f) eAf) - \ L r J ( f ) W - « a * - • -ICO*. (4.26) If N (f) is the number of the data points at a given channel, the average duration d u rav( /) in this channel is N(f) dura ■ w = jvjT) E dur’ W ’ < 4-3) and the stan d ard deviation of the data points is 163 where n(f) is the number of non-zero coefficients in the model Eq. (4.1), as it follow from SVD solution of the regression problem. M inim um least square fit allows one to find the unknown coefficients a tfc ( /) in Eq. (4.1) together w ith their variances a2 k (/) (Appendix C), so th a t the result of the regression analysis can be presented as a “o-interval” : T he confidence interval for a coefficient, a t / f c ( /), based on N% level of confidence, is given by {«<*(/) }< r = au(/)±cr«'fc(/)- (4.5) (/)}accepted (-0 ( / ) ’^N%,[N(/) — n(/)] > (^’®) where t is the Student T distribution function based on N% confidence and — n (/)] degrees of freedom. In this study, = 95% level was chosen, so th a t We next consider several specific models of the scaling laws for th e duration of strong ground m otion—specific forms of the Eq. (4.1). 4.2 M o d e l dur = dur(M,M2, A) We first discuss the case when no inform ation on the propagation p ath is available, except the epicentral distance. The scaling Eq. (4.1) in this case is: dur(f) = ai(f) + a2{f)-M + a3(f)-M2 + a4(f)-A, (4.7) 164 where M is the m agnitude of the earthquake and A is the epicentral distance. Eq. (4.7) was first fit into d ata for horizontal and for vertical com ponents sepa rately. No significant discrepancies in the behavior of a t ( /) , i = 2 -f- 4, for the horizontal and for vertical m otion were discovered. However, constant coefficient ai{f) was found to be different in these two cases. Thus, in the final model, a t(/)>* = 2-t-4, were obtained using data from all components com bined, while ol ( /) was calculated separately for horizontal and for vertical m otion (the details are discussed in Appendix C.) D uration of strong ground motion has to be monotonically increasing func tion of m agnitude (see Eq. (3.9)): dur(M) — aex p (/3M), (4.8) where a and /? are some constants (different at various channels). A pproxim ation of Eq. (4.8) by a parabolic function dur(M ) = ai + a2-M + az-M 2 is not valid when M is less than some critical value M m\a = —a2 /2az- Concluding the above considerations, the final equation for the model can be w ritten as: ( duAh^{f) ) f a[h\ f ) ) 2 \< iUr W ( / ) } = { (!('')( / ) ) + 0 2 ( / ) M + “3 ( / ) 'M + a 4 < /) ' A ’ <4-9a) where the epicentral distance, A, is measured in kilometers and ( M , if M > M min(f), M = m ax{M ,M min(/)} = < (4.96) I d**rnin\j) otilCrWlSGj where M min(f) - 2a* ffy (4 -9c) 165 D uration of the horizontal com ponent of strong motion, d u r ^ { f ) , differs from its vertical counterpart, duAv\ f ), because different constant coefficients a[h\ f ) and a[v\ f ) result from the regression analysis. Eq. (4.9a,b,c) was fit to the data in two steps. In the first iteration, Eq. (4.9a) w ith M = M was considered, so th a t the set {at '(/)>* = 4}first and M min( /) were com puted. In the second interaction, M min( /) , obtained earlier, was used to fit Eq. (4.9a,b,c) to the data. The set { a ,(/), t = 1 4 } seCond practically coincides w ith {<*»(/), t = 1 -f- 4}first, so either of them can be used as the final o u tp u t from the regression analysis. The fact th a t the fitted coefficients of both iterations are so similar is due to the small num ber of the d a ta points w ith M < M min « 3.5 (see distribution on Fig. 2.1). Table 4.2 and Figs. 4.1, 4.2 and 4.3 display the results of the regression analysis of Eq. (4.9). Table 4.2 gives the fitted regression coefficients of Eq. (4.9) as ot (/) ± < r,(/), where o f ( /) are the variances (the measure of reliability) of th e values found. Zero values for a coefficient and its variance correspond to the cases when it was not possible to calculate a ,( /) with reasonable accuracy at a given frequency band (i.e |cr,/a,| > 1.) The number of the available data points N ( f ) is very different at each channel, reflecting the statistical reliability of the regression analysis performed. The average duration, durav( /) (Eq. (4.3)) and the standard deviation of the data points from the model, aaur (Eq. (4.4)), are also listed in Table 4.2. The average duration of strong earthquake ground shaking sharply decreases as frequency of the m otion increases. Fig. 4.1 gives the graphical representation of the frequency dependent coefficients a,-(/) (solid lines); “o-interval” (Eq. (4.5)) and 95% confidence interval (Eq. (4.6)) are plotted w ith dashed and dotted lines to display the reliability of the solution obtained. T ab le 4.2 Results of the regression analysis of Eq. (4.9). Channel fo # o f data points C oefficients a; and their accuracy ("a-interval") Mmin 0dur durav number (H z) N (f) aj(h) +C1O 1 ) a j(y) ±0109 a2 ±02 a3 ± 0 3 34 ±04 (sec) (sec) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 10.2 38.3 2 0.12 311 19.1 ±1.1 19.4 ±1.6 .0 .0 .191 ±018 10.2 28.3 3 0.21 962 11.9 ±0.6 13.6 ±0.7 .0 .0 .215 ±012 8.1 21.4 4 0.37 1499 7.8 ±3.2 8.2 ±3.2 .84 ±.52 .0 .191 ±008 7.4 2 1 .0 5 0.63 2029 1 .8 ±2.0 3.7 ±2.1 1.51 ±.35 .0 .184 ±.007 7.8 18.7 6 1.1 2618 -3.6 ±1.3 -1.0 +1.4 2.10 ±.23 .0 .149 ±005 6.9 15.6 7 1.7 3099 -4.1 ±0.8 -1.6 ±0.8 1.90 ±14 .0 .123 ±004 5.3 12.4 8 2.5 3359 7.1 ±1.7 8.7 ±1,7 -2.67 ±62 .41 ±06 .084 ±003 3.26 3.7 9.0 9 4.2 2714 8.7 ±1.4 1 0 .0 ±1-4 -3.91 ±.54 .57 ±05 .071 ±003 3.43 3.2 7.6 10 7.2 2553 9.6 ±1.1 1 0 .1 ±1-1 -4.68 ±42 .66 ±04 .064 ±003 3.55 2.6 6.4 11 13 1572 6.2 ±1.1 6.3 ±1.1 -3.33 ±42 .52 ±04 .055 ±003 J 3.20 2.0 5.1 12 21 724 1 0 .1 ±2.1 1 0 .1 ±2.1 -4.68 ±84 .62 ±08 .056 ±.007 3.77 1.8 4.2 1 horiz 1 vert M M 2 A C orresponding parameters [co n sth] [co n stv] 40 40 f (Hz) 30 30 20 20 -2 10 - 4 10 f (Hz) f (Hz) -10 -1 0 a 8 .20 6 4 2 .05 f (Hz) f (H z) 0 .00 10 F ig . 4.1 The coefficients a ,( /) in Eq. (4.9), plotted versus the central frequency of the channels (solid lines). T he coefficients are bounded by their V -intervals” (dashed lines) and by their 95% confidence intervals (dotted 1 6 8 As it follows from Table 4.2 and Fig. 4.1, the duration of strong ground m otion does not depend on the earthquake m agnitude if the frequency of shaking, / , is less th an some critical value: / < f cr « 0.25 Hz (first three channels). This fact has simple physical explanation. If the period of the waves employed to “m easure” the duration is longer than the characteristic tim e of the ru p tu re process in the source, To, the dependence of the duration of strong m otion on To cannot be resolved. As a consequence, the dependence of duration on the length of the source, and likewise, on the seismic moment and m agnitude, disappears. T he critical frequency f cr depends on the quality of the d ata base. Suppose th a t th e d a ta base has statistically sufficient number of strong earthquakes w ith the source corner frequency / , and ju st a few records of very strong earthquakes w ith corner frequency < / . Then the dependence dur — dur(M) can be followed up to / , th a t is, through the incoherent p art of the source spectra. The incoherent radiation occurs where the waves have the length shorter th an the size of the fault, so th a t different parts of the fault em itt radiation in different phase. In this case, f cr « / . If the central frequency of the channel jo is sm aller than the critical frequency (/o < frc « / ) , the waves belong to the coherent p art of the source radiation, and then no connection of the duration observed and the characteristic tim e of the source can be made. The more “com plete” d ata base is, the sm aller f cr will be. However, f cr always exists due to the fact th a t there is an upper limit on the m agnitude of the earthquakes th a t can happen in a region. One of the largest earthquakes observed (and recorded) so far was Chilean (1960) earthquake, w ith M s = 8.3, w ith fault plane 800 x 200 km 2 and ru p tu re velocity « 3.5 km /sec (Kanamori and Cipar, 1974). Corner frequency of this event could be estim ated as 0.004 hz. O ur data base all comes from the W estern U nited States and, in particular, from California. Since the thickness of the seismogenic zone in California is less than about 20 km, large earthquakes ru p tu re narrow and long segments of the earth crust. In such conditions, the largest events rarely exceed M s = 7.7 (Kern County, 1952 earthquake). T he largest observed local m agnitude is M l « 6.5 (Trifunac, 1991a). We designate the corner frequency of the source spectrum of the k— earth quake from the d ata base as /* , and move towards the high frequency end of the spectrum . As frequency / increases, progressively more cases w ith fh < f can be observed. M ore earthquakes participate in the d ata base w ith the m otion pro duced by their incoherent p art of the spectrum , the dependence dur = dur(M) becomes m ore prom inent and a 2(/) increases. Further increase of frequency ( / > 1.7 Hz) results in fk < f for all sources. This makes it possible to resolve the coefficient as(f) which is responsible for the quadratic form of the depen dence dur = dur(M). More detailed discussion on the nature of the dependence of the duration of strong ground shaking on the m agnitude of an earthquake can be found in C hapter 6 . Next we consider the dependence of the coefficient representing the dispersion and scattering term , a 4, on frequency. The value of 04 gives the increase in the duration (in seconds) per each kilometer traveled from the epicenter of the earthquake to the recording station. As it was noticed earlier (C hapter 2), for low and interm ediate frequencies, this increase can be explained by the dispersion of the surface waves, travelling through irregular, bu t generally layered stru ctu re of the upper crust. The equation 170 for the physical explanation of this coefficient was suggested. Here cm- m( /) and cm ax(/) are the effective m inim um and maximum phase velocities of the surface waves in the region under consideration. The effective variety of the phase veloc ities can be m easured as the w idth of the shaded stripe in Fig. 3.3. Notice th a t a 4 (/)» as shown by Eq. (3.15), also is related to the variety of the phase velocities of the surface waves. Compare now the behavior of the coefficient 0.4 (f) and the w idth of the shaded stripe in Fig. 3.3 as frequency changes from the lowest (long waves) to the highest (short waves). For very long surface waves, only one mode of propagation is available, and almost no dispersion occurs; thus, 0 4 (f) — ► 0 as / — ► 0. Increasing / slightly, we introduce new modes of propagation and the variety of phase velocities available increases. As a result, 0 4 (f) grows and reaches its maximum . Further increase in frequency causes the effective concen tratio n of the phase velocities of different modes at the sm allest shear velocity of th e region. cm- m( /) does not change w ith frequency any more, b u t cmax (/) decreases. Of course, the speeds of propagation greater than cmax( /) also exist (see Fig. 3.3), however, the relative am ount of energy, transm itted by the corre sponding (high) modes is relatively low. The values of phase velocities, th a t are greater th a n cmax(/) are distinct, so the pulses of energy carried by the corre sponding modes arrive at the station one at a tim e and very soon after the strong S wave. These pulses will m ost probably not be noticeable on the background of various m ultiply reflected body waves which form the “tail” of the S wave. The modes th a t have phase velocities smaller than cmax( /) , arrive very soon one after another, may be several at a time. Moreover, those arrive relatively late, when th e body waves already exhausted their energy, so the presence of these (low) m odes can be easily recognized on the accelerogram and they may be counted 171 as contributing to the increase of the duration of strong m otion. Thus we can think about “effective” decrease of the span of values of the phase velocities; and this effect is shown as the decrease of the w idth of the shaded stripe on Fig. 3.3 as / — ► oo. In agreement w ith our expectation on the nature of the coefficient a 4 (/)> its value decreases w ith increasing frequency, for m oderate frequencies. For high frequencies ( /> 5 -r 10 Hz) practically does not depend on fre quency. The nature of the broadenning of strong m otion pulses w ith distance differs here from the dispersive nature of the low-frequency wave propagation. At high frequencies, the strong m otion consists prim arily of body waves (with some contribution of the surface wave energy). In this case the increase of duration of pulses m ight be attributed to the scattering on the long period com ponent of the random velocity fluctuations in the lithosphere (Sato, 1989), and this increase m ight appear to be independent on frequency. We note th a t at its m aximum (frequency about 0.2 hz), the value of a\ corresponds to the increase of duration by 2 sec per each 10 km of epicentral distance, and at / & 15-^20 Hz this value drops to 0.5 sec per each 10 km. More detailed consideration, pertinent to the physics of the increase in the duration of strong ground motion due to dispersive properties of the m edium between the source and the station can be found in C hapter 7. The constant coefficients a[h^ and have different meaning for different frequency bands. Thus, for channel # 1 ( /0 = 0.075) Hz, being the only coeffi cients distinct from zero, a[h^ and give the direct estim ate of duration for horizontal and vertical components. This estim ate does not change w ith magni tude or epicentral distance. For the second and third channels (/o = 0.12 -f- 0.21 Hz), a[h^ and a [v^ give the value of the duration at the epicenter of the earth- 172 quake and this value does not depend on the m agnitude of the event. For higher frequencies, ai(f) may not have any physical m eaning, because it effectively serves as a constant term in the truncated Taylor expansion of the equation dur(M) = aexp(/3M). However, comparing and the difference in duration of horizontal and of vertical motion can be studied. For very low fre quency , but this result is not reliable due to the small num ber of data points used in the analysis a t these frequencies. For the interm ediate frequen cies (0.21 -r 4.2 Hz), vertical com ponent possesses longer duration. This can be explained by more prominent participation of the P-wave in the vertical m otion. Analysis (see C hapter 8) shows th at for those frequencies, the origin of the first tim e interval, giving contribution to duration (see definition Eq. (2.2)) is different for horizontal and for vertical components. For vertical m otion, the first strong pulse usually corresponds to the arrival of the P-wave, while P-arrival does not produce noticeable strong motion on the horizontal com ponents. In the latter case, the first strong pulse can usually be related to the S-arrival. Thus, vertical com ponent seems to have more waves, producing strong m otion, and this results in the increase of the duration of the vertical motion com pared to the horizontal. A t high frequencies ( / > 8.5 Hz), the difference between vertical and horizontal m otions disappears due to multiple scattering w ith mode conversions th a t occur at such frequencies in highly inhomogeneous upper crust and in the sedim entary layers. W hat are the specific values of the duration of strong ground shaking, as predicted by Eq. (4.9)? Fig. 4.2 shows the dependence of the duration of strong m otion as obtained from Eq. (4.9) for various frequency bands. Numbers in the Horizontal com ponent f0 =.37 Hz 173 0 20 40 60 80 100 120 140 160 8 R T T 7 50 128 79 ,34 6 5 0 3 o to o 0 3 4 ,1 11 1 f I I I I l\l H 3 3 20 40 60 80 100 120 140 160 Epicentral distance (km) Horizontal com ponent f0 =.63 Hz 0 20 40 60 80 100 120 140 160 8 7 6 5 4 3 0 20 40 60 80 100 120 140 160 Epicentral distance (km) F ig . 4 .2 a Channels # 4 and # 5 : isolines of the duration (in seconds) of the horizontal com ponent of strong earthquake ground motion as defined by the Eq. (4.9). Numbers in the dashed “boxes” designate the num ber of the d ata points used in the analysis for the particular range of m agnitudes and epicentral distances. Dependence on epicentral distance is always linear for displayed channels. The parabolic dependence on m agnitude can be seen for high frequencies. For low frequencies, only linear dependence on m agnitude can be resolved. 0) 3 • rH 0 c ti s 8 7 ,378 190 20 6 5 ^ 3 > a > 4 i _ l l E 3 Horizontal com ponent f0 =1.1 Hz 174 20 40 60 80 100 120 140 160 C L ) TJ C J bfl 8 7 1 0i 406 29 27 6 5 \ 266 4 3 3 0 20 40 80 60 100 120 140 160 Epicentral distance (km) Horizontal com ponent f0 =1.7 Hz 20 40 60 80 100 120 140 160 0 ) '•d E 3 hfi 8 7 112 - 102' 36 < 2 0 6 <39 5 = 41 4 3 0 20 40 60 80 100 120 140 160 Epicentral distance (km) F ig . 4 .2 b Channels # 6 and #7 : isolines of the duration (in seconds) of the horizontal com ponent of strong earthquake ground motion as defined by the Eq. (4.9). Numbers in the dashed “boxes” designate the num ber of the d ata points used in the analysis for the particular range of m agnitudes and epicentral distances. Dependence on epicentral distance is always linear for displayed channels. The parabolic dependence on m agnitude can be seen for high frequencies. For low frequencies, only linear dependence on m agnitude can be resolved. Horizontal com ponent f0 = 2.5 Hz 20 40 60 80 100 120 140 175 160 ill I ill T T T T T T T 12 441 4 0 4 1 1 5 21 1 8 6 64 1 5 G bfl cd S 4 ^ 4 4 3 0 20 60 40 80 100 120 140 160 Epicentral distance (km) Horizontal com ponent f0 =4.2 Hz 20 40 60 80 100 120 140 160 < D rd 2 • r H tm c d 8 7 33i 234 6 - 189 44 5 5 5 2 3 5 4 77 1 111111 II111111 1 1 1 1 60 3 F t i i i I I i i i 0 1 -1 .1 -L L I I l-B q 160 20 40 80 100 120 140 Epicentral distance (km) F ig . 4 .2 c Channels # 8 and # 9 : isolines of the duration (in seconds) of the horizontal component of strong earthquake ground m otion as defined by the Eq. (4.9). Numbers in the dashed “boxes” designate the num ber of the d ata points used in the analysis for the particular range of m agnitudes and epicentral distances. Dependence on epicentral distance is always linear for displayed channels. The parabolic dependence on m agnitude can be seen for high frequencies. For low frequencies, only linear dependence on m agnitude can be resolved. Horizontal com ponent f0 =7.2 Hz 176 2 0 40 60 80 100 120 140 160 0 ) Ti 2 • rH 3 bfi c tf s 8 7 300 6 — i 30 5 553 4 3 3 0 20 40 60 80 100 120 140 160 Epicentral distance (km) Horizontal com ponent f0 =13 Hz 0 20 40 60 80 100 120 140 160 53 00 42 ;87 26 87 3 0 20 40 100 60 80 120 140 160 Epicentral distance (km) F ig . 4 .2 d Channels # 1 0 and #11: isolines of the duration (in seconds) of th e horizontal com ponent of strong earthquake ground m otion as defined by the Eq. (4.9). Numbers in the dashed “boxes” designate the num ber of the d ata points used in the analysis for the particular range of m agnitudes and epicentral distances. Dependence on epicentral distance is always linear for displayed channels. The parabolic dependence on m agnitude can be seen for high frequencies. For low frequencies, only linear dependence on m agnitude can be resolved. 177 Horizontal com ponent fo -2 1 Hz 0 20 40 60 80 100 120 140 160 8 7 21 6 13 H O 5 r-\ 4 47 3 160 20 40 60 80 120 140 0 100 Epicentral distance (km) F ig . 4 .2 e Channel #12: isolines of the duration (in seconds) of the horizontal com ponent of strong earthquake ground motion as defined by the Eq. (4.9). Numbers in the dashed “boxes” designate the num ber of the d ata points used in the analysis for the particular range of m agnitudes and epicentral distances. Dependence on epicentral distance is always linear for displayed channels. The parabolic dependence on m agnitude can be seen for high frequencies. For low frequencies, only linear dependence on m agnitude can be resolved. 178 “boxes” for ( Aj < A < A ;+ i, A / = /• 20 km, / = 0 8, { (4.10) I M k < M < M k+u M k = k + 3, k — 0 -r 4 represent the num ber of d a ta points used in the analysis, for the range of pa ram eters defined by Eq. (4.10). Em pty “boxes” correspond to the pairs of values (/, k ) where no d ata are available. At each frequency band the model presented can be considered to be reliably defined only for the range of m agnitudes and epicentral distances where substantial num ber of data points is available. Fig. 4.3 presents the statistics of the residuals of the model. For each d ata point, the absolute, £j(f) (Eq. (4.2a)), and the relative, £yel (Eq. (4.2b)), resid uals are plotted in Figs. 4.3a and 4.3b, versus frequency band (channel) num ber A finite interval of abscisae was assigned to each channel, and the x-coordinates of all residuals, corresponding to this channel, were chosen random ly w ithin this interval, w ith their y-coordinate (which gives the values of £j(f) or £yel(/)) un changed. As a result, the set of dots, representing the distribution of the residuals at each channel, forms a vertical stripe, and the variation of the density of this stripe w ith y-coordinate corresponds to the change in the distribution function of the residuals. Solid lines in Fig. 4.3 give the levels of the residuals, £±s%{f) and £r ±s%[f)- The level £+g%(f) is defined by the following: num ber of points with positive residuals and |e (/)| < k+ 6% (/)| ^ y total number of points w ith positive residuals ’ (4.11a) and e_s% (/) is the level where num ber of points w ith negative residuals and |e (/)| < \^-s%{f)\ _ ^ y total num ber of points w ith negative residuals (4.116) 179 40 30 20 10 0 3 1 2 5 4 6 7 9 10 12 3 0 0 250 200 150 100 50 0 - 5 0 -100 (%) b ) •* ..' ± i iii• ■ £ itSlfc.l l ^Wzi V '|^ '« - J < g ^ t ji y&^v 5$ 4 5 6 7 8 Channel n um ber F ig . 4 .3 T he residuals (dots) of Eq. (4.9) are plotted versus channel num ber. Model prediction corresponds to the horizontal thick solid line. T hin solid lines are m arked by the values of Eg%{f) an< i e5% (/) f°r several values of 6 : (1) 6 % = ±50% ; (2) 6 % = ±75% , (3) 6 % = ±90% , (4) 6 % = ±95% . For each < 5 , these lines bound the region where 6 % of all residuals (dots) are located. a) absolute residuals ery, see Eq. (4.2a). b) relative residuals Syel, see Eq. (4.2b). 180 Four different values of 6 % were considered: 50%, 75%, 90%, and 95%. Defini tions, sim ilar to Eq. (4.11) were adopted for T he distribution of the residuals can serve as a measure of the quality of the m odel. In this case, the actual duration of the strong ground m otion appeared to be “predicted” by the model with the error less than 50% for the absolute m ajority of d a ta points. We would like to add, however, th a t this so-called “error” is not ju st “the error of the model.” The large scatter in d a ta points also reflects the variety of phenom ena not included in our analysis and not known or understood a t present. 4 .3 . M o d e l dur — dur(M,M 2 , A , h , R , h R , R 2 ,h 2 ,<p) We now consider the most complete model among all studied in this work. It includes the dependence on m agnitude M; on epicentral distance A; on the depth of sediments at the recording site h; on R — the characteristic horizontal distance to the nearest rock, capable of producing reflections; and on the angle ip, subtended at the station by the surface of the rock, where reflection occurs. The equation of this model is: ( durW (f) | = f «<*>(/) U « r » ( / ) J \ «<•>(/) + 4 h)(f)-R + a{ rh)(f)-hR+ 4 v)(f)-h + a( 6v)(f)-R + a ^ \ f ) - h R + (4-12a) + 4 h\ f ) - R 2 + a ^ \ f ) - h 2 + 4o)(f)-'P + 4 v)(f)-R 2 + 4 v)(f)-h 2 + 4o{f)-lP "b a2{f)'M + + < J4 (/)'A + + + + 181 where the epicentral distance A, depth of sediments h and distance to the re flecting rock R are measured in kilometers, angle M = m ax{M ,M min(/)} = < (4.126) ^ M m\n (/] otherwise, where M min( /) — • (4' 12c) T he subscript “+ ” corresponding to the last 6 terms in Eq. (4.12a) designates th a t the expression in square brackets should be taken into account only if the quantity inside (we designate it as rregion in Eq. (3.26)) is positive: [7 "reg io n (/)] 4. — m a x { 0 , r reg i o n ( / ) } — i I 0 1 ( /). if ^region ( / ) > 0 , otherwise. (4.12d) T he values of R, h and < p are assumed to be zero if the station is located on rock. The dependence of the duration of strong motion on param eters h, R and < p is different for horizontal and for vertical components. Thus, we consider two different sets of coefficients, i — 5 -h 10} for the horizontal com ponents and ( a ^ ( / ) , i — 5 -rl0 } for the vertical ones. For the calculation of the duration of horizontal com ponents, d u r ^ ( f ) , only term s with superscript (k) should be included. For the estim ate of the duration of the vertical com ponent, dur^v\ f ) , only term s w ith superscript (v) should be considered. T he set {a,j(/), i < 4} was discussed already when the model Eq. (4.9) was introduced, and all the considerations (including two-step fitting, first to get M m- m ( /) and second to check if the introduction of M m- in changes coefficients) are applicable to the present model. However, the third iteration is required 182 here. T he first two iterations do not take Eq. (4.12d) into account by ignoring the subscript “+ ” in Eq. (4.12a). The result of these interaction is the set of coefficients ( a ,( /) , t = 5 v 1 0 } se c ond - This set was used in (4.12d) during the th ird iteration for estim ate of r re g i o n ( / ) at each data point. The resulting set, { a ,(/), t = 5 -r 10}third, was found to be almost the same as the previous two, because the m ajority of points have r reg ;on > 0 (see later Fig. 4.5). Table 4.3 presents the results of the regression analysis of Eq. (4.12). All notation here is the same as in Table 4.2 which described Eq. (4.9). Num ber of d ata points available, N(f), is reduced for some channels due to the lack of inform ation on the depth of sediments under some recording sites. As a result, the instability of the problem slightly increased, causing small increase of M mm in the corresponding frequency bands. Fig. 4.4 displays the dependence of the coefficients of the regression model on frequency (solid lines) and shows their reliability in the form of “cr-interval” defined in Eq. (4.5) (dashed lines), and in the form of the 95% confidence interval defined by Eq. (4.6) (dotted liens). No significant change can be found in the set ( a ,( / ) , i — 1 t 4 } as compared with the previous model (Eq. (4.9)). This is im portant, and shows, in particular, th at param eter R is not correlated w ith epicentral distance and indeed represents an independent characteristic of the propagation path. The coefficients ( a ,( /) , t = 5 -r 10} describe the influence of the geom etry of the sedim entary basin where the recording station is located. The coefficient aio(f) represents the “strength” of horizontal reflections (measured by angle <p), and it is positive. The coefficients scaling the contribution of the quadratic term s, R 2 and h 2, are negative as expected. Thus, the description of the prolongation T ab le 4.3 Results of the regression analysis of Eq. (4.12). Channel number fo (Hz) # of data points N (f) C o e f f i c i e n t s a; and t h e i r a c c u r a c y ( " t r - i n t e r v a l " ) Mmin °dur (sec) d urav (sec) a,O’) ±ai0 0 ±ai<v) a2 ±02 a3 ±03 34 ±04 a5(h) ± C T 5 < h) ± a 6(h) a7< > » ± a 7(h) a 8(h) ± a 8d> ) ash1 ! ±ag(h) aio<w ±Oio< h ) a5lv> ± a 5(W a6< v> ±ct6(v) a7< v> ± 0 7(v) a8< v> ± a 8< v> agM ± a 9< v> aio(v) ± a l0(v) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 19.1 ±1.1 19.4 ±1.6 .0 .0 .191 ±.018 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10.2 28.3 3 0.21 962 11.9 ±0.6 13.6 ±0.7 .0 .0 .215 ±.012 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 8.0 21.4 4 0.37 1381 5.8 ±3.4 6.8 ±3.5 .84 ±■54 .0 .196 ±.009 .0 .0 .0 .0 .0 .0205 ±0045 .0 .0 .0 .0 .0 .0169 ±.0075 7.3 20.9 5 0.63 1219 -5.8 ±2.8 -2.4 ±3.1 2.04 ±.46 .0 .200 ±009 .0 .458 ±.057 .0389 ±0111 -.0075 ±0008 -.42 ±08 .0174 ±0061 .59 ±1.15 .245 ±109 .0638 ±.0192 -.0054 ±0016 -.65 ±17 .0224 ±.0107 7.9 19.7 6 1.1 1550 -7.2 ±1.7 -4.2 ±1.9 2.01 ±•32 .0 .166 ±.006 .56 ±■54 .382 ±.051 .0348 ±0089 -.0070 +.0007 -.41 ±.09 .0224 ±.0044 .95 ±■78 .142 ±075 .0607 +.0143 -.0039 ±0011 -.59 ±13 .0209 ±.0073 6.7 16.3 7 1.7 1865 -4.3 ±1.0 -1.9 ±1.2 1.63 ±.21 .0 .134 ±.005 .89 ±.41 .201 ±.038 .0232 ±0068 -.0040 ±.0006 -.34 ±■07 .0082 ±.0032 1.52 ±60 .105 ±.055 .0267 +.0106 -.0024 ±.0008 -.44 ±10 .0 5.4 13.3 8 2.5 2005 6.3 ±1.7 7.4 ±1.8 -2.51 ±.66 .36 ±.06 .095 ±.003 .0 .142 ±019 .0142 +.0041 -.0025 ±0003 -.11 ±.03 .0053 +.0020 1.42 ±.37 .089 +.034 .0083 ±0063 -.0016 ±.0005 -.28 ±06 .0 3.49 3.5 9.6 9 4.2 2283 8.4 ±1.5 9.9 ±1.5 -3.74 ±.56 .52 ±.05 .076 ±.003 .0 .059 ±.012 .0 -.0006 ±0002 .0 .0027 ±.0014 .96 ±26 .0 .0 .0 -.12 ±.05 .0 3.60 3.1 7.6 10 7.2 2244 9.2 ±1.1 9.6 ±1.2 -4.52 ±.44 .65 ±.04 .062 ±.003 .0 .0 .0 .0 .0 .0 .32 ±•22 .0 .0 .0 -.04 +.04 .0 3.48 2.6 6.5 1 1 13 1572 6.2 ±1.1 6.3 ±1.1 -3.33 ±■42 .52 ±04 .055 ±003 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 3.20 2.0 5.1 12 21 724 10.1 ±2.1 10.1 ±2.1 -4.68 ±.84 .62 ±.08 .056 ±.007 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 3.77 1.8 4.2 I horiz 1 vert M M2 A il R hR R 2 h 2 h R hR R2 h2 h o r i z o n t a l c o m p o n e n t v e r t i c a l c o m p o n e n t C o r r e s p o n d i n g p a r a m e t e r s 0 0 C O [c o n sth] 40 30 20 10 -1 0 [co n stv] 40 30 20 10 f (Hz) -10 f (Hz) -2 - 4 -6 .6 .4 2 f (Hz) 0 .1 1 10 .20 .15 .10 f (H z) .05 .00 F ig . 4 .4 a The coefficients a ,(/), i = 1 -r 4 in Eq. (4.12), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “c-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). [hR] .10 .5 .08 .4 .06 .3 .04 2 10 .02 .1 f (Hz) % f (Hz) f (Hz) - l 0 .00 10 10 f (Hz) .000 J I I 1 , 1 ____L 10 f (Hz) -04 10 -.2 -.002 .03 -.4 -.004 .02 -.6 -.006 .01 -.8 % f (H z> -.008 -1.0 .00 F ig . 4 .4 b The coefficients a t ( /) , * = 5 -r 10, horizontal component, in Eq. (4.12), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 185 •'10 -1 .5 ,4 .3 .2 .1 (Hz) 0 1 1 10 .08 06 ,04 02 f (Hz) .00 v 04 ,03 .02 .01 f (Hz) .00 .000 10 -.0 0 2 -.004 -.006 -.008 f (H z) -.2 -.4 -.6 -.8 -1.0 F ig . 4.4c The coefficients a ,(/), i = 5 -r 10, vertical component, in Eq. (4.12), plotted versus the central frequency of the channels (solid lines). The coefficients axe bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 186 187 of duration due to the presence of the limited sedim entary basin by Eq. (4.12) appears to be meaningful, as it results in the increase of duration for the inter m ediate values of R and h and in no increase for small or large R and h (see discussion in C hapter 3). Fig. 4.5 shows the positive contribution to the overall duration, predicted by Eq. (4.12), m ade by the sum of term s involving R and h: ( d u r ^ ( R ,h ) 1 ( a ^ - h + a ^ - R + a[h^-hR + a ^ - R 2 + a ^ - h 2 1 { dAir{v){R,h) ) ~ { + a£o)-R + a( 7 v) -hR + a ^ - R 2 + a<v) -h2 ) ' The top row in Eq. (4.13) corresponds to the horizontal com ponent, and the bottom row describes the vertical component of motion. Similar to the case of Fig. 4.2, the num ber of the d a ta points, used in the analysis is presented for each range of param eters f Ri < R < R l+1, Rt = l-10, / = 0 t 7, I (3.27) I hk < h < hk+i, hk = k, k = 0 -r- 6, for each pair of values (Z , k ) in the corresponding “box” in R — h space. Em pty “boxes” mean th a t no d ata points are available for this range of pa ram eters. Notice th at for both horizontal and vertical com ponents practi cally all d ata points fall in the area where dur(R . h) > 0. Consequently, ^region = dur(R, h) + {contribution due to <p} is positive for even greater num ber of d ata points. This not only explains why the third iteration of the model (which treated the points where rregion < 0 in a special way, see (Eq. 4.12d)) and the second iteration (which did not consider condition rreg!on>0) gives practi cally the sam e results, but also verifies our assumptions about how the presence of a sedim entary basin can influence the duration of strong ground m otion (see discussion in C hapter 3). Horizontal co m p o n en t f0 =.63 Hz 188 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 g i ii 111111 1 1 1 ii 1 1 1 1 1 ii 1 1 1 1 ii 11 in 1111111111111111111111111111 n 111111 -p n 111111 R (km ) Vertical co m p o n en t f0 =.63 Hz 0 10 20 30 40 50 6 0 7 0 80 0 10 20 30 40 50 60 7 0 80 R (km ) F ig . 4,.5a Channel # 5 : isolines of the additional (relative to the basem ent rock sites) duration (in seconds) of strong ground m otion due to specific geom etry of a sedim entary basin, as predicted by Eq. (4.12). The positive contribution of th e term s dur^ (R,h) and dur^ (R,h) defined by Eq. (4.13) only is considered. N um bers in dashed “boxes” designate the number of the d a ta points used in the analysis for the particular range of R and h. See Fig. 3.15 for comparison with the raw data. H orizontal co m p on en t 10 2 0 30 4 0 50 f 0= l.l Hz 1 8 9 60 70 80 7 rriin 6 5 __ 4 27 3 2 5 9 1 30 22 0 0 0 10 20 30 40 50 60 70 80 R (km ) Vertical co m p o n en t f0 =1.1 Hz o 10 20 30 40 50 60 70 80 7 6 5 4 54 3 2 2 4 1 0 10 20 30 50 40 60 70 80 R (km ) F ig . 4 .5 b Channel # 6 : isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground motion due to specific geome try of a sedim entary basin, as predicted by Eq. (4.12). The positive contribution of the term s dur^h\ R ,h ) and dur^(R ,h) defined by Eq. (4.13) only is consid ered. Numbers in dashed “boxes” designate the number of the d ata points used in the analysis for the particular range of R and h. See Fig. 3.15 for com parison with the raw data. H orizontal co m p o n en t f0 =1.7 Hz 1 0 2 0 3 0 4 0 5 0 6 0 7 0 190 80 A 7 12 6 5 86 4 2 8 111 3 '4 2 < V 2 12| 2 4 1 1 3 6 / < 2 4 I I I I I 111 I 11111 I I I I 1 1 I . 1 1 . 1 . 1 0 0 0 10 20 30 40 50 60 70 80 R (km) Vertical co m p o n en t 1.7 Hz A 10 20 30 40 60 80 50 70 6 5 4 5 7 3 2 2 8 1 0 10 20 30 40 50 R (km) 60 70 80 F ig . 4 .5 c Channel # 7 : isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to specific geome try of a sedim entary basin, as predicted by Eq. (4.12). The positive contribution of th e term s dur^h\ R , h) and dur^ (R. h) defined by Eq. (4.13) only is consid ered. Num bers in dashed “boxes” designate the number of the d ata points used in th e analysis for the particular range of R and h. See Fig. 3.15 for com parison with the raw data. Horizontal co m p o n e n t f0 =2.5 Hz 1 9 1 10 20 30 40 50 60 70 80 7 T T T T 12 6 10 5 2 8 66 4 3 8 27 12 100 3 '45 85, 2 2 4 59 1 2 8 3 3 7 2 4 I I 11 I r 20 ............ 0 10 30 40 50 60 70 80 R (km) Vertical co m p o n en t f0 =2.5 Hz o 10 20 30 40 50 60 70 80 7 6 5 3 ^ 5 5 3 3 4 17 54 3 2 1 0 0 10 20 30 40 50 60 70 80 R (km ) F ig . 4 .5 d Channel # 8 : isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground motion due to specific geome try of a sedim entary basin, as predicted by Eq. (4.12). The positive contribution of the term s dur^h\R ,h ) and dur^ (R.h) defined by Eq. (4.13) only is consid ered. Numbers in dashed “boxes” designate the number of the d ata points used in the analysis for the particular range of R and h. See Fig. 3.15 for comparison with the raw data. Horizontal co m p o n en t f0 = 4.2 Hz 192 0 10 20 30 40 50 6 0 7 0 80 7 11111II11111 6 5 2 8 110 6 4 4 8 4 7 2 3 3 2 7 0 2 4 2 1 4 5 3 1 7 1 3 2 2 4 3 7 2 8 M .i 1 1 .ft I n 111 m ill i H o 70 0 20 0 10 30 40 50 60 80 R (km ) Vertical co m p o n e n t f 0 = 4 .2 Hz 10 20 30 40 50 60 70 80 a 7 + + 4-H IH -H -H 6 5 5 0 3 6 4 3 6 3 2 9 2 1 1 1 3 0 10 20 30 40 50 0 60 80 70 R (km ) F ig . 4 .5 e Channel ^=9:isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to specific geome try of a sedim entary basin, as predicted by Eq. (4.12). The positive contribution of the term s dur^ ( R . h ) and dur^ (R.h) defined by Eq. (4.13) only is consid ered. Numbers in dashea “boxes” designate the number of the d ata points used in the analysis for the particular range of R and h. See Fig. 3.15 for comparison with the raw data. 193 Vertical co m p on en t f0 =7.2 Hz 10 20 30 40 50 6 0 7 0 80 S 4 M 4 E 11111111 1 I 11 I 11 1 I 1 1 1 1 1 1 1 1 ! ' m m n i TlTTin IT ITI1 1 1 1 1 1 4 ( t i n m n I I I I I I I I T T ■& .5------- 0 t- - - - - 3 L - - __ 4 r~— V.5~ — = E 2 6 5 6 4 0 4 1 9 1 l = 4 2 7 7 4 3 6 E 1 1 6 4 7 0 5 ---- 5 2 2 L5-------- 2 9 ------- 0 1 3 7 1 6 1 3 1 1 1 4 1 3 1-L .l LLJ LL.I 2 _L1U 111 1 .1 4 III! Ill 1 1 5 J.l LI 1 1 1 1 1 2 J..LLL.U.I U .l 9 III......... J U..1U I I P 10 2 0 30 40 50 R (km) 60 70 80 F ig . 4 .5 f Channel # 10: isolines of the additional (relative to the basement rock sites, s = 2) duration (in seconds) of strong ground motion due to specific geometry of a sedimentary basin, as predicted by Eq. (4.12). The positive con tribution of the terms d u r^ (R ,h ) and dur^ v\R ,h ) defined by Eq. (4.13) only is considered. Numbers in dashed “boxes” designate the number of the data points used in the analysis for the particular range of R and h. See Fig. 3.15 for comparison with the raw data. 1 9 4 The coefficients responsible for the description of the prolongation of duration of strong ground shaking due to the specific geometrical properties of the limited sedim entary basin, are distinct from zero in the interm ediate frequency range only. This can be explained in term s of a simple physical model proposed by Trifunac (1990b) for the description of the effects produced by the geological sedim ents and local soils on th e amplification of strong motion. He noticed th a t a sedim entary (or soil) layer of thickness h works as a band-pass am plification filter w ith a specific high, / j , and low, / 2, cut-off frequencies. The low cut-off is the frequency for which the corresponding quarter wavelength in sedim ents (or soil) coincides w ith the thickness h of sediments (or soil layer). For high frequencies, the amplification effects can be overshadowed by inelastic attenuation which results in the dependence of f\ on, among all other param eters, the attenuation factor Q, typical for the sediments (or soil) considered. In the interm ediate range, / 2 > / > / i , the average amplification practically does not depend on frequency. Amplification effects can be explained by the m ultiple reflections in the sedim entary basin (of course, the increase in am plitude upon entering soft layer also takes place). The physical reason that stands behind the prolongation of duration in a sedim entary valley can be assumed to be the sam e m ultiple reflection phenom enon (see later C hapter 7). It seems logical then to consider T rifunac’s model for the explanation of the duration related effects. A t low frequencies ( / < 0.3 Hz), all prolongation coefficients {a»(/), i = 5 -r 10} are equal to zero and no influence of sedim entary basin on the duration of m otion can be noticed. At channel # 4 (/o = 0.37 Hz) the prolongation of duration is expressed by the term involving tp only (see Table 4.3). Angle of ef fective reflection, < p , happens to be more sensitive to long waves than param eters 195 R and h because it measures the overall “strength of reflection,” w ithout going into details of the geometry. A t / = 0.63 - T - 2.5 Hz the geometrical properties of the sedim entary basin “work” in full strength, and all term s in rregion have non-zero values. It is in teresting to notice, th a t the range of param eters R and h where dur(R. h) > 0 (the area inside zero isoline in Fig. 4.5), being slightly different for horizontal and vertical com ponents, preserves itself for both com ponents in the frequency range / = 0.63 -r 2.5 Hz. This effect may be similar in nature to the absence of dependence of the amplification factor on the frequency of m otion inside Tri- funac’s > / > / j . Inside the frequency range where dur(R , h) > 0, the value of dur(R, h). however, does depend on frequency. The m axim um possible contri bution of dur(R, h) changes from 7.5 sec to 2.5 sec for the horizontal com ponents and from 5 sec to 3.5 sec for the vertical component when frequency changes from 0.63 Hz to 2.5 Hz. The difference in prolongation for different frequencies m ight naturally come from the assum ption th at, making the same amplification effect, longer waves will cause more prominent prolongation of duration simply due to their long period nature: each additional cycle takes more tim e when long waves are considered. A fter the transition range (4.2-f7.2 Hz), where some effects of the geom etry of the sedim entary basin can still be noticed, the short wave range / > 5.0-=-8.5 Hz sets in. For these frequencies, no influence on duration of strong ground m otion by the presence of a sedim entary basin w ith specific geometry can be observed. M ost probably this can be explained by high inelastic attenuation, typical for the upper hundreds of meters of soil and sediments (Trifunac, 1993). In Pliocene- Pleistocene sediments of the Los Angeles basin, Hauksson et al. (1987) estim ated 196 Q for shear waves to be about 25 for the depths between 420 m and 1500 m, and about 100 between the surface and 420 m. M alin et al. (1988) observed an average Q = 9 for shear waves in an ophiolite complex at Oroville valid for the depths from 0 to 500 m. Thus, for high frequencies, m ultiple reflections, causing prolongation of duration, have to com pete w ith inelastic attenuation, and the resulting effect appears to become negligible. One of the “geology” models, used by Trifunac (1990b) for the description of am plification effects has th e following param eters: depth of sedim ents h — 2 km , shear wave velocity (3 = 2 km /sec, inelastic attenuation factor Q = 100. The estim ates of fi and fi come out to be: for high cut-off frequency fi — 5 Hz, for low cut-off fi = 0.24 Hz. A good agreement can be found between these values of / i and /2 and the frequencies th a t bound the range where the prolongation of d uration due to the presence of a sedim entary basin can be noticed. We exam ine now the differences between the sets of coefficients obtained for the horizontal {a\h\ f ) , i — 5 — 10} and for the vertical {a,-"^(/), t = 5-1- 10} com ponents. Table 4.3, and Figs. 4.4-4.5 show th at, in general, vertical compo nent is more sensitive to the values of h—vertical characteristic dimension of the valley, and horizontal component is more sensitive to the horizontal characteristic dim ension— distance to the reflecting rocks, R. Coefficients and < 4 ^ (/), which scale the influence of the param eter R on the horizontal com ponent of m otion, are better defined and can be followed in the wider frequency range th a n their vertical counterparts <4v\ f ) and ag> \ f ) . Conversely, the coefficients th a t describe the contribution of the depth of sedim ents h to the duration of m otion on horizontal com ponent, < 4 ^ (/) and a[h\ f ) , have higher variances and are distinct from zero in a narrower frequency range com pared to < 4 ^ (/) and 1 9 7 a 7^ ( / ) i which are responsible for the similar description in the case of vertical com ponent. As a result, the duration of strong m otion on horizontal com ponents appears to be more sensitive to the horizontal characteristic dim ension R, while duration on vertical component “feels” the depth of sediments under th e station, h, better, than it “feels” R. If we assume th a t Love waves and the m ultiply scattered S H waves, give higher contribution to the horizontal com ponent than to vertical, and th a t the Rayleigh waves do produce a substantial vertical mo tion, then this can be explained at follows. Param eter R describes the geom etry of the basin on a large scale, while h gives more local description in term s of depth of sediments right under the recording station. According to Levander (1990), Love wave dispersion perhaps is much more sensitive to irregularities in otherwise layered media, than Rayleigh waves. O ur irregularly shaped reflect ing rocks, positioned at the distance R from the station, can be interpreted as such irregularities. Rayleigh wave dispersion perhaps does not “feel” the sm ooth irregularities in a wave guide th a t much, and this wave can be considered as a representative of the mean structure beneath the receiving station. In our term s, the description of this structure can be done by the depth of sedim ents at the recording site, h. So, simplifying the situation, we assume th a t Love waves ap pear to carry inform ation about param eter R, and this inform ation is recorded by the horizontal components. Rayleigh waves have substantial vertical compo nent and “rem em ber” the depth of sedim entary deposits, h, causing the duration of vertical m otion to be sensitive to h. There is also the second reason why duration of strong motion on vertical com ponents is more sensitive to the depth of sediments than the duration on the horizontal com ponents. As it was mentioned earlier and will be dem onstrated in 1 9 8 C hapter 8, duration of strong motion of vertical com ponent has some contribu tion from the body P-wave. The P-arrival is seen on horizontal com ponent also, b u t it does not bring enough energy to be counted as strong m otion. As a result, the inform ation about the depth of sediments, carried by P -wave and the waves scattered and reflected several times in the sedim entary layer ju st underneath the station, can be recognized prim arily by the vertical com ponent of m otion. T he last param eter, the angle < p with which the reflecting rocks are seen from the station, is a measure of the intensity of the duration of the horizon tal reflections. The characteristic dimension of these reflections is described by R. Recalling the fact th a t the horizontal com ponent is more sensitive to the value of R, th a n the vertical, we should not be surprised by the sim ilar behavior of com ponents w ith respect to the value of p. Actually, param eter < p is “sup plem entary” to R as its value shows how noticeable the effect of R having the p articular value can be. Both < p- and R-related coefficients are b etter defined on horizontal com ponents. The typical values, obtained for a io (/) , give the increase of duration by about 2 sec for / « 0.37 t 1.1 Hz and by about 0.5 sec and less for / « 2.5 -f 4.2 Hz per each 100° of (p. Fig. 4.6 shows the statistics of residuals of Eq. (4.12,) for absolute sy (Eq. (4.2a)) and relative £yel (Eq. (4.2b)) residuals. All notation coincides w ith the one introduced earlier for Fig. 4.3, which displays the residual statistics of Eq. (4.9). T he “quality” of a model can be measured, as it was m entioned already, by the value of \e±s%{f)\ and |e±$% (/)|, defined by Eq. (4.11). T he sm aller these values are, the better the model is. Comparing the Fig. 4.3b, which corresponds to the model w ithout R, h and < p, and Fig. 4.6b, which describes the present model Eq. (4.12), some decrease in |£±5% (/)| can be noticed for the m ore com- 199 40 30 20 10 -1 0 -2 0 1 2 3 4 5 6 7 8 9 10 11 12 300 250 200 150 100 50 -5 0 -1 0 0 12 Channel num ber F ig . 4 .6 The residuals (dots) of Eq. (4.12) are plotted versus channel num ber. Model prediction correspond to the horizontal thick solid line. T hin solid lines are marked by num bers and show £$%(/) and £$%(/) for several values of 6: (1) 6% = ±50% ; (2) 6% = ±75% , (3) 6% = ±90% , (4) 6% = ±95% . For each 6, these lines bound the region where < 5 % of all residuals (dots) are located. a) absolute residuals Ej, see Eq. (4.2a). b) relative residuals £yel, see Eq. (4.2b). 200 plete model. However, the introduction of new param eters results in disregarding of some d ata points where at least one of the new param eters is not available for some reason. Smaller num ber of d ata points leads to less flexibility (smaller num ber of degrees of freedom) and, consequently, to the lower accuracy of the solution of the least square fit. So, comparison of the two models w ith the help of the residual statistics is not straightforward. We may add here th a t the main reason why Eq. (4.12) is considered, is not the reduction in the relative errors of the prediction of the duration it gives, but the fact th a t the new model provides better understanding of the physics of the problem and shows the way how the mechanism of prolongation of duration of strong ground motion might work. Further considerations regarding this model and the nature of the phenom ena involved can be found in C hapter 7. 4.4 M o d e ls dur = dur (M ,M 2, A , R , R 2,(p) a n d dur = dur (M , M 2,A,h, h2) It is not always possible to obtain complete inform ation about all param eters involved in the description of a sedim entary basin in Eq. (4.12). We will describe here two models th a t can be used if only h or only R and + a,2(f)-M + az(f)-M + a 4 ( / ) - A + «V" '(/) i + a( 6h) (f)-R + a( 8 h) (f)-R2 + a[^ a 6W )( / ) -^ + a 8V)( / ) - ^ 2 + a io (f)-’ P + + (4.14a) 201 where the epicentral distance A and the horizontal characteristic dim ension R are m easured in kilometers, _ { AT, if M > M min(f), M = rnax{M ,M min( / ) } = ’ f (4.146) -iWniiDy/) otherwise, M m in(/) = 2 ^ ( f ) ’ (4’14C) and H + = m ax{0,[-]} = { 1 '1 ’ ‘f ['l > 0 , (4.14d) 1 0 otherwise. T he values of R and < p are assumed to be zero if recording station is located on the basem ent rock. As before, term s with superscript (h ) correspond to th e horizontal components and term s with superscript (v) describe the vertical m otion. Eq. (4.14) was fitted to the data in three steps identical to those described earlier for the model Eq. (4.12). The results of this regression analysis are pre sented in Table 4.4 and in Figs. 4.7 and 4.8. Table 4.4 gives the stan d ard (see Tables 4.2 and 4.3 earlier) representation of the numerical results, showing the num ber of d ata points available, N(f), all fitted coefficients a ,( /) and their vari ances (/*), M mm, average duration durav and the standard deviation from the model a dur for each frequency band. All notation was defined in the discussion of Table 4.2. Fig. 4.7 presents the frequency dependent coefficients of Eq. (4.14). Com par ison w ith Fig. 4.4 shows th a t all coefficients, involved in this “trun cated ” model, behave very similarly as the corresponding coefficients of the “com plete” model Eq. (4.12). Again, duration of strong ground motion of the horizontal compo nents appears to be more sensitive to the value of the horizontal characteristic T ab le 4.4 Results of the regression analysis of Eq. (4.14). Channel number fo (H z) # o f data points N (f) C o e f f i c i e n t s a; a n d t h e i r a c c u r a c y ( " a - i n t e r v a l " ) Mmin <7dur (sec) durav (sec) a j(h) ±0 1 ® a i ® ±0 1 ® 3 -2 ±< 72 33 ± 0 3 34 ± 0 4 36(W ±0 6 ® a s ® ±08® 310® ±(7l0® 36® ±0 6 ® 38® ±08® 310® +O10® 1 0.075 37 40.8 ±2 .0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 .0 .0 1 0 .2 38.3 2 0 . 1 2 311 19.1 ±1.1 19.4 ±1.6 .0 .0 .191 ±.018 .0 .0 .0 .0 .0 .0 1 0 .2 28.3 3 0 .2 1 962 11.9 ±0 .6 13.6 ±0.7 .0 .0 .215 ± .0 1 2 .0 .0 .0 .0 .0 .0 8 .1 21.4 4 0.37 1381 5.2 ±3.4 5.8 ±3.6 .94 ±.54 .0 .195 ±.009 .0 .0 .0206 ±.0045 .093 ±.066 -.0013 ± .0 0 1 0 .0144 ±.0079 7.3 20.9 5 0.63 1891 -6.2 ±2.2 -4.8 ±2.3 2.01 ±.36 .0 .198 ±.007 .313 ±.033 -.0045 ±.0005 .02 0 1 ±.0039 .285 ±.056 -.0037 ±.0008 .0254 ±.0065 7.3 18.4 6 1 .1 2431 -5.2 ±1.3 -2.8 ±1.4 1.61 ±.23 .0 .169 ±.005 .322 ±.024 -.0050 ±.0004 .0216 ±.0029 .204 ±.040 -.0027 ±.0006 .0271 ±.0047 6 . 2 15.3 7 1.7 2892 -3.0 ±0.8 -0.9 ±0.9 1.27 ±•15 .0 .137 ±.004 .189 ±.018 -.0028 ±.0003 .0095 ±.0021 .165 ±.027 -.0019 ±.0004 .0084 ±.0031 5.0 1 2 .2 8 2.5 3132 6.4 ±1.6 7.8 ±1.6 -2.44 ±.60 .35 ±.06 .094 ±.003 .106 ±.012 -.0015 ±.0002 .0047 +.0014 .122 ±.018 -.0015 ±.0003 .0020 ±.0021 3.49 3.4 9.0 9 4.2 2608 8.5 ±1.4 10.5 ±1.5 -3.88 ±.55 .54 ±.05 .076 ±.003 .060 ±.012 -.0007 ±.0002 .0022 ±.0014 .0 .0 .0 3.59 3.1 7.6 10 7.2 2553 9.6 ±1.1 1 0.1 ±1-1 -4.68 ±.42 .66 ±.04 .064 ±.003 .0 .0 .0 .0 .0 .0 3.55 2.6 6.4 11 13 1572 6.2 ±1.1 6.3 ±1.1 -3.33 ±.42 .52 ±.04 .055 ±.003 .0 .0 .0 .0 .0 .0 3.20 2.0 5.1 12 21 724 1 0.1 ±2.1 1 0.1 ±2.1 -4.68 ±.84 .62 ±.08 .056 ±.007 .0 .0 .0 .0 .0 .0 3.77 1.8 4.2 1 horiz 1 vert M M 2 A R R 2 < P R R 2 9 horizontal vertical C o r r e s p o n d i n g p a r a m e t e r s 202 [co n sth] [co n stv] 40 40 f (Hz) 30 30 20 20 -2 -4 ± f (Hz) f (Hz) -10 -10 -6 a .8 .20 6 .15 4 .10 2 .05 f (Hz) f (Hz) 0 .00 1 1 10 10 F ig . 4 .7 a The coefficients a t ( /) , i = 1 -j- 4, in Eq. (4.14), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 203 .000 1 L I I I _ _ _ _ _ _ _ _L * 10 f (Hz) -04 .5 -.0 0 2 .4 .03 .3 -.004 .02 .2 -.006 .01 .1 (H z) .0 -.008 .00 10 (y) V .000 i n » I ____ l 10 f (Hz) -04 5 -.002 .4 .03 3 -.004 .02 .2 -.0 0 6 .01 1 (Hz) .0 -.008 .00 10 F ig . 4 .7 b The coefficients o e (/), a8(/) and aio(/)- (top row—horizontal component, bottom row—vertical component) in Eq. (4.14), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “<7-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 204 205 dimension R , than the duration of the vertical motion. Possible explanations of this phenom enon were already discussed earlier in this C hapter. Fig. 4.8 displays the positive contributions of the term s a6(f)-R + a9(f)-R2 to the total duration, predicted by Eq. (4.14), for horizontal and for vertical com ponents. In the “truncated” model, where the geom etry of th e basin is m odel), the result appears to be “averaged” over the depth of sedim ents h. This can be seen from comparison of Fig. 4.8 (“truncated” model, Eq. (4.14)) w ith Fig. 4.5 ( “com plete” model, Eq. (4.12)). Second “truncated” model, considered in this section, addresses the case when only the depth of sediments, h, is available: described by R only (instead of by R and h, as it is done in the “com plete” (4.15a) where the epicentral distance A and the depth of sediments under the recording site, h, are measured in kilometers, M - m ax{M ,M mm(/)} M min(/) if M > M min( /) , otherwise, (4.156) M min(/) and Superscripts (h ) and (v) in Eq. (4.15a) correspond to the horizontal and to the vertical com ponents of motion respectively. max 206 a S ,h)(f)*R+akh)(f)*Ra (sec) - .37 Hz (C h.4) .63 Hz (C h.5) - 1.1 Hz (C h .6) 1.7 Hz (C h .7) 2.5 Hz (C h .8) \ \ ---------------- 4.2 Hz (C h .9) a( 6 v)(f)*R+akv)(f)*R2 (sec) 6 5 4 3 2 1 K 0 0 10 20 30 40 50 60 70 80 R (km ) F ig . 4 .8 A dditional (relative to the basement rock sites) duration (in seconds) due to the specific horizontal characteristic dimension of a sedim entary valley, R, as predicted by Eq. (4.14): a) horizontal com ponent; b) vertical com ponent. This “truncated” model preserves the m ain features of the “com plete” model (Eq. (4.12), Fig. 4.5) regarding the behavior of the term s which describe the prolongation of duration due to the specific geometry of a sedim entary basin. 207 The regression analysis was completed in three steps, the way it was done for Eqs. (4.12) and (4.14). The results of the regression analysis are shown in Table 4.5 and in Figs. 4.9 and 4.10. As before (see “complete” model Eq. (4.12)), ver tical com ponent is more sensitive to the depth of sediments, than the horizontal. 4 .5 M o d e ls dur = dur A,s) a n d dur = dur (M , A ,s,s£,) D epth of sediments at a recording site is the type of inform ation th a t is not always easy to obtain. The same can be said about horizontal characteristic dimension of a sedim entary basin. Therefore, it is useful to consider the model, which utilizes simplified description of the geological conditions at the station. T he classification of sites in term s of param eter s was described earlier (C hapter 3), so we will ju st recall here these definitions, s = 2 is assigned to the sites, located on basem ent rock, s = 0 correspond to the sites on sedim ents, and interm ediate (or hard to determine) sites are designated as s = 1. According to the discussion in C hapter 3, two independent coefficients are needed to describe the influence of param eter s on duration of strong ground m otion, because, unlike other param eters involved in the model, s is a qualitative variable. Thus, we consider the model d u r ^ i f ) j f a[h){f) d u r ^ ( f ) J " 1 „<">(/) + a 13( /) - S (1)+ a i 4( /) - S (0), where the epicentral distance A and the depth of sediments under recording site are m easured in kilometers, M = m ax{M , M min(/)} M min ( / ) otherwise if M > M min( /), (4.166) T ab le 4.5 Results of the regression analysis of Eq. (4.15). Channel f 0 # o f data points C oefficients a; and their accuracy ("a-interval") Mmin <^dur durav number (H z) N (f) aiW iOjCh) ajC h) ± G 100 a2 ±02 &3 +03 34 ± 0 4 a5W ± a 5(h) a9(h) +0 9 ® a5(v) +O5W a9(v> ± C T 9(V ) (sec) (sec) 1 0.075 37 40.8 +2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 19.1 ± 1.1 19.4 ± 1.6 .0 .0 .191 +.018 .0 .0 .0 .0 10.2 28.3 3 0.21 962 11.9 ±0.6 13.6 ±0.7 .0 .0 .215 ±•012 .0 .0 .0 .0 8.1 21.4 4 0.37 1499 7.8 ±3.2 8.2 ±3.2 .84 ±•52 .0 .191 ±.008 .0 .0 .0 .0 7.4 21.0 5 0.63 1796 1.3 ±2.1 3.3 ±2.4 1.60 ±•37 .0 .182 ±.008 .0 .0 1.43 ±.84 -.29 ±.14 7.9 18.8 6 1.1 1600 -1.3 ±1.8 0.9 ±1.9 1.63 +.32 .0 .154 ±.007 1.09 ±.45 -.20 ±.08 1.40 +.66 -.24 ±.12 7.5 16.6 7 1.7 1916 -3.0 ±1.0 -1.1 ±1.1 1.63 +.20 .0 .126 ±.005 1.12 ±■33 -.18 ±.06 1.68 +.47 -.26 ±.09 5.8 13.5 8 2.5 20 59 6.0 ±1.8 6.6 ±1.9 -2.23 +.68 .36 ±.06 .089 ±.003 .22 ±.21 -.02 ±.04 1.21 ±.30 -.18 ±.05 3.10 3.9 9.8 9 4.2 23 89 8.5 ±1.4 9.1 ±1.5 -3.73 ±.55 .55 ±.05 .070 ±.003 .0 .0 .95 ±.28 -.12 ±.05 3.39 3.2 7.6 10 7.2 224 4 9.2 ±1.1 9.6 ±1.2 -4.52 +.44 .65 +.04 .062 ±.003 .0 .0 .32 ±.22 -.04 ±.04 3.48 2.6 6.5 11 13 1572 6.2 ±1.1 6.3 ±1.1 -3.33 ±.42 .52 ±.04 .055 ±.003 .0 .0 .0 .0 3.20 2.0 5.1 12 21 724 10.1 ±2.1 10.1 ±2.1 -4.68 ±.84 .62 ±.08 .056 ±.007 .0 .0 .0 .0 3.77 1.8 4.2 1 1 M M 2 A h h2 h h2 horiz vert horizontal vertical C o r r e s p o n d i n g p a r a m e t e r s 208 [co n sth] [co n stv] 40 40 f (Hz) 30 20 20 - 2 -4 /•A f (Hz) f (Hz) - 1 0 -10 - 6 a a .8 / .20 .6 .15 4 .10 2 .05 f (Hz) f (Hz) 0 .00 1 1 10 10 F ig. 4 .9 a The coefficients a t ( /), i = 1 -r 4, in Eq. (4.15), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “o-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 209 10 f (Hz) - l io f (Hz) .0 f (Hz) -.2 - .4 - . 6 -.8 - 1.0 .0 f (Hz) -.2 - .4 - .6 -.8 - 1.0 F ig . 4 .9 b The coefficients 0.5(f) and 0 9(f) (top row—horizontal component, bottom row—vertical component) in Eq.(4.15), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “cr-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 210 211 - .63 Hz (C h.5) -- 1.1 Hz (C h.6) - 1.7 Hz (C h.7) 2.5 Hz (C h.8) - 4 . 2 Hz (C h.9) - 7.2 Hz (C h .10) a ^ h ^ ( f ) * h + a ^ 9 h ) ( f ) * h 2 (sec) 3.0 2.5 2.0 1.5 1.0 3 . 0 r a ^ v) ( f ) * h + a ^ v) ( f ) * h 2 (sec) 2 .5 - 2.0 1.5 \ 5 10 "-Os //;/ h (km) F ig . 4 .1 0 A dditional (relative to the basement rock sites, s — 2) duration (in seconds) due to the depth of sedim ents under the recording site, h, as predicted by Eq. (4.15): a) horizontal component; b j vertical com ponent. This “truncated” model preserves the m ain features of the com plete model (Eq. (4.12), Fig. 4.5) regarding the behavior of the term s which describe the prolongation of duration due to the specific geometry of a sedim entary basin. 212 M min( /) - 2a* ffy (4' 16c) and s (1) = • I 1' if 5 = 1, 1 0, if 5 ^ 1 , s W = < f 1’ if 5 = 0, l o , if 5 ^ 0 . As before, superscript (h) corresponds to horizontal and (u) to vertical compo nent. Coefficients ( a t ( /) , i = 1 -f- 4} were discussed earlier. Two new coefficients ®i3 ( /) and ai 4 (f) describe the influence of the geological conditions on the dura tion of strong ground motion. The first one, Oi3 (/), shows the change in duration for interm ediate (5 = 1) sites compared with stations, located on the basem ent rock. The second coefficient, a i4(/), scales the change in duration for sites on sedim ents (s = 0) com pared to the basement rock sites, taken as a reference. Prelim inary investigation shows th a t Oi3(/) and a 14(/) do not differ for hori zontal and for vertical components. As we saw before, coefficients, scaling the influence of the depth of sedim ents, h, and the influence of the distance to the nearest reflecting rock, R, were different for horizontal and for vertical m otion. This difference in the behavior of the 5-related coefficients and h- or iE-related co efficients can probably be explained by the “roughness” of 5-classification alone. This classification is unable to pass the information necessary for distinguishing betw een horizontal and vertical directions, to the regression coefficients a j 3 and a 14. Num erical results of the regression analysis of Eq. (4.16) are sum m arized in the Table 4.6 (all notation coincides with th at discussed earlier). G raphical representation of the regression coefficients, plotted versus the central frequency T ab le 4.6 Results of the regression analysis of Eq. (4.16). Channel fo # o f data points C oefficients aj and their accuracy ("a-interval") Mmin Odur durav number (H z) N (f) a iW + 0 )(hl ai(W ±C](h) a 2 ±(j 9 a3 + 0 3 34 ± 0 4 313 ±013 314 ±0]4 (sec) (sec) 1 0.075 37 40.8 ±2 .0 32.5 ±3.1 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 19.1 ± 1.1 19.4 ± 1.6 .0 .0 .191 ±.018 .0 .0 10.2 28.3 3 0.21 962 11.9 ±0 .6 13.6 ±0.7 .0 .0 .215 ± .0 1 2 .0 .0 8.1 21.4 4 0.37 1179 7.2 ±3.4 7.3 ±3.4 .60 ±.55 .0 .184 ±.009 .0 3.06 ±.51 7.7 20.5 5 0.63 1641 -5.3 ±2.3 -2.9 ±2.4 1.89 ±.37 .0 .175 ±.008 3 .0 2 ±.85 6 .0 2 ±•74 8.1 18.3 6 1.1 2171 -6.3 ±1.5 -3.8 ±1.5 2.08 ±.25 .0 .140 ±.006 1.39 ±.63 3.85 ±•54 7.0 15.1 7 1.7 2625 -6 .2 ±0.9 -3.9 ±0.9 2 .0 0 ±.15 .0 .1 2 0 ±.004 .81 ±.44 2.33 ±.38 5.5 12.2 8 2.5 28 72 5.7 ±1.7 7.1 ±1.7 -2.61 ±.63 .42 ±.06 .082 ±.003 .0 1.30 ±.16 3.11 3.7 8.9 9 4.2 2439 7.8 ±1.5 9.0 ±1.5 -3.86 ±.55 .57 ±.05 .070 ±.003 .0 1.14 ±.14 3.39 3.2 7.4 10 7.2 2351 9.3 ±1.1 9.9 ±1.1 -4.67 ±.43 .6 6 ±.04 .063 ±.003 .0 .49 ± .1 2 3.54 2 . 6 6.3 11 13 1488 6.1 ± 1.1 6 .2 ± 1.1 -3.41 ±.43 .54 +.04 .054 ±.003 .0 .38 ± .11 3.16 2 . 0 5.1 12 2 1 683 1 1.6 ±2.3 11.7 ±2.3 -5.43 ±.91 .70 ±.09 .056 ±.007 .0 .42 ±.15 3.88 1 .8 4.4 1 horiz 1 vert M M 2 A s (» S(0 ) C o r r e s p o n d n g p a r a m e t e r s [c o n s th] 40 30 20 J Q . - 1 1 \ I I ___ J P i > » < ■ ■ % M f (Hz) -10 [c o n s tv] 40 20 <s S > f (Hz) - 1 0 f (Hz) -2 -4 -6 8 6 4 2 f (Hz) 0 .20 % % .15 .10 .05 f (Hz) .00 F ig . 4 .1 1 a The coefficients a ,( /) , i = 1 - r 4 in Eq. ( 4 . 1 6 ) , plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “o-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 6 5 4 3 2 1 0 .1 1 10 f (Hz) [S(0)] f (Hz) F ig . 4 .1 1 b The coefficients ai3 (/) and a ^ f ) in Eq. (4.16), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “< 7 -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 215 216 of the channels, is given in Fig. 4.11. The coefficients { a » (/),t = 1 -i- 4} and M min(/)> representing the dependence of duration on m agnitude and epicentral distance, rem ain practically the same as in all previously considered models. Two new coefficients, ais(f) and a i 4( /) are of special interest here. Notice th a t both of them are significantly different from zero in the sam e frequency range, where the coefficients, scaling the influence of h , R and < p were well defined (i.e. ^ 0). D uration of strong motion is longer on sedim entary sites then on rock sites (see coeff. a ^ ) by about 6 sec at frequency 0.63 Hz (channel # 5 ); the m axim um prolongation of duration in a sedim entary basin scaled by param eters R and h , is 7.5 -f-5 sec (Fig. 4.5) at the same frequency 0.63 Hz. D ata for channel # 8 (fo = 2.5 Hz) give the following results: additional duration, scaled by s, is about 1 sec, while scaled by R and h is less than about 2.5 -h 3.5 sec. This sim ilarity confirms th a t s-site classification and more detailed description of the site properties by param eters R and h are the two different interpretations of the influence of geology surrounding the site on the duration of strong ground m otion. Com parison of ais(f) and a i 4 (/) also makes some physical sense. Excluding high frequencies, where ai 3 (f) is not well defined, a 14( /) is greater than ai 3 (f) and has smaller variances. It corresponds to the duration on sedim ents (s = 0, coefficient 014) being longer, than duration at interm ediate sites (s = 1, coef ficient 013). Explanation may lay in the assumption th a t interm ediate sites do not have well defined multiple-layered sedim entary structure underneath them , as the sites on sedim ents. As a result, the multiplicity of reflections, constructive interference, and, therefore, duration are reduced. 217 The last model, considered in this C hapter, examines the influence of local soil conditions on the duration of strong ground motion. The classification of sites (C hapter 3) is accomplished by assigning a value to the soil param eter Sl - Deep soil sites have sl = 2, stiff soil is designated as sl = 1, and sl = 0 stands for a “rock” site. According to the discussion in C hapter 3, both qualitative indicator variables should be included in the model equation in the form given by Eqs. (3.28)-(3.3l), th a t is a n ( /) - S ,£ ( 1^ + a i 2(/)-S ,£ ,2^ + + ai4(/)--Si,0\ where and S are indicator variables for s = 1 and s = 0, defined by Eq. (4.16d), and and S ^ are analogous indicator variables for sl = 1 and sl = 2 (see Eq. (3.29)). However, substantial reduction in the num ber of d ata points (due to the lack of inform ation about sl for many sites) causes instability of the regression analysis. We decided, although, strictly speaking, this is not a correct procedure, to treat param eter s in this model as a regular quantita tive “continuous” variable. This reduces the number of unknown coefficients in the model and “improves” numerical stability. Note, th at consideration of s as a quantitative variable is equivalent to the assumption th a t Ci4( /) = 2a i 3 ( /). Fig. 4.11 shows, th a t the last equality is not far from being correct. Thus, the disregarding of the qualitative nature of s dose not constitute a m ajor m isrepre sentation. U nfortunately, this is not enough for improving the stability of the inversion problem . T he next step we considered was to disregard the term az(f)-M2. The 2 1 8 model we obtained looks is then: — max + a4 ( / ) ’A + aiB (/)•( 2 s) + a ii'l where A is expressed in kilometers and s i l) = ( 1, if sl = 1, L I 0 , if sl # 1, s l 2) = { 1, if sl = 2 , L I 0 , if sl # 2. + i 1) + a 1 2 ( f ) - S ^ \ (4.17 a) (4.176) To avoid negative values of the duration at locations close to a source of small m agnitude (which results from the model due to the lack of d ata at small M), we assum e th a t the duration at the source should not be less th a t 1 sec. A d a ta base w ith b etter coverage of small earthquakes is necessary to tre a t cases w ith small m agnitude in more realistic way. The reason to consider the term a is (/)-(2 — s) instead of a i 5( /) - s is th a t we prefer to have basem ent rock as a reference and to deal w ith positive 015 if the duration on sediments is longer th an th a t on the rock sites. The results of the regression analysis of the model Eq. (4.17) are shown in Table 4.7 and Fig. 4.12. Substantial reduction of the num ber of d ata points, N (f), can be noticed from comparison of Table 4.6 (model w ithout s^) and Table 4.7 (model w ith sl). Coefficient 0.4 (f), responsible for scaling the influ ence of the epicentral distance A, did not change. Coefficients a i( f) and 0 2 (f) have different meaning now due to consideration of linear approxim ation of ex ponential function aexp(f)M) « const 1 + const2 -A/ instead of the quadratic approxim ation a e x p(PM) « consti + const2-M + const3 -M 2. T ab le 4 .7 Results of the regression analysis of Eq. (4.17). Channel fo # o f data points C oefficients a, and their accuracy ("a-interval") (7dur durav number (H z) N (f) aj(h) ±cti< W aiCh) ±ci(h ) a2 ±02 04 ±04 a is ± 0 1 5 a n ±<7 11 a i2 ±Oi2 (sec) (sec) 1 0.075 37 40.8 ±2 .0 32.5 ±3.1 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 19.1 ± 1.1 19.4 ± 1.6 .0 .191 ±.018 .0 .0 .0 10.2 28.3 3 0.21 850 9.5 ±1.0 11.3 ± 1.0 .0 .204 ± .0 1 2 1.32 ±.49 .0 .0 8.2 20.7 4 0.37 1179 1 0 .0 ±0.7 1 0 .2 ±0 .8 .0 .187 ±.009 1.87 ±.35 .0 .0 7.7 20.5 5 0.63 1139 4.6 ±0.7 6.9 ±0 .8 .0 .213 ±.009 1.50 ±.53 2.17 ±.89 4.47 ±1.01 7.7 19.0 6 1.1 1376 2.9 ±0 .6 5.8 ±0 .6 .0 .184 ±.007 .36 ±.42 3.83 ±.70 7.26 ±.80 7.0 16.7 7 1.7 1555 0 .2 ±1.0 2 .6 ±1.0 .40 ±•15 .156 ±.005 .0 4.01 ±.43 5.70 ±.42 5.6 14.0 8 2.5 1522 -9.2 ± 1.0 -7.7 ±1.1 1.97 ±.17 .106 ±.004 .0 2.64 ±.30 3.57 ±.29 3.9 10.3 9 4.2 1134 -1 0 .0 ±1.1 -8 .8 ±1.1 2.17 ±.19 .097 ±.004 .0 2.52 ±•31 2.87 ±.31 3.7 9.3 10 7.2 1070 -11.5 ±0.9 -1 1 .0 ±0.9 2.62 ±.16 .076 ±.004 .0 1.51 ±.28 1.33 ±.27 3.1 8.4 11 13 616 -1 0.6 ±1.0 - 1 0.8 ± 1.0 2.56 ±.18 .057 ±.006 .0 .77 ±.28 1.12 ±.27 2.5 7.2 12 21 273 -9.9 ±1.5 - 1 0 .0 ±1.5 2.39 ±.27 .060 ± .0 1 1 .0 .0 .75 ±.29 2.4 6.5 1 horiz 1 vert M A S SiO) S l< 2> C o r r e s p o n d i n g p a r a m e t e r s [c o n sth] [co n stv] 4 0 4 0 3 0 20 20 f (Hz) l-U-d_ _ _ _ _ _ _ _L 10 f (Hz) -1 0 -1 0 [M] [A] 3 .0 r a 2 2 . 5 .20 \ 2.0 . 1 5 .10 1.0 . 0 5 f (Hz) f (Hz) .00 F ig . 4 .1 2 a The coefficients a i ( / ) , a 2(/) and a 4(/) in Eq. (4.17), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 220 2.5 r- 2.0 - 1.5 - 1.0 - .5 - [s] - .1 ■ ■ 10 , f (H z) 7 6 5 4 3 2 1 0 1 1 10 f (Hz) 7 6 5 4 3 2 1 1 10 f (Hz) F ig . 4 .1 2 b The coefficients a 15(/), c u ( / ) and o12( /) in Eq. (4.17), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “c-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 222 Coefficient ais(f), showing the influence of the geological site conditions, is sim ilar to a 13( /) and a 14( /) , which have the same role in the model Eq. (4.16). In the low frequency channel (/o = 0.37 Hz), the duration of m otion on the sedim entary sites (s = 0) is about 4 sec longer, than on the rock sites, when predicted by model Eq. (4.17). The previous model, Eq. (4.16), gave about 6 sec of prolongation for the same conditions. At high frequencies, ai$(f) is not well defined (condition |0 i s ( / ) / a i 5 ( /) | < 1 is hardly satisfied). We consider now the “soil” coefficients a u (f) and a i 2(/). The first one, a n ( / ) j com pares the duration of strong motion on stiff soil sites (si, = 1) and on “rock” sites (sl = 0). The second coefficient, a i 2( /) , shows the prolongation of duration on deep soil sites (sl = 2), taking “rock” sites as a reference. As expected, a j 2( /) > a u ( / ) > 0, which shows th a t the duration of strong ground m otion is th e longest at the stations, located on deep soils, and shortest at the “rock” sites (with all other factors kept constant). Also, the influence of the local soil conditions on the duration can be noticed at higher frequencies, com pared to the range where influence of geological conditions is noticeable. Consider again the sim ple physical model (Trifunac, 1990b), developed for the explanation of the am plification effects by geological and by local soil conditions, and discussed in C hapter 4 pertinent to the prolongation of duration, caused by geological site conditions. The range of frequencies, where the effect of local soils on the duration is noticeable, is about 0.63 -r 21 Hz. The corresponding param eters of the layer of soft soil on a more stiff half-space, which would produce the effect of a band-pass filter, predicted by Trifunac’s model, could be the following: the depth of local soft soil 50 m, the shear wave velocity in the soil 100 m /s, inelastic attenuation Q = 100. In this case, the low cut-off frequency, / 2, is about 0.5 Hz. 223 Lower frequency waves are too long to “notice” the existence of a soil layer and to be disturbed by it. The high cut-off, predicted by the attenuation m odel, is fi = 25 Hz. High inelastic attenuation overshadows any (positive) am plification or prolongation effects possibly produced by multiple scattering in the soil layer at frequencies f > f\. The range f i > f > f-i, i.e. / « 0.5-7-25 Hz, obtained from the “band-pass model” for amplification, is similar to the frequency band where th e presence of soft local soils prolongs the duration of strong ground m otion. This supports the assumption that the physical processes, causing am plification of strong m otion and the prolongation of it are similar in nature. In the present data base, any attem pt to include si in the equations dealing w ith param eters h, R and ip, failed due to instability of the solution of th e regres sion equations. For the time being, all we can say is th at param eters R and sl seem to be “related” to each other. From the processes of sedim entation, erosion, flush floods etc., we expect deeper soils in wider valleys. Filling the gaps in the inform ation about local soil conditions could allow one to include param eters si, h, R and < p in the duration model simultaneously. 2 2 4 5. S C A L IN G T H E D U R A T IO N O F S T R O N G G R O U N D M O T IO N IN T E R M S O F M O D IF IE D M E R C A L L I I N T E N S I T Y A N D O T H E R P A R A M E T E R S 5.1 M o d e ls d u r = d u r{lM M , A ', I m m A') a n d d u r = d u r [Im m ) We next tu rn to the problem of prediction of the strong ground m otion in a noninstrum ented region. This means th at the m agnitude of an earthquake is not known, and the rough description of the intensity of shaking in term s of the Modified M ercalli Intensity (MMI) scale will be used instead. The term s to be included in the models which use this description were specified already (C hapter 3), so we can proceed directly to the discussion of some specific models. The first two models we consider are the simplest ones, which do not include any detailed param eters like those th at describe geology, configuration of the valleys or the soil conditions. Equation ■ w ( / ) ' . . + G19(/)-Im M + G 4(/)-A ' + ’(/) J (5.1) + a 2o ( /) ■ I m m A ') , 1] gives the scaling of the duration of strong motion in term s of the num erical value assigned to the Modified Mercalli Intensity at the site, I m m > and the hypocentral distance A', m easured in kilometers. As before, (and in all models introduced in this C hapter later on), superscript (h ) corresponds to the horizontal com ponent of m otion, and superscript (v) describes the vertical com ponent. Note th a t the d a ta base used in this work is the one almost entirely obtained in California, and hence, the inform ation about the dispersion and attenuation laws typical for this region are “built in” the coefficients resulting from the regression analysis. Note 2 2 5 also th a t we are using hypocentral distance instead of epicentral because about 2/3 of the d ata about I m m , used in the regression, were not directly observed a t a site, bu t were instead estim ated using Eq. (2.1) (see Lee and Trifunac, 1985). This equation uses the specific definition of distance, which is closer to the hypocentral than to epicentral distance. Given th a t the m ajority of California earthquakes are shallow events, the hypocentral depth of H = 5 km was assigned to th e events which did not have this param eter in the catalogue. Sim ilar to the case w ith m agnitude-scaled duration (Eq. (4.9)), prelim inary calculations shows th a t the coefficients 04, a i9 and 020 are practically the same for the horizontal and for the vertical motions. Coefficient a 1 ? conversely, has different values for the different com ponents. Eq. (5.1) assumes the duration to be > 1 sec to avoid small and negative values of dur th a t may result at high frequency channels if the simple equation dur — (•) is applied to sm all distances and very small intensities (I m m « 2 -r 3). This is caused by the lack of d a ta w ith low intensities (see distribution of d ata on Fig. 2.6). Eq. (5.1) was fit to the d ata in two steps. In the first iteration, instead of the form ula dur = max[(-), 1], the simple equation dur = (•) was considered and th e coefficients ( a i ( / ) , a4 (f), a 19( /) , a 2o(f)} first were obtained. This first set was used in the second iteration for the evaluation of the quantity (•). The d a ta points for which (•) < 1 sec were not included in the second iteration of the regression analysis. As expected, the set { a i ( / ) ,a i 9( /) , a 4( / ) ,a2o(f)}aecond is alm ost the sam e as { a j( /) , aig(f), a 4( /) , 020 (/ ) } /t r a t > and either of them can serve as the solution of the least square fit problem. The coincidence of these two sets of coefficients follows from the fact th a t the d ata base does no t include m any cases for which (•) < 1 sec. 226 The results of the regression analysis are sum m arized in Table 5.1 (all no tatio n used coincides with the one introduced earlier for the models w ith earth quake m agnitude, see C hapter 4). Graphical representation of the regression coefficients, plotted the versus the central frequency of the channels, is shown in Fig. 5.1. As it could be expected from the results of the previous C hapter, and is explicitly shown in Fig. 5.1 and Table 5.1, the duration of strong ground m otion does not depend on the Modified Mercalli Intensity level for the low frequency of m otion (/< 0 .1 Hz). The dependence dur = d u r [I\x\A, A') for higher frequencies is shown in Fig. 5.2, where the isolines of the strong m otion duration, as pre dicted by Eq. (5.1), are shown for the horizontal com ponent of m otion. For every fixed intensity, duration grows with distance, in accordance w ith w hat should be expected from the results of C hapter 4, where this growth was explained by the dispersion and scattering phenomena. The behavior of the duration of strong m otion as a function of intensity, for a fixed distance, is m ore complex. Intensity by itself is a function of the earthquake m agnitude and the distance to the source. Being directly proportional to the magnitude, intensity grows when m agnitude increases. Intensity also grows w ith the decrease of the distance to the source. These two facts result in w hat appears as a “contradictory” behavior of the du ration of strong m otion as a function of intensity. On one hand, duration should increase when intensity increases, because it could correspond to the increase of duration w ith the increase of m agnitude. On the other hand, duration should decrease w ith increasing intensity, because the increase of intensity could cor respond to the shorter distance (with no change in m agnitude). The resulting picture depends on which of those two effects prevails. One also should remember th a t the intensity at a site is being assert, primarily, by estim ating the dam age T ab le 5.1 Results of the regression analysis of Eq. (5.1). Channel fo # o f data points C oefficients a, and their accuracy ("a-interval") <7dur durav number (H z) N(f) a [(h) +ai(h) a j(v) ±CTi(v) a ig ±<719 34 ± < 7 4 320 ±<720 (sec) (sec) 1 0 .0 7 5 37 40.8 ±2 .0 32.5 ±3.1 .0 .0 .0 1 0.2 3 8 .3 2 0 .1 2 311 27.7 ±5.4 28.2 ±5.7 -1.30 ±.75 .182 ±.019 .0 10.1 2 8 .3 3 0.21 9 6 2 33.3 ±2.7 35.3 +2.7 -3.17 ±.37 .195 ± .0 1 2 .0 7.8 2 1 .4 4 0 .3 7 1499 23.8 ±2 .6 24.2 ±2 .6 -1.73 ±.39 .084 ±.040 .018 ±.007 7 .4 2 1 .0 5 0 .6 3 2 0 3 5 13.7 ±2.1 15.6 ±2.1 -.62 ±.32 .134 ±.033 .0 1 2 ±.006 7 .8 18.7 6 1.1 2 6 3 6 1 0 .0 ± 1.6 1 2 .8 ±1.6 -.44 ±.25 .089 ±.025 .016 ±.004 7 .0 1 5 .6 7 1.7 3 1 1 9 5.1 ± 1.0 7.8 ± 1.0 -.03 ±.16 .046 ±.018 .021 ±.003 5 .4 1 2.4 8 2.5 3 4 1 8 4.4 ±0.7 6 .2 ±0.7 -.1 1 ± .1 1 -.018 ±.013 .025 ± .0 0 2 4 .0 9.1 9 4 .2 2 7 3 9 1.7 ±0 .6 3.1 ±0 .6 .16 ± .1 0 -.043 ±.013 .030 ± .0 0 2 3.3 7 .6 10 7 .2 2 5 7 6 1.0 ±0.5 1.6 ±0.5 .18 ±.08 -.070 + .0 1 2 .035 ± .0 0 2 2.8 6 .4 11 13 1584 - 1.1 ±0.5 - 1.0 ±0.5 .46 ±.08 -.028 ±.017 .027 ±.003 2 .3 5.1 12 21 735 -3.4 ±0.7 -3.3 ±0.7 .75 ± .1 2 .118 ±.038 .005 ±.006 2 .0 4 .2 1 horiz 1 vert Imm A' ImmA' C orresponding parameters 227 [c o n s th] [c o n s tv] 40 40 30 30 f (Hz) -1 20 20 -2 10 0 - 3 (Hz) f (H z) -10 - 1 0 - 4 a .20 .03 /W .15 .02 .10 .05 .01 .00 f (Hz) .00 V -.0 5 -.1 0 -.0 1 F ig . 5.1 The coefficients a ,( /) in Eq. (5.1), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “er-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 228 Horizontal com ponent fo^.12 Hz 229 0 20 40 60 B O 100 120 140 160 VIII VIII 67 VII III III 0 20 40 60 80 100 120 140 160 H ypocentral d istan ce (km ) H orizontal co m p o n en t f0 =.21 Hz 0 20 40 60 B O 100 120 140 160 VIII VIII VII ■ s®*' 'sfl- ' ■ ^ ■ ■ s8" ^ hi 0 20 40 60 80 100 120 140 160 H ypocentral d ista n ce (km ) F ig . 5 .2 a Channels # 2 and # 3 : isolines of the duration (in seconds) of the hor izontal com ponent of strong earthquake ground m otion as defined by Eq. (5.1). N um bers in dashed “boxes” designate the number of the d ata points used in the analysis for the particular range of intensities, Im m > and hypocentral distances, A '. For each fixed Im m , duration grows w ith A t The dependence of duration on intensity for fixed hypocentral distance depends on frequency and on distance: the duration grows w ith Im m for high frequency channels and decreases for low frequency channels. A sm ooth transition from one p attern to another can be seen in interm ediate channels, where dur grows with increasing Im m f°r large A ', and decreases w ith increasing Im m f°r small A'. See Fig. 3.6 for com parison w ith the actual data. Horizontal com ponent f0 =.37 Hz 0 20 40 60 80 100 120 140 160 VIII VIII VII 197 26 — V C O 6 » III III o 0 20 40 60 80 100 120 140 160 H ypocentral d istance (km ) H orizontal com p on en t f0 =.63 Hz 0 20 40 60 80 100 120 140 160 VIII VIII I o VII 35 VII 57 09 52 ,30 l C P 29 39 IN to C O III 0 40 20 60 80 100 120 140 160 H ypocentral d istance (km ) F ig . 5 .2 b Channels # 4 and # 5 : isolines of the duration (in seconds) of the hor izontal com ponent of strong earthquake ground motion as defined by Eq. (5.1). N um bers in dashed “boxes” designate the number of the d a ta points used in the analysis for the particular range of intensities, Im m , and hypocentral distances, A '. For each fixed Im m > duration grows w ith A'. The dependence of duration on intensity for fixed hypocentral distance depends on frequency and on distance: the duration grows w ith Im m for high frequency channels and decreases for low frequency channels. A sm ooth transition from one p attern to another can be seen in interm ediate channels, where dur grows with increasing Im m for large A ', and decreases w ith increasing Im m for small A '. See Fig. 3.6 for com parison w ith the actual data. Horizontal com ponent f0 =1.1 Hz 0 20 40 60 80 100 120 140 160 VIII VIII T, VII 140 VII 29' 260 166 20 4 ) 40 C P o III III 140 160 80 100 120 40 60 20 0 H ypocentral distance (km ) H orizontal com p on en t f0 =1.7 Hz 0 20 40 60 80 100 120 140 160 VIII VIII VII 310 41 VII 171 56 >79 26 341 48 115 III III cn 160 100 120 140 80 40 60 0 20 H ypocentral distance (km ) F ig . 5.2c Channels $ 6 and $7: isolines of the duration (in seconds) of the hor izontal com ponent of strong earthquake ground motion as defined by Eq. (5.1). N um bers in dashed “boxes” designate the number of the d ata points used in the analysis for the particular range of intensities, I m m , and hypocentral distances, For each fixed I m m , duration grows w ith A '. The dependence of duration on intensity for fixed hypocentral distance depends on frequency and on distance: the duration grows w ith I m m for high frequency channels and decreases for low frequency channels. A sm ooth transition from one p attern to another can be seen in interm ediate channels, where dur grows with increasing Im m for large A , and decreases w ith increasing I M m for small A '. See Fig. 3.6 for comparison w ith the actual data. Horizontal com ponent f0 =2.5 Hz 232 0 20 40 60 80 100 120 140 160 ^ i - ^ T r r r r r VIII VIII <"aj.2 2 0 ; 199 12 VII 429 !72 56 184 66 72 1 5 1 2 - III III 20 0 40 60 80 100 120 140 160 H ypocentral d istan ce (km ) H orizontal co m p o n en t f0 =4.2 Hz 0 20 40 60 80 100 120 140 160 VIII VIII 1 8 1 VII 222 rf > . 4 0 ta in 20 0 40 60 80 100 120 160 140 H ypocentral d istance (km ) F ig . 5 .2 d Channels # 8 and # 9 : isolines of the duration (in seconds) of the hor izontal com ponent of strong earthquake ground motion as defined by Eq. (5.1). Numbers in dashed “boxes” designate the number of the d ata points used in the analysis for the particular range of intensities, I m m - , and hypocentral distances, A '. For each fixed Im m , duration grows with A t The dependence of duration on intensity for fixed hypocentral distance depends on frequency and on distance: the duration grows w ith Im m for high frequency channels and decreases for low frequency channels. A smooth transition from one p attern to another can be seen in interm ediate channels, where dur grows with increasing Im m f°r large A ', and decreases with increasing Im m for small A '. See Fig. 3.6 for com parison w ith the actual data. Horizontal com ponent f0 = 7.2 Hz 233 0 20 40 60 80 100 120 140 160 50' VIII VIII 64 VII 44 221 57 III III 40 60 80 100 0 20 120 140 160 H ypocentral d istan ce (km ) H orizontal co m p o n en t ^ = 1 3 Hz 0 20 40 60 80 100 120 140 160 VIII 38 VIII _<-x \t2 — — 104 VII VII 273' 25 19 158 iO- III III 80 120 20 40 60 100 140 160 0 H ypocentral d istan ce (km ) F ig . 5.2 e Channels # 1 0 and #11: isolines of the duration (in seconds) of the hor izontal com ponent of strong earthquake ground m otion as defined by Eq. (5.1). N um bers in dashed “boxes” designate the number of the d ata points used in the analysis for the particular range of intensities, Im m , an< l hypocentral distances, A '. For each fixed Im m - , duration grows with A t The dependence of duration on intensity for fixed hypocentral distance depends on frequency and on distance: the duration grows w ith Im m for high frequency channels and decreases for low frequency channels. A sm ooth transition from one p attern to another can be seen in interm ediate channels, where dur grows w ith increasing Im m for large A', and decreases w ith increasing Im m for small A '. See Fig. 3.6 for com parison w ith the actual data. 2 3 4 H orizontal co m p o n en t f0 = 2 1 Hz 0 20 40 60 80 100 120 140 160 VIII 24 VIII 43 VII 130 .75 H H III III 0 20 40 60 00 100 120 140 160 H ypocentral d istan ce (km ) F ig . 5 .2 f Channel #12: isolines of the duration (in seconds) of the horizontal com ponent of strong earthquake ground motion as defined by Eq. (5.1). Numbers in dashed “boxes” designate th e num ber of the d ata points used in th e analysis for the particular range of intensities, Im m i and hypocentral distances, A '. For each fixed Im m > duration grows with A '. The dependence of duration on intensity for fixed hypocentral distance depends on frequency and on distance: the duration grows w ith Im m for high frequency channels and decreases for low frequency channels. A sm ooth transition from one pattern to another can be seen in interm ediate channels, where dur grows with increasing Im m for large A ', and decreases w ith increasing Im m for small A'. See Fig. 3.6 for com parison w ith the actual data. 2 3 5 to structures, sensitive to the short period part of the spectrum at the site. Long and short period waves attenuate w ith different rates, so th a t a severe earthquake felt at a long distance may have short period am plitudes sm aller and long period am plitude higher than a smaller shock recorded at a short distance. As a result the behavior of the intensity scale at low frequencies may appear contradictory. For long period waves (channels # 2 and # 3 , /o = 0.12 -r 0.21 Hz), the influence of the earthquake m agnitude is not felt (see C hapter 4), and as a result of complex interference of the factors described above the increase in Imm (for fixed distance) causes the decrease of duration. For the high frequencies (/o > 2.5 Hz), the dispersion does not play so critical role, as it does for low and interm ediate frequencies (recall the behavior of the coefficient a4(f) in Eq. (4.9), Fig. 4.1). As a result, duration increases with the increasing intensity, because the latter is caused prim arily by the growing m agnitude. In the interm ediate frequency range (channels # 4 -r 7, /o = 0.37 4- 1.7 Hz), the behavior of dur = dur(lMM) is of interm ediate and dual nature. For the long distances it resembles th e behavior, typical for the high frequency channels, and for the short distances dur{I\AM) appears to be similar to dur(Imm) for the low frequencies. This might naturally come from the fact th a t the definition of “long” and “short” should be “scaled” by the wave length of the channel in question. Once this “scaling” is taken into account, it is easy to understand why the “transition” distance (where “short” distance borders with “long” distance) moves towards the source when the frequency of vibration becomes higher. T he numbers in the dashed the “boxes” in the Fig. 5.2 give the number of the d ata point available for the analysis for the specific range of distances and Modified Mercalli intensities. The results of the model should be treated 236 as reliable only inside the range of distances and intensities where the sufficient am ount of d ata points is available. T he model discussed above has the distance to the earthquake source as one of th e param eters. We assume th a t the depth of the hypocenter can be roughly obtained (from the teleseismic records, for exam ple), or assum ed using som e geological stadies. The location of the epicenter can also be found, at least roughly, if the intensities are known at many locations surrounding the epicenter. We understand, however, th a t the model which does not include any distance to th e source (epi- or hypo-central) as a param eter, may appear to be more useful. Such a model would also be less region dependent, as it does not assume (explicitly at least) any region specific dispersion or attenuation law. We next consider such a model: duAh)(f) ) d u r ^ ( f ) J max (5.2) Table 5.2 and Fig. 5.3 show the results of the regression analysis of this model, which was perform ed in two steps, similar to those of used for Eq. (5.1). The behavior of the coefficient flig(/) can easily be explained now by comparison w ith th e Fig. 5.2 (model Eq. (5.1)) and by recalling the discussion on the nature of th e dependence dur = dur (Im m , A7 ) in the previous model. T he duration decreases w ith the grow th of intensity at low and interm ediate frequencies, where the increase of intensity corresponds to the decrease the of distance. In the high frequency range (/o > 2.5 Hz), the duration is longer when the intensity increases. For these frequencies, intensity is governed more by m agnitude, than by the distance to the source. T ab le 5.2 Results of the regression analysis of Eq. (5.2). Channel fo # o f data points C oeff. aj and their ("a-interval") Odur durav number (H z) N (f) a . O ') +ai(h) a,M +a](v) a i9 ±019 (sec) (sec) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 10.2 38.3 2 0.12 311 54.1 ±5.2 53.6 ±5.6 -3.88 ±.79 11.5 28.3 3 0.21 9 6 2 52.3 ±2.7 54.2 ±2.8 -4.74 ±.40 8.7 21 .4 4 0 .3 7 1499 42.3 ±1.8 43.2 ±1.8 -3.33 ±.27 8.4 21 .0 5 0.63 2035 35.8 ±1.6 37.9 ±1.7 -2.75 ±.25 9.2 18.7 6 1.1 26 36 27.7 ±1.2 30.9 ±1.3 -2.05 ±.19 8.6 15.6 7 1.7 3119 15.9 ±0.9 18.8 ±0.9 -.71 ±.14 7.1 12.4 8 2.5 3418 9.2 ±0.6 11.2 ±0.6 -.12 ±.10 5.3 9.1 9 4.2 27 39 3.1 ±0.6 4.7 ±0.6 .66 ± 1 0 4.9 7.6 10 7.2 25 76 -0.3 ±0.5 0.5 ±0.6 1.06 ±.09 4.5 6.4 11 13 1584 -2.5 ±0.5 -2.0 ±0.5 1.22 ±08 3.5 5.1 12 21 735 -3.2 ±0.6 -2.8 ±0.6 1.19 ± 1 0 2.9 4 .2 1 horiz 1 vert Imm Corr parameters 237 [c o n s th] 60 50 40 30 20 10 f (Hz) - 1 0 [c o n stv] 60 50 40 30 20 10 f (Hz) -10 f (Hz) - l -2 - 3 - 4 - 5 F ig . 5.3 The coefficients a,-(/) in Eq. (5.2), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 239 5.2 M o d e ls dur = dur(lMM,A\ iMM&'ih, R,hR, R 2 ,h 2 ,<p) a n d dur = dur(/jvfMj h, R , hR , R 2, h 2 tp) T he next two intensity based models include the geometrical description of a sedim entary basin (where the recording station is located) in term s of (a) the depth of sedim ents under the recording site, h; (b) the distance to the rocks th a t can reflect the waves (coming from the source) in the direction of the station, R\ and (c) the angle subtended at the station by the surface of th e rocks, capable of producing effective reflections. These models are, in many ways, sim ilar to the m ost com plete model, used in this work— Eq. (4.12). In equation ( durW (f) 1 _ \ d u r(u)( /) J max + < * 1 9 ( f ) ' l M M + G4( /) - A '+ + 4 k) (/).h + 4 h) {f)-r + 4 h) if) ■ hR + <4o) ( /) •h + 4 v) ( /) r + 4 v) ( /) -hR+ + 4 h){f)-R 2 + 4 h)(f ) ' h 2 + 4o + 4 v)(f)-R 2 + 4 v)(f)-h 2 + a[v J(f)-<p + (5.3a) w here all distances are measured in kilometers, the sum of the term s involving h, R and < p (and designated earlier as r r e g i o n ) should be considered only if it is greater than zero: [^region ( /)]4 _ — max { O ,r reg i o n ( / ) } • (5.36) The values of h, R and < p are assumed to be zero if the site is located on rock. 2 4 0 Eq. (5.3) was fit to the d ata in tree steps. First two steps are sim ilar to w hat was used in the analysis of the models Eq. (5.l)-(5.2), and the subscript “+ ” in Eq. (5.3a) was not taken into account at these stages. During the th ird step, ^region was estim ated using the results of the second interaction, and the final set of coefficients was obtained. As before, all three sets of coefficients—the results of three step fitting— are very similar to each other. T he numerical results of the regression analysis are sum m arized in Table 5.3, where all notation coincides with the one previously used. Fig. 5.4 displays th e regression coefficients, involved in the model, as functions of the central fre quency of the channels. The set of coefficients, responsible for the scaling the duration in term s of the intensity and distance, {ai (/), aig(f), ( 1 4(f), a^o(/)} , is very sim ilar to w hat was obtained in Eq. (5.1), which did not include any geolog ical param eters (com pare Figs. 5.2 and 5.4a). Coefficients ( a 5( /) -f-a10(/)} are sim ilar to their counterparts in the “m agnitude model” Eq. (4.12), which scales the duration as dur — dur{M,M 2 ,& ,h,R ,hR ,R 2 ,h 2 ,<p). The same sim ilarity can be noticed in com parison of the Fig. 4.5 (“m agnitude model” ) and Fig. 5.5 ( “intensity model” ), which show the isolines of the additional duration of strong ground m otion (in seconds) defined by the term s, involving R and h (positive contribution only): j d u r ^ ( R , h) ) f <4^ •h + .R + ■ hR + a ^ -R2 + ag1 ^ •h2 ) \ d u r (R, h) J ( ag^ ■ h + •R + a 7^ -hR + •R 2 + a9^ •h 2 J One of the features, common for both models, is the higher sensitivity of the horizontal m otion to the horizontal dimension of the sedim entary valley, R, and the greater influence of the depth of sediments (vertical dimension), h, on the vertical com ponent of motion. Q uantitatively (see Fig. 5.5), the predictions of T ab le 5.3 Results of the regression analysis of Eq. (5.3). Channel number fo (Hz) # of data points N(f) Coefficients a; and their accuracy ("a-interval") 0dur (sec) d urav (see) a,O') lOiOO ± 0 ,0 ) ai9 ±019 44 ±04 420 ±020 a sO O ±05(h) a^> ±06< h) a/W ±a7(h ) ag(h ) ± 0 8(h) a9(h > ±cr9< h> a1 0 W ±0]O(h) a5< v> ± 0 5< v) a6(v) ± 0 6(v) a7(v> ±0?(v) asiv> ± 0 8(v) a9|v) ± 0 9(y) aio"') ±0io(v) I 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10.2 3S.3 2 0.12 311 27.7 ±5.4 28.2 ±5.7 -1.30 ±.75 .182 ±.019 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10.1 2S.3 3 0.21 962 33.3 ±2.7 35.3 ±2.7 -3.17 ±■37 .195 ±.012 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 7.8 21.4 4 0.37 1364 17.1 ±2.8 17.9 ±2.9 -1.25 ±,40 .169 ±.044 .012 ±.007 .0 .0 .0 .0 .0 .0185 ±.0044 .0 .0 .0 .0 .0 .0162 ±.0072 7.1 20.7 5 0.63 1182 14.8 ±4.1 18.5 ±4.3 -1.89 ±.63 .215 ±.067 .013 ±.011 .0 .343 ±.056 .0265 ±.0110 -.0055 ±.0008 -.32 ±.08 .0282 ±.0062 1.49 ±1.14 .101 ±.107 .0657 ±.0186 -.0037 ±.0016 -.75 ±.17 .0265 ±.0105 7.6 19.4 6 1 .1 1472 6.0 ±3.2 9.5 ±3.3 -1.06 ±.52 .156 ±.069 .018 ±.011 .84 ±.56 .351 ±.053 .0213 ±.0096 -.0061 ±.0008 -.35 ±.09 .0360 ±.0047 1.16 ±81 .107 ±.076 .0520 ±.0159 -.0032 ±.0012 -.54 ±.14 .0310 +.0074 64 15.7 7 1.7 1879 7.2 ±1.5 10.4 ±1.7 -.83 ±.26 .010 ±.023 .027 ±.004 1.53 ±41 .181 ±.039 .0233 ±.0068 -.0037 ±.0006 -.41 ±.07 .0139 ±.0032 2.41 ±.60 .042 ±.056 .0246 ±.0106 -.0016 +.0008 -.51 ±.10 .0 5.4 13.4 8 2.5 2053 6.7 ±1.0 8.1 ±1.1 -.81 ±.17 -.037 ±.016 .028 ±.003 .0 .146 ±.019 .0053 ±.0019 -.0022 ±.0003 .0 .0105 ±.0021 1.99 ±.40 .075 ±.037 .0070 ±.0068 -.0015 +.0006 -.32 ±.06 .0 3.S 9.S 9 4.2 2295 1.6 ±0.6 2.8 ±0.7 .05 ±.11 -.026 ±.014 .027 ±.002 .0 .074 ±.013 .0 -.0009 ±.0002 .0 .0046 ±.0015 1.30 ±■27 .0 .0 .0 -.16 ±■05 .0 3.2 7.6 10 7.2 2576 1.0 ±0.5 1.6 ±0.5 .18 ±.08 -.070 ±.012 .035 ±.002 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 2.8 6.4 1 1 13 1584 -1.1 ±0.5 -1.0 ±0.5 .46 ±.08 -.028 ±.017 .027 ±.003 .0 •0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 2.3 5.1 12 21 735 -3.4 ±0.7 -3.3 ±0-7 .75 ±.12 .118 ±.038 .005 ±.006 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 2.0 4.2 1 horiz 1 vert Imm A’ ImmA ' h R hR R2 h2 < P h R hR R2 h2 9 horizontal component vertical component Corresponding parameters [co n sth] [con stv] 19 40 40 30 30 -1 20 20 - 2 - 3 f (Hz) f (Hz) -10 - 1 0 - 4 .20 .03 .15 .02 .10 .05 .01 .00 f (Hz) .00 - .0 5 -.10 -.01 F ig . 5 .4 a The coefficients / ) , c i 9( /) , a4(/) and a 2o (/) in Eq. (5.3), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 242 ■ ■ / 1 10 f (Hz) -1 .000 -.0 0 2 -.0 0 4 - .0 0 6 -.0 0 8 f (Hz) 10 ■W :'///: '///•' .5 .4 .3 .2 1 (Hz) o l l 10 10 f (Hz) -.2 - . 4 -.6 - . 8 - 1.0 [hR] .10 .08 .06 .04 .02 .00 .04 .03 .02 .01 N&. f (Hz) .00 F ig . 5.4b The coefficients a ,( /) , t = 5 -r 10, horizontal component, in Eq. (5.3), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 243 [hR] 3 r .10 r .5 .08 .4 .06 3 .04 .2 10 .02 .1 f (Hz) f (Hz) -1 .0 .00 10 f (Hz) V .000 r -.2 -.0 0 2 \. — .03 ///. - .4 -.0 0 4 .02 - .6 -.0 0 6 .01 -.8 f (Hz) -.0 0 8 - 1.0 .00 F ig . 5.4c The coefficients ot ( /) , t = 5 -r 10, vertical component, in Eq. (5.3), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 244 H orizontal co m p o n e n t f0 =.63 Hz 2 4 5 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 7 12 6 5 02. 4 26 22 3 2 1 0 H I I I I I I I I I L . I . l . l . U a o 20 30 40 50 60 70 80 R (km ) Vertical co m p o n e n t f0 =.63 Hz 10 20 30 40 50 6 0 70 80 A a 7 g r 6 5 22 4 3 2 1 3 6 Ill 0 10 20 30 40 50 60 70 80 R (km ) F ig . 5 .5 a Channel # 5 : isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to the geom etry of a sedim entary basin, as predicted by Eq. (5.3). The positive contribution of the term s dur} '(R,h) and dur'v> (R. h) defined by Eq. (5.4) only is considered. N um bers in dashed boxes” designate the number of d ata points used in the analysis for the particular range of R and h. See Fig. 3.15 for com parison with the raw d ata and Fig. 4.5 for com parison w ith the “m agnitude model.” Horizontal c o m p o n e n t f 0 =1.1 Hz 246 0 10 20 30 40 50 60 70 80 c l A 7 6 5 4 29 24 3 2 65 10 1 24 I L L 11111 a 0 20 10 30 40 50 70 80 0 60 R (km ) Vertical c o m p o n e n t a 10 20 30 40 50 60 70 80 0 7 7 6 i __ 5 29 4 54, 3 2 27 1 0 10 20 30 40 50 60 70 80 R (km ) F ig . 5 .5 b Channel # 6 : isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to the geom etry of a sedim entary basin, as predicted by Eq. (5.3). The positive contribution of th e term s d u r(/l)(iE, h) and d u r ^ [R, h) defined by Eq. (5.4) only is considered. N um bers in dashed ^boxes’ ’ designate the num ber of d a ta points used in the analysis for the particular range of R and h. See Fig. 3.15 for com parison with the raw d ata and Fig. 4.5 for com parison w ith the “m agnitude m odel.” Horizontal co m p o n en t f0 =1.7 Hz 2 4 7 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 7 6 12 5 31 430- 66 4 28 3 '42 2 C O 72 1 24 £3 6 1 1 1 1 I 1111111 m i l l 0 70 20 40 50 60 80 0 30 10 R (km ) Vertical co m p o n en t f0 =1.7 Hz 10 20 30 40 50 60 70 80 c l A 7 I III I u T T T T 6 I 5 33 4 57 12 3 21 2 30 1 0 70 80 50 60 20 30 40 0 10 R (km ) Fig. 5.5c Channel $7: isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to the geom etry of a sedim entary basin, as predicted by Eq. (5.3). The positive contribution of the term s d u r(/l)(iE,/i) and durW(R,h) defined by Eq. (5.4) only is considered. N um bers in dashed ‘ 'boxes” designate the num ber of d ata points used in the analysis for the particular range of R and h. See Fig. 3.15 for comparison with the raw d ata and Fig. 4.5 for com parison w ith the “m agnitude model.” B M H orizontal co m p o n e n t 10 2 0 30 40 50 fn — 2.5 Hz 60 70 2 4 8 8 0 7 TTTTTT 12 6 10 5 31 129 88 66 4 38 100 3 45 10 2 12 24 71 1 E 283 I = 1 1 I I I I ill 37 24 i l l 1111 a o 80 0 0 10 20 30 40 50 60 70 R (km ) Vertical co m p o n e n t f0 =2.5 Hz 10 2 0 30 40 50 60 70 80 a 7 6 5 i 55 35 33 4 17 54 3 21 2 33 - 3 1 0 0 0 10 20 30 40 50 60 70 80 R (km ) F ig . 5 .5 d Channel # 8: isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to the geom etry of a sedim entary basin, as predicted by Eq. (5.3). The positive contribution of th e term s d u r(fc)(J?,/i) and d u r(u)(jR,/i) defined by Eq. (5.4) only is considered. N um bers in dashed boxes” designate the number of d a ta points used in the analysis for the particular range of R and h. See Fig. 3.15 for com parison w ith the raw d ata and Fig. 4.5 for com parison w ith the “m agnitude model.” Horizontal co m p o n e n t f0 = 4.2 Hz 249 0 10 20 30 40 50 60 7 0 80 7 6 5 2 8 110 6 4 4 8 4 7 2 3 3 2 1 6 7 0 2 5 0 1 4 5 3 3 2 1 7 1 2 4 3 7 2 8 1 .1 1 1 1 1 . 0 0 20 10 30 4 0 50 60 70 80 R (km ) 7 6 ^ 5 Z 4 A 7 3 2 1 0 Vertical co m p o n en t f0 =4.2 Hz 10 20 30 40 50 60 70 80 4 1 1 1 1 1 1 1 1 p r ii iinr 1 1 1 1 1 1 1 1 1 p 1 1 1 1 1 1 1 1 i - - - - - - - - - - - - - 9 - - - - - - - - - - - - - - 1111111111 irn 11111 1 4 - - - - - - - - - - - - - -l- - - - - 2 - - - - - prm 1 itt 1 1 1 1 I T T 1 4 4 1 = = 2 7 5 0 3 6 ! 5 C O 1 1 1 = 4 2 7 8 ! 3 3 6 | 1 1 4 4 5 5 2 i 2 2 9 j l l l l l l l l i j 1 ^ 8 2 3 1 ......^ J 4 = 1 1 3 = 1 1 1 1 1 1 1 1 1 1 1 I I I I I I I 1 1 1 3 _ L 1 . . 1 . 1 I I I I I 4 I 1 1 I I 1 I I I . . . . . . T 4 ' 2 . . . . . . . . . .. . . . . . .. . . . .. . . . . . .. . . 1 0 I I 1 1 I I I 1 11 1 1 1 1 1 1 1 1 = 10 20 60 70 80 30 40 50 R (km ) F ig . 5 .5 e Channel # 9 : isolines of the additional (relative to the basem ent rock sites, s = 2) duration (in seconds) of strong ground m otion due to th e geometry of a sedim entary basin, as predicted by Eq. (5.3). The positive contribution of the term s dwr '{R,h\ and dur. {R,h) defined by Eq. (5.4) only is considered. N um bers in dashed boxes” designate the number of d ata points used in the analysis for the particular range of R and h. See Fig. 3.15 for com parison with the raw d ata and Fig. 4.5 for com parison w ith the “m agnitude model.” 2 5 0 the current model regarding the additional duration of strong m otion due to the specific shape of the sedim entary basin is practically the sam e as in the case of the “m agnitude model” (Eq. (4.12)) and varies from the m axim um of 8 sec in channel # 5 (horizontal component) to about 1 sec in channel # 9 . As in Eq. (4.12), the isoline of zero additional duration practically coincides w ith the boundary in the R — h plane which separates the area of “no data” from “a lot of d ata.” This rem arkable stability in the behavior of the coefficients of Eq. (4.12) and (5.3) and the consistency in preserving of the main qualitative and quantitative features of the predictions, resulting from these two models, give us additional confidence in our assum ptions and in the choice of the models. We next tu rn to a simplified companion of the model ju st discussed: f 4 % ) ( d u r W ( / ) | \d u r(« > (/) J = max 4 “ ’(/I + + aiH)(f)-h + 4 h) (f)-R + 4 h){f)-hR+ 4 v){f)-h + 4V){f)-R + 4v){f)-hR+ +4 h){f)-R2 + 4 h)[f)-hi + 4o )(f)-h2 + 4o (5.5a) + + where all distances are m easured in kilometers, and dur(R,h,<p) accounted for only if it is positive: ^region IS [^region ( /) ] + — m ax{0>7region(/)}' (5.56) As before, the values of h, R and < p are assumed to be zero if the site is located on rock. Exactly as in the case of the previous model, Eq. (5.5) was fit to the d a ta in tree steps, results of which appear to be very similar to each other. T able 5.4 Results of the regression analysis of Eq. (5.5). Channel number fo (H z ) # o f data points N (f) C o e f f i c i e n t s aj a nd t h e i r a c c u r a c y ( " c r - i n t e r v a l " ) <7dur (sec) d u rav (sec) a,<h> +CJi(h) a ](vl ±<?i<v> a 19 ±<7l9 35^ ) ±CT5< h) afiW ±C T6(h) a7(h) ± a 7< h> a8(h) ± a 8(h) a9(W ± a 9< h) a,o(h) ± 0 ,0® a5M ±Ct5(v) afiM ± c t6(v) a7w ±ct7(v) agM ± a s(v) a9M + a 9M a 1 0 M ± O |0(v) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 54.1 ±5.2 53.6 ±5.6 -3.88 ±■79 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 11.5 28.3 3 0.21 962 52.3 ±2.7 54.2 ±2.8 -4.74 ±.40 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 8.7 21.4 4 0.37 1399 42.8 ±1.9 43.9 ±1.9 -3.45 ±.28 .0 .0 .0 .0 .0 .0010 ±.0051 .0 .0 .0 .0 .0 .0 8.2 20.7 5 0.63 1657 35.3 ±2.0 37.6 ±2.5 -2.94 ±.29 .0 .0 .0 .0 .0 .0163 ±.0050 1.40 ±1.18 .0 .0234 ±.0111 .0 -.48 ±.16 .0078 ±.0124 9.3 18.4 6 1 .1 1472 25.3 ±2.3 30.2 ±2.5 -2.86 ±■37 1.81 ±74 .304 ±.069 .0243 ±.0126 -.0041 ±.0010 -.63 ±.12 .0394 ±.0060 2.61 ±.91 .0 .0254 ±.0092 .0 -.59 ± 13 .0264 ±.0096 8.4 15.7 7 1.7 1879 17.9 ±1.3 23.1 ±1.5 -1.94 ± ■ 2 2 1.93 ±.55 .204 ±.052 .0005 ±.0091 -.0019 ±.0008 -.41 ±.09 .0293 ±.0042 3.04 ±.66 .0 .0170 ±.0068 .0 -.61 ± ■ 1 0 .0062 ±.0062 7.3 13.4 8 2.5 2836 9.9 ±0.7 11.7 ±0.8 -.76 ±.11 .0 .108 ±.017 .0 -.0006 ±.0003 .0 .0187 ±.0019 2.95 ±.42 .0 .0088 ±.0043 .0 -.48 ±.07 .0062 ±.0039 4.9 9.1 9 4.2 2567 3.7 ±0.6 5.2 ±0.6 -.39 ±.10 .0 .0 .0 .0 .0 .0158 ±.0020 .0 .0 .0 .0 .0 .0165 ±.0030 4.7 7.6 1 0 7.2 2576 -0.3 ±0.5 0.5 ±0.6 1.09 ±.09 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 4.5 6.4 1 1 13 1584 -2.5 ±0.5 -2.0 ±0.5 1.22 ±.08 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 3.5 5.1 1 2 21 735 -3.2 ±0.6 -2.8 ±0.6 1.19 ±.10 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 2.9 4.2 1 h o riz 1 v e rt I m m h R hR R 2 h2 < P h R hR R2 h2 < P h o r i z o n t a l c o m p o n e n t v e r t i c a l c o m p o n e n t C o r r e s p o n d i n g p a r a m e t e r s [co n sth] 60 50 40 30 20 10 f (Hz) -1 0 [co n stv] 60 .7— ; 50 40 30 20 10 f (Hz) -1 0 10 f (Hz) - l -2 - 3 - 4 - 5 F ig . 5 .6 a The coefficients a\ (/) and 019( /) in Eq. (5.5), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “c-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). to C n to [hR] .10 .5 .08 .4 .06 .3 .04 2 .02 .1 f (Hz) f (Hz) f (Hz) - i ,0 .00 10 10 f (Hz) .000 10 10 10 f (Hz) -04 -.2 -.0 0 2 .03 - .4 - .0 0 4 .02 -.6 -.0 0 6 .01 -.8 f (Hz) -.0 0 8 - 1.0 .00 F ig . 5.6b The coefficients a t (/), i — 5 ~ 10, horizontal component, in Eq. (5.5), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “<r-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 253 [HR] .10 .5 .08 4 .06 .3 .04 .2 10 .02 .1 f (Hz) f (Hz) f (H z) -1 0 .00 10 10 f (Hz) v .000 10 10 f (Hz) -04 - .2 -.0 0 2 .03 - .4 - .0 0 4 .02 - .6 -.0 0 6 .01 - .8 -.0 0 8 - 1.0 .00 F ig . 5.6c The coefficients a ,( /) , t = 5 -j- 10, vertical component, in Eq. (5.5), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “c-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). plotted 254 2 5 5 The calculated regression coefficients and other inform ation about this model are presented in Table 5.4 and in Fig. 5.6. Recall, th a t the param eter R is not “felt” well by the vertical com ponent of motion in the previous model. In the current equation, this param eter cannot be “detected” at all by the vertical com ponent, and coefficients a,Q> \ f ) and a ^ \ f ) are zero. In other respects, the set {<i5( /) -r a io (/)} from the Eq. (5.5) is similar to the set {as(f) aio(f)} from th e Eq. (5.4), bu t the coefficients are well defined in som ew hat sm aller frequency range. This is w hat should be expected, because the removal of the distance to the source from the set of the param eters of the model leads to the higher scattering of d ata around the model’s prediction and to the additional uncertainties in the calculation of very sensitive coefficients {a5(/)-i-a io (/)} . The first three coefficients, a[h\ f ) , a i ^ ( / ) , and aig(f) practically coincide w ith their counterparts from the simplest Eq. (5.2), which scaled duration as dur(lM\f)- 5.3 M o d e ls dur = dur(IMM, A ' , R , R 2,<p), dur = dur(IMM, R , R 2,<p), dur = dur(IMM, A ', IMMA ',h,h2) a n d dur = dur(lMM,h,h2) This group of models is similar to those discussed in C hapter 4, section 4 and deals w ith cases when the depth of sediments, h, or the inform ation about the configuration of the basement rocks on the surface (in term s of R and < p) is not available. 256 The equation f durw lf) 1 \ dur(vI (/) J max A k\ f ) 4 ”)</) + «19 + 0 4 (7 )- A '+ + +O 2o (/) •Im m A ;) , l] + 4 h) (/) R + a( gh) [f) -R2 + aSP (/) -< P «6W ) ( /) R + ° 8U ) ( /) •R 2 + «10 if) '< P where all distances are m easured in kilometers and [']+ = m ax {(),[•]} + + (5.6a) (5.66) is the first one in this sequence. The results of the regression analysis of Eq. (5.6) are presented in Fig. 5.7 and in the Table 5.5. As expected, the set of coefficients { a x ( /) ,a 19( / ) , a 4( / ) , a 2o (/)} is sim ilar to w hat we saw in the model w ithout R, h and ip param eters (Eq. (5.1)). The set { a e (/), as{f), a io (/)} repeats the m ain features of the same set from the model Eq. (4.14), which scaled duration as dur(M ,M 2, A,R, R 2,<p). The positive contribution (additional duration) due to the specific shape of the valley, as predicted by Eq. (5.6), is shown in Fig. 5.8. The com parison of the last figure w ith Fig. 4.8 displays rem arkable sim ilarity in the behavior of dur(R) in the models involving intensity (Eq. (5.6)) and m agnitude (Eq. (4.14)). The simplified version of Eq. (5.6) is f ( durW (f) | < . , > = max \r f « r W ( / ) J + + < aih)(f)-R + aih)(f).R2 + a[h 0)(f)-<p 4v)(f)-R + 4 v){f)-R2 + 4 o { f W + + (5.7a) T ab le 5.5 Results of the regression analysis of Eq. (5.6). Channel fo # o f data points Coefficients ai and their accuracy ("ff-interval") C Jdur durav number (H z) N(f) ±ai(h ) ± a i(v) aj9 +019 34 ±04 320 +C20 36(h) ±C6(h) a 8W ± 0 8(h) a i 0(h) +Oio(h) 3 6 ^ + o 6(v) a 8(v) + o 8(v) a io M ±Oio(v) (sec) (sec) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 27.7 ±5.4 28.2 ±5.6 -1.30 +.74 .182 ±.019 .0 .0 .0 .0 .0 .0 .0 10.1 28.3 3 0.21 962 33.3 ±2.7 35.3 ±2.7 -3.17 +.37 .195 ±.012 .0 .0 .0 .0 .0 .0 .0 7.8 21.4 4 0.37 1364 17.1 ±2.8 17.9 ±2.9 -1.25 ±.40 .169 ±.044 .012 +.007 .0 .0 .0185 ±.0044 .0 .0 .0162 ±.0072 7.1 20.7 5 0.63 1897 7.8 ±2.2 9.2 ±2.3 -.56 ±.32 .096 ±.032 .023 ±.005 .288 ±.033 -.0042 +.0005 .0211 ±.0039 .262 ±.055 -.0035 ±.0008 .0274 ±.0066 7.4 18.5 6 1.1 2443 6.1 ±1.5 8.7 ±1.6 -.61 +.23 .049 ±.024 .026 +.004 .328 ±.024 -.0051 +.0004 .0251 ±.0029 .217 ±.040 -.0029 ±.0006 .0297 +.0047 6.2 15.4 7 1.7 29 06 4.6 ±1.0 7.0 ±1.0 -.35 +.15 .033 ±.017 .023 ±.003 .208 ±.018 -.0031 +.0003 .0132 ±.0021 .172 ±.027 -.0021 +.0004 .0109 ±.0031 5.0 12.3 8 2.5 3185 4.7 ±0.7 6.4 ±0.7 -.41 +•11 -.020 ±.013 .025 ±.002 .141 ±.013 -.0020 +.0002 .0078 ±.0014 .141 ±.019 -.0018 +.0003 .0048 ±.0022 3.7 9.1 9 4.2 2567 1.9 ±0.6 3.8 ±0.6 -.03 +.10 -.032 ±.013 .029 +.002 .076 +.012 -.0009 +.0002 .0044 ±.0014 .0 .0 .0065 ±.0021 3.2 7.6 10 7.2 25 76 1.0 ±0.5 1.6 ±0.5 .18 +.08 -.070 ±.012 .035 ±.002 .0 .0 .0 .0 .0 .0 2.8 6.4 11 13 1584 -1.1 ±0.5 -1.0 ±0.5 .46 +.08 -.028 ±.017 .027 ±.003 .0 .0 .0 .0 .0 .0 2.3 5.1 12 21 735 -3.4 ±0.7 -3.3 ±0.7 .75 +.12 .118 ±.038 .005 ±.006 .0 .0 .0 .0 .0 .0 2.0 4.2 1 1 Imm A' ImmA' R R2 < P R R2 < P horiz vert horizontal vertical Corresponding parameters 257 [co n sth] [co n stv] 19 40 40 30 30 A / / f (Hz) 20 -1 20 - 2 - 3 f (Hz) f (Hz) -10 -1 0 - 4 f .20 .03 .15 ■ I ■ II ■ I ■ 7 I '■ I U ■ II: .02 .10 .05 .01 f (Hz) .00 m .00 -.0 5 •\v -.10 -.01 F ig . 5.7 a The coefficients a i ( / ) , a.1 9(f), o4(/) and a 2o (/), in Eq. (5.6), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 258 .000 .5 f f (Hz) -04 -.002 .4 .03 .3 - .0 0 4 .02 ,2 - .0 0 6 .01 (Hz) .1 .0 -.0 0 8 .00 10 v .000 10 f (Hz) -04 .5 -.0 0 2 .4 /• .03 3 - .0 0 4 .02 .2 -.0 0 6 .01 .1 f (Hz) (Hz) .0 - .0 0 8 .00 F ig . 5 .7b The coefficients a6(f), a 8(/) and aio(f). (top row—horizontal component, bottom row—vertical component) in Eq. (5.6), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 259 260 akh,(f)*R+akh)(f)*Rs (sec) 6 .63 Hz (C h.5) 1.1 Hz (C h.6 ) 1.7 Hz (C h .7) 2.5 Hz (C h .8 ) 4.2 Hz (C h.9) 5 . s. 4 3 2 9___. 1 0 0 10 2 0 30 40 50 60 70 80 «r a ( 6 v ) (f)*R+akv ) (f)*R2 (sec) 5 - ___ 4 3 2 / / 1 0 0 2 0 10 30 40 50 60 70 80 R (km ) F ig . 5.8 A dditional (relative to the basement rock sites) duration (in seconds) due to the specific horizontal characteristic dimension of a sedim entary valley, R, as predicted by Eq. (5.6): a) horizontal component; b) vertical com ponent. This “truncated” model preserves the main features of the “com plete” model (Eq. (5.3), regarding the behavior of the term s which describe the prolongation of au ratio n due to the specific geometry of the sedim entary basin. This prolongation can also be com pared with the corresponding term s from the sim ilar “m agnitude” model, Fig. 4.8, Eq. (4.14). 261 where all distances are m easured in kilometers and [']+ = max {0 , [•]} (5.76) T he results of the regression analysis of this model are given in Table 5.6 and in Fig. 5.9. The coefficients a x (/) and aig(f) behave as expected, i.e. are similar to their counterparts from the model Eq. (5.2), where ai and a i 9 are the only coefficients present. The rest of the coefficients in the present model Eq. (5.7), however, does not quite follow the behavior of the corresponding coefficients from the m ore com plete “intensity model” Eq. (5.6) or “m agnitude m odel” Eq. (4.14). T he lack of the inform ation about the distance to the source makes the equa tion very approxim ate, and the results of the regression analysis becomes more unstable. T he models I durm {f) | max 4 hHf) I « iv ,(/) + <*20 ( / ) -1m m A ') , 1] + a ^ \ f ) - h + a,gh\ f ) - h 2 a[v\ f ) - h + 4 w)(/)-/i2 + (5.8a) + where and f durW(f) ) \ d u A v)(f) J = max •]+ = max{0 , [•]}, ' ( .!*> (/) (5.86) + US v)if) ° 7^ {f)'h + o!g\f)-h2 a 7 ^ ( / ) ‘^ + a9V \ f ) ' h 2 + + + (5.9a) T ab le 5.6 Results of the regression analysis of Eq. (5.7). Channel number fo (H z) # o f data points N (f) C o e f f i c i e n t s a) a n d t h e i r a c c u r a c y ( " a - i n t e r v a l " ) Ctdur (sec) durav (sec) a,(h) ±Ci(h) aj(v) ± C J i (v) ai9 ±019 a6(h) ± a 6(h) a8(h) ±ag(h) aio(h) ±(T;0® a6M ±C6(v) a8(v) ±ag(v) aio(v) ±ctio69 1 0 .075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 1 0 . 2 38.3 2 0 . 1 2 311 54.1 ±5.2 53.6 ±5.6 -3.88 ±.79 .0 .0 .0 .0 .0 .0 11.5 28.3 3 0 . 2 1 962 52.3 ±2.7 54.2 ±2.8 -4.74 ±.40 .0 .0 .0 .0 .0 .0 8.7 21.4 4 0.37 1399 42.8 ±1.9 43.9 ±1.9 -3.45 ±.28 .0 .0 .0010 ±.0051 .0 .0 .0 8 . 2 20.7 5 0.63 1851 33.0 ±1.8 35.3 ±2.0 -2.58 ±•27 .0 .0 .0170 ±.0049 .0 .0 .0147 ±.0080 9.1 18.3 6 1.1 23 44 20.9 ±1.4 24.2 ±1.5 -1.99 ±.21 .213 ±.030 -.0022 ±.0005 .0386 ±.0037 .130 ±.050 -.0006 ±.0007 .0391 ±.0059 7.7 14.9 7 1.7 2805 12.3 ±0.9 15.0 ±1.0 -.87 ±.14 .126 ±.022 -.0008 ±.0003 .0259 ±.0027 .104 ±.034 -.0002 ±.0005 .0254 ±.0039 6 . 2 11.8 8 2.5 3185 10.2 ±0.6 12.4 ±0.7 -.82 ±.10 .107 ±.016 -.0005 ±.0003 .0188 ±.0019 .095 ±.025 -.0003 ±.0004 .0151 ±.0028 4.8 9.1 9 4 .2 2567 3.7 ±0.6 5.2 ±0.6 .39 ±.10 .0 .0 .0158 ±.0020 .0 .0 .0165 ±.0030 4.7 7.6 1 0 7.2 2576 -0.3 ±0.5 0.5 ±0.6 1.06 ±.09 .0 .0 .0 .0 .0 .0 4.5 6.4 11 13 1584 -2.5 ±0.5 -2.0 ±0.5 1.22 ±.08 .0 .0 .0 .0 .0 .0 3.5 5.1 1 2 2 1 735 -3.2 ±0.6 -2.8 ±0.6 1.19 ±.10 .0 .0 .0 .0 .0 .0 2.9 4.2 1 horiz 1 vert Imm R R2 9 R R2 9 horizontal vertical C o r r e s p o n d i n g p a r a m e t e r s 262 [con stv] 60 50 > S s 40 30 20 10 f (Hz) -1 0 [co n sth] 60 30 20 f (Hz) -1 0 f (Hz) -1 -2 - 3 - 4 - 5 F ig . 5 .9 a The coefficients a \ (/) and aig(f) in Eq. (5.7), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “cr-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 263 .000 5 f (Hz) 04 .7//->\V . :ll: :'///■ '■ > ) -.002 ,4 .03 .3 -.0 0 4 .02 v. .2 .v. -.0 0 6 .01 .1 f (Hz) .0 -.0 0 8 .00 10 .000 i-l-LLl_ _ _ _ _ _ _ _L 10 f (Hz) -04 5 -.0 0 2 .4 .03 3 - .0 0 4 .02 .2 f (Hz) -.0 0 6 .01 .1 - .0 0 8 .0 .00 F ig . 5.9b The coefficients a6(/), a 8(/) and aio(/)- (top row—horizontal component, bottom row—vertical component) in Eq. (5.7), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “<r-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 264 2 6 5 where again [■]+ = max{0,[-]} (5.96) give the prediction of the duration of strong ground motion when the depth of sedim ents, h, is available and the param eters R and < p are not known. T he results of the regression analysis, performed on the first model, Eq. (5.8), are shown in Table 5.7 and in Figs. 5.10-5.11. The results for the second model, Eq. (5.9), can be found in Table 5.8 and in Fig. 5.12. In the previous section, we m ade some rem arks about the sim ilarity of the models Eq. (5.6)-(5.7) and their “m agnitude counterpart” dur = dur(M,M2, A , R , R 2,tp) (see Eq. (4.14)), Figs. 4.7-4.8). The sam e rem arks are applicable regarding the consistency of the current models Eq. (5.8)-(5.9) w ith their “m agnitude counterpart” dur — dur(M ,M 2, A , h , h 2), Eq. (4.15), Figs. 4.9-4.10. The only difference is th a t Eq. (5.9) behaves more stable, than the corresponding Eq. (5.7) which includes the “R — (p” description. 5.4 M o d e ls dur = dur(lM M ,A\lM M A,,s), dur = dur ( /m m , - s ) , dur{IMM, A \ I MMA ',s,sL) a n d dur = dur(IMM,s,sL) We now tu rn to the models which describe duration of strong ground m otion in term s of, among other param eters, geological param eter s and local soil pa ram eter s l. The first two models include geological param eter s only. Similar to Eq. (4.16), which scaled duration as dur(M, M 2, A ,s), we use two independent coefficients to describe the influence of param eter s on duration (see discussion T ab le 5.7 Results of the regression analysis of Eq. (5.8). Channel fo # o f data points C oefficients a; and their accuracy ("a-interval") °dur durav number (H z) N (f) aj(h) + a i(h) ai(h) ±Oi(h) a i9 ±Cfl9 34 ±04 320 ±c?20 a 5(h) ± a 50i) a 9(h) ± a 9(h) a5M ■ ±<75^ a9M ± a 9(v) (sec) (sec) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 .0 1 0 . 2 38.3 2 0 . 1 2 311 27.7 ±5.4 28.2 ±5.7 -1.30 ±.75 .182 ±.019 .0 .0 .0 .0 .0 1 0 .1 28.3 3 0 . 2 1 962 33.3 +2.7 35.3 ±2.7 -3.17 ±•37 .195 ±.012 .0 .0 .0 .0 .0 7.8 21.4 4 0.37 1499 23.8 ±2.6 24.2 ±2.6 -1.73 ±.39 .084 ±.040 .018 ±.007 .0 .0 .0 .0 7.3 2 1 . 0 5 0.63 2035 13.7 ±2.1 15.6 ±2.1 -.62 ±.32 .134 ±.033 .012 ±.006 .0 .0 .0 .0 7.8 18.7 6 1 .1 1612 9.9 ±2.7 12.4 ±2.8 -.55 ±.43 .122 ±.035 .009 ±.006 1.73 ±.45 -.30 ±.08 1.83 ±.67 -.28 ±.12 7.6 16.7 7 1.7 1930 4.8 ±1.5 7.0 ±1.6 -.12 ±.25 .080 ±.023 .013 ±.004 1.76 ±.33 -.26 ±.06 2.07 ±.48 -.30 ±.09 5.9 13.6 8 2.5 2107 4.7 ±1.0 5.6 ±1.1 -.26 ±•17 -.001 ±.016 .021 ±.003 .85 ±.22 -.08 ±.04 1.70 ±•32 -.22 ±.06 4.2 1 0 . 0 9 4.2 2411 1.5 ±0.6 1.5 ±0.7 .22 ±.10 -.032 ±.013 .028 ±.002 .0 .0 1.43 ±.29 -.18 ±.05 3.3 7.6 1 0 7.2 2 5 7 6 1.0 ±0.5 1.6 ±0.5 .18 ±.08 -.070 ±.012 .035 ±.002 .0 .0 .0 .0 2.8 6.4 11 13 1584 -1.1 ±0.5 -1.0 ±0.5 .46 ±.08 -.028 ±.017 .027 ±.003 .0 .0 .0 .0 2.3 5.1 1 2 2 1 735 -3.4 ±0.7 -3.3 ±0.7 .75 ±.12 .118 ±.038 .005 ±.006 .0 .0 .0 .0 2 . 0 4.2 1 1 Imm A' Imm A' h h^ h h2 horiz vert horizontal vertical C o r r e s p o n d i n g p a r a m e t e r s 266 [co n sth] [con stv] 40 40 30 30 f (Hz) -1 20 20 - 2 - 3 f (Hz) f (Hz) -1 0 -1 0 - 4 t a .20 .03 .15 .02 .10 .05 .01 .00 f (Hz) .00 - .0 5 -.1 0 -.01 F ig . 5 .1 0 a The coefficients a i ( / ) , a 19( /), a 4(/) and a 2o(f) in Eq. (5.8), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “c-intervals” (dashed lines) and by their estim ated 95% confidence intervals. f (Hz) -.2 - .4 -.6 X 10 -.8 f (Hz) -1 -1.0 f (Hz) -.2 - .4 -.6 -.8 f (Hz) -1 - 1.0 F ig . 5.10b The coefficients 0.5(f) and 0.9(f) (top row—horizontal com ponent, bottom row—vertical component in Eq. (5.8), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals. 268 269 a ^ h)(f)*h+ a ^ h)(f) *h2 (sec) 4 .0 1.1 Hz (C h .6 ) 1.7 Hz (C h.7) 2.5 Hz (C h .8 ) 4.2 Hz (C h .9) 3.5 3 .0 2 .5 2.0 1.5 1.0 a k v)(f)*h+akv)(f)*h2 (sec) 4 .0 3 .5 3 .0 2.5 2.0 1.5 * N A x '/■ ■ / 1.0 '/ ■ / h (km) F ig . 5 .1 1 A dditional (relative to the basement rock sites, s = 2) duration (in seconds) due to the depth of sediments under the recording site, h, as predicted by Eq. (5.8): a) horizontal component; b) vertical com ponent. T his “truncated” model preserves the m ain features of the “com plete” model (Eq. (5.3)), regarding the behavior of the term s which describe the prolongation of duration due to the specific geometry of the sedim entary basin. This pro longation can also be com pared w ith the corresponding term s from the similar “m agnitude” model, (Fig. 4.10, Eq. (4.15)). T ab le 5.8 Results of the regression analysis of Eq. (5.9). Channel fo # o f data points C oefficients a; and their accuracy ("cr-interval") Gdur durav number (H z) N (f) ai(h) ±CT](h) ai(h) ±d(h) aig ± <719 a 5(h) ± a 5(h) a 9(h) ±cr9(h) a5M ±G5(V ) a 9<» ±g9(v) (sec) (sec) 1 0.075 37 40.8 ±2.0 32.5 ±3.1 .0 .0 .0 .0 .0 10.2 38.3 2 0.1 2 311 54.1 ±5.2 53.6 ±5.6 -3.88 ±.79 .0 .0 .0 .0 11.5 28.3 3 0.21 962 52.3 ±2.7 54.2 ±2.8 -4.74 ±.40 .0 .0 .0 .0 8.7 2 1 .4 4 0.3 7 1499 42.3 ±1.8 43.2 ±1.8 -3.33 ±.27 .0 .0 .0 .0 8.4 21 .0 5 0.63 1756 34.6 ±1.9 36.1 ±2.3 -2.59 ±.29 .0 .0 2.37 ±1.01 -.41 ±.17 9.3 18.6 6 1.1 1522 2 2 .1 ±2.1 25.6 ±2.2 -1.63 +.33 4.00 ±.54 -.70 ±.10 3.17 ±.81 -.48 ±.14 9.0 16.0 7 1.7 1930 14.9 ±1.2 18.1 ±1.4 -1.06 ±.20 4.19 ±•41 -.64 ±.07 3.59 ±.61 -.52 ± .1 1 7.5 13.6 8 2.5 21 07 8.3 ±0.8 9.7 ±0.9 -.35 ±•14 2.74 ±.29 -.36 ±.05 3.23 ±.42 -.44 ±.08 5.5 10.0 9 4.2 1667 1.5 ±0.8 2 .6 ±0.9 .56 ±.14 2.68 ±.29 -.36 ±.05 3.33 ±.43 -.48 ±.08 4.8 8.4 10 7.2 2 5 7 6 -0.3 ±0.5 0.5 ±0.6 1.06 ±.09 .0 .0 .0 .0 4.5 6.4 11 13 1584 -2.5 ±0.5 -2.0 ±0.5 1.2 2 ±.08 .0 .0 .0 .0 3.5 5.1 12 21 735 -3.2 ±0.6 -2.8 ±0.6 1.19 ±.10 .0 .0 .0 .0 2.9 4.2 1 1 Imm h h2 h h2 horiz vert horizontal vertical C o r r e s p o n d i n g p a r a m e t e r s 270 [c o n s th] 60 50 40 30 20 10 f (Hz) -1 0 [c o n stv] 60 50 40 30 20 f (Hz) -1 0 f (Hz) - l -2 - 3 - 4 - 5 F ig . 5 .1 2 a The coefficients a i ( / ) and a 19(/) in Eq. (5.9), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “o-intervals” (dashed lines) and by their estim ated 9 5 % confidence intervals (dotted lines). 4 3 2 1 f (Hz) 0 1 10 1 .0 f (Hz) -.2 - .4 -.6 -.8 - 1.0 0 H; f (Hz) -.2 - .4 -.6 -.8 - 1.0 4 3 2 1 f (Hz) 0 .1 1 10 F ig . 5 .1 2 b The coefficients a 5 (/) and ag(f) (top row—horizontal component, bottom row—vertical component) in Eq. (5.9), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 272 273 in C hapter 3). J d u r (fc)( /) 1 I d u r ^ i f ) } ^ +®2o (/)-Im m A /) , 1] + a ^ / ) - ^ 1^ + a i4 ( /) ’S '^ , where A ' is m easured in kilometers and S (0) = / 1' if s = 0- I 0 , if s / 0 . Recall here, th a t s = 0 corresponds to the sites on sedim ents, s = 2 stands in term s of h, R and (p is not available. The results of the regression analy sis of Eq. (5.10) are shown in Fig. 5.13 and Table 5.9. T he set of coefficients {®i(/)> a i9 (/)> ° 4 (/)> a 2o (/)} is similar to the same set from Eq. (5.1). The last two coefficients, (/) and a i4( /) , behave as their counterparts from the model Eq. (4.16) (see Fig. 4.11). On sediments, the duration of strong m otion is generally longer, than on rock. Thus, a i4 (/) shows th at, for the sites on sed im ents, this prolongation can be as much as about 5 sec (channel # 6 , / 0 = 1.1 Hz), and o i3 (/) gives the prolongation for the interm ediate sites of about 2.5 sec for the sam e frequency (channel # 6). T he simplified version of Eq. (5.10) appears to be too unstable in regression analysis. To improve the stability we consider the geological param eter s as a “regular” quantitative variable. The corresponding equation is then: for the sites located on basem ent rock and s = 1 corresponds to the interm edi ate sites. This model can be used when the detailed description of site geology durW{f) durW(f) = max + « 19(/)'^ m m I > 1 + a i S(/}{2 — s). (5.11) T a b le 5.9 Results of the regression analysis of Eq. (5.10). Channel fo # of data points Coefficients a; and their accuracy ("o-interval") Odur durav number (Hz) N(f) ^ 00 KTlOO a, 00 ±O l(h) a i9 ±019 a4 +C4 320 ±<720 a o ±013 314 ±014 (sec) (sec) 1 0.075 37 40.8 ±2 .0 32.5 ±3.1 .0 .0 .0 .0 .0 10.2 38.3 2 0.12 311 27.7 ±5.4 28.2 ±5.7 -1.30 ±.75 .182 ±.019 .0 .0 .0 10.1 28.3 3 0.21 850 26.6 ±3.0 28.5 ±3.1 -2.51 ±•41 .191 ±.013 .0 .0 2.83 ±.64 7.9 2 0 .7 4 0.37 1179 20.3 ±2.9 20.5 ±3.0 -1.51 ±.43 .088 ±.042 .016 ±.007 .0 2.98 ±.51 7.7 20.5 5 0.63 1647 8.5 ±2.4 1 1.0 ±2.5 -.55 ±.35 .127 ±.035 .0 1 2 ±.006 2.84 ±.85 5.76 ±•74 8.1 18.3 6 1.1 2 1 89 5.8 ±1-7 8.5 ±1.8 -.23 ±.26 .093 ±.026 .014 ±.004 1.44 ±.63 3.83 ±•54 7.1 15.1 7 1.7 2645 3.8 ± 1.1 6.3 ±1.1 -.09 ±.17 .039 ±.018 .021 ±.003 1 .1 0 ±.44 2.36 ±.38 5.6 12.2 8 2.5 2931 4.2 ±0.7 5.7 ±0.7 -.19 ± .1 2 -.026 ±.013 .026 ± .0 0 2 .0 1.28 ±•17 4.1 9.0 9 4.2 2 4 6 4 1.2 ±0 .6 2.4 ±0 .6 .16 ± .1 0 -.045 ±.013 .030 ± .0 0 2 .0 .8 8 ±•15 3.4 7.4 10 7.2 2 5 76 1.0 ±0.5 1.6 ±0.5 .18 ±.08 -.070 ± .0 1 2 .035 ± .0 0 2 .0 .0 2.8 6.4 11 13 1584 - 1.1 ±0.5 - 1 .0 ±0.5 .46 ±.08 -.028 ±.017 .027 ±.003 .0 .0 2.3 5.1 12 21 735 -3.4 ±0.7 -3.3 ±0.7 .75 ± .1 2 .118 ±.038 .005 ±.006 .0 .0 2.0 4 .2 1 horiz 1 vert Imm A’ ImmA' s o S(0) Corresponding parameters 274 INVESTIGATION OF DURATION OF EARTHQUAKE RELATED STRONG GROUND M OTION Volume II by Novikova Elena Igorevna A D issertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIV ERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree D O CTO R OF PHILOSOPHY (Civil Engineering) May 1995 Copyright Elena I. Novikova [c o n sth] [con stv] 40 40 30 f (Hz) -1 20 20 -2 -3 (Hz) f (Hz) -1 0 -1 0 -4 a .20 .03 .15 7 ;'/ 7/ :l. ■■II III: 11: .02 .10 .05 .01 .00 f (Hz) .00 -.05 -.10 -.01 F ig . 5 .13a The coefficients a i ( / ) , a ig( /), a 4(/) and a 2o (/) in Eq. (5.10), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “<r-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 5 4 3 2 1 0 .1 1 10 f (Hz) 6 5 4 3 2 1 1 10 f (Hz) F ig . 5 .13b The coefficients a 13(/) and a i4(/) in Eq. (5.10), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “cr-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 276 277 We use here the term a is (/)-(2 — s) instead of a is ( /) - s because we w ant the coefficient ais(f) to be positive when it shows the additional duration on sed im ent (s = 0 or, for interm ediate sites, s = 1) as com pared w ith duration for th e basem ent rock sites (s = 2). The results of the analysis of this model are given in Table 5.10 and Fig. 5.14. All the results agree w ith the previously discussed models: first three coefficients are like in Eq. (5.2), and the last one Oi5( /) , which scales the influence of param eter s on duration, is very close to { ° 13( /) + 0 .5ai4(/)} -0 .5 , where a i3(/) and a i4( /) are taken from the model Eq. (5.10). The last relationship, (/) — 0 -5'{ai3(/) + 0.5-a14(/)} , is w hat m ight be expected when the effect of the geology is described in two different ways; one way accounts for the qualitative nature of the param eter s (and gives coefficients 013 and a i4), and the second way, which ignores this qualitative nature and treats param eter s as a quantitative param eter (the second way results in bringing up the coefficient ais). We now consider the influence of the local soil conditions at the recording site. U nfortunately, the instability of the regression analysis (due to the small am ount of d ata on the geological rock sites) does not allow us to include the qualitative consideration of the param eter s into this model. So, we are left w ith I i u r «■>(/) 1 = I d « r ™ (f) J max °i^(/) 1 , . f + ®19(/)'^MAf + o4 ( / ) > A ,+ ° l ( / ) J (5.12a) + a 20(/)'/m m A ') , 1] + a 15(/)- (2 - s) + a n ( f ) - S ^ + a i 2( / ) - S ^ L ’ T ab le 5.10 Results of th e regression analysis of Eq. (5.11). Channel number fo (H z) # o f data points N (f) C oefficients a; and their accuracy ("a-interval") Odur (sec) durav (sec) ai(b) ±<jl(h) a ! ® ±C ?i (h) a ig ± 0 1 9 ai3 ±013 1 0 .0 7 5 37 40.8 ±2.0 32.5 ±3.1 .0 .0 1 0 .2 3 8 .3 2 0 .1 2 311 54.1 ±5.2 53.6 ±5.6 -3.88 ±.79 .0 11.5 2 8 .3 3 0.21 8 5 0 44.4 ±3.3 46.3 ±3.4 -4.10 ±.46 1.92 ±.53 8.9 2 0 .7 4 0 .3 7 1179 37.2 ±2.2 38.0 ±2.2 -3.20 ±.31 2.60 ±.40 8.8 2 0 .5 5 0 .6 3 1647 28.7 ±1.9 31.3 ±2.0 -2.58 ±.27 3.52 ±.38 9 .7 18.3 6 1.1 2 1 8 9 21.2 ±1.4 24.4 ±1.5 -1.79 ±.21 2.73 ±.29 8.7 15.1 7 1.7 2 6 4 5 12.7 ±1.0 15.4 ±1.0 -.69 ±.15 1.83 ±.22 7 .3 12.2 8 2 .5 2931 7.5 ±0,7 9.3 ±0.7 -.14 ±.11 1.12 ±.16 5 .5 9 .0 9 4 .2 2 4 6 4 1.8 ±0.6 3.3 ±0.7 .64 ±.10 .74 ±.14 4 .9 7 .4 10 7 .2 2 3 7 4 -0.8 ±0.6 -0.2 ±0.6 1.07 ±.09 .21 ±.13 4 .4 6 .2 11 13 1500 -2.4 ±0.5 -2.0 ±0.6 1.20 ±.09 .04 ±.13 3 .4 5.1 12 21 7 3 5 -3.2 ±0.6 -2.8 ±0.6 1.19 ±.10 .0 2 .9 4 .2 1 horiz 1 vert I mm S C orresponding parameters 278 [c o n s th] [c o n s tv] 6 0 6 0 5 0 \Ny 4 0 3 0 30 20 20 10 f (Hz) f (H z) -1 0 - 1 0 f (Hz) - l -2 -3 -4 -5 15 4 3 2 1 0 F ig . 5.14 The coefficients a ,( /) in Eq. (5.11), plotted versus the central frequency of the channels (solid lines). The coefficients axe bounded by their V -intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 279 280 w here the hypocentral distance A ' is m easured in kilometers and S<‘> & 1, if sl = 1, 0 , if s l ^ 1, 1, if sl = 2 , 0 , if s l 2 . (5.126) Recall here, th a t s l = 2 stands for the soft soil sites, s l = 1 designates stiff soil sites and s l = 0 stands for the “rock” sites. All term s which include s or s £ , were chosen in such a way, th a t the positive corresponding coefficients show th e prolongation of duration on sedim entary and soft soil sites, as com pared w ith basem ent rock locations or “rock” sites. The results of the analysis w ith Eq. (5.12) are shown in Table 5.11 and in Fig. 5.15. F irst six coefficients have th e functional forms sim ilar to those found in the previous models. T he new coefficients o u ( / ) and a i 2( /) can be com pared w ith their counterparts from Eq. (4.17), which scaled duration as dur = d u r (M ,M 2,A,s,SL) (Fig. 4.12). The behavior of o u ( / ) and a 12(/) in the current model and in the model Eq. (4.17) are rem arkably similar. The additional duration (in both models) on the soft soil sites (com pared w ith “rock” ) is about 7 sec at the frequency about 1 Hz, and on stiff soil sites it is about 3.5 -j- 4 sec at the frequencies 1 -f- 2 Hz. The influence of the geological param eter is more prom inent at lower frequencies, w ith m axim um at channel # 4 (/o = 0.37 Hz). T he last model to consider here is the simplified form of the previous one, and the equation describing it is f < £ \ f ) | d u r W (f) ) = { d u r M (f) J m ax + (5.13) + a 1 5 (/)’(2 — s) + ai6{f)-SL- T ab le 5.11 Results of the regression analysis of Eq. (5.12). Channel number fo (H z) # o f data points N (f) C oefficients a; and their accuracy ("O-interval") <7dur (sec) durav (sec) aiCh) ±Oi(h) a i( h) ±Oi(h) a i9 ±<*19 a± ±(74 0 2 0 ±<720 a is ±<715 a n ±<7n a i 2 ±<712 1 0.075 37 40.8 +2 .0 32.5 ±3.1 .0 .0 .0 .0 .0 .0 1 0 .2 38.3 2 0 . 1 2 311 27.7 +5.4 28.2 +5.7 -1.30 ±.75 .182 ±.019 .0 .0 .0 .0 1 0 .1 28.3 3 0 . 2 1 850 27.5 +3.2 29.3 +3.2 -2.58 ±•42 .194 ±.013 .0 .95 ±.48 .0 .0 8 . 0 20.7 4 0.37 1179 19.3 +3.0 19.5 +3.0 -1.47 ±.43 .090 ±.043 .016 ±.007 1.81 ±.35 .0 .0 7.7 20.5 5 0.63 1139 15.1 +3.1 17.5 +3.2 -1.73 ±.46 .059 ±.040 .027 ±.007 1.29 ±.53 1.99 ±.89 4.42 ± 1.01 7.7 19.0 6 1 .1 1376 12.1 +2.5 15.1 ±2 .6 -1.51 ±.38 .075 ±.032 .019 ±.006 .23 ±.42 3.80 ±.70 7.22 ±.80 7.0 16.7 7 1.7 1550 8 .0 +1.9 10.4 +1.9 -.97 ±.29 .048 ±.025 .0 2 0 ±.004 .0 3.78 ±.44 5.47 ±•42 5.6 14.1 8 2.5 1574 5.8 +1.4 7.4 ±1.4 -.65 ± .2 1 -.008 ±.019 .023 ±.003 .0 2.40 ±.33 3.45 ±•31 4.2 10.5 9 4.2 1159 1.2 + 1.4 2.5 ±1.4 .04 ± .2 2 .005 ± .0 2 1 .0 2 2 ±.004 .0 2 .0 2 ±.32 2.36 ±•30 3.7 9.3 10 7.2 1093 2 .2 +1.3 2 .8 ±1.3 .04 ± .2 0 -.056 + .0 2 2 .029 ±.004 .0 .92 ±.29 .74 ±•27 3.2 8.3 11 13 628 -2.5 +1.8 -2 .6 +1.8 .80 ±.27 .025 +.045 .015 ±.007 .0 .27 ±.31 0.92 ±•29 2.7 7.1 1 2 2 1 284 -4.7 +1.5 -4.7 ±1.5 1 .0 2 ± .2 1 .034 ± .0 1 0 .0 .0 .0 0.96 ±.30 2.5 6.3 1 horiz 1 vert Im m A' Im m A' S S i P S l® C o r r e s p o n d i n g p a r a m e t e r s [c o n s th] [c o n stv] 19 40 40 30 30 -1 20 20 -2 - 3 (Hz) f (Hz) M M -10 -1 0 - 4 t a .20 .03 .15 .02 .10 .05 \Y .01 f (Hz) .00 f (H z) .00 - .0 5 -.10 -.01 F ig . 5 .15a The coefficients ai ( /), a 19( /) , a 4(/) and a 2o (/) in Eq. (5.12), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 282 2 .5 2.0 1 .5 1.0 7 r a H 6 5 4 3 2 1 0 10 • f (Hz) 1 10 f (Hz) F ig . 5.15b The coefficients a 15( /), au (f) and a^U) in Eq. (5.12), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 283 T ab le 5.12 Results of the regression analysis of Eq. (5.13). Channel number fo (H z) # o f data points N (f) C oefficients a; and their accuracy ("a-interval") Odur (sec) durav (sec) aj(h) ±C Tl(h) ajCh) ±Gl(h) aig ±019 a is ±Ol5 a n ±^13 1 0 .075 37 40.8 ±2 .0 32.5 ±3.1 .0 .0 .0 10.2 38.3 2 0.12 311 54.1 ±5.2 53.6 ±5.6 -3.88 ±•79 .0 .0 11.5 28.3 3 0.21 850 44.4 ±3.6 46.3 ±3.6 -4.10 ±.49 1.92 ±•57 .0 9.6 20.7 4 0.37 1179 37.2 ±2.5 38.0 ±2.5 -3.20 ±.35 2.60 ±•45 .0 9.9 20.5 5 0.63 1647 28.7 ±1.9 31.3 ±2 .0 -2.58 ±•27 3.52 ±.38 .0 9.7 18.3 6 1.1 1376 28.7 ±1.8 31.8 ±1.9 -2.96 ±.27 3.17 ±.46 .84 ±.42 8.4 16.7 7 1.7 1550 24.3 ±1.4 26.5 ±1.4 -2.06 ± .2 1 .0 1 .6 6 ±.26 7.3 14.1 8 2.5 1574 15.5 ±1.0 17.0 .± 1.1 -1.07 ±16 .0 .96 ±.19 5.5 10.5 9 4.2 1159 10.5 ±1.2 11.9 ±1.2 -.35 ±.18 .0 .43 ± .2 0 5.2 9.3 10 7.2 2 5 7 6 -0.3 ±0.5 0.5 ±0 .6 1.06 ±.09 .0 .0 4.5 6.4 11 13 1584 -2.5 ±0.5 -2 .0 ±0.5 1 .2 2 ±.08 .0 .0 3.5 5.1 12 21 735 -3.2 ±0 .6 -2 .8 ±0 .6 1.19 ± .1 0 .0 .0 2.9 4 .2 1 horiz 1 vert Imm s SL C orresponding parameters 284 [c o n s th] [c o n stv] 60 60 50 7 /A 40 f 50 40 -1 30 30 - 2 20 20 - 3 - 4 [!mm1 f (Hz) f (Hz) -1 0 - 5 3 3 2 2 /*V 1 1 f (H z) ■ A 0 1 10 F ig . 5.16 The coefficients at(/j in Eq. (5.13), plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 285 286 T he results of th e regression analysis of this model are presented in Table 5.12 and in Fig. 5.16. Notice th a t this “rough” model is hardly able to detect the influence of th e local soil conditions on the duration of strong ground m otion, although th e previous model Eq. (5.12) allows one to detect this influence. It would be interesting to consider a model which includes the detailed geo logical description of the region (in term s of R, h and < p) and the local soil site conditions simultaneously. Unfortunately, all our attem pts to construct such a model failed due to the instability of the regression solution. It seems th a t the only way to achieve th a t goal is to obtain the values of the param eter sl for all th e sites involved in the d ata base and thus substantially increase the num ber of available d a ta points. 287 6. FR EQ UEN CY D E PEN D EN T DURATION OF STRO NG G R O UN D M OTION, DURATIO N OF THE R U PTU R E AT THE SOURCE A N D DURATIO N A N D THE NU M BER OF THE SEPARATE STRONG M OTION PULSES: THEIR D EPEN D EN C E ON M A G N ITU D E Previous two Chapters were devoted to selection of the regression models of the duration of strong earthquake ground motion. In this C hapter, we study th e earthquake source, using our definition of the strong m otion duration, by extracting the duration of the rupture at the source from the total duration of strong ground shaking. We will also establish the link between the num ber of strong m otion pulses contributing to the total duration and th e m agnitude of the earthquake. 6.1 Duration of Strong Ground M otion as a Function of M agnitude for Different Frequencies T he duration of the rupture process at the source can be approxim ated, as we saw earlier, by Eq. (3.9): r0 = aexp(/3Af). (3.9) However, the duration of the rupture can appear to be different from To when observed through “windows” of various narrow frequency bands. Let us designate this “observed” (at some frequency) duration as durf(M). For each frequency 288 / , only the source w ith the length L> v/f, were v is the velocity of the rupture, can be seen as having durf(M) = t0. If the source dim ension L < v/f, then th e frequency / belongs to coherent p art of the source spectrum (all the points a t th e source plane appear to radiate “in phase” ). As a result, the wave w ith wave length A> 1/ / “sees” the source as a point, and, therefore, cannot carry inform ation neither about source length L, nor about the ru p tu re tim e To. The m agnitude of the source M and the rupture length L are correlated. Thus, for every frequency / , there exists the value of m agnitude, M /, such th a t f dury(M ) = t0 = aexp(fiM), if M > M j, | dur j{ M ) = const. ~ j > T n. if M < M /. ^ ^ T he graphic representation of Eq. (6.1) is shown in Fig. 6.1. Consider now durj(M) for different frequencies. W hen / increases, the cor responding critical dimension L decreases, and so does M f. T he family of curves (6.1) for various values of / is presented in Fig. 6.2a. As it can be seen, lim dur AM) = To = a exp {(3M) for VM, f — * o o and lim durt(M ) = tq for V /. M— * o o Fig. 6 .2b shows the displacement source spectrum of an earthquake of some m ag nitude M. The corner frequency of this spectrum is designated as / . Consider now th e m otion along the line AA' in Fig. 6.2a and 6.2b. At high frequencies, / > / (incoherent p art of the source radiation), the wave length is sm all enough to recognize dimension of the source and the “observed” duration is equal to th e “tru e,” physical duration of the rupture. Once the frequency becomes small ( / < / , coherent radiation), no dependence of the “observed” duration dur j on Source duration at fixed frequency f: durf (M) ( o b s e r v e d ) T q = aexp(/3M) ( a c t u a l ) - l Mf is such that: Magnitude F ig . 6 .1 “A ctual” duration of the rupture process in the source, To, as a function of m agnitude (dashed line) and the “observed” duration of strong motion (at zero epicentral distance) as a function of m agnitude for a M < M f and M > M f for a fixed frequency / (solid line). (3S is the velocity of shear waves. 289 low f -1 f = f 'h ig h f - i ~(high Magnitude log Q log f F ig . 6 .2 D uration of the source rupture at different frequencies. a) D uration of strong ground m otion, “observed” at various frequencies, durf(M). as function of magnitude (solid line) and the “actual” duration of the rupture process at the source, tq (dashed line). Shaded area represents the range of M and / where d ata are available. b) Displacement source spectrum fi, of an earthquake of m agnitude M. While moving along line AA! in both figures (fixed m agnitude M ), one can observe the change in behavior of dur j{M). when the corner frequency of the spectrum , / , is crossed. 290 291 M can be noticed, because dur j is proportional to 1/ / and is not related to L or to M any more. Each curve in Fig. 6.2a can be approxim ated by durf (M) = a (/) exp { /? (/)-M } , (6 .2) w here a(f) and /3(f) depend on frequency. Assume th a t the source radiates high frequencies during all the tim e while the rupture takes place (we will discuss this assum ption later). It follows from the above discussion, th a t lim a = a, lim /?(/) = /?, (6.3a) / — *oo / — ► oo jirn a(f) — 0, Hrn (3(f) - oo, (6.36) where a and (3 are constant coefficients in the expression To = aexp((3M). T here is one more consideration th a t can change the picture in Fig. 6.2a slightly, and th a t is the influence of the limited w idth of the source W for large earthquakes. This lim itation comes from the finite size of the seismogenic zone (about 5 -r 20 km) for the strike-slip events on San A ndreas fault system (see Fig. 3.1). The change in geometry of the fault (large sources are m ore prolonged) influences param eter 7 in Eq. (3.6) and gives slightly different dependence of M versus log10 L for small and for large earthquakes. T he weaker dependence of M on L for small M and L results in possible increase of param eter (3 in the expression r0 = aexp(/3M) for small m agnitudes and for sm all source dimensions (see Eq. (3.8)). We next present some results related to the above discussion. F irst, we w ant to test the assumed dependence durf(M) with the help of linear models. Consider the model dur = dur(M, A) which is a simplified version of the model 292 Eq. (4.9). T he particular equation is and all notation is the same as before. This model was fit to the d a ta in three different ways: 1) for all available data, 2) for the records w ith M < 5 only, and 3) for th e records w ith M > 5 only. The frequency dependent coefficient a2(f) is presented in Fig. 6.3a, 6.3b and 6.3c for the above three cases. According to Fig. 6.2a, no dependence on m agnitude can be noticed for small earthquakes a t low and at interm ediate frequencies, and this is w hat we see in Fig. 6.3b. A t high frequencies, this dependence is weak. For larger earthquakes, M > 5 (Fig. 6.3c)), 0.2(f) is substantial and it grows w ith / . This is w hat should be expected from Fig. 6.2a: approxim ation of durj(M) for large M w ith straight lines results in the growth of the slope of these lines when / increases. The failure of this rule occurs at channels # 1 1 and # 1 2 (/o = 13 -j- 21 Hz). T here can be a t least two reasons for this. First, the data are not evenly available for all frequencies and m agnitudes. Table 6.1 illustrates th a t by showing the number of d a ta points (horizontal and vertical components combined) considered in the above regression model at each channel for various m agnitudes. The shaded area in Fig. 6.2a corresponds to the range of M and / were the num ber of d a ta points is sufficient to give a stable regression analysis. At high frequencies, th e lack of d a ta for large M can disturb the coefficient a2(f). The second reason for the decreases of a2(f) at high frequency channels can be the failure of the assum ption th a t the high frequencies are continuously radiated during the process of rupture. a) all M b ) M<5 f (H z) - l f (Hz) - l c) M >5 f (H z) - l F ig . 6.3 The coefficient 0.2(f) in Eq. (6.4), plotted versus the central frequency of the channels (solid lines). The coefficient is bounded by its “a-interval” (dashed lines) and by its estim ated 95% confidence interval. a | All data points included in regression; b) D ata w ith M < 5 only; c) D ata w ith M > 5 only. It is seen th a t the dependence of duration on m agnitude is much stronger for large magnitudes. 293 2 9 4 T a b le 6 .1 Number of d ata points (vertical and horizontal com ponents com bined) available for the analysis at each frequency band for different m agnitudes (model in Eq. (6.4)). Channel number fo (H z) M a g n i t u d e M 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 1 0 .0 7 5 0 0 0 0 0 0 0 31 2 4 2 0 .1 2 0 0 0 0 0 10 2 271 14 14 3 0.21 0 0 0 0 0 64 34 821 11 32 4 0 .3 7 0 0 0 0 24 127 20 0 1100 16 32 5 0 .6 3 0 0 0 57 151 2 0 9 2 7 7 1271 27 37 6 1.1 0 0 35 132 264 281 299 1520 4 2 4 5 7 1.7 3 0 122 2 3 9 356 330 327 1634 4 3 45 8 2 .5 3 0 208 3 3 0 437 377 . 335 1619 43 4 4 9 4 .2 10 0 272 355 423 301 2 3 0 1063 30 30 10 7.2 27 0 318 358 419 2 5 4 195 924 29 29 1 1 13 29 0 230 277 266 145 128 479 12 6 12 21 16 0 115 151 106 63 62 2 0 8 3 0 2 9 5 We will discuss this last statem ent in connection w ith the next, nonlinear, model: 4ML{f) = ai{f) + a(f)exp{f3(f)-M} + a4(f)-A + a16{f)-sL (6.5) T he m eaning of £ ( /) and /?(/) is the same as in Eq. (6.2). As only a linear (at most) dependence on M can be detected for the low frequencies, only high frequency channels (/o > 1.1 Hz) are considered here. For stability, a ^ \ f ) and a[V\ f ) are combined in one coefficient a i ( / ) . This coefficient is expected to be equal to 1/ / , bu t due to disturbances imposed by various filters employed in the calculation of the “observed” duration (see A ppendix A), O i(/) is of the order of t' = l / / c, where f c is th e corner frequency of the sm oothing filter introduced for the reasons described in C hapter 2. The local soil condition param eter s l is employed here to improve the accuracy of the model. As it was shown earlier (C hapters 4 and 5), the contribution of the local soil condition term is significant at high frequencies, which we are going to explore, while geological condition term plays role at interm ediate frequencies only. The soil conditions param eter s l is treated here as a quantitative variable for simplicity and for numerical stability in regression. This nonlinear model was analyzed in the following m anner. At each fre quency band the linear regression of Eq. (6.5) was performed for various assumed values of /?. The standard deviation of the d ata points from the model, o^ur (see Eq. (4.4)), was com puted for each value of /?. The trial th a t gave the sm allest Odur was adopted as the final result of the regression analysis. The fitted coeffi cients and other param eters of Eq. (6.4) are given in Table 6.2. Fig. 6.4 presents the graphs of 3 ( /) and of (3(f). In agreement w ith Eq. (6.3a), 3 ( /) — > 0 and T able 6.2 Results of the regression analysis of Eq. (6.5). Channel fo # o f data points C oefficien ts and their accuracy ("a-interval") CJdur dur:iv number N (f) a i(f) ± a i(f ) a ( f ) ±cr-(f) lo g io «( f ) m a4(f) ±C4(f) a 16(f) ±cf 16(f) (sec) (sec) 6 1.1 1358 4.3 ±0.5 3 .0 0 e -ll ±2 .0 0e-ll -1 0 .5 2 3.23 .183 ±.007 3.66 ±.27 7.1 16.7 7 1.7 1530 4 .0 ±0.4 1.87e-05 ± .52e-05 -4.73 1.59 .153 ± .005 2.5 0 ±.21 5.8 14.0 8 2.5 1552 2.7 ±0.3 2.7 0 e-0 4 ± .2 4 e-0 4 -3.57 1.35 .103 ± .004 1.67 ± .15 4 .2 10.5 9 4.2 1134 2.3 ±0.3 9.9 0 e-0 4 ± .5 7 e-0 4 -3 .00 1.24 .089 ± .0 04 1.06 ± .14 3.6 9.3 10 7.2 1070 2.4 ±0.2 3.54e-03 ± .1 5e-03 -2.45 1.09 .067 ± .0 04 .28 ±.12 2.8 8.4 11 13 616 1.5 ±0.3 1.16e-02 ± .6 9 e-0 2 -1.93 .93 .054 ±.005 .34 ±•12 2 .4 7.2 12 21 7 2 4 1.1 ±0.1 5.71e-03 ± .3 0e-03 -2 .24 1.04 .058 ± .0 06 .0 1.8 4 .2 1 M a g n i t u d e M A SL C o r r e s p o n d i n g p a r a m e t e r s 2 9 7 — 2 -4 -6 - 8 - 1 0 -1 2 10 f (Hz) 3.5 3.0 2.5 2.0 10 f (Hz) F ig . 6 .4 Frequency dependent coefficients a(f) and (3(f) of the nonlinear Eq. (6.5) plotted versus the central frequency of the channels. As / — ► oo, a(f) and 0(f) approach their limits a and 0 from the equation of the source duration To — a exp (0M). 298 0(f) — > oo as / — > 0 . According to Eq. (6.3b), a (/) and (3(f) should approach their finite limits a and /?(« 1), as / — ► oo. These limits can be treated as the constants from the expression of the source duration T0 = aexp(0M). (3.9) Recall th a t B rune’s formula Eq. (3.2) and the assum ption th at high frequency waves are continuously radiated by the source during the rupture process, imply L = a 'e x p ((3M), (3-H ) i.e. the sam e constant 0 appears in the dependence of the source length on earthquake m agnitude, as in the dependence of duration of the source radiation on earthquake m agnitude. Jovanovich et al. (1974), considering the results of C hinnery (1969), Gibowicz (1973), T hatcher (1972), Hanks and Wyss (1972), and Wyss and M olnar (1972), gives the upper estim ate if the source length as £max = 0 .2 0-10°-4OM. (6 .6) The recent work of Trifunac (1993), where he considers the results of seismolog- ical and of field estim ates of perm anent ground displacements, the independent estim ates of seismic moment and the empirical equation for scaling the Fourier am plitude spectra of strong ground motion, gives the lower estim ate of L : £min = 0.01T 0°-5M, (6.7) which is in agreement w ith the works sited above. O ur value (0 = 1), translated into the logarithm of base 10 units, gives L ~ 100-43M. Note th a t the estim ates of other authors come from the m easurem ent of the source length L from the field d ata or from the low frequency end of the spectrum , 299 and our results come from the analysis of the duration of strong m otion at high frequencies. Com parison of these two estim ates relies on the assum ption th a t the source radiates high frequency continuously during the whole process of rupture, which is, of course, questionable. Also, high frequencies attenuate quickly so th a t a t strong m otion station near fault only not too distant fault segm ents may contribute. 6.2 Duration of the High Frequency Radiation from the Source and the Corresponding Source Dim ensions As B rune (1991) notices in his review on source dynamics, “recent inversions for slip distribution using strong m otion records have verified th a t m oderate size earthquakes on and in the San Andreas fault system tend to be complex, w ith m ultiple centers of high frequency radiation” (Trifunac, 1974; H eaton, 1990; M endoza and Hartzell, 1989; Doser, 1990a,b; Hartzell and H eaton, 1983, 1986; H artzell, 1989). Thus, we may assume th a t the high frequency radiation comes from ju st a portion (may be a small portion) of the source in the form of several pulses. A t least two major types of models of complex earthquakes were proposed: th e asperity and the crack-type events. T he asperity model was introduced by Kanam ori and Steward (1978), and M adariaga (1979) and developed further by Rudnicki and K anam ori (1981). Mc- G arr (1981) also considers a fault model where a circular asperity is surrounded by a previously faulted annular region of some bigger radius. R autian (1991) analyses the source spectra of the earthquake of Soviet C entral Asia, extracted 300 from coda (R auitan and K halturin, 1978; R autian et al., 1981), and concludes th a t the asperity model can explain many relationships and correlations found in the data. Das and Kostrov (1983, 1986) continue the m athem atical developm ent of the asperity model, and Boatwright (1988) and Gusev (1989) look into the com posite asperity model, where a complex faulting is considered as a random superposition of sm aller “asperity type” subevents. The crack model differs from the asperity model by the boundary conditions, imposed on the area, surrounding the subevent. In the crack m odel, the area around subsource does not slip, so th a t the source plane has a net of unbroken barriers. T his model was developed by Das and Aki (1977), Aki (1979), Hadley and Helm berger (1980) and Papageorgiou and Aki (1983a,b). T here are at least two arguments in favor of the asperity type m odel. F irst, the barrier-crack model is not satisfactory from the point of view of tecktono- physics: the unbroken barries should finally break in some unclear way to allow plates to slip one w ith respect to another (Gusev, 1989). Second, as Boatw right (1988) points out in his investigation of both models, the barrier models is either deficient in the low-frequency radiation, or requires m ultiple ruture of the sam e fault area, which is hard to imagine dynamically. Having in m ind the asperity model, we will address the question of the dim en sions of the fault area, responsible for emission of the high frequency radiation, the duration of which we can measure. The results of the analysis of Eq. (6.5) suggest th a t the last three channels can be considered together in the description ( V of th e high frequency radiation of the source, because a(f) and /?(/) practically do not depend on frequency for the highest frequency channels (/o = 7.2 -r 21 3 0 1 Hz). We consider here the following three models of the high frequency duration: dur = o i + a 0-10^oM + a4-A + ai6-sz,, (6.8 a) dur = oi + ao-10^°M + a4-A, (6 .86) dur = ao-10^°M + a4-A. (6 .8c) T he term s otQ-10P°M in Eq. (6 .8) can be considered as the duration of th e high frequency radiation from the source, which we designate here as r'. T he first m odel, Eq. (6 .8a), is identical to Eq. (6.4), but expressed in term s of the log arith m w ith base 10 instead of the base e. The second model, Eq. (6 .8b), was introduced to increase the num ber of available d ata points by excluding the soil condition term (local soil conditions are not available for the substantial am ount of stations). The third models, Eq. (6 .8c), appears to be the only model stable enough to allow one to com plete the nonlinear regression analysis not only for all available d ata in wide range of m agnitudes, bu t also separately for large and for sm all earthquakes. The regression analysis of Eqs. (6 .8), accom plished in the sam e m anner, as the analysis of Eq. (6.4), gives the following results: dur = 2.10 + 0.003M 0-48M + 0.066-A + 0.28-sL, (6.9a) dur = 1.28 + 0.014-10'4°M + 0.062-A, (6.96) dur = 0.093-10-29M + 0.061-A. (6.9c) T he last model also gives: dur = 0.28-10’20M + 0.050-A for M < 5, (6.9c') dur = 0.058-10'32M + 0.062-A for M > 5. (6.9c") 3 0 2 In all these models, the d ata for the channels # 1 0 , # 1 1 and # 1 2 only were considered. All these d ata were assumed to be one homogeneous d a ta set, char acterizing the high frequency radiation ( /o = 7.2 -r 21 Hz). Eqs. (6.9a,b,c) lead to the estim ate of the duration of rupture, during which the high frequency radiation was released: t ' = 0.0031-10-48M, (6.10a) t' = 0.014-10-40M, (6.106) t ' = 0.093-10-29M, (6.10c) and t' = 0.28-10-2OM for M < 5, (6 .10c') t ' = 0.058-10'32M for M > 5. (6.10c") We will now assume th a t the “length” V and the “w idth” W' of the high- frequency-release area (may be consisting of several disjoint portions) can be related to tim e t' by the formula, similar to B rune’s (1970): Tj t ' Kf — A for M < 5 (6.11) v for sm all earthquakes, and by Trifunac’s (1993) Eq. (3.10), applied to r ', L' and W'\ L' W ' t' a — A + — for M > 5 (6 .12) v 2pa - K 1 for large earthquakes. Here v is velocity of rupture, and (3a is shear wave velocity. We will take v = 2.2 km /sec and v — 0.6•/?<,. We will also assum e th a t W '/L ' = W / L and will consider (Trifunac, 1993) W/L » 0 .7 , 0.3, 0.15 for M = 5, 6, 7. (6.13) 303 F urther, we consider two possibilities: unilateral (A = 1) and bilateral (A = 0.5) faulting, which will give the minimum, L'mm, and the m axim um , L'max, bounds on the length of the high-frequency-release area. Com bination of Eqs. (6.10)- (6.13) lead to several estim ates of L'max and L'mXn, which are presented in Fig. 6.5, together w ith th e bounds on the total length of the source, L, given by Eqs. (6 .6) and (6.7), and all the estim ates of L from the Fig. 3.1. The overall trend of the models Eq. (6.10)-(6.13) regarding to the estim ates of L suggests th a t the high- frequency-release area practically coincides w ith the total area of th e source for sm all (M < 4.5) earthquakes and is somewhat sm aller than this area for large earthquakes. Note th a t our estim ate of V should be taken w ith caution due to th e lim itations imposed by filters, used in the com putation of strong m otion duration. It appears (Appendix A), th at all of our duration models cannot recognize the pulse of energy w ith duration sm aller than about 2 -f- 3 sec, hence, th e value of V can be (and is) overestimated. We notice th a t our last model of the high frequency source duration, Eq. (6 .10c), distinguishes between two different geometries of the fault plane which characterize small and large earthquakes. For small earthquakes, the area of the source A ~ LW « L 2, but for large shocks the w idth W is “fixed” and A ~ L 1. As we saw earlier (Eqs. (3.6), (3.8)), this causes param eter /?o in t ~ 10^°M to be different for small (f3'0) and for large (/%') earthquakes. M ore over, the ratio of these two values, according to Eq. (3.8), should be Pq/P'q = 2/3. If our assum ption of the geometrical sim ilarity of asperities and the whole source plane {W /L — W '/L') holds, then the same ratio should be tru e for the high frequency radiation. Our result (see Eqs. (6.10c') and (6.10c") gives C D T3 P -P • p H P c d 8 - 7 - 6 - 5 - 4 - 3 - F a u lt le n g th , estimates shown - Jovanovich L mjn ~ Trifunac L ength L * 10c') and (6.10c " ) T 0 l o g i o l o g i o L f ( k m ) F ig . 6.5 An estim ate of the length of the source area, responsible for the release of high-frequency radiation, L', which is obtained from Eqs. (6.11)-(6.12), where A = 0.5 -f 1 and t ' is given by Eq. (6.10). The estim ates of the overall length of the source, L (filled circles and the shaded zone between L max and Amin lines) are shown for comparison. 304 3 0 5 Pol Pa = 20/32 « 2/3. The assum ption W /L = W '/L ' means th a t larger earth quakes are capable of breaking larger asperities, which physically seems logical. 6.3 Duration and the Num ber of the Separate Pulses Contributing to Strong Ground M otion as a Function of M agnitude This section is devoted to further discussion of the relations one can establish between the duration of strong ground motion, as defined in this work, and some properties of the earthquake source. In the beginning we will clarify some notation we use. We defined the duration of strong ground m otion as the sum of tim e in tervals, later called pulses, during which the integral of acceleration, velocity or displacem ent squared has the steepest slope (see definition Eq. (2.1)). For each record (component of acceleration, velocity or displacement), the num ber of those pulses, m j, and the duration of every one of them , ^dur^u,se^ will be different at each frequency band. T he notation used in the rest of this C hapter (and in the next C hapter also) is similar to w hat was introduced earlier (see beginning of C hapter 3.) In particular, when we discuss the total frequency dependent duration of strong m otion, the duration of one pulse or the num ber of pulses, we use respectively d u rf , dur^ulse and m / (or dur(f), dttrpulse( /) and m(f)). If we are concerned prim arily w ith the functional form of the dependence of those quantities on some earthquake and site param eters, we use the same “nam es” of variables, bu t we underline them and show explicitly specific param eters we are interested in. For exam ple, the functional dependence of durj , dur^uIse and my on m agnitude of 306 th e earthquake is designated as durf(M), d u rpulse(M ) and m /(M ) respectively. We w ant to emphasize, th a t all “regular” (not underlined) notation is intended to stan d for the “real” or “measured” quantities, or th e quantities predicted by m ore or less a complete model. All underlined notation stands for explicit dem onstration of the functional dependence of some quantities on one or two param eters, and this dependence is not m eant to represent a com plete model. The total frequency dependent duration, d u r/, is, according to (2 .1), mt dur/ — ^ ^dur^uIse^ . (6-14) i =i 3 We next consider some regularities in the behavior of the quantities m / and d u rpulse and the physical reasons th at might explain them . This section, in particular, deals w ith the dependence of m and durpulse on the m agnitude of an earthquake. T he way how these param eters depend on the epicentral distance is considered in C hapter 7. From previous discussion, it follows th a t we may assum e th a t the separate pulses, which give contribution to the duration of strong ground m otion, can be radiated by different “asperities” on the fault surface (suppose we corrected for the propagation effects and, thus, consider the location “ju st above the source” ). We observe the rupture process through several narrow frequency bands, one at a tim e. A t each channel (frequency band), the “roughness” of the source can be only resolved down to some limited characteristic length i{f). In our case, t(f) ~ vw/ f et (6.15) where f c is the corner frequency of the sm oothing filter used in the calculation of d uration (Appendix A, Table A .l), and vw is the wave velocity. On the other 3 0 7 hand, the waves, th a t carry inform ation about the source, have to be short enough to “see” separate asperities. Thus, if the characteristic size of an asperity is p, and the typical distance between them is r > p, the frequency / > v/r is needed to recognize the existence of the separate asperities, and even higher frequency / > v /p is necessary for correct identification of the size of those asperities. Suppose now th a t we are not limited by the “filter constraint” Eq. (6.5). W hat would then be the correspondence between the dependence durf(M) and d « r^ul3e(M )? A t low frequencies, where the wave length is so long th a t even the inform ation about the size of the source cannot be kept, strong m otion consists of one pulse only (rnj(M) — 1) and dur^(M) = dur^ulse(M ) (Fig. 6 .6a)). In this case, the critical m agnitude Mf, which is the border between the sm all event (the one th a t the wave w ith frequency / “sees” as a point source) and an event w ith distinguishable source dimension (see Eq. (6 .1) and Fig. 6 .2a), is outside the available range of m agnitudes. For the m oderate frequencies, M f is inside this range, and, therefore, f dur AM) - d u ryUlse(Af). m/ « 1, if M < Mf, { , (6 -16) ^ durf(M) & m f dur1 p ae(M). m,f > 1, if M > M f. T his effect is schematically illustrated in Fig. 6 .6b. At high frequencies, when M f moves tow ards small m agnitudes, the difference between the total duration of strong m otion, dur f(M). and the duration of one pulse, dur^ulse(M ), becomes m ore pronounced (Fig. 6 .6c). T he exact appearance of the dependencies dur^ulse(M ) and rn,f{M) is very com plex, especially for high frequencies. The duration of one pulse th a t we can really m easure, is the result of processes at the source and along the propagation path. W hich of those two influence the high frequency content of th e observed t o t a l d u r a t i o n d u r a t i o n o f o n e p u l s e a) low frequency b) moderate frequency c) high frequency + -> cti u 2 Q ~mf (M) tim es > ► Magnitude Magnitude Magnitude F ig . 6 .6 Comparison of the dependence of the total duration, durf(M). (solid lines) and the duration of one pulse, durpulae(M), (dashed lines), on m agnitude at a) low, b) m oderate and c) high frequencies. M f designates the frequency dependent critical m agnitude from Eqs. (6 .1) and (6.16) (such th a t the dimension of the source of the event w ith magnitude M f is approxim ately equal to the wave length of the wave with the frequency / ) . At very low frequencies, M f is bigger th an the m agnitudes represented in the d ata base, so the real dimension of the source cannot be resolved and only one pulse of strong motion is seen. M oderate and high frequencies allow to resolve the dimensions of the source and the dimension and the number (my) of subsourses, which may have produced separate pulses of strong motion. 309 signal in a more radical way? The latter question is debated now (Anderson, 1991). Some investigators believe th a t the high frequency cut-off, f ma.x , exists, and th a t it results from attenuation effects (Hanks, 1982), while others attrib u te it to th e lack of high frequency generation at the seismic source due to the existence of a non-linear cohesive zone at the tip of a propagating crack (Papa- georgiou and Aki, 1983a,b; Papageorgiou, 1988). O thers find th a t this cut-off disappears if one plots the logarithm of spectral am plitudes versus frequency on a linear scale (Anderson and Hough, 1984). Analysis of em pirical equations which describe th e strong m otion acceleration spectra near source show no evidence of /max (Trifunac, 1993). If / max exists, and if it is caused by the process at the source, m /(M ) may decrease at high frequencies. Some researches consider the seismic rupture as a fractal process (Aviles el al., 1987; Frankel, 1990; Okubo and Aki, 1987). In this case, down to the scales where self-similarity may prevail (down to the certain high or m oderate frequency), the num ber of pulses m /(M ) should be a nondecreasing function of / . Recall now the lim itations imposed by the filters, which are used to calculate duration (see Eq. (6.15) and Appendix A). W hatever the true dependence m y(M) is, it will be destroyed in the process of calculating duration for frequencies higher th an 0.5 Hz (Table A .l in A ppendix A). We will see only sm oothed, averaged and the m ost pronounced features of the functions rrif(M) and dur^ulse(M). Fig. 6.7 displays the distribution of the actual d ata points. For each fre quency band, the total duration dur^ulse (big dots) and the duration of separate pulses duryU ,3e (small dots) are plotted versus the m agnitude of an earthquake. T here are several small dots per each big one, as there may be several tim e in tervals th a t contribute to the total duration of strong ground m otion. For the 310 clarity of the figure, small dots are slightly shifted to the left along m agnitude axes (by 0.02 units of m agnitude), and big dots are shifted to the right by the sam e am ount. Comparison of the last two figures shows th a t our expectations (sum m arized on Fig. 6 .6 ) are not contradicted by the actual d ata (displayed in Fig. 6.7), provided the distortions coming from filters are taken into account. Fig. 6.8 shows the actual dependence of the num ber of strong m otion pulses on m agnitude. For the clarity of the figure, some finite interval on the vertical axis is assigned to each value of m y, and the data points which correspond to each my are arbitrarily (with respect to the vertical axis) located w ithin this strip. T he abscissae of all d ata points were preserved. For th e quantitative description of this phenomenon, we consider two simple regression equations. For the duration of one pulse, we assume d u r ^ l3e( f ) = b 1(f) + b2(f)-M + bA(f)-A, (6.17) and for the num ber of those pulses in one record, we take M f ) = ci (/) + M f ) ' M + c4(/)-A . (6.18) Here, as in Eq. (6.14), the total frequency dependent duration of strong ground m otion for each com ponent of acceleration, velocity or displacem ent is ju st the sum of m ( /) term s: dur(f) = <*urpulse(/). m (/)te rm s T he frequency dependent coefficients, 6i( /) and M f ) , are num bered according to the corresponding a ,( /) , which were used to scale the total duration (see Table 3.2). In this representation, at every frequency band, each com ponent of acceleration, velocity or displacement gives one d ata point for the regression 311 075 Hz 50 40 30 20 10 O CD i f0 =.12 Hz 50 40 O 30 a 50 Q 40 30 20 10 M a n i t u d e F ig . 6 .7 a Channels # 1 to # 3 : obsreved duration of separate pulses (small dots) and the observed total duration of strong ground m otion (big dots) are plotted versus m agnitude of an earthquake. For the clarity of the figure, sm all dots are shifted to the left along the m agnitude axis (by 0.02 units of m agnitude), and big dots are shifted to the right by the same amount. The trends of these two distributions can be com pared w ith those in Fig. 6 .6 . 312 50 40 30 20 10 CD a f0 =.63 Hz 50 40 30 20 50 Q 40 30 20 10 n i t u d e M a F ig . 6 .7 b Channels # 4 to # 6 : obsreved duration of separate pulses (small dots) and the observed total duration of strong ground m otion (big dots) are plotted versus m agnitude of an earthquake. For the clarity of the figure, sm all dots are shifted to the left along the m agnitude axis (by 0.02 units of m agnitude), and big dots are shifted to the right by the same am ount. T he trends of these two distributions can be com pared w ith Fig. 6 .6 . 313 50 40 30 20 10 0. < D 3 4 5 6 7 8 50 40 30 20 + •> 10 3 f n =4.2 Hz 50 Q 40 30 20 10 M a g n i t u d e F ig . 6 .7 c Channels # 7 to # 9: obsreved duration of separate pulses (small dots) and the observed total duration of strong ground m otion (big dots) are plotted versus m agnitude of an earthquake. For the clarity of the figure, sm all dots are shifted to the left along the m agnitude axis (by 0.02 units of m agnitude), and big dots are shifted to the right by the same am ount. T he trends of these two distributions can be com pared w ith Fig. 6 .6 . 3 1 4 50 40 30 20 10 O < D 50 40 30 20 + * 10 2 f0 =21 Hz 50 Q 40 30 20 10 M a g n i t u d e F ig . 6 .7 d Channels # 1 0 to #12: obsreved duration of separate pulses (small dots) and the observed total duration of strong ground m otion (big dots) are plotted versus m agnitude of an earthquake. For the clarity of the figure, small dots are shifted to the left along the magnitude axis (by 0.02 units of m agnitude), and big dots are shifted to the right by the same am ount. T he trends of these two distributions can be com pared w ith Fig. 6 .6 . 10 315 8 - f0 = . 0 7 5 Hz m 6 - < v 4 - 2 - -H i i ! i i 1 i 1 1 i • ■ i 3 4 5 6 7 8 10 p Oh 8 - f0 = . 1 2 Hz 6 - o 4 - 2 _ : i • : | i . • • i u i i ! < 1 1 1 1 : : 1 i i 0 3 4 5 6 7 8 10 & a 8 f0 = . 2 1 Hz 0 6 4 :! i • -ill** I 2 i [ ; ’ •ill : i ;::; :! 1 1 1: !; s « !, i •; 111 i • 1 I 2 4 5 6 7 8 M a g n i t u d e F ig . 6 .8 a Channels # 1 to # 3 : observed number of separate pulses of strong m otion, m y, is plotted versus the m agnitude of an earthquake. For th e clarity of the figure, a finite interval of the vertical axis is assigned to each value of m y, and th e ordinate of each d ata point, corresponding to this m /, is chosen arbitrarily w ithin this interval. The grow th of my w ith M could be explained, in part, by the increase of the num ber of “patches, broken during the rupture w ith the increase of m agnitude. T he “true” dependence m y(M ) may be disturbed by the filters which we used for frequencies greater than about 0.5 Hz. M a g n i t u d e F ig . 6 .8 b Channels # 4 to # 6 : observed num ber of separate pulses of strong m otion, mf, is plotted versus the m agnitude of an earthquake. For th e clarity of the figure, a finite interval of the vertical axis is assigned to each value of mf, and the ordinate of each d ata point, corresponding to this mf, is chosen arbitrarily w ithin this interval. The grow th of m / w ith M could be explained, in part, by the increase of the num ber of “patches, broken during the rupture w ith the increase of m agnitude. T he “tru e” dependence rrif(M) may be disturbed by the filters which we used for frequencies greater than about 0.5 Hz. i o r f0 =1.7 Hz 317 m 6 « 4 C O 2 r-H 2 Ph 0) 10 rD S 0 2 6 * 4 2 I ! I ; j : ; I ; j . i ; : i • I : I i i M i ! I I 11 i 1 1 1 1 ! ! ! I : I I ! i I ! 1! h i I • ! i I I 10 2.5 Hz 8 6 4 2 I I I I 3 4 5 6 7 8 f0 =4.2 Hz ill; ! ! I I : ; . I I ; : M i l l ! I I I i I M 3 4 5 6 7 8 M a g n i t u d e F ig . 6 .8 c Channels # 7 to # 9 : observed number of separate pulses of strong m otion, m /, is plotted versus the m agnitude of an earthquake. For the clarity of the figure, a finite interval of the vertical axis is assigned to each value of m /, and th e ordinate of each d ata point, corresponding to this m /, is chosen arbitrarily w ithin this interval. T he grow th of m t w ith M could be explained, in part, by the increase of the num ber of “patches, ’ broken during the rupture with the increase of m agnitude. T he “true” dependence rrif(M) may be disturbed by the filters which we used for frequencies greater than about 0.5 Hz. 03 03 03 10 318 7.2 Hz 8 4 2 3 4 5 6 7 0 10 f0 =13 Hz q-i 6 — O 4 - i i 2 i « 1 1 i i 1 ; 1 i !! * ' lii !:!■:!•! 1 • 1 l 1 • , Jh 1 ! .11 i. : 1 11111 i i i i ; i 1 i 1 1 i ! 1 1 ! 1 . i • i 03 3 4 5 6 7 8 10 j— pO a 8 - f o = 2 1 Hz 0 6 - £ 4 - l i 2 i 1 • 11 •; | " * 1 * * i 1 1! i i ;; = ■ ! ■ . : i I | j ! j ' : ! i : i i ; ! ! 1 : i I i i 3 4 5 6 7 8 M a g n i t u d e F ig . 6 .8 d Channels # 1 0 to #12: observed num ber of separate pulses of strong m otion, mf, is plotted versus the m agnitude of an earthquake. For the clarity of the figure, a finite interval of the vertical axis is assigned to each value of m / , and the ordinate of each data point, corresponding to this mf, is chosen arbitrarily w ithin this interval. T he growth of m / w ith M could be explained, in part, by the increase of the num ber of “patches, broken during the rupture w ith the increase of m agnitude. The “tru e” dependence m.f(M) may be disturbed by the filters which we used for frequencies greater than about 0.5 Hz. 319 Eq. (6.18) and one or more d ata points for the regression Eq. (6.17). Thus, the statistics in Eq. (6.17) is much better than in Eq. (6.18) and in all previously considered regression equations. T he prelim inary fit of the Eq. (6.17) and Eq. (6.18) separately for the hor izontal and vertical components shows little difference between com ponents of different orientation. Thus, we decided to consider the d a ta for horizontal and vertical m otions as one uniform data set. Tables 6.3 and 6.4 present the results of the linear least square fit of Eq. (6.17) and (6.18) respectively. The quantities am and m av in Table 6.4 are defined sim ilarly to the definition of < T d ur and durm (see Eq. (4.3)-(4.4)). Fig. 6.9 (top row) shows the frequency dependent coefficient b2(f) in the model th a t scales the duration of one strong motion tim e interval. This coefficient reflects the growth of the duration of one strong motion pulse w ith m agnitude. Three cases of the dependence b2 (/) are presented in Fig. 6.9: a) when all available d ata are used in the analysis, b) when d ata points with M < 5 are only used, and c) is the case w ith M > 5 d ata only. The growth of the duration of one strong motion interval w ith m agnitude is much more pronounced for large earthquakes. High energy release (large m agnitude) is related to large slip, so th at, on the average, the slip, capable of breaking an asperity of a particular strength, occurs throughout big area. This may cause the duration of a pulse of energy, em itted by the asperity during large earthquake to be longer, th an in the case of a small shock. Low energy earthquake may just not be able to break the same asperity com pletely (Trifunac, 1993). Due to the lim itations imposed by the filters, used in the calculation of d ura tion (A ppendix A), it is too speculative to assume reliability of the quantitative T ab le 6 .3 Results of the regression analysis of Eq. (6.17) (duration of a separate strong motion pulse, all available # of data Coeff. bi and their Channel fo "a-interval" <3dur durav number (Hz) points b i ±ai b2 ±02 b 4 ±04 (sec) (sec) 1 0 .0 7 5 38 37.32 ±1.92 .0 .0 1 1 .8 6 3 7 .3 2 2 0 .1 2 4 7 0 16.79 ±.87 .0 .0364 ±.0135 1 0 .7 0 1 8 .7 0 3 0.21 18 3 5 10.39 ±.33 .0 .0179 ±.0063 6 .8 8 11 .2 0 4 0 .3 7 3 7 5 3 8.19 ±.18 .0 .0041 ±.0034 5 .9 3 8 .3 7 5 0 .6 3 5 7 4 5 1.87 ±.86 .72 ±•14 .0054 ±.0025 5 .3 7 6 .5 9 6 1.1 7 1 2 3 .95 ±.61 .70 ±.10 .0098 ±.0019 4 .8 0 5 .7 3 7 1.7 7281 -.67 ±.48 .89 ±.08 .0136 ±.0019 4 .5 7 5 .2 6 8 2 .5 7 2 1 6 .13 ±.34 .63 ±.06 .0101 ±.0015 3 .5 6 4 .2 9 9 4 .2 4 8 5 4 -1.17 ±.37 .87 ±.07 .0116 ±.0020 3 .4 6 4 .2 5 10 7 .2 4 0 5 2 -2.41 ±.36 1.06 ±.07 .0134 ±.0021 3 .3 0 4 .0 6 11 13 2 2 4 9 -3.02 ±•39 1.14 ±.08 .0105 ±.0031 2 .7 6 3 .5 5 12 21 9 4 2 -1.98 ±.53 .85 ±.11 .0328 ±.0067 2 .4 2 3.25 1 M A Corr. parameters T ab le 6.4 Results of the regression analysis of Eq. (6.18) (number of strong motion tim e intervals, all available data) Channel number fo (H z) # o f data points C oeff. Cj and their "a-interval" Am mav Cl ±ffi C2 ± 0 2 C4 ± 04 1 0 .075 37 .99 ±.07 .000 ±.000 .0007 ±0011 0 .1 6 1.03 2 0.12 311 1.20 ±.07 .000 ±.000 .0066 ±0011 0 .6 6 1.51 3 0.21 962 1.29 ±.06 .000 ±.000 .0148 ±0013 0 .8 7 1.91 4 0.37 1499 1.63 ±.05 .000 ±.000 .0218 ±0011 1.05 2.5 0 5 0.63 2029 1.88 ±.33 .006 ±.057 .0239 ±0012 1.28 2.83 6 1 .1 2 5 2 2 1.44 ±.26 .045 ±.048 .0279 ±0015 1.28 2.6 7 7 1.7 3099 1.22 ±.17 .104 ±.031 .0142 ±.0008 1.11 2.35 8 2.5 3396 .83 ±.13 .151 ±.025 .0121 ±.0007 0.99 2.12 9 4.2 2 7 14 . .35 ± .1 1 .210 ±.022 .0090 ±0007 0.83 1.79 10 7.2 2553 .15 ±.09 .221 ±019 .0081 ±.0007 0 .7 0 1.59 11 13 1572 .26 ±.10 .184 ±021 .0089 ± 0010 0 .5 9 1.43 12 21 7 2 4 .07 ±.13 .223 ±027 .0040 ±0018 0 .5 0 1.30 1 M A Corr. parameters b ) M <5 1.5 1.5 1.5 A 1.0 1.0 1.0 10 10 10 f (Hz) f'-.(Hz) f (Hz) -.5 -.5 -.5 d ) all M e ) M <5 f) M >5 _ 10 f (Hz) f (Hz) f (Hz) -.1 -.1 F ig . 6 .9 The frequency dependent coefficients scaling durpulse (62(7)5 top row) and m (c2(/) bottom row) in Eq. (6.17) (duration of one strong m otion pulse) and Eq. (6.18) (number of strong motion pulses in a component) with respect to m agnitude are plotted versus the central frequency of the channels (solid lines). The coefficients are bounded by their “o-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines): a) and d) all available d ata points included; b) and e) data points with M < 5 only; c) and f) data points with M > 5 only. D uration of one pulse and the num ber of pulses grow mainly when M > 5. 322 3 2 3 behavior of th e coefficient f » 2(/)- However, we may expect th a t 62 ( /) grows w ith frequency if the process in the source is a self-similar chaos (fractal). The ab sence of this growth in high frequencies may be an indicator of the fact th a t some m inim um size of asperities capable of storing any considerable stress exist in the crust. T he increase of the filters resolution and more careful study will probably give some clues. T he bottom row of the Fig. 6.9 shows the frequency dependent coefficient c2( /) in Eq. (6.18), which scales the number of strong m otion pulses. Again, three cases were considered: d) all data, e) data with M < 5 only, and f) d ata w ith M > 5. Small earthquakes, M < 5, have one pulse of strong m otion energy only: c2(/) « 0 in Fig. 6.9e. The growth of the num ber of strong m otion tim e intervals as a result of increasing m agnitude happens prim arily in the range M > 5, Fig. 6.9f (see also the raw data, Fig. 6 .8). The recent work by Trifunac (1993) supports this conclusion. One more comment can be made here. Comparison of Fig. 6.9a and Fig. 6.9d shows th a t 62 (/)j responsible for the dependence of the duration of one pulse on m agnitude, and c2( /) , which scales the growth of the num ber of strong motion pulses w ith respect to m agnitude, have different behavior. “One pulse duration” coefficient, 62 (/"), is positive for / > f u where fi « 0.5 Hz. “N um ber of pulses” coefficient, c2( /) , significantly differs from zero for / > / 2, and / 2 is higher than fi. Possible explanation may be as follows. At low frequencies, f < fi, the wave length is too long to recognize the dimension of the source, so we see only one pulse (c2 = 0), the duration of it is « 1/ / , and the duration of this single pulse is not related to the m agnitude (62 = 0 ). In the interm ediate frequency range, f\ < f < / 2, the physical length of the source exceeds the wavelength of the 3 2 4 waves we consider, bu t separate asperities still cannot be resolved because they are sm aller th an the corresponding wave length. As a result, is still around zero (one pulse only), bu t 62 7 ^ 0 , which means th a t the duration of the only pulse we see does depend on m agnitude. Finally, at high frequencies ( / > / 2) separate asperities can be resolved and the number of them increases w ith m agnitude (c2 > 0 ). T he duration of one strong motion tim e interval, durpulse (Fig. 6.9a), model Eq. (6.17), and the total duration of strong motion, dur (Fig. 6.3a), model Eq. (6.4)), show similar dependence on m agnitude: the coefficient a2( /) from Fig. 6.3a differs from the coefficient 62 (/) from Fig 6.9a in scale only. It appears th a t separate pulses may represent some kind of subevent, the superposition of which turns out to be the strong earthquake (Boatwright, 1988). However, we are obviously not able to distinguish between the crack and the asperity type subevents. 6.4 The Number of the High Frequency Patches and the “Asperity Size” We will specify now w hat we mean by the asperity model in the application to the strike-slip events in California, and suggest some relationships th a t might characterize the dimensions, involved in this model. We consider th e following scenario (Trifunac, 1993). The San Andreas fault system is loaded by some strain and a certain rate, which occurs due to the plate tectonics. Some p arts of the fault plain are creeping continuously (local stress is small), some rem ain locked and are released during an earthquake only. The vertical dimension of the region 3 2 5 w here this locking is possible is about 5 -j- 20 km—the size of seismogenic zone. T he n atu ral fluctuations in local rigidity, geom etry of the plate boundary and some other reasons may cause uneven distribution of stresses along the fault. Especially high stress consecrations areas may be associated w ith “asperities.” D uring an earthquake w ith small energy release (M< 5), only one of such as perities is ruptured, causing a slip over somewhat larger area, surrounding this broken hardening. If the stress, previously supported by th e broken asperity was too high, the redistribution of forces among remaining asperities, together w ith th e dynam ic im pact caused by the failure of the first asperity, can overload some of th e nearby patches. As a result, one or more subevents may be triggered. The to tal area of slip may come out to be substantially larger, than the area of the effective stress support, provided by the asperities. We do not imply th a t this is the only model th a t can explain the observations, however, our studies indicate th a t m odel is w orth considering. In our previous discussion we used the following notation. The length and the w idth of the total fault area were designed by L and W respectively. The “length” and “w idth” of (may be disjoint) area th a t constitute the asperities was designated by V and W '. We now will look at the characteristic size of the asperity p (which may be of the order of W', because it is not probable th a t several asperities can be “fit” into a very limited vertical dim ension of the seismogenic zone). We also will consider the average length of the slip area th a t corresponds to one patch, r, which is of the same order as the distance between the centers of asperities. We now assume th at, in the record of a big earthquake, we can “see” substantially separated asperities as pulses of high frequency energy. If several patches are too close to each other to be resolved, we can count them 3 2 6 as one big patch (one “asperity” ). In this case, we may write: L = m 0r (6.19) and V = mop, (6.20) where mo is the num ber of patches, or asperities, broken during the rupture. We will assum e th a t the num ber of strong motion pulses recorded in high frequency channels (bands # 1 0 , # 1 1 and #1 2 , fo = 7.2 -j- 21 Hz), can be described as m — a m-10/?"'A * + c4-A, (6.21) where the first term gives the number of strong energy pulses, em itted by the source, so th a t m0 = . (6 .22) Eq. (6.21) was fit to the d ata in the same manner as Eq. (6.4). Only the d ata for the three highest frequency channels were considered, and these d a ta were assum ed to represent a homogeneous data set. The results of the regression analysis cam e out to be m = 0.85-10 O2M + 0.014-A for M < 5, (6.23a) m = 0.32-10n M + 0.0067-A for M > 5. (6.236) Eqs. (6.21)-(6.23) give for the number of high frequency strong energy pulses, radiated from the source and recorded “ju st at the epicenter” : m0 = 0.85-10°2 A * for M < 5 (6.24a) m0 = 0.32-10 UM for M > 5 (6.246) 3 2 7 Suppose now th a t each such pulse corresponds to an asperity (or to a com pact group of asperities). Then the length of the slipped area, corresponding to one asperity, r, can be obtained from Eqs. (6.24) and (6.19), where L m\n < L < L max, and Z » min an(i -^max are given by Eqs. (6.7) and (6 .6). T he result is shown in Fig. 6.10 by the inclenly hatched area, where the lines Lmin and Z/max, and the d ata on the source w idth (Chinnery, 1969) are shown for comparison. A nother estim ate of the length r can be m ade by fitting Eq. (6.21) to the d ata when the value of /?m is fixed and is equal to /?£, from L = a L- 1 0 ^ M , (6.25) and considering several narrow intervals on m agnitude scale, one at a tim e. We use two estim ates of L : L m\n— Eq. (6.7) and Lma,x—Eq. (6 .6 ). T he result is: for M = 3 t 4 , (6.26a) and ( m 0 = 0.0108-10-5M m Q = 0.0049-10-5M m0 = 0.0019-10-5M m0 = 0.0009-10-5M m0 = 0.027-1040M m0 = 0.015-1 0 40M m0 = 0.007-10'40M m0 = 0.004-10-4OM for M = 6 -j- 7. The use of Eq. (6.19) gives the estim ate r: rmin < r < rmaz when i min < L < L max and mo is given by Eq (6 .20). These estim ates are plotted in Fig. 6 .10. The estim ates of the asperity size p can be evaluated in a similar m anner. Instead of Eq. (6.25), we use t ' = a 0 -10PoM for M = 4 -r 5, for M — 5 - T - 6 , for M — 6 -f- 7, for M = 3 - T - 4, for M = 4 -r 5, for M — 5 -f- 6 , (6.266) 8- 7 - T3 R _i 3 6H G M 5-j co 4 - 3 - F a u lt le n g th , L - Jovanovich (1974) - Trifunac (1992a) F a u lt w id th , W Chinnery (1969) \ \ \ N ssssss NTVn L ength Eq. (6 Eq. (6 L ength p from Eq. (6.10a) from Eq. (6.10b) from Eq. (6.10c) B from Eqs. (6.10c') and (6.10c ") 0 1 2 log10 L, log10 W , log1 0 p, log1 0 r (km) F ig . 6 .1 0 Estim ates of the asperity size p, and of the length of the area involved in slipping due to the rupture of one asperity, r, versus the m agnitude. The estimates of p and r are obtained through the num ber of high frequency pulses in a strong motion record. The data on the fault w idth (Chinnery, 1969) and the estim ate of the total fault length, L max (Jovanovich, 1974) and Lm;n (Trifunac, 1993), are shown for comparison. 329 and fix 0m = 0O during regression analysis of Eq. (6.21). T hen, using our models of the duration of the high frequency radiation from the source—Eq. (6 .10) and Eq. (6 .20) and also assuming for simplicity V — v-t* , where v = 2.2 m /sec, we get four estim ates of the size of one asperity, p, separately for several ranges of m agnitude. For Eq. (6 .10a) these estim ates are p = .53, 1.1, 2.9 and 5.5 km for M = 3 -j-4 ,4 -r5 , 5 - 7-6 , and 6 -f-7. For Eq. (6.10b) they are respectively p = 1.1, 2.1, 4.3 and 7.4 km. For Eq. (6.10c): p = 2.7, 4.2, 6.5 and 9.7 km. Eqs. (6.10c') and (6.10c") give p = 3.6, 4.8, 6.0 and 9.4 km. Those estim ates are presented in Fig. 6.10 by thick vertical lines of different patterns. Note th a t the dependence of the asperity size p on the m agnitude of earth quake repeats the p attern of the dependence W (M ), where W stand for the fault w idth. Being generally somewhat smaller than W, p practically reaches the size of th e seismogenic zone (the maximum size of this zone is not more th a n 20 km). We now can estim ate the ratio r/p , using Fig. 6 .10. M cC arr (1981) obtained 1 < r /p < 10 and did not notice any dependence on m agnitude. O ur d a ta indi cate some dependence on magnitude: for M = 3.5, r/p « 1.5, and as m agnitude increases, r/p grows, reaching r/p « 4.5 for M = 6.5. The grow th of r /p is in agreem ent w ith the observation th at small earthquake are prim arily crack type event (i.e. r/p « 1), and large shocks can be represented as superposition of several asperity type events, for which r/p should be greater th an 1 (Boatw right, 1988). One should remember, however, th a t our assum ptions involved in this dis cussion, do not necessarily hold. Das and Kostrov (1988) note th a t the duration of the subpulses, related to the asperity fracture is not determ ined by he size of the asperity, b u t rather by the overall physical conditions in the area ajacent to 330 asperity. Also, due to attenuation effects, we may “miss” some pulses th a t are ra diated by the distant parts of the fault area. For this m atter, may be some other definition of the distance to he source should be considered (Trifunac, 1991a), so th a t this effect can be somehow included into the attenuation term 04-A. 3 3 1 7. THE PROPAGATION EFFECTS: A CLOSER LOOK 7.1 Dependence of the Dispersion on the Propagation Path Characteristics We tu rn now to the discussion of the dispersion phenomenon and its influence on the duration of strong earthquake ground motion. We m entioned already in C hapter 3 th a t the prolongation of duration due to dispersion of surface waves (for the long and interm ediate wavelengths) and due to scattering on the random velocity inhomogeneities (for the high frequency waves) can be described by the term = a 4 (/)> A , where A is the epicentral distance and the coefficient a 4( /) can be determ ined from the regression analysis. We also noted th a t this coefficient could be bounded by ®4 < — --------- — , (7.1) ^min ^max where umjn and umax are the minimum and the m aximum wave velocities along the p ath from the source to the station. It was assumed, th a t umax can stand for the approxim ation of the shear wave velocity in rocks at some depth, and um;n is close to the shear wave velocity in sedim entary layer, typically covering those rocks. A sim ple explanation of the dependence of a 4 on frequency was discussed in C hapter 4 (see model Eq. 4.9, Figs. 4.1 and 3.3). We will show here one more argum ent in support of the above assumptions. For each pair source - station we wish to introduce a param eter which will characterize the path from source to the station as “soft” or “h ard.” Using the S m ith’s (1964) m ap, we can estim ate w hat part of its travel tim e a direct (surface) wave spends in sediments. The m ap shows the distribution of basem ent 332 rocks, appearing on the earth ’s surface, but does not always allow the estim ation how shallow or deep the sedim entary layer is. Thus, we may only approxim ately judge if there is a really representative sedim entary layer above the basem ent rock. We choose to define the path from the source to the recording site as “h a rd ” if the straight line from the epicenter to he station, when draw n on the S m ith ’s m ap, goes through the basement rocks, appearing on the surface, 30% or m ore of its length. If the line “spends” less than 30% of its’ length in the rocks, we call the p ath “soft.” By this classification, we assigned 319 of our records to belong to “soft” propagation path, 167 records were classified as “hard,” and 9 out of 496 records could not be assigned to any of these categories because the corresponding source a n d /o r stations have coordinates, not covered by the S im ith’s m ap. We now make the following assumptions. Let umax in Eq. (7.1) represent the shear wave velocity in the basem ent rock at some depth (say, 3 -j-7 km ), and umin be the “effective” shear wave velocity in the “mixed” top layer, which consists of rocks of various rigidities and sizes, embedded into surface sedim ents from below. This stru ctu re is very similar to the “layer-on-a-half-space” model. We assume th a t umax should be the same for both the “soft” and the “hard”path categories, while umin is sm aller for “soft” paths, then for “hard” paths. E stim ating now the corresponding a 4 from Eq. (7.1), we expect a4[“soft”] > a 4[“hard” ] for low and m oderate frequencies. To test the above hypothesis we again consider the em pirical model equation of duration of strong ground m otion, discussed earlier in C hapter 4: f duAh)(f) ) ( a\h\ f ) ) __ __2 \d u r< " > (/) j = {„<” > (/) j +“ *(/)•«+ « 3 (/)-M + « 4 (/)-A , (4.9a) 333 w here the epicentral distance, A, is measured in kilometers and M = m ax{M , M min(/)} , (4.96) where = 2 ^ u i ‘ (4,9c) T he regression analysis of this equation is performed (as described in C hapter 4) three tim es: a) for all available data, b) for cases w ith “soft” propagation paths only, c) for cases w ith “hard” paths only. All the regression coefficients come out sim ilar in all three cases, with the exception of the coefficient 0 4 (f). T he dependence of 0,4 on frequency is shown on Fig. 7.1 for all three cases. As expected, for low and m oderate frequencies, where the nature of the prolongation of the duration is expected to be dispersive, a 4[soft] > a 4[all cases] > a 4[hard]. To be m ore specific qualitatively, we assume umax to be 4 km /sec. Then, for case a) 0.4(f) < 0.21 sec/km , and umin can be estim ated (from Eq. (7.1)) as about 2.2 km /sec. For case b), a 4 < 0.27 sec/km , and um;n « 1.9 km /sec. For case c), a4 < 0.12 sec/km , which gives umjn « 2.7 km/sec. Due to the difference in a 4, longer durations of strong ground m otion can be expected if the path from source to the recording site consists prim arily of sedi m ents in th e upper p art of the earth ’s crust. For “soft” paths, the prolongation due to dispersion is about 2.2 -j- 2.7 sec per each 10 km of epicentral distance at the frequency of about 0.1 -f- 0.6 Hz. For “hard” paths, the prolongation under the sam e conditions can be as little as 0.9 -f- 1.2 sec per 10 km. T he differ ence gradually diminishes with increasing frequency and practically disappears a) all data b ) "soft" path c ) "hard" path .30 .25 .20 .15 .10 .05 .00 f (H z) .30 .30 .25 .25 .20 .20 .15 .15 .10 .10 .05 .05 .00 .00 10 f (Hz) f (Hz) F ig . 7.1 The coefficient a^[f) in Eq. (4.9), plotted versus the central frequency of the channels (solid lines). The coefficient is bounded by its “cr-interval” (dashed lines) and by its estim ated 95% confidence interval (dotted lines). a.) All available d ata included; b | Cases w ith “soft” propagation paths only; c) Cases w ith “hard” propagation paths only. Dispersion is more pronounced for “soft” propagation paths, where the gradient of velocities available is larger, than along “hard” paths. 334 3 3 5 a t f > 5 -r 10 Hz. At such high frequencies, a4 is, probably, not so much the function of vm\n and umax in the sense discussed above, b u t rather describes the propagation path on some smaller scale, indistinguishable by the S m ith’s m ap. As we m entioned earlier, the scattering of the waves on the random velocity stru ctu re of the lithosphere might cause the increase of the duration of strong m otion a t high frequencies (Sato, 1989). 7.2. Num ber o f Strong M otion Time Intervals and their Duration as a Function o f Epicentral D istance We next consider the evolution of the com pact pulse of energy which was em itted by the source and travels in a layered m edium in a form of surface waves subjected to dispersion. For each fixed frequency (or a narrow frequency band), several speeds of propagation (several modes) exist (see C hapter 3, Fig. 3.3). Schematically, an example of the evolution of a pulse, em itted by the source and observed through a narrow frequency band, is presented on Fig. 7.2. Three modes of propagation are shown, the fastest is m arked as pulse 1, the slowest corresponds to pulse 3. At distance d j, the apparent length of th e pulse is relatively long, because different modes of surface waves did not have enough tim e to get separated. Distance ^2 corresponds to some transition range, where the distinct modes are getting separated, so th a t the num ber of pulses increases and the length of every one of them decreases. A t the distance < £ 3 from the source, all three pulses of strong motion can be seen—in the order # 1, # 2 , and # 3 . All of these pulses are sharing the energy th at initially belonged to the only pulse em itted by the source. 336 p u l s e a t th e s o u r c e Time (from the source time) p u l s e 1 p u l s e 2 p u l s e 3 F ig . 7.2 “G radual” separation of different modes of surface waves, as those travel through layered medium. The arrival tim e and the duration of different pulses are shown as a function of distance from the source, where all pulses originate in the form of one strong motion pulse. Pulse 1 corresponds to the fastest (the highest) mode, pulse 3 corresponds to the slowest (the lowest) mode. A t sm all distances, the duration of the pulse seems to increase w ith distance, at m oderate distances this duration decreases due to pulse separation and, for big distances, duration of one pulse remains practically the sam e (if the frequency band is infinitely narrow). 337 As a result of this “gradual" separation, first, for small epicentral distances, the length of one pulse grows with distance from the source. Then, when the distance increases further, the duration of a pulse decreases a bit. For large distances from the source, the duration of one pulse is a slowly increasing function of distance. The schematic distribution of the pulses lengths, for a fixed frequency band, is given in Fig. 7.3. T he pulses of small duration exist due to the procedure of identifying strong m otion intervals in the accelerogram (see Fig. 2.10: the level po. 9 0 can cut an arbitrary small tim e interval on the graph of the derivative of the sm oothed integral of the squared function of m otion). Therefore, the distribution of possible durations of strong m otion intervals in Fig. 7.3 is bounded by zero from below and by some finite function, shown by the solid line, from above. The steep slope of this line at small distances corresponds to a rapid increase of pulse duration. At big distances, where individual modes are separated, the upper bound of the pulses duratons still slightly increases because some dispersion exists inside a frequency band of a finite w idth for each of the separated (in time and space) modes. The distribution of the actual d ata is displayed in Fig. 7.4, where the dura tion of one pulse of strong ground motion is plotted versus epicentral distance, separately for each frequency band. At low frequencies, the duration of a sepa rate pulse is too long, so greater distances are needed to separate different modes. As a result, the effects discussed above, cannot be detected. At m oderate and high frequencies, the agreem ent between the actual distribution of d u rpulse w ith respect to epicentral distance, and the distribution we expect from our simplified physical consideration (Fig. 7.3), is satisfactory. For very high frequencies, there are not enough d ata at long distances to make any definite conclusions. Fig. 7.5, 3 3 8 d i s t a n c e a t w h i c h E picentral d ista n ce p u l s e s a r e g e t t i n g s e p a r a t e d F ig . 7 .3 A schematic representation of the range of d u rpul3e (duration of one strong m otion pulse) as a function of epicentral distance for a narrow frequency band channel. Before th e modes (subject to dispersion), get completely separated, the length of a pulse grows (see Fig. 7.2). During the process of separation, the length of the pulse decreases. At large distances each mode is travelling alone and the length of th e corresponding pulse slightly grows due to dispersion inside the frequency band of a finite w idth. 339 o =.075 Hz 50 40 30 20 O < D 40 20 60 80 100 120 140 160 180 o = .12 Hz 50 40 30 20 10 100 120 140 160 180 20 40 60 80 50 40 30 20 10 0 f0 =.21 Hz i i : JL 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7 .4 a Channels # 1 to # 3 : obsreved duration of separate pulses of strong ground m otion plotted versus epicentral distance. At m oderate and high frequencies, the distribution of pulses’ duration agree qualitatively w ith our expectations summarized in Fig. 7.3. o 0 m o 50 40 30 20 10 0 50 40 f0 =.37 Hz 3 4 0 • ' • *3il'■ ** •H .* V |« L l »2( IS i: ijil .I' ili'i: i i ii J Lu L I 0 20 40 60 80 100 120 140 160 180 f 0 = . 6 3 Hz 'lliji *r- liciti-ill i aj , • i : . 1 B r ' ' ; I _ L I I 100 120 140 160 180 f 0 = 1 .2 Hz :• ! . : ; :: i i i ! • • ■ -i j I i i I i » i ! i 1 i ii 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7 .4 b Channels # 4 to # 6 : obsreved duration of separate pulses of strong ground m otion plotted versus epieentral distance. At m oderate and high frequencies, the distribution of pulses’ duration agree qualitatively w ith our expectations summarized in Fig. 7.3. o 0 ) CO eJ o • r-H -p c d P P Q 50 40 30 2 0 10 0 50 40 30 2 0 10 0 f0 =1.1 Hz 3 4 1 i< i H i t 40 80 100 120 140 160 180 f0 =2.5 Hz .i — i l ' ' ' iililiiliii Ii nini 1 H i; , 40 60 80 100 120 140 160 180 f0 =4.2 Hz 50 — 40 - 30 - 2 0 - 10 - j i i °() 2 0 k Mili i i j Sili :i L : 1 . i 1 i I. _u_ 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7.4c Channels # 7 to # 9 : obsreved duration of separate pulses of strong ground m otion plotted versus epicentral distance. At m oderate and high frequen cies, the distribution of pulses’ duration agree qualitatively w ith our expectations sum m arized in Fig. 7.3. o < D C O 50 40 30 2 0 10 0 f0 = 7.2 Hz 342 I . : i ' . - J . - l 1 i! 'I M i ii I * .I ■ x 0 2 0 40 60 80 100 120 140 160 100 50 40 30 o 20 • rH 10 c d ° ( 3 50 Q 40 30 20 10 uc f0 =13 Hz a M B & iJB -i x X 2 0 40 60 80 100 120 140 160 180 f0 =21 Hz x X 2 0 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7 .4 d Channels $ 1 0 and $12: obsreved duration of separate pulses of strong ground m otion plotted versus epicentral distance. A t m oderate and high frequencies, the distribution of pulses’ duration agree qualitatively w ith our ex pectations sum m arized in Fig. 7.3. 10 343 m 0) 40 2 0 60 80 100 120 140 160 100 P 10 O h f0 =.12 Hz o 0 2 0 40 60 80 100 120 140 160 180 10 f0 =.21 Hz 0 2 0 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7 .5 a Channels # 1 to # 3: observed number of separate pulses of strong m otion, m /, is plotted versus epicentral distance. For the clarity of the figure, each value of m / is assigned some final interval on the vertical axis, and the ordinate of each d ata point, corresponding of this m /, is chosen arbitrarily w ithin this interval. In agreem ent w ith Fig. 7.3, the number of pulses mainly grows during th e first tens of kilometers. 10 8 m 6 CD 4 tn 2 r — H 2 10 Ph 8 < 4 - 1 6 O 4 2 u < u 10 a 8 6 4 2 3 4 4 f0 =.37 Hz ± ± 0 2 0 40 60 80 100 120 140 160 180 :: f0 =,63 Hz '< • = i- : j": : .v ^ 0 20 40 60 80 100 120 140 160 180 f0 = 1.2 Hz '* -i: • “ • 1’ 1 ' • 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7 .5 b Channels # 4 to # 6 : observed 111101661 of separate pulses of strong m otion, m /, is plotted versus epicentral distance. For the clarity of the figure, each value of m / is assigned some final interval on the vertical axis, and the ordinate of each d ata point, corresponding of this mt, is chosen arbitrarily w ithin this interval. In agreem ent w ith Fig. 7.3 the number of pulses m ainly grows during the first tens of kilometers. 10 0 G O 6 C D C O O h 4 2 1 0 8 3 4 5 f0 =1.1 Hz • ’ * ' r! 'i' • : 'u ■ i i » I. f ! ! '• i : : : I : • • • I Vh 6 ° 4 2 u CD 10 & £ 8 0 6 * 4 2 Iiw w .-M .il : > • ■ if- 1 1 0 20 40 60 80 100 1 1 1 1 120 140 160 180 - f0 = 2.5 Hz - • ■ : ; ■ ■ ! '! ir= 1 : i : • • ■ ' - ;■ : • • ! : • I : , ' . ; • • : • ! • . * i ; . • • • • i: • * : ! ■, ! , f \ * • 1 1 • 0 20 40 60 80 100 120 140 160 180 - f0 =4.2 Hz — ‘ . 1 - - i V:i. :l\: ' . i : = : ; I . ■ : ’ . . ,i : - i '= • - ^ * ■ i i : : ■ : * . * * :'i, • i i ; i i 0 20 40 60 80 100 120 140 160 180 Epicentral distance (km) F ig . 7.5 c Channels # 7 to # 9 : observed number of separate pulses of strong m otion, m /, is plotted versus epicentral distance. For the clarity of th e figure, each value of m / is assigned some final interval on the vertical axis, and the ordinate of each d ata point, corresponding of this m /, is chosen arbitrarily w ithin this interval. In agreem ent with Fig. 7.3 the number of pulses m ainly grows during the first tens of kilometers. cn cn 10 8 6 4 2 346 f0 =7.2 IIz X 0 20 40 60 80 100 120 140 160 180 P h 10 8 % - i < U f 0 =13 Hz x X 0 20 40 60 80 100 120 140 160 180 10 fn =21 Hz 0 20 40 100 120 140 160 180 60 80 Epicentral distance (km) F ig . 7 .5 d Channels # 1 0 to #12: observed num ber of separate pulses of strong m otion, m /, is plotted versus epicentral distance. For the clarity of th e figure, each value of m / is assigned some final interval on the vertical axis, and the ordinate of each d a ta point, corresponding of this m /, is chosen arbitrarily w ithin this interval. In agreem ent w ith Fig. 7.3 the number of pulses m ainly grows during th e first tens of kilometers. 3 4 7 which shows the distribution of the observed num ber of pulses versus the epicen tra l distance, also supports our assum ptions about the nature of the dependence m f { A ). To get some quantitative ideas, we turn back to the models d u rpulse( /) - 6i ( / ) + 62(/)-M + 64(/)-A , (6.17) and ™(f) = ci ( /) + c2(/)-M + c4(/)-A , (6.18) which scale the duration of separate tim e intervals, considered as strong m otion, and the num ber of those intervals per one component of acceleration, velocity or displacem ent. The results of the linear least square fits are presented in Table 6.3 and Fig. 7.6a-c for Eq. (6.17) and in Table 6.4 and Fig. 7.6d-f for Eq. (6.18). We are now prim arily concerned w ith the behavior of the coefficients & 4 (/)> responsible for the duration of one pulse as a function of the epicentral distance, and c4( /) , which scales the number of strong m otion intervals w ith respect to epicentral distance. Three cases are shown in Fig. 7.6: a) and d) results of the regression analysis when all available d ata points are considered, b) and e) where the d ata w ith epicentral distance A < 30 km only are included, and c) and f) when d ata w ith A > 30 km only are taken into account. T he top row of graphs in Fig. 7.6 clearly shows th a t the duration of one pulse of strong m otion grows fast in the very beginning of the propagation p ath (couple of dozen kilometers) and practically does not depend on epicentral distance, A , when A > 30 km. The num ber of pulses (Fig. 7.6, bottom row) also grows prim arily in the first tens of kilometers of the propagation path bu t still continues to grow later on. Comparing Figs. 7.1a and 7.6a we can notice th a t the absolute ' ‘ ’ I nl < .06 .04 _ C 1 f (Hz) 10 d) all A [A ] • ’ / .04 :ll r .02 V' 1 : : | o o -.0 2 .06 b) A<30 km ' A '[ A ] c) A»30 km [A] / 1 f (Hz) 10 .02 -.0 2 L C .04 .02 .00 e) A<30 km 06r c 4 [A] 1 f (Hz) 10 f) A>30 km [A] .04 - .02 - 1 ---1 — 1 I 1 I III d 'QQ { Z j___ |__i I t I I 1 1 1 I I t I ■ I 1 ■ I F ig . 7.6 The frequency dependent coefficients scaling durpulse (64( /) , top row) and m (c4( /) , bottom row) in Eq. (6.17) (duration of one strong m otion pulse) and Eq. (6.18) (number of strong motion pulses in a compo nent), w ith respect to epicentral distance are plotted versus central frequency of the channels (solid lines). The coefficients are bounded by their “< 7-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines): a) and d) all available d ata points included; b) and e) data points w ith A < 30 km only; c) and f) data points w ith A > 30 km only. D uration of one pulse and the num ber of pulses grow mainly when A < 30 km. 348 349 value of the coefficient 64(7 ), which scales the prolongation of duration of one pulse of strong m otion is approxim ately 10 times smaller th an the value of 0 4 (7 ), which describes th e sam e effect for the total duration. Hence, the dependence of th e duration of one strong motion pulse on epicentral distance is, in average, m uch weaker, th a n the dependence of the total duration on this distance. To sum m arize our conclusions, we note th a t in the beginning of th e propa gation p ath the duration of strong ground motion grows due to increase of the length of pulse em itted from the source. This pulse later separates into differ ent modes. The greater the distance is (from the source), the more m odes get separated and can be distinguished. So, at greater distances, the total duration grows prim arily due to the increase of the number of pulses (modes, separated in tim e and space)—one more pulse every 50 km (for frequencies 0.3 -f 1.5 Hz). 7.3 Geom etry of a Sedimentary Basin and Separate Pulses of Strong Ground M otion O ur next goal is to analyze the increase in the duration of strong ground m otion as a result of the geom etry as a sedim entary basin. We earlier consid ered Eq. (4.12), which scales the total duration w ith respect to, am ong other param eters, depth of sedim ents under the recording site, h, and the distance to th e reflecting rocks, R. As it was shown (see Fig. 4.5), the duration of strong ground m otion increases w ith R and h, if R and h are not too big. A sugges tion was m ade, th a t these prolongations can be explained by m ultiple reflections of waves in vertical and in horizontal directions. To test this assum ption, four 350 sim ple regression equations can be considered: d urpulse( /) = bi{f) + be(f)-R, (7.2) d u r ^ lae( f ) = b 1( f)+ b 5(f).h, (7.3) and m{f) = c i( / ) + ce (/)-fl, (7.4) m{f) = ci ( / ) + c5 [f)'h. (7.5) Here, as before, d u rpuIse( /) designate frequency dependent duration of one tim e interval, giving contribution to the total duration of strong m otion and m(f) is the num ber of these intervals in one component of acceleration, velocity or displacem ent. For simplicity, horizontal and vertical m otions are considered to gether. Frequency dependent coefficients, c ,(/) and b{(f), i — 5,6, scale the influence of the geom etry of the sedim entary basin on m (/) and on d u rpulse(/). T he dependence on R and h was studied separately to make the inversion prob lem more stable. In Eqs. (7.2) and (7.4), only the data points w ith R < 35 km (the increasing p art of dur(R), see Fig. 4.5) were considered. The Eqs. (7.3) and (7.5) were fit to the d ata with h < 3.5 km only. The plots of the frequency dependent coefficients th a t resulted from the regression analysis, are presented in Fig. 7.7. We first consider the dependence of durpuIse(/) and m(f) on the horizontal characteristic dimension of the sedim entary basin, R —the left column of graphs in Fig. 7.7. We recall, th a t the total duration of strong m otion increases with R, for R < 35 km. It is seen from Figs. 7.7a,c th a t at low frequencies ( / < 2 Hz) this increase happens due to the increase of the number of separate pulses 1.2 .10 7 V / / .05 .00 J. 10 f (Hz) -.0 5 - .2 c ) .4 .06 .04 .2 .02 .1 .00 f (Hz) f (Hz) - .0 2 .0 F ig . 7.7 The frequency dependent coefficients scaling durpulse(/) (top) and m (f ) (bottom ) in term s of R and h, for R < 35 km and h < 3.5 km (Eq. (7.2)-(7.5)) (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines), a) Eq. (7.2); b) Eq. (7.3); c) Eq. (7.4); d) Eq. (7.5). As a result of the specific geometry of a sedim entary basin, at low frequencies, the increase of total duration (if it happens) occurs due to the increase of the number of strong motion pulses. For high frequencies, the length of each pulse also increases. 351 352 of strong m otion, arriving to the station. For high frequencies, th e length of each pulse also increases. There are at least two mechanisms th a t could be responsible for these results 1) Consider the scheme in Fig. 7.8a. T he rocks, surrounding the station, reflect some additional energy in the direction of the station, and this energy, travels through the layered m edium in the form of the surface waves. These waves come to the station later than the direct waves, and, thus, spend more tim e travelling greater distance. This results in the additional dispersion, the mechanism of which is the same as for dispersion due to increase in epicentral distance, A , when A is large. Indeed, one can notice, th a t th e increase of th e to tal duration of strong ground motion happens due to the increase in the num ber of pulses rather than due to the prolongation of each pulse, for both large distances A (see Figs. 7.6c and 7.6f) and small R (see Figs. 7.7a and 7.7c). T he fact th a t “i? dispersion” is alike “A dispersion for large A ” can probably be explained by noting th a t, in average, the total distance travelled by th e reflected wave, is about A + R, and it sums up to be more than 30 km, so th a t the m echanism of prolongation of duration typical for long distances, is prevailing. 2) Consider now the second scheme—Fig. 7.8b. We assumed, th a t the irregular rocks, surrounding the station, reflect some waves towards it, which otherw ise would never come to the recording site. Thus, in addition to the “dispersive” natu re of the increase in the number of pulses of strong m otion, (see above) we would like to be able to detect this “reflective” nature also. We expect the coefficient ce(f) in Eq. (7.4), scaling the influence of R on the num ber of pulses, to be bigger, than 0 4 (f) in Eq. (6.18), which scales the influence of epicentral distance A on the same number. This “extra” increase should come from this “reflective” nature. As we saw before (Figs. 7.6a or 7.6c), one additional pulse 3 5 3 a ) indirect dispersed arrival SOURCE STATION direct dispersed arrival original pulse ROCK reflected pulse SOURCE STATION sum o f direct and reflected pulses direct pulse original pulse F ig . 7 .8 Two mechanisms for increasing the number of pulses, m: a) via th e in crease of the distance travelled— “dispersion,” b) by m ultiplication of the num ber of pulses, m , by the factor of 2 , because of reflection from a rock— “reflection.” 3 5 4 appears every 50 km of epicentral distance A , for f « 0.3-M.5 Hz. From Fig. 7.7c, each 20 km of R gives rise to an additional strong m otion pulse. We can assume th a t each 1 km of R really means <2 additional km of the distance the reflected wave has to travel in the path source-rock-station. In this case, we get <40 km per each pulse. Therefore, the number of strong motion tim e intervals grows slightly faster than in the case of pure “dispersion,” or propagation, case, and this additional increase of m (/) is nothing more bu t the “count,” coming from th e fact th a t the reflecting rock “m ultiplied” the num ber of pulses by two. Due to the overlapping and attenuation, we do not see all of those additional pulses, and the real “m ultiplication” is by a constant smaller, than 2 . At high frequencies, the number of pulses grows slower than the duration of one pulse due to overlapping of a great number of propagating pulses into ju st a few relatively long strong m otion time intervals. The dependence of durpulse and m on the depth of sedim ents, h, is very sim ilar to the dependence on R. The corresponding plots are shown on Figs. 7.7b and 7.7d. 7.4. Soil and Geological Site Conditions and Separate Pulses of Strong Ground M otion T he last question we would like to discuss in this C hapter deals w ith the dependence of the duration of one pulse of strong ground m otion, d u rpulse, and the num ber of such pulses per one com ponent, m, on geological and local soil site conditions. 16 1.0 1.0 f (HZ) -.2 -.2 - .4 - .4 c ) d) 6 m ' 1 • l f e V : f (Hz) -.2 - .2 F ig . 7.9 The frequency dependent coefficients scaling durpulse(/) and m[f) in term s of s and sl , Eq. (7.6)- (7.9) (solid lines). The coefficients are bounded by their “a-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines), a) Eq. (7.6); b) Eq. (7.7); c) Eq. (7.8); d) Eq. (7.9). As a result of a specific geological and soil site conditions, the prolongation of the total duration of strong ground motion (when it takes place) happens mainly via an increase of the number of strong motion pulses. 355 3 5 6 We consider here the following correlations, where, for simplicity, s, the geological conditions indicator, and s^, the soil type indicator are considered as quantitative variables: durpulse(/) = brif) + 615( / H 2 - s), (7.6) durpu,se( /) = 61( / ) + 616(/)-s i , (7.7) and m(f) = ci{f) + ci5( / ) - ( 2 - s), (7.8) m (/) = cx(/) + Ci6( /) - s L. (7.9) T he coefficients &,-(/) and C j(/), i ~ 15,16, show the change of durpulae and of m respectively due to the presence of sediments or soil deposits under the recording site. In all cases, the rock sites (geological rock and local “rock” soil) are chosen as the reference. T he results of the linear regression fit of the d ata to Eqs. (7.5)-(7.8) are presented in Fig. 7.9. Recall the increase of the total duration of strong ground m otion (for some frequencies) due to the presence of sediments and soil deposits in Eq. (7.8), Fig. 4.12. We see now th at this prolongation was caused prim arily by the increase in the number of time intervals, giving contribution to the total duration. Similar to the case of Eq. (4.17), the influence of the geological site conditions on the num ber of strong motion pulses (coefficient Cis(f)) is more pronounced at lower frequencies, than the influence of the type of soil on the same num ber (coefficient cie(f)). This can be explained, as in the case of Eq. (4.17), by th e different characteristic depths of sedimentary and of soil deposits. 357 8. SCATTERING AND ATTENUATION EFFECTS A N D THE NATURE OF STRONG GROUND M OTION 8.1 The Distribution of the Strong M otion Pulses along the Record and their Relative “Strength” In this session, we try to identify some statistical features of the distribution of strong m otion tim e intervals along the time axis in the recorded acceleration, velocity and displacement. We also address the question of the relative “stren g th ” of the separate pulses of strong motion. We choose the arrival of the S wave as a reference time. Accelerographs of analog type, which recorded the m ajority of the d a ta from our d a ta base, have a vertical starter. As a result, in many cases, the accelerome ter is triggered by the P- wave, which usually has a strong high frequency vertical com ponent. In some cases, the transducer remains inoperational until S-wave arrival, or even longer, especially in the case of a weak or distant earthquake. M any interm ediate cases are possible, i.e. triggering reflected waves, arriving between the P and S waves. Due to the lack of the inform ation about the first arrival, the correct identification of phases on strong motion accelerogram is not always possible. The task is simplified when the same earthquake is recorded by m any instrum ents, and comparison of wave forms on different stations can give additional inform ation. Fig. 8.1 shows the correlation between the tim e of S'-wave arrival as m easured from the beginning of the recorded accelerogram, t a, and the distance to the source, d. The distance is taken to be the epicentral, or hypocentral. T he latter 358 2 0 - All points O 15f < D m lOh < n • * • M . *• • • •• • _» • • • _ » • v il ■ ar • Y'-s, " • • ’ ' j • •* * • * ■ • • • . „ . •• • “ r O Jr 50 100 150 200 D i s t a n c e ( k m ) F ig . 8 .1 a The correlation between the tim e of the S -wave arrival, as m easured from the beginning of the recorded accelerogram, and the distance to the source (epicentral, or hypocentral, where available), with all d ata points included. 359 20 _ s = 2 b a s e m e n t r o c k - S L = ° l o c a l " r o c k " 15 - - • 10 - - 5 •• . 0 .y \"f * i i i i .■ ..V 1 i , i i o 20 _ S = 1 i n t e r m i d i a t e - S L = 1 s t i f f s o il C D 15 _ • _ W 10 - - « m 5 0 - * • • • ■ * * • ••• • i i i i _ • i • . ;l .. 1 i i i 20 s = 0 s e d im e n t s y — i II C O 1 d e e p s o il 15 • 10 • • • 5 °c ■ • \ • • ■ i i . i * • • * ' / v • • • ■ 1 i i i 50 100 150 200 0 50 100 150 200 D i s t a n c e ( k m ) F ig . 8 .1 b The correlation between the time of the S-wave arrival, as m easured from the beginning of the recorded accelerogram, and the distance to the source (epicentral, or hypocentral, where available) for different geological and local soil site conditions. 3 6 0 C f i +■> 10 5 0 20 15 10 5 0 M<5.5 _ h<0.3 km - — • M ■ ,• ■ • , Jit'."' V , 1 . 1 I 1 _ 5.5<M<6.5 _ 0.3<h<3.5 k m • • . • ” . * i “ ■ v ■ • . » r • • vr-,r. . • . ■ . - p . - . m ! • r ■ r • Jf • • , * ' 1 ______ 1 1 1 _ M>6.5 _ h>3.5 k m • - - ‘ • • • * ■ • • , • / • • • • ‘ .i i i i »• * • • cT.-i . * 1. ' 1 . _.l 1 0 50 100 150 200 0 50 100 150 200 D i s t a n c e ( k m ) F ig . 8.1c The correlation between the tim e of the 5-wave arrival, as m easured from th e beginning of the recorded accelerogram, and the distance to th e source (epicentral, or hypocentral, where available) for different m agnitudes and depth of sedim ents under the recording site. 361 was chosen if th e hypocentral depth for the earthquake was available. Fig. 8.1a gives the distribution t a = ta(d) for all data points. The dots positions right on the x-axis correspond to the triggering of the instrum ent by the 5 wave. The upper bound of the region, covered by the dots, corresponds to the triggering by the P wave. This upper bound can be approxim ated by a straight line, tilted by an angle r/j w ith respect to the x-axis. This angle is related to the P- and 5-wave velocities, a p and (3a respectively, in the following manner: For the Poisson ratio v = 0.25, the above equation leads to In soft m edium (where /3a is small), angle tp is big, and, conversely, sm all ip corresponds to stiff “fast” medium. Comparing tp for different site classifications in Fig. 8.1b, one may notice sm aller ip on basement rock sites, than on sedim ents. T he difference between various soil types (right hand side of Fig. 8.1b) is not seen well, may be due to the small thickness of the soil layers, so th a t effective velocities of P- and 5-waves are not much altered by the presence of soil). In Fig. 8.1c, th e com parison of ta = ta(d) for different m agnitudes is shown. It is known th a t big earthquakes tend to take place at greater depths (see Fig. E.3), so the waves, em itted by the source spend more time in “fast” deep layers of the crust. Indeed, the angle ip in Fig. 8.1c is greater for small M , and this corresponds to lower (in average) velocities of propagating waves. For completeness, Fig. 8.1c also shows the travel tim e dependence on the depth of sediments under the station. The p attern of d a ta points for various h agrees w ith the pattern for various geological site conditions. 362 Having verified th a t our determ ination of the 5-wave arrival tim e does not contradict the basic physical considerations, we can use this tim e as a reference point on the tim e axis in all subsequent analyses. We next consider the distribution of strong motion pulses along the tim e axis. T he duration of one strong m otion pulse, cfurpulse, as a function of tim e when it was recorded by the station (the zero-time corresponds to the 5-wave arrival) is shown on Fig. 8.2 for all frequency bands, separately for horizontal and for vertical com ponents and for different ranges of magnitude. Each strong m otion pulse, which started at tim e and ended at tim e w ith respect to the 5 -wave arrival tim e, is shown as a dot, w ith its ordinate equal to du rpulse = t and its abscissa corresponding to the middle of the time interval when it was observed, (t - M ^ ) / 2. Three distinct groups of pulses can be identified on Fig. 8.2 (m oderate and high frequencies). The first group is seen at negative tim es (hence, before the 5 -wave arrival) and can probably be associated w ith the energy brought by the P-waves. The second group, w ith t > 0 are 5-waves mixed w ith the fastest modes of the surface waves. The third group occupies the broadest range of arrival times, is the m ost irregular one and could be associated w ith the energy brought by the surface waves, direct and reflected. As it can be seen from the figures, the proportion of the P -wave arrivals, being counted as strong m otion, grows w ith frequency and is also greater for higher m agnitudes. T he latter fact may be related to the late triggering of the transducers by weak earthquakes, so th a t P wave is not recorded. The growth of the participation of the P-wave in the strong m otion can be explained by the well known property of the P-wave spectra: they have substantial am plitudes in higher (com pared to 5- and surface waves) frequencies. Notice also the difference between the horizontal 40 20 0 S 40 m ^ 20 a; w 3 o a g 40 TJ 20 0 40 20 0 - M = 3 ^ 5 i , i , i , i f 0 =.075 Hz H o rizo n ta l 1 , 1 , . M = 3 - § - 5 i , i , i , f 0 =.075 Hz Vertical - M = 5 ^ 6 i , i , i , i i , i , . M = 5 + 6 i . i . i , . M = 6 * 6 . 5 • ■ • • _ « • • * .*% _ # 1 , 1 . 1 . 1 , • i , i - M = 6 + 6 . 5 • • » I . I . I , - M > 6 . 5 .. ; _ _ 1 , 1 > I , 1 i , i , . M > 6 . 5 .. i , j.. i , i . ., ... 1 .. — L _. 1 -20 0 20 40 60 80 -20 0 20 T i m e ( s e c ) 40 60 80 F ig . 8 .2 a Channel #1: the duration of a strong motion pulse, which started at the time and finished at the tim e t (2), du rpulse = t ^ — t^l\ is plotted versus the middle of the tim e interval when it was observed, 0.5-(tM + 1^2)). T he time is m easured from the moment of the 5-wave arrival. This plot allows one to see the relative significance of P -, 5 - and surface waves as the contributors to the duration of the strong ground motion. co 0 > 40 20 0 O 4 0 C O w 20 < u 0 5 3 o p < £ 40 'd 20 0 40 20 0 - M=3*5 f0 =.12 Hz H o rizo n ta l i , i , i , i , i , i - M=3+5 f 0 -.1 2 Hz Vertical - M = 5 - r 6 1 i 1 i 1 i 1 i 1 i 1 i - M = 5 - t - 6 1 . 1 . 1 , I , 1 , 1 - M = 6 - r 6 . 5 . • • : . •• I \ . . I V : i ' . *1. . ; I . M = 6 - t- 6 . 5 , * ’ - • & / • • • . * * '/' * r .< ■ •« . . i i i i i * • " , • i • , • i i - M > 6 . 5 • V-: *. • ' . > r •••• •.• • -r#1 ’ •* • — m • v * * . • * * • • .. i . , *•_.! .* -i , * i i . M > 6 . 5 • • ; . 1 . . . 1 . L , 1 , t * , ] -2 0 20 40 60 80 T i m e -2 0 0 20 ( s e e ) 40 60 80 F ig . 8 .2 b Channel # 2 : the duration of a strong motion pulse, which started at the tim e and finished at the tim e t (2), dur^nlae = tW - t ^ , is plotted versus the middle of the tim e interval when it was observed, The tim e is measured from the moment of the S -wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. w 0 5 o < L > m < D tn 2 A 2 'd 40 20 0 40 20 0 40 20 0 40 20 0 • M=3+5 f0 =.21 Hz H o rizo n ta l 1 , 1 , 1 , 1 , 1 , 1 M=3*5 f0 =.21 Hz Vertical 1 1 1 1 I 1 1 1 1 t 1 1 - M=5-6 - .A .1 * ‘* : • ■ * > * * •» , 1 i ’I * . * i • • • * 1 • i « * .l .• i* '1 i 1 i - M=5-6 • • •• * • • i . r , i * i , i , i - M=6*6.5 1 , W • ,7 • < : - M=6t -6.5 • • Jjf'k' v i , -*}/ 'fto r a y & i* ’ '. * .* * r • » .• i - M>6.5 i n? -in -tft. in ', i: . i - M>6.5 * v 4 •* V ‘ A t * i f 9' '*'? f’ N * » . ' i , * , .A» i : l- , i • , i -2 0 20 40 60 80 T i m e -2 0 0 20 ( s e c ) 40 60 80 F ig . 8.2c Channel # 3 : the duration of a strong motion pulse, which started at the tim e and finished at the tim e , durPulse = tW - t W , is plotted versus the middle of the tim e interval when it was observed, 0.5- ( f ^ + t ^ ) . The time is m easured from the moment of the S-wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. 40 20 0 S 40 C f i ^ 20 a > 0 3 3 o Q < § 40 'd 20 0 40 20 0 - M =3-s-5 f 0 = . 3 7 Hz H o r iz o n ta l 1 i 1 j. 1 i 1 i 1 i 1 i M=3-h5 f 0 =.37 Hz Vertical - M =5-j-6 — ^ i t ■ M =5-j-6 1 , •if • • * . . 1. * 1 , 1 , 1 . M=6*6.5 — • • • ’ * * * * • • - M=6-^6.5 ; •' * . 1 "I. . * . J i 1 . M >6.5 • * • - M >6.5 i r . \ i r -20 20 40 60 80 T i m e -2 0 0 20 ( s e c ) 40 60 80 F ig . 8 .2 d Channel # 4 : the duration of a strong m otion pulse, which started at the time and finished at the tim e t^2\ durpulse = t is plotted versus the middle of the tim e interval when it was observed, 0.5-(t^^ + f (2)). The time is m easured from the moment of the S-wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. w C i 40 20 0 8 40 m 20 « m 3 o p . £ 40 nd 20 0 40 20 0 M=3-h5 f 0 = .6 3 Hz H o rizo n ta l _ \ j f v i i n , i , i - M=3-5 f 0 = .6 3 Hz Vertical • i , - r \‘ * , i • , * i , i , i - M=5-s-6 , . . , , , - M=5-r-6 I , VosO• * { > * l I - M = 6 - j - 6 . 5 * * * * * . M = 6 - r 6 . 5 . M > 6 . 5 _ 1 •. ./• I.‘ - M > 6 . 5 • i _ , i . t. . , -2 0 0 20 40 60 80 T i m e -2 0 0 20 ( s e c ) 40 60 80 F ig . 8.2e Channel # 5 : the duration of a strong motion pulse, which started at the tim e and finished at the time t ^ , durPulse = t™ - i W , is plotted versus the middle of the tim e interval when it was observed, 0.5-(ft1) + tW ) . The tim e is measured from the mom ent of the S -wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. co 0 5 40 20 0 S 40 G O ^ 20 4 ) tn 3 o a £ 40 X) 20 0 40 20 0 M=3-5 f„ = l . l Hz H o rizo n ta l ~ ~ J i . # .< !< * .-i i i M =3-5 f o = 1.1 Hz Vertical - M=5^6 • * i iffr i * if i v } * i , v , ,• i i - M=5+6 r . I , * . • ! ’ . l . * , i . , i . M=6-t -6.5 ■ , . . . ........ - M=6-r6.5 - M>6.5 % • _ » (1 * . M>6.5 -2 0 0 20 40 60 80 T i m e - 2 0 0 20 ( s e c ) 40 60 80 F ig . 8 .2 f Channel # 6 : the duration of a strong motion pulse, which started at the tim e and finished at the tim e tW , dur?ulse = tW - , is plotted versus the middle of the tim e interval when it was observed, 0.5-(f I1) + t ^ ) . The time is measured from the moment of the 5 -wave arrival. This plot allows one to see the relative significance of P -, S- and surface waves as the contributors to the duration of the strong ground motion. 368 40 20 0 S 40 0 2 ^ 20 C J m 3 o & . £ 40 Tl 20 0 40 20 0 - M=3+5 f 0 =1.7 Hz H o rizo n ta l • * 1 . i * a * 1 1 1 - M=3+5 f 0 =1.7 Hz Vertical 0 • i . r • , .. i i i - M =5-5-6 i , « . i. i i - M=5-6 i , m & im&h\'.j j .. i i, i. i . M =6-j-6.5 , i . k. l , . - M=6-j-6.5 - M >6.5 i , ... i i . M >6.5 • ----1 ----1 A. ► i i -2 0 0 20 40 60 80 T i m e o -2 0 ( s e c ) 20 40 60 80 F ig . 8.2 g Channel # 7 : the duration of a strong m otion pulse, which started at the tim e and finished at the time d u rpulse = t ^ — iW , is plotted versus the middle of the tim e interval when it was observed, 0.5-(t^^ + t ^ ) . The time is measured from the moment of the 5-wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. 369 40 20 0 % 40 73 ^ 20 ( U m 3 o ft § 40 20 0 40 20 0 - M=3-s-5 f Q = 2 .5 Hz H o r i z o n t a l # / 1 i J * « • • « i 1 i 1 i I i ^ M=3-r5 f Q = 2 .5 Hz V e rtic a l t - M=5+6 .... L - - i — * ' J ■ 1 I I - M=5-r6 l 1 i* . ... . .i 1 1 - M=6-6.5 t *• 1 ■ . ' « > J . 1 . M=6-r6.5 J M - - M>6.5 ... l . i. ihi» 1 1 i ...i . M>6.5 i . -i. i i -2 0 20 4 0 60 80 T i m e - 2 0 0 20 ( s e c ) 40 60 80 F ig . 8.2 h Channel # 8 : the duration of a strong motion pulse, which started at the time and finished at the tim e t& , durPulse = i<2) - 1W , is plotted versus the middle of the tim e interval when it was observed, + t ( 2)). The time is measured from the moment of the 5-wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. w 40 20 0 S 4 0 m w 20 < u cn 3 o ft £ 40 20 0 40 20 0 - M=3-s-5 f 0 = 4 .2 Hz H o rizo n ta l i . i , . M = 3 4 - 5 f 0 = 4 .2 Hz Vertical i . i , - M = 5 - r 6 - M = 5 + 6 1 1 | | i i i , l , • i i - M = 6 - j - 6 . 5 - M=6t-6.5 1 » it .1 - ». ( . M > 6 . 5 . M > 6 . 5 ..1 1 • I T lV W i ‘ .1 1 1 . 1 , . * 1 i i. >i 1 i. 1 i I i -2 0 0 20 40 60 80 T i m e -20 o 20 ( s e c ) 40 60 80 F ig . 8.2i Channel # 9 : the duration of a strong motion pulse, which started at the tim e and finished at the tim e t(2\ durpulse = t ^ — 1 is plotted versus the middle of the tim e interval when it was observed, + t(2l). The tim e is m easured from the moment of th e 5-wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. w 40 20 0 S 40 G O ^ 20 I D tn 3 o a £ 40 'd 20 0 40 20 0 1 1 1 1 1 £ I I C O • I * c n t - f 0 = 7.2 Hz H o rizo n ta l i . i . - M =3-s-5 f 0 =7.2 Hz Vertical i , i , i , - M =5-s-6 i i .i i i , i . . M=5h -6 i .1*, i . i , i i | i | i £ I I C • 05 - i * i , i , - M=6^-6.5 i . i . i , . M >6.5 _ • m" * * , i , i , - M >6.5 _ i . , i . i . i , -2 0 20 40 60 T i 80 m e -20 o ( s e c ) 20 40 60 80 F ig . 8.2j Channel #10: the duration of a strong motion pulse, which started at the tim e and finished at the time durPulse = - fW , is plotted versus the middle of the time interval when it was observed, O .S -^ 1) + 1(2)). The time is measured from the moment of the 5-wave arrival. This plot allows one to see the relative significance of P -, 5 - and surface waves as the contributors to the duration of the strong ground motion. 372 40 20 0 S 40 U l ^ 20 m 3 o a g 40 20 0 40 20 0 - M=3-r5 f 0 =13 Hz H o rizo n ta l i . i , - M=3-5 f 0 =13 Hz V ertical 1 . 1 , 1 , - M=5*6 1 1 > ll r 0^ a » . i . 1 i i , i - M = 5-r6 1 , 1 , 1 , - M=6-r6.5 i • i i , i , - M=6+6.5 i , . i 1 . 1 , 1 , . M >6.5 1 . i .1 * * . r i i , i i - M >6.5 — j — .. i 1 , 1 , 1 -2 0 20 40 60 80 T i m e -2 0 0 20 ( s e c ) 40 60 80 F ig . 8.2k Channel # 1 1 : the duration of a strong motion pulse, which started at the tim e and finished at the tim e it2), durpulse = t t2) —i t 1), is plotted versus the middle of the tim e interval when it was observed, ( ^ • ( it1) + it2)). The tim e is measured from the moment of the S'-wave arrival. This plot allows one to see the relative significance of P-, S- and surface waves as the contributors to the duration of the strong ground motion. G O O GO f0 =21 Hz H o rizo n ta l f0 =21 Hz Vertical 40 20 M=5-?-6 S 40 g o ^ 20 c o 3 o a g 40 20 M>6.5 M>6.5 40 20 -2 0 20 40 60 80 -2 0 20 40 60 80 T i m e ( s e c ) F ig . 8.21 Channel #12: the duration of a strong motion pulse, which started at the tim e t ^ and finished at the tim e t^2\ du rpulse = t ^ is plotted versus the middle of the tim e interval when it was observed, + f(2)). The time is measured from the moment of the 5-wave arrival. This plot allows one to see the relative significance of P -, 5- and surface waves as the contributors to the duration of the strong ground motion. 374 375 and the vertical com ponents of the signal. The P -wave is more pronounced on the vertical com ponent. The surface waves do not distinguish between the vertical and the horizontal motions. We next discuss the statistical properties of the quantity th a t m easures the relative strength of the different pulses which give contribution to the total du ration of the strong ground motion. We will measure the “weakness” of a pulse by com paring it w ith the strongest pulse on the particular com ponent of accel eration, velocity or displacement. For th a t m atter, we consider the ratio ,-th , \ ®rms (strongest pulse) PU,Se) “ " a m , (gfc pulse) ’ I8-1” ' where the root mean square am plitude of an i— strong m otion interval, of the signal f ( t ) is defined as: ®rm8 (*') — (2 ) T I /2 f t[i) f 2{r)dr ' t W - t W (8.1b) All notation here coincides w ith the one from the definition of duration, Eq. (2.2). Fig. 8.3 presents the distribution of data points in the plane “q versus the m iddle point of the strong motion tim e interval,” where the 5-wave arrival time is taken as a reference. The d ata are grouped w ith respect to the m agnitude of the earthquake and the distance to the source. There is no distinction m ade between the horizontal and the vertical motion. The schematic graph in Fig. 8.4 sum m arizes the m ajor trends th a t can be seen in Fig. 8.3. Fig. 8.5 shows the second and the third coefficients in the regression equation: q(f) ~ consti(f) + const2(f)-M + const4(f)-A, (8.2) A=(R20 km M =3-i-5 A=20+40 km M =3-s-5 A=40+70 km M=3+5 Channel 1 f0 =.075 Hz 6 q 4 2 A=0+20 km M =5-i-6 A=20-f-40 km M=5+6 A=40+70 km M =5-i-6 6 q 4 2 A=0-5-20 km M=6-6.5 A=20-r40 km M =6-i-6.5 A=40-70 km M =6-s-6.5 6 q 4 2 A=0-^20 km M >6.5 A=20h-40 km M >6.5 A=40-s-70 km M >6.5 A>70 km M >6.5 6 q 4 2 ) 80 -2 0 0 Time (sec) F ig . 8 .3 a Channel # 1 : the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the 5-wave arrival. 376 A=0-i-20 k m M =3+5 A = 2 0 + 4 0 k m M =3+5 A = 4 0 + 7 0 k m M=3-5-5 Channel 2 f0=.12 Hz 6 q 4 2 A = 0+ 20 k m M=5-i-6 A = 20 + 4 0 k m M=5-^6 A=40-i-70 k m M=5-5-6 6 q 4 2 A=0-j-20 k m M=6-s-6.5 A=20-i-40 k m M=6-j-6.5 A=40-j-70 k m M=6-i-6.5 6 q 4 2 A=0-^20 k m M>6.5 A = 20-f40 k m M>6.5 A =404-70 k m M>6.5 A>70 k m M>6.5 6 q 4 2 -20 0 80 Time (sec) F ig . 8 .3b Channel # 2 : the relative weakness of the strong motion time interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the S -wave arrival. 377 A=0-i-20 k m M =3+5 A=20-5-40 k m M =3^5 A=40-s-70 k m M=3-i-5 C h a n n el 3 f0 =.21 Hz 6 q 4 2 A = 0 + 2 0 k m M = 5 + 6 A =20-i-40 k m M=5-i-6 A=40-i-70 k m M=5-s-6 6 q 4 2 A=0-5-20 k m M=6-=-6.5 A = 2 0 -r4 0 k m A = 4 0 -r7 0 k m M = 6 -r6 .5 6 q 4 2 A=0-=-20 k m M>6.5 A = 20-^40 k m M>6.5 A =40-j-70 k m M >6.5 A>70 k m M>6.5 6 q 4 2 -2 0 0 Time (sec) F ig . 8.3c Channel # 3 : the relative weakness of the strong motion time interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. co o o A =0-i-20 k m M = 3 + 5 1 , L 1 , 1 , 1 , 1 , A = 2 0 + 4 0 k m M =3-i-5 i i i . t . i . i . i , A = 4 0 + 7 0 k m M=3-5-5 Channel 4 ; f0=.37 Hz i , i , i , i , i , i , A = 0-^20 k m M=5-5-6 \ T ' V • . — , • • • 1 , 1 -V 'L , • 1 , 1 , 1 , A = 2 0 + 4 0 k m M = 5 * 6 ' - V . s - ' . . f. . i . • ‘. A = 4 0 + 7 0 k m M =5-:-6 . * ' : i ? ' ‘ i . -i . p '-.- i , i , i , A>70 k m M=5-i-6 A = 0 -r2 0 k m M = 6^-6.5 « % • ?»'*• •' 1 , f . • yui • , 1 , 1 , 1 , A = 2 0 -r4 0 k m M = 6 ^ 6 .5 .* * • * * '• • A =40-s-70 k m M =6-r-6.5 i , '* , r , A>70 k m M=6-=-6.5 • . * J ’ ni :Vv . ‘ - y a % * i r . i . v .-r- i *.>• -i . * i r , A=0-h20 k m M >6.5 .1 , .1_i.- -i’ ■ i . 1 . 1 , 1 , A = 2 0 + 4 0 k m M >6.5 * ' ■ i , i , ■ A =40-j-70 k m M >6.5 A>70 k m M >6.5 i . ' i r ’ i i V i - V - i \ , -2 0 0 2 0 4 0 6 0 80 -2 0 0 2 0 4 0 6 0 8 0 - 2 0 0 2 0 4 0 6 0 80 Time (sec) - 2 0 0 2 0 4 0 6 0 80 F ig . 8.3 d Channel # 4 : the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the 5-wave arrival. A = 0+ 20 k m M =3+5 i , i • i i i i i i A=20-1-40 k m M=3-i-5 i , i :»* , i , i , i , A = 40 + 7 0 k m _ M =3*5 i , i i i , i , i , i , Channel 5 ; f0=.63 Hz i , i , i , i , i , i , A=0-!-20 k m M =5*6 '3.^° . Tor;. • v ' 1 , A = 20 + 4 0 k m M=5-i-6 ,i It* ? i . ‘ft v v h ' . i . i . i A=40-i-70 k m M=5^-6 ■ •' ' • i , i , i . i . i , A>70 k m M=5-r6 — , \ : i , i* . ,. i , i . i , i , A=0-r20 k m M=6-e-6.5 Ai'An'l*.. :. I.I. ’ T V * < . 1 . 1 . 1 . A =20-r40 k m M=6-i-6.5 — S t A =40-r70 k m • M=6-r-6.5 “ J ? * * . A>70 k m M = 6 - s - 6 . 5 : i A=0-5-20 k m M>6.5 — i —i — 1- .i. r * t 11_ij— i __i — i __i — A = 2 0 * 4 0 k m M>6.5 i , t « i , ’ i , i , A =40-r70 km M>6.5 A>70 km M>6.5 ■ ■ • T Jd:."- i ' ' ' i... i.v*y:r , m , i , -2 0 0 20 40 60 80 -2 0 0 20 40 60 80 -2 0 0 20 40 60 80 T im e (se c ) -2 0 0 20 40 60 80 F ig . 8.3e Channel # 5 : the relative weakness of the strong m otion time interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. A = 0 + 2 0 k m M=3-5-5 ‘ < * * .V j C »• • i , ■ '; • n , i , i , A = 2 0 + 4 0 k m u M =3+5 _ l , ! * * i A = 4 0 + 7 0 k m M=3-s-5 i , t , i , i , i , i , Channel 6 ; f 0 = i . i Hz A = 0 + 2 0 k m M = 5*6 'y • i , , i , i , A = 2 0 + 4 0 k m M =5+6 A=40-i-70 k m M=5-i-6 . ' v-:‘ . . . 'I sty • * > • ! ifl ?'M 1 * L L ^ c _ i 1 i 1 . 1 . A>70 k m M=5-i-6 A =0-r20 k m M=6-=-6.5 • i * i , n fiiti . i . i . i , A=20-r-40 k m M=6-f-6.5 A=40-5-70 k m . M=6-j-6.5 A>70 k m M=6-i-6.5 A=0^-20 k m M>6.5 '" • .. >■*.. .. .. * : - - i , A*,-'.'.i- : i * , i , i , A=20-i-40 k m M>6.5 1 , • ,1 , 1 , 1 , A =40-r70 k m M>6:5 A>70 k m M>6.5 i , ► ' • ! * / 1*1, i , i , -20 0 20 40 60 80 -2 0 0 20 40 60 80 -2 0 0 20 40 60 80 Time (sec) -2 0 0 20 40 60 80 F ig . 8 .3f Channel # 6 : the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the S-wave arrival. A = 0+20 k m M =3+5 J-. J i , i J i . ? ' i , i , i , A= 2 0 + 4 0 k m M =3+5 '.A*- i , i f i ’ r , i , i , i , A = 40+ 70 k m M =3+5 Channel 7 : f0 =1.7 Hz 1 . 1 . 1 . 1 , 1 . 1 , A = 0+20 k m M=5 + 6 ■ • i - ‘ • 1 — l k BT-- t. - 1 ■ 1 1 1 1 1 1 A = 20+ 40 k m M =5+6 ■ ; 4 i , : i , i , i , A = 40+ 70 k m M =5+6 ■ . t:! I . i i . i: is i , i , i . i . A>70 k m M=5-5-6 — t i . i * , i . i . i . i A = 0+20 k m M =6+6.5 _ 1 — i 1 i 1 i ! i 1 i A = 20+ 40 k m M =6+6.5 — \ ’ ; , m - : (s A = 40+ 70 k m M =6+6.5 A>70 k m . M =6+ 6.5 A = 0+20 k m M>6.5 ; \ !' ; • .j-j-.r'-Z.'a'.o i -i.. i .i i , A = 20+ 40 k m M>6.5 ■V*. *. ^ i v 1 ’ ---1 ---1 C'Hsil.-Tl < __1 __|__|__i __1 __i_ A = 40+ 70 k m M>6.5 J j f c h ' i , , , A>70 k m M>6.5 i i i - ,: If1 ........................ - 2 0 0 20 40 60 80 Time (sec) - 2 0 0 20 40 60 80 F ig . 8.3g Channel # 7 : the relative weakness of the strong motion time interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. A=0-s-20 k m M =3+5 : I i isft- 1 i 1 i_l l 1 i A = 2 0 + 4 0 k m [ _ M=3-f-5 1 1 * ' • . ! i • • * * i , l ‘ ' l , 1 , 1 , 1 , A = 40 + 7 0 k m M=3-i-5 i . i , i , i , I , i . Channel 8 ; f 0 = 2.5 Hz i , i , i , i , i . i , a * . i . i . i A = 2 0 + 4 0 k m M =5+6 * i t * . a f t ................. A = 40 + 7 0 k m M=5-^6 - - M i , r. V • i , i , i , i , A>70 k m M =5+6 i . i *'1 i . i , i , i , A=0-h20 k m M=6-f-6.5 i , V-A 1 i , i , i , i , A = 20-f40 k m M=6-i-6.5 ” 1 L, i f e - , ______ A =40-r70 k m M=6-j-6.5 i . . . . . . . A>70 k m M =6-r6.5 1 l\: . fJ.K jrfln •.•1 . 1 , 1 . A=0-s-20 k m M>6.5 . '.. - i , i .s ii ’ , , i , i , i , A = 20 h -40 k m M>6.5 •* y . * * "| ................ , A=40-i-70 k m M>6.5 # • A>70 k m M>6.5 •i'.'.iXnA - 2 0 0 20 4 0 60 80 - 2 0 0 20 40 60 80 - 2 0 0 20 4 0 60 80 - 2 0 0 2 0 40 60 80 Time (sec) F ig . 8 .3 h Channel # 8 : the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the 5-wave arrival. o o A=0-5-20 k m M =3+5 i ,* . i , «*?■ i , i , i , i , A = 20+ 40 k m M=3-s-5 1 , , 1 , 1 , 1 , A = 40 + 7 0 k m M=3-a-5 i . i , i , i , i , i , Channel 9 ; f 0 = 4 . 2 H z 6 I I I I . i . i . i A=20-i-40 k m M =5+6 i , , i , i , i , A = 40+ 70 k m M =5+6 - i- . ? A>70 k m M =5+6 / 1 . 1 * . 1 . 1 . 1 , 1 . A=0-r20 k m M =6h -6.5 1 , tiSi' 1 1 It]-.! 1 I A =20-r40 k m M =6h -6.5 . % i , i , i , i , A=40-=-70 k m M = 6*6.5 * A>70 k m M = 6 -6 .5 i . i . i . i . A=0-^20 k m M>6.5 1 , 1 , 1 , 1 , A =20-r40 k m M>6.5 _ > ? « . ■ 1 , 1 ......................... A = 4 0 -7 0 k m M>6.5 < • i , ” ,' i , i , i , A>70 k m M>6.5 'y -' r'i'-L ' • • 1 , .' * 1 • 1 i’jrf"? * 1 , 1 , 1 , -20 0 20 40 60 80 -20 0 20 40 60 80 -20 0 20 40 60 80 -20 0 20 40 60 80 Time (sec) F ig . 8.3i Channel # 9 : the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the 5-wave arrival. o o A = 0+ 20 k m M =3+5 £ i , i , i , i , i , A =20-r40 k m M =3+5 1 , • Lf'i 1 . 1 . 1 . 1 . A = 40 + 7 0 k m M=3-i-5 Channel 10 ; f 0 = 7 . 2 H z A=0-i-20 k m M =5+6 — « > i , Jf*r, i , i , i . i , A = 20 + 4 0 k m M=5-f-6 • s - A = 40 + 7 0 k m M=5*-6 ■ ■ ■ 1 1 . 1 , 1 . 1 . A>70 k m M=5-s-6 A=0-r20 k m M=6-j-6.5 — \t i . S ■ i ................... ..... A = 20 h -40 k m M=6-j-6.5 1 & k ' ' i . i , i , i , i A = 4 0 -r7 0 k m M=6^-6.5 ■ " * i , i i i , i A>70 k m M =6- t -6.5 “ '.'H f * ' • - I .<• | | , | , | A=0-e-20 k m M>6.5 ' : 1 . L | .............................. A = 2 0 -r4 0 k m M>6.5 - - .v.r. • • i* A = 4 0 * 7 0 k m M>6.5 - J # - i ................................................. A>70 k m M>6.5 ~ i , , , , , , - 2 0 0 20 40 60 80 - 2 0 0 20 40 60 80 - 2 0 0 20 40 60 80 - 2 0 0 20 40 60 80 Time (sec) F ig . 8.3j Channel #10: the relative weakness of the strong motion time interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the S -wave arrival. o o A = 0+ 20 k m M=3-i-5 A = 20 + 4 0 k m M=3-i-5 A = 4 0 + 7 0 k m M =3+5 Channel 11 f0=13 Hz 6 q 4 2 A = 0+ 20 k m M=5-5-6 A = 2 0 + 4 0 k m M=5-5-6 A = 4 0 + 7 0 k m M=5-s-6 6 q 4 2 A=0-s-20 k m M = 6*6.5 A =20-r40 k m M =6h -6 .5 A=40-=-70 k m M =6h -6.5 6 q 4 2 •vc*-' i A = 0 * 2 0 k m M>6.5 A = 2 0 * 4 0 k m M>6.5 A = 4 0 -7 0 k m M>6.5 A>70 k m M>6.5 6 q 4 2 _ i— . —L_£U— i —I —i — i — i i — i — i — i .]uz i i I i i i I i_l I ' ir. i , i , i . i , I i , i :? i I i . i , i , - 2 0 0 20 4 0 60 80 - 2 0 0 20 40 60 80 - 2 0 0 20 40 60 80 - 2 0 0 20 40 60 80 Time (sec) F ig . 8.3k Channel #11: the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the mom ent of the 5-wave arrival. A=0-5-20 k m M = 3 + 5 A = 2 0 + 4 0 k m M=3-5-5 A=40h-70 k m M = 3 * 5 C h a n n el 12 f0 =21 Hz 6 q 4 2 A = (H 2 0 k m M=5-f-6 A = 2 0 + 4 0 k m M=5-!-6 A = 4 0 + 7 0 k m M = 5 * 6 6 q 4 2 A =0-h2 0 k m M = 6 -r6 .5 A = 2 0 -r4 0 k m M =6-i-6.5 A = 4 0 -r7 0 k m M =6-j-6.5 6 q 4 2 A=0-s-20 k m M >6.5 A = 2 0 -f4 0 k m M>6.5 A =40-i-70 k m M>6.5 A>70 k m M>6.5 6 q 4 2 -2 0 0 8 0 Time (sec) F ig . 8.31 Channel #12: the relative weakness of the strong motion tim e interval, q (Eq. 8.1), is plotted versus the middle point of this interval, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. 3 8 8 s h o r t d ista n c e s or s m a ll m a g n itu d e s m o d e ra te d ista n c e s *(pr m a g n itu d e s \ :: I 5'?K !>':5 i5 5 r.2 5iS *5i? 5i5 5i?!2ia-^ s ! s { a I S & $ [ s ! s long d is ta n c e s or large m a g n itu d e s tim e F ig . 8 .4 Overall trends of the distribution q — q(time) w ith respect to epicentral distance and earthquake m agnitude. The param eter q, defined by Eq. (8.1) shows the “relative weakness” of a strong motion pulse, when com pared w ith the strongest pulse in the record. .008 .006 -.1 .004 -.2 .002 - .3 .000 f (Hz) - .4 -.0 0 2 \ \ - .5 - .0 0 4 F ig . 8.5 The frequency dependent coefficients, scaling the relative weakness of a strong motion pulse with respect to M and A (Eq. (8.2)) are plotted versus central frequency of the channels (solid lines). The coefficients are bounded by their “cr-intervals” (dashed lines) and by their estim ated 95% confidence intervals (dotted lines). 389 390 were only d ata with g / 1 were considered (q = 1 corresponds to the strongest pulse). As we see, both const 2 {f ) and consts(f) are negative. W ith the increase of epicentral distance, q decreases, because the “weakest” strong m otion intervals are “eaten up” by attenuation and cannot be qualified as “strong” any more. As far as m agnitude is concerned, the diversity of the relative strength of pulses gen erally decreases with it. Why is it so? One possible explanation is th a t asperities, responsible for em itting separate pulses, work differently for different m agnitude. Big earthquake crashes them completely, and different asperities release all the stress they have stored (and there might exist an upper bound for this value). As a result—similar pulses are em itted. Small earthquake does not have enough energy to crash all asperities completely. Possibly, it crashes asperities som ewhat partially, and this results in relatively weak pulses of energy being em itted, so th a t the diversity of pulses’ “strength” (or “weakness” ) increases. O ur results may be related to the observations of R autian et al. (1981) m ade in the conti nental C entral Asia. They noted, th a t the upper lim it of the observed apparent stress, 7 7 < 7, does not depend on the seismic moment of the source, Mq. At the sam e tim e, the lower limit of i/o tends to increase w ith Mo, causing sm aller scat tering of d ata points for big seismic moments. For California data, however, we are not aware of any results th a t would show such trends. 8.2 Scattered Energy in the Strong M otion Record— the “Strong M otion Coda” T his section is devoted to the study of the scattered energy recorded by a strong m otion device. As we saw above, strong motion pulses can be identified as 391 P- or S-waves, or as various surface wave modes, direct or reflected from horizon tal and vertical boundaries. The energy coming between and after those pulses cannot be related to any specific wave package and may be considered as being produced by scattering of body and surface waves by irregular inhomogeneities in the E a rth ’s crust. T he scattered incoherent motion, usually called coda, has been studied by m any investigators (see the review by Herriaz and Espinosa, 1987 and report by Aki, 1991). T he work began from pioneering papers of Aki (1969) and Aki and Chouet (1975), where only the single weak scattering was considered and the use of the Born approxim ation for the description of the coda waves was introduced. R autian and K halturin (1978), and R autian et al. (1978, 1981) used the model of Aki and Chouet in their experimental studies of coda. It was found th a t coda envelope as function of time m ost likely should be described by several segments w ith different Q coefficients. These segments could be the result of the layering of the m edium or of the progressive increase in the order of scattering at longer lapse tim es. The role of layering was also emphasized by Wang and H errm ann (1988). Kopnichev (1977) considered double an triple isotopic strong scattering, and Gao et al. (1983) studied multiple isotopic weak scattering, using (locally) Born approxim ation. Sato (1988), however, questioned the inevitability of the consideration of the high order scattering as the explanation of the coda excita tion a t large lapse time. Sato (1977) considered the influence of mode conversion during single isotopic scattering on the formation of the coda envelope and Sato (1982b) studied effects of nonisotropic scattering and nonspherical source radia tion. Wu an Aki (1985) derived the expressions of the mean-square am plitudes of the scattered field in the case of elastic (vector) waves and small scale in 392 homogeneities using Born approxim ation. Wu (1985) used tran sp o rt theory to describe the propagation of energy in the scattering m edium and employed it to separate scattering effects from intrinsic attenuation. Zeng (1991) and Zeng et al. (1991) provided a complete formulation of scattered wave energy propagation in a random isotopic scattering medium which unified all the existing theories on seismic scattering, including the reactive transfer theory and m ultiple scattering theories based on the ray method. On the basis of this unified theory a new powerful m ethod for separating the scattering attenuation from the intrinsic ab sorption was developed (Mayeda et al. 1992), and it was found th a t the intrinsic absorption dom inates over the scattering attenuation for frequencies higher than about 6 Hz (and vice versa for lower frequencies). A variety of numerical m eth ods was developed to study the effects th a t cannot be still described by theory and to investigate the realistic crust models (see review by Frankel, 1989). Despite the substantial progress in the development of theory, many discrep ancies still exist between the theory and the observations. Thus, sim ultaneous determ ination of intrinsic Q~l and scattering Q~l is sometimes nonunique (Wu and Aki, 1988). On the other hand, the simple equation for the coda spectral density envelope for single scattered waves (time t is m easured from the “source origin tim e” ) P(u>,t) ~ t~ 7 exp(— ujQc 1^ (8.3) for body (-7 = 2) or surface (7 = 1) waves, based on Born approxim ation (Aki and C houet, 1975), fits the observations relatively well, allowing one to determ ine so-called “coda Q ,” (Jc, as a function of frequency w. The same equation holds for “diffusion” scattering—when coda is assumed to be the result of chaotic propagation of energy which can be described as the diffusion of body waves. In 393 this case 7 = 1.5. The real meaning of Qc and its relationships w ith intrinsic Q and scattering Q still rem ains somewhat questionable and some details are still to be clarified (M ayeda et al., 1992). Eq. (8.3), however, is still used intensively for practical purposes, such as determ ination of the source param eters from coda waves (R autian and K halturin, 1978; R autian, 1991; R autian et al., 1981) and studies of spatial and tem poral variations of coda Q (Jin and Aki, 1989; Peng, 1989). In this work, we will determ ine Q~l (or, more precisely, Q ^ 1) from the strong m otion records, w ithout using any usual seismic d ata (low energy and long lapse tim e). We understand th a t our determ ination will not provide new inform ation for seismologists, because much more detailed work was already carried out in the California region (Peng, 1989, for example). However, our results, determ ined directly from strong m otion accelerogram may come in handy for earthquake engineers, providing them w ith the practical knowledge about the rate of decay of the STRONG p art of the scattered energy filed, coming to the recording site. These scattered waves, which cannot be identified w ith any of the known phases and are coming in between and after well defined strong m otion pulses, we will call strong motion coda. The energy, carried by those waves, may play a significant role in the response of an engineering structure. If the dam ping of a stru ctu re is not large enough (which is the fact in m ost cases), the strong m otion coda m ay prolong the duration of “strong response” of the structure by “filling the gaps” between strong m otion pulses and by continuing to “pum p” enough energy to the structure at a rate greater than the dissipation rate, after the last well defined strong m otion pulse arrived to the site. 3 9 4 We consider the normalized envelope of coda, which we define as the square root of the ratio of the coda power spectral density to the spectral density of the prim e waves, coming to the site. We assume th a t the p a rt of the strong m otion coda spectral density (decreasing with tim e ) can be expressed by Eq. (8.3), and the spectral density of the direct waves is proportional to n 'exp(— w<0Q _1)» w here to is the tim e, needed for the direct waves to travel from the source to the station. G eometrical spreading param eter 7 * is equal to 2 for the body waves and to 1 for the surface waves. The difference between 7 and 7 ' is small, if any. We disregard the geometrical spreading altogether, for the following reasons. F irst, in strong m otion seismology, the distances to the source are of the order of the source size. Thus, not much of the geometrical spreading is taking place anyway (direct energy from the source is often coming to the site from a wide angle). Second, as we will see later, our m ethod of obtaining Q is not sensitive to the p articular values of 7 and 7 '. The expression for the normalized coda am plitude envelope is then exP i-n ft/Q ) ( - * f t \ f n f t o \ fo ^ ~ e x p(—n fto /Q ) = C X P ( - 9 - ) - e X P ( I T ) ■ (M We understand th a t we are mixing here the intrinsic Q, the scattering Q and the coda Q. The result we are going to get will give some “effective” Q, which can surve for engineering purposes, showing the rate, w ith which the flux of strong ground m otion energy is dying out w ith time. We next describe the method used in this work to extract the envelope of scattered incoherent motion from a record obtained by a strong m otion ac celerom eter triggered by an earthquake. We use here the concept of root mean-square am plitude, a rm3, of th e function f(t) in a tim e interval [t'\t" ] as a representation of the average power, or energy 395 flux, in this interval. Here, as before, f{t) stands for acceleration a(t), velocity v(t), or displacem ent d(t). Recalling the definition of duration, Eq. (2 .2), and expression for a rm3, Eq. (8.1b), we observe: ~ H { I [ I ! < 8'5) Fig. 8.6 shows how the envelope of scattered m otion can be constructed utilizing th e values of a 2m3 observed between and after strong m otion pulses. Cases of several different frequency bands are shown. Fig 8.6 (1) displays the band-pass filtered strong m otion function f(t) (acceleration in our case). Fig. 8.6(2) gives derivative of the sm oothed integral of this function squared (in full scale), w ith th e area under this function being shaded for the clarity of the figure. Fig. 8.6(3) reproduces the above graph in different scale and shows the procedure of con structing the envelope. We recall th at several threshould levels p^ were used as possible alternatives in the definition of duration of strong ground m otion (see Eq. (2.2) and Fig. 2.10). During the tim e when the expression in the figure brackets in Eq. (8.5) is grater than ({•} > p^), the integral J* / 2 (r)d r gains p portion of its final value / 0 < e “d / 2(r)d r, where tend is the length of the record. T his tim e was called the duration of strong ground m otion, corresponding to p.-100% of energy gain. Only p = 0.9 was considered in the previous C hapters of this work. Here we make use of /x=0.99, p= 0.975, /x=0.95, p= 0.9 and ^= 0.85. We are concerned now w ith the scattered motion, so we consider the portions of f(t) th a t lie between strong motion pulses, or at the very beginning or very end of th e record. T he quantity {•} from Eq. (8.5) is displayed in Figs. 8.6(3) and 8.6(2) (thin solid line). T he dashed horizontal line represents the different threshold levels crossings of which w ith {•} define the locations of the starting and of 396 10 5 0 - 5 -10 30 o a > c n 20 C M a o a w h T CM O a e d 30 4 0 50 Time (sec) 60 7 0 8 0 10 20 F ig . 8 .6 a C onstruction of an envelope of the scattered energy, a^ms(0 i shown for the record of the Borrego M ountain earthquake, April 8 , 1968, in the channel # 3 ( / 0 - 0.21 Hz). 1) The band-pass filtered record (thin line) and the envelope of the incoherent m otion, a rm3(t) (thick line). 2) The derivative of the sm oothed integral of the acceleration squared and the first five threshold levels p 3) Same as in 2 ), bu t amplified. Thick horizontal lines represent aj?m3, aver aged over “weak” m otion intervals, which are defined through various Black solid circles show the middle points of those intervals, and the line connecting them stands for «rms(0 - 397 20 C M a < u 0 3 10 s o -10 ^ -2 0 150 I 100 CM a 50 o a ( 0 20 h 'h C M 15 o CM 10 20 40 50 T im e (se c ) 60 70 80 30 F ig . 8 .6 b Construction of an envelope of the scattered energy, a^.ma(t), shown for the record of the Borrego M ountain earthquake, April 8 , 1968, in the channel # 5 ( / 0 = 0.63 Hz). 1) T he band-pass filtered record (thin line) and the envelope of the incoherent m otion, a rms(£) (thick line). 2) The derivative of the smoothed integral of the acceleration squared and the first five threshold levels 3) Same as in 2), bu t amplified. Thick horizontal lines represent a^ms* aver aged over “weak” m otion intervals, which are defined through various p/i. Black solid circles show the middle points of those intervals, and the line connecting them stands for a^m3(t). 398 20 10 0 -1 0 tM - 2 0 100 50 C M a o a n 15 h T3 P o 10 ( 0 a N ^ d 10 20 30 40 Time (sec) 50 60 70 80 F ig . 8 .6 c C onstruction of an envelope of the scattered energy, a*mg(t), shown for th e record of the Borrego M ountain earthquake, April 8 , 1968, in the channel # 7 ( / 0 = 1.7 Hz). 1) T he band-pass filtered record (thin line) and the envelope of the incoherent m otion, < Z r r n s (<) (thick line). 2) T he derivative of the sm oothed integral of the acceleration squared and the first five threshold levels p 3) Sam e as in 2), b u t amplified. Thick horizontal lines represent a% ms, aver aged over “weak” m otion intervals, which are defined through various Black solid circles show the middle points of those intervals, and the line connecting them stands for a^ms(t). 399 30 C M o < D GO a o 200 150 \ 100 CM 0 0 c n ¥ ' P 20 C M O 15 ^ 10 CM a 10 20 30 40 Time (sec) 50 60 70 80 F ig . 8 .6 d C onstruction of an envelope of the scattered energy, arma(t), shown for the record of the Borrego M ountain earthquake, April 8 , 1968, in th e channel # 9 ( / 0 = 4.2 Hz). 1) T he band-pass filtered record (thin line) and the envelope of th e incoherent m otion, a rm3(<) (thick line). 2) The derivative of the sm oothed integral of the acceleration squared and the first five threshold levels 3) Same as in 2), bu t amplified. Thick horizontal lines represent Orma, aver aged over “weak” m otion intervals, which are defined through various Black solid circles show the middle points of those intervals, and the line connecting them stands for a^m3(t). 4 0 0 the ending times (vertical solid lines) of strong and weak m otion intervals. T he levels of fljmg, corresponding to different /x, are shown as bold solid horizontal intervals on Fig. 8.6(3). The black circles show the m iddle tim e point in each of those intervals. The envelope of o,^ma(t) can be defined now as the line, con necting all these middle tim e points, in the order of increasing tim e. The curve ar m s (0 is shown as a bold line in Fig. 8.6(3). The envelope of incoherent m otion corresponds to arma(t) = \/a%ms(t), and this is the line shown on Fig. 8 .6 (1) as a thick solid line. The specific set of values of fi, used in the construction of arms(f), was chosen from the initial set {0.99, 0.975, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7} by trial and error. We w anted to get the function th a t would describe the “weak” m otion, and the last three values of fi from the above set, as we believe, correspond to m otion, “interm ediate” in strength. We recognize th a t the definition of the envelope of weak incoherent m otion, as outlined above is too rough. May be the better candidate for the function th a t can serve as the definition of the am plitude envelope of the scattered field is so called “instantaneous envelope” of the weak portion of the record. The “instanta neous envelope” can be obtained as (Re2( /) + Im 2( / ) ) 1/ 2, where R e (/) = 0.5 •/ and Im (/) is the Hilbert transform of f(t) (Keilis-Borok, 1989; Novikova and R autian, 1991). However, this would take additional com puter tim e, because then the tim e consuming com putations of H ilbert transform should be com pleted for all 12 channels of each of 3 com ponents of 492 records of acceleration, velocity and displacem ent (12 * 3 * 492 * 3 cases). 4 0 1 Earthquakes of different strength have different levels of the envelope of weak m otion. To eliminate this difference, we consider the normalized envelope , . envelope of weak m otion w t : ------:------------7T-, 8.6 average arms during strong motion w here the construction of the envelope of weak m otion is described above, and the strong m otion mentioned in the denom inator corresponds to 90% (/z = 0.9) gain of th e integral / 0 t' “d / 2(r)d r. The specification of the mom ent which corresponds to t = 0 is not critical at this point, but, for definiteness, we note here th a t we consider th e mom ent of 5 -wave arrival to be the “zero tim e” on the seismogram. T he norm alization Eq. (8 .6) was chosen because of the approxim ately linear dependence of log10 |/ o 'nd / 2(T)^r on m agnitude (Trifunac and Brady, 1975b) and on the fact th a t the source spectrum , seismic moment Mo (and, consequently, m agnitude M ) in particular, can be reconstructed from the frequency dependent level of coda waves (R autian and K halturin, 1978; R autian et al., 1981). We are using here the basic property of coda waves—the possibility to decouple source and propagation effects (Aki and Chouet, 1975). The curves w(t), Eq. (8.5) are plotted for all 12 frequency bands in Fig. 8.7 versus tim e (zero tim e corresponds to the m om ent of 5-wave arrival), separately for various ranges of distances and m agnitudes. Horizontal and vertical components of m otion are considered together for simplicity. It can be seen th a t the level and the shape of iu(t), on the average, does not depend on the m agnitude of earthquake (as expected), but changes w ith distance. This justifies the consideration of the averaged normalized envelope, w(t), defined in the following. The average (taken over all m agnitudes) is com puted separately for different ranges of epicentral distances. At each fixed Channel 1 f0=.075 H z : M =3+5 - A=20-i-40 k m : M =3*5 - A=0-i-20 k m : M =3+5 - A=40-s-70 k m .01 W I M =5+6 - A = 0 + 2 0 k m : M =5+6 - A = 4 0 + 7 0 k m I M =5+6 - A = 20 + 4 0 k m : M =5+6 - A>70 k m .01 W : M =6+6.5 - A = 0 + 2 0 k m : M=6-!-6.5 - A = 20 + 4 0 k m : M=6-5-6.5 - A = 40 + 7 0 k m : M =6+6.5 - A>70 k m .01 : M>6.5 - A=0-i-20 k m : M>6.5 - A = 20+ 40 k m : M>6.5 - A=40-5-70 k m : M>6.5 - A>70 k m .01 J i — l — i__ l— i_ I — i___ i— i I _ _ _ _ _ _ i I i I i I _ _ _ i i_ _ _ _ i i ■ i i i . i . i i i i i i i i i , i . 0 20 40 60 80 0 20 4 0 60 80 0 20 4 0 60 80 0 20 4 0 60 B O Time (sec) F ig . 8 .7 a Channel #1: the normalized envelopes of scattered waves, w(t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the S -wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, bu t changes w ith distance. w .1 .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 M =3+5 A=0-f-20 k m i . i . i . i M=3-f-5 A=20-s-40 k m J i l i i ■ i M =3*5 A = 4 0 + 7 0 k m _l I I I l_x -1— I 1 Channel 2 f0=.12 Hz j i i i i . i M=5-s-6 A = 0+ 20 km i . i i i . i M =5+6 A = 20+ 40 k m _ J I I I I L ■ * ■ M =5+6 A = 4 0 + 7 0 k m _l i l . i i i M =5+6 A>70 km i I __ J i i M =6+6.5 A = 0+ 20 km i , i . i . i I -1 I M = 6*6.5 A = 20+ 40 k m _ J i 1 i I , L M=6-i-6.5 A = 4 0 + 7 0 k m J i L l . i M =6+ 6.5 A>70 k m _i i I i I . i M>6.5 A = 0+ 20 km _J i i . I i I i_ M>6.5 A = 20+ 40 k m _ i I I . L _ l j i i i _ M>6.5 A = 4 0 + 7 0 k m i . i i . i J L _ M>6.5 A>70 km _i ■ i . i J i L 20 4 0 60 80 0 20 40 60 80 0 20 40 60 80 Time (sec) 0 20 40 60 80 F ig . 8 .7 b Channel # 2 : the normalized envelopes of scattered waves, w(t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, bu t changes with distance. 403 1 w .1 .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 F ig . 8.7c Channel # 3 : the normalized envelopes of scattered waves, w(t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The time is m easured from the moment of the S -wave arrival. The level and the shape of w (t), on average, does not depend on the m agnitude, bu t changes w ith distance. x M =3*5 A = 0+ 20 k m M =3+5 A = 20+ 40 k m _l i l i i . i M =3+5 A = 40 + 7 0 k m _J I I I I L Channel 3 f0=.21 Hz J i L M =5+6 A = 0+ 20 k m J i - i 70 k m M =6^6.5 A = 0+ 20 k m M =6+6.5 A = 20+ 40 k m M=6-f A = 40+ 70 k m M=6-f-6.5 A>70 k m M>6.5 A = 0+ 20 k m _ J i l i l_ M>6.5 A = 20+ 40 k m M>6.5 A=40-i-70 k m M>6.5 A>70 k m _i . i i 0 20 4 0 6 0 80 0 20 4 0 60 80 0 20 40 60 80 Time (sec) 20 40 60 80 1 w .1 .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 F ig . 8.7d Channel # 4 : the normalized envelopes of scattered waves, iu(t), plotted versus tim e, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the mom ent of the 5-wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, but changes w ith distance. ^ M =3+5 A = 0 + 2 0 k m i , i i , i M =3+5 A=20-«-40 k m _ J I I ! I I L M =3+5 A = 4 0 + 7 0 k m Channel 4 f0 =.37 Hz J . I i I _ _ j L M =5+6 A = 0+ 20 k m M =5+6 A = 2 0 + 4 0 km M=5-t-6 A = 4 0 + 7 0 k m J L - M =5+6 A>70 k m M =6+6.5 A=0-5-20 k m M =6+ 6.5 A = 2 0 + 4 0 k m M=6-5-6.5 A=40-i-70 k m M =6-i-6.5 A>70 k m -J .__1 __, __L M>6.5 A = 0+ 20 k m _ ! i 1 ___ .___ 1 ___ i___ L M>6.5 A = 2 0 + 4 0 k m _J I 1 I 1 L M>6.5 A = 4 0 + 7 0 k m _i i l . i . M>6.5 A>70 k m _ l i I , L _ L -L 0 20 4 0 60 80 0 20 40 60 80 0 2 0 40 60 80 Time (sec) 20 40 60 80 A = 0+ 20 k m W .01 1 W .1 .01 1 W .1 .01 1 W .1 .01 - M =3+5 J i L _ i i i_ A= 2 0 + 4 0 k m r M =3*5 _l L_J L_J ■ I - M =3+5 -J I L _ A = 40 + 7 0 k m J i I i I i L Channel 5 f0=.63 Hz i . i _ J_ _ I _ _ L . J i L A = 0+ 20 k m M =5+6 A = 20+ 40 k m A = 40 + 7 0 k m A>70 k m M =5+6 J i _ J i I i _ _ J i L - M =5+6 J i 1 _ _ _l I L _ J I L M =5+6 _ 1 i I i I i L M =6+6.5 A = 0+ 20 k m A = 2 0 + 4 0 k m M =6+6.5 A = 40+ 70 k m A>70 k m M =6+6.5 -I I I L__l ■ I M =6+ 6.5 _ J I I L. J I L A = 0+ 20 k m M>6.5 _ 1 i L A = 2 0 + 4 0 k m A = 40+ 70 k m A>70 k m j i [_ J i L M>6.5 _ l i L » » ■ - M>6.5 J . i _1 I L. J . I M>6.5 _ l i 1 ___ ,___ L J i L 0 20 4 0 60 80 0 20 40 60 80 0 20 4 0 60 80 Time (sec) 20 40 60 80 F ig . 8.7e Channel # 5 : the normalized envelopes of scattered waves, w(t), plotted versus tim e, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the m om ent of the 5-wave arrival. The level and the shape of iv(t), on average, does not depend on the m agnitude, but changes w ith distance. 4 0 6 A=0-i-20 k m W .1 .01 1 M =3+5 J i I i L _l I L. - M=3-s-5 J i I i I i L A=20-:-40 k m -i t — i [- M =3+5 J i L_ A = 4 0 + 7 0 k m J i L Channel 6 f0 =1.1 Hz j i i i i . i A=0-h20 k m W .01 1 W .1 .01 1 w .1 .01 M =5+6 A =20-r40 k m - M =5+6 J i 1 i l__i L M =5+6 A=40-!-70 k m _ i i i_ M =5+6 _ J . I ■ I A>70 k m J i L A=0-^20 k m M=6-i A =20-r40 k m A = 2 0 + 4 0 k m A = A n - = - 7 n V m M=6-:-6.5 _l i I . L M =6-i-6.5 _ J i i . i A>70 km J . L A = 0 + 2 0 k m A = 4 0 + 7 0 k m M>6.5 _! i I i I i L M>6.5 i . i - M>6.5 i . i i . i . i M>6.5 _! i 1__, I A>70 km J . L 0 20 4 0 60 80 20 40 60 80 0 2 0 40 60 80 Time (sec) 0 20 40 60 80 F ig . 8 .7 f Channel # 6 : the normalized envelopes of scattered waves, u>(t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the S -wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, b u t changes w ith distance. 407 1 w .1 .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 F ig . 8.7g Channel # 7 : the normalized envelopes of scattered waves, w(t), plotted versus tim e, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, but changes w ith distance. - M =3+5 = • A = 2 0 + 4 0 k m U M =3+5 i , i , i . r . i , - A = 4(H 70 k m ' % \- M =3+5 r Channel 7 f 0 =1.7 Hz i i i i i . I . i . - M =5+6 i , i , i , i , i , r k m - M =5+6 I , i , i , I , i , r A>70 k m - M=5-s-6 i , i , i , i , i , r A = 0 ^ 0 k m - M =6+6.5 i , i , i , i , i , r k m - - M =6+6.5 i , i , i , i , i , r ____ A>70 k m - M =6+6.5 i , i , i , i , i r A = 0+ 20 k m - M>6.5 i , i , i , i , i , r ^ ^ ^ ^ ^ ^ A = ^ + 4 0 k m ^ i , i , i , i , - A = 4(H 70 k m - M>6.5 i . i . .............................. - M>6.5 _l... I _ .i i.. i i i I i J______I 1_1____1 . 1 I 1----1___I-----1_____1_____I _1 . 1____ l - i I * 1 i I 1 t 1 t 1 i _1_ i I i 1 I . I _i I i 1 i I i 0 20 4 0 60 80 0 20 40 60 80 0 20 40 60 80 0 20 4 0 60 80 Time (sec) w .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 r A = 0+ 20 k m - i , i i i • i , i ■ r A =20-r40 k m - M =3+5 i • i > i • i . i . r A=40-i-70 k m [ M y - M =3+5 ! i 1 i 1 i 1 i 1 i C h a n n e l 8 f 0 = 2 . 5 H z i . i . i , i , i r A=0-r20 k m - A =20-r40 k m r A =40-r70 k m i , i , i , i , i , r A>70 k m - M =5+6 i , i . i , i , i r A = 0- h20 k m ^ ? P* , i . i . i , r ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 4 0 k m r _ A=40-5-70 k m - M =6+ 6.5 i . i , i , i , i , - M=6-!-6.5 i ■ i , i . i , i - M>6.5 i , i A = 0+ 20 k m r ^ 4 = 2 0 + 4 0 k m ^ , '^i , i . i , i , r A = 40 + 7 0 k m - M>6.5 i , i i i , i , i , k m j_ _ _ _1 _ _ _ _ _i. _ L . u . 1 . L- M>6.5 i — l — i — l — i— i _ _ _ _ _ _ i _ _ _ _ _ _l _ _ _ _ _ _ i _ _ _ _ _ _l _ _ _ _ _ _ i _ _ _ _ _ 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Time (sec) 0 20 4 0 60 80 F ig . 8 .7 h Channel # 8: the normalized envelopes of scattered waves, w(t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, but changes with distance. 4 0 9 w .1 .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 - A=0-i-20 km M=3+5 A=20-5-40 km M=3*5 _ 1 I u J I L A = 40+ 70 km M=3+5 J i l . i . i Channel 9 f0 =4.2 Hz J__ J i L A = 0 -r2 0 km M =5-r6 A = 2 0 -r4 0 km M = 5 + 6 A=40-5-70 km M = 5 + 6 J . I . __1 __, __L -1- I— ■ I A>70 km M =5*6 J i I i I i I , L A=0-h 20 km M=6-s-6.5 A =20-j-40 km M=6-s-6.5 A= 4 0 7 0 km M=6h-6.5 A>70 km M =6-t -6 .5 t * 1 » I _ i I i \ A = 0 + 2 0 km M>6.5 A = 2 0 + 4 0 km M>6.5 A = 4 0 + 7 0 km M>6.5 i , i . i . i . i J i l i l i i . i J i l i l i 1 i i J i L A>70 km M>6.5 J I I L _ 0 2 0 4 0 6 0 80 0 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 80 Time (sec) 0 2 0 4 0 6 0 80 F ig . 8.7i Channel # 9 : the normalized envelopes of scattered waves, w(t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the 5-wave arrival. The ievel and th e shape of iv(t), on average, does not depend on the m agnitude, b u t changes w ith distance. 410 w .01 1 w .1 .01 1 w .1 .01 1 w .1 A=0+20 km M=3-i-5 A = 20-f40 km M=3-i-5 r/ \ i . i . i . i . i A = 40 + 7 0 km M=3-5-5 J i L J I I- Channel 10 f0 =7.2 H z J i I . L - j L A=0-s-20 k m M =5-r-6 A = 2 0 -r4 0 k m M=5-=-6 _ J I L - A=40-7-70 k m M = 5 = 6 i . i J i L A>70 k m M = 5-r6 J . L A = 0-^20 k m M = 64-6.5 A=20-^40 k m M =6-r-6.5 J i I i I i I i L A = 4 0 * 7 0 k m M = 6 - 6 .5 J ■ l i l . i . i A>70 k m M =6-j-6.5 J , l , i . i A = 0+ 20 k m M>6.5 .01 - A = 2 0 + 4 0 k m M >6.5 A = 40 + 7 0 k m M >6.5 _ I I L i j i i_ J i I i I i I . L A>70 k m M>6.5 J i I i I i I . L j i i 0 2 0 4 0 6 0 80 0 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 8 0 Time (sec) 0 2 0 4 0 6 0 80 F ig . 8.7j Channel #10: the normalized envelopes of scattered waves, w(t), plotted versus tim e, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the mom ent of the S-wave arrival. The level and the shape of tu(t), on average, does not depend on the m agnitude, bu t changes w ith distance. 411 1 w .1 .01 1 w .1 .01 1 w .1 .01 1 w .1 .01 F ig . 8.7 k Channel #11: the normalized envelopes of scattered waves, w(t), plotted versus tim e, separately for various ranges of m agnitude M and epicentral distance A. The time is measured from the moment of the S -wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, but changes w ith distance. - A = 0+ 20 k m : M=3 + 5 i , i , i , i , i , r A = 2 0 + 4 0 k m : M=3*5 1 ■ 1 , 1 , 1 . 1 . t- A = 4 0 + 7 0 k m E M=3-s-5 : i , i , I , i , i , r Channel 11 f 0 = 1 3 Hz i , i , i , i . i , r A = 0 * 2 0 k m 1 . 1 j l . i 1 , r A = 2 0 * 4 0 k m r ^ -4-40 k m r A =40-r70 k m ' M=5^-6 i i i i i , i , i , r A>70 k m : M=5*6 i , i , i , i , i , i . ................................... ..... i , i , i , i , i , k m r A>70 k m M =6+ 6-5 ; 1 , 1 , 1 . 1 , 1 r A = 0 + 2 0 k m i , i , i , i , i , . i i i , i , i , i , r ^ ^ ^ ^ ^ ^ 0 + 7 0 k m i , i , i , i , i , r A>70 k m .. 1 , ! i 1 , 1 , 1 , J 1-----1___ I-----1___I___I-----1____1 ------1 I I - 1 I ' I » i a I i I 1 i I . > I . t I i 1 _i I _ 1 i 1 , 1 i I i | , 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Time (sec) w .0 w .0 w .0 w A=0+20 km M=3-i-5 J i L A=20h -40 km M=3-i-5 i , i , i , i , i A=40+70 km M=3-i-5 J i L _l I L- Channel 12 f0 =21 Hz i . i -J I 1 _ _ _ _ _ _ 1 _ _ _ _ _ _ I _ _ _ _ _ _I _ _ _ _ _ _ L A=0-=-20 km M=5-=-6 A=20-=-40 km M=5-r6 A=40h-70 km M=5-=-6 J ! __1 __I __1 __I __1 __1 __L J I I I I . I I L A>70 km M =5*6 J i l i I . i ■ i A=0-s-20 km M =64-6.5 A=2(H 40 km M =6h -6.5 A =40-5-70 km M=6-:-6.5 J . i . i . i . i A>70 km M = 6 -6 .5 J . __1 __. i . i ■ ■ ■ 01 - A =0+20 km M>6.5 A=20-s-40 km M>6.5 A =40+70 km M>6.5 i . i . i . i . i J i I i I i I i L A>70 km M>6.5 0 20 40 60 80 0 20 40 60 80 0 20 4 0 60 80 Time (sec) 0 20 40 60 80 F ig . 8.71 Channel #12: the normalized envelopes of scattered waves, w{t), plotted versus time, separately for various ranges of m agnitude M and epicentral distance A. The tim e is measured from the moment of the S -wave arrival. The level and the shape of w(t), on average, does not depend on the m agnitude, bu t changes with distance. 4 1 4 tim e t, w(t) is defined as logioN ^)] = TT7~r loSioM *)]> (8.7) ' r' l=i w here N(t) is the num ber of normalized envelopes th a t are defined a t tim e t for the range of distances considered, and we(t) is the value the t— norm alized envelope taken a t tim e t = t. The averaging is “logarithm ic” because we expect the exponential dependence as a result of our calculations (see Eq. (8.4)). O ur goal is to obtain Q — Q(f) by comparing w(t) for different ranges of distances at each frequency band (channel). For a given channel num ber, designated as i, we should choose n ranges of distances , where j = 1 ,2 ,...,n is the range num ber. The criteria in choosing djiym, d£)ax and n are as follows: (l) n should be as big as possible, so th a t any p attern of changing «>,y(<) with distance can be noticed (here u7,-y(<) stands for the averaged envelope at channel number i and at distance range num ber j); (2) the distance ranges should not be too wide, so th at various norm alized envelopes wi{t) from Eq. (8.7) correspond to approxim ately the sam e distance; (3) the num ber of com ponents m ,j, falling into the range j at every fixed frequency band t, should be approxim ately the same for all j; (4) all m,y should be big enough to provide good statistics and sm ooth resultant uJ,y(t). Not all of these requirem ents can be satisfied. O ur choice of all of the m entioned param eters is sum m arized in Table 8 .1. As it is seen, n = 4 for m ajority of channels (n = 2 for channel # 1 and n = 3 for channel # 2 ). Columns a) and c) in Table 8.1 show the ranges of distances, jdjym -r djjyax , and the number of com ponents of acceleration velocity or displacem ent, m,y, available in the d ata base for each range; column b) gives T ab le 8.1 Param eters, chosen for calculations of average normalized envelope, at each channel number i and each range of epicentral distances num ber j : a.) ranges of epicentral distances; b) average epicentral distance in the range considered; c) number of components (horizontal and vertical combined) of acceleration, velocity and displacement, available in the data base for the range considered. R a n g e n u m b e r , j j = l j = 2 j = 3 j - 4 Channel number, i Central freq. fo a) dj! niin j. + d jimax (km) b ) d ;iav (km ) c) m il a) di2min * f di2max (km ) b ) di2av (km ) c) mj2 a) dj3min + + di3max (km ) b ) di3av (km ) c) mj3 a) di4min + + di4max (km ) b) di4av (km) C) m ;4 i = 1 0 .0 7 5 Hz Of- 40 30.0 9 4 0 f 70 53.4 24 o • f * o o - 0 1 0 0 + 180 - 0 i = 2 0.1 2 Hz O f 35 24.7 120 35 f 70 45.1 135 7 0 f 100 79.3 14 1 0 0 + 180 - 0 i = 3 0.21 H z O f 30 22.0 223 30 f 55 40 .4 599 55 f 70 59.9 38 -J 1 o j - i - 1 co 1 ° 105.5 56 i = 4 0.37 H z O f 30 18.2 4 0 2 30 f 50 4C.0 783 50 f 70 5 6 .6 132 70 f 130 99.5 71 i = 5 0 .6 3 Hz O f 20 10.6 483 20 f 40 32.6 653 4 0 + 65 4 6 .9 662 65 + 130 95.5 105 i = 6 1.1 H z O f 20 9.8 702 20 f 40 32.5 793 4 0 + 65 47.5 733 65 f 120 85.0 144 i = 7 1.7 H z O f 17 8.6 925 17 f 35 2 7 .4 561 35 f 60 4 3 .6 1015 6 0 + 110 74.2 209 i = 8 2 .5 H z O f 12 7.1 977 12 f 35 24.0 773 35 + 60 4 3 .9 1026 60 + 100 70.6 194 i = 9 4 .2 H z O f 12 7.0 1011 12+35 23.5 634 35 + 60 43.5 656 60 f 100 71.1 123 i = 10 7 .2 H z O f 12 6.9 1030 1 2 + 30 20.7 4 7 4 3 0 + 55 39.9 641 55 + 90 65.7 152 i = 11 13 H z O f 10 5.7 631 10 f 25 15.2 351 2 5 + 40 33.2 28 6 4 0 + 80 49.3 184 i = 12 21 Hz O f 10 5.4 348 10 f 25 14.7 162 25 + 40 31.9 128 4 0 + 80 49 .2 60 4 1 6 the average epicentral distance for each range and channel, dfj, where ^av _ 1 y -' / epicentral distance A for the \ tJ m. v ^ \ d ata point in the j — channel / 3 d“in<A<d“a lt • J — — *J are ju st the averages of epicentral distance among all the cases from the d ata base, belonging to the range dfjm d ^ ax| . For each i and j (i = 1 -j-12, j = 1 -r 4), the norm alized averaged envelopes Wi.it) were calculated using Eq. (8.7). To com pare for different j (different distances), some common reference tim e should be accepted. (Recall th a t up to now we use the tim e of S-wave arrival as the “zero tim e”). Taking the “source tim e”— the m om ent when the rupture started—as such a reference tim e, and knowing th e effective epicentral distance for each range, dfj , all the curves «J,y(t) can be translated into the new reference tim e using tran sit times for S-wave in Southern California (slightly corrected version of Richter (1958): his chart assumes focal depth 25 km, average focal depth for the records in our d ata base is about 8 km ). Thus, from this moment on, we will express all time dependent functions using the origin tim e as the reference (zero) time. Fig. 8.8 shows two exam ples of Wij(t): a) channel i = 6 (/o = 1.2 Hz), range j = 3 (dav = 47.5 km) and b) channel i = 8 (/o = 2.5 Hz), range j = 2 (dav = 24.0 km ). We assume th a t the two pronounced peaks occurring in u7,-y(t) can be associated w ith two groups of well defined strong m otion pulses— P-waves and com bined S- and surface waves (see discussion in previous section). T he P- wave is coming first, it has relatively small am plitude, and the scattered energy arriving to the station right after and together w ith the P-wave can be interpret as being “lost” by the P-wave due to scattering on random inhomogeneities. This scattered energy arrives in the station rightr after the P-wave. This energy 417 S a n d s u r f a c e .01 10 100 1 5 * a n d s u r f a c e .1 .01 10 100 F ig . 8 .8 T he averaged normalized envelopes of the incoherent scattered m otion, recorded by strong m otion accelerograph, «7,y(t); t = 0 corresponds to th e “source tim e” : a] frequency channel i — 6 , distance range j = 3; b) frequency channel i = 8, distance range j = 2 (see Table 8 .1). The first “bum p” on w ,y(t) may be attributed to the energy “lost” by the P- wave due to scattering by random inhomogenities and arriving to the site right after the P-wave. The second “bum p” may correspond to S- and surface waves’ scattered energy. 418 has low intensity. Thus, the first peak in w»y(t) is small; however, it has greater relative intensity on high frequency channels. The second “bum p” on Wij{t) is b etter pronounced than the first one. According to the arrival tim es, this could be the scattered energy “lost” by S'— and by surface waves. T he sm ooth p a rt of Wij(t) curves corresponds to the tim e intervals where m any of wi{t) are available (large N(t) —see Eq. (8.7)). Irregular “bum py” parts indicate the lack of data. Now we can com pare the data—envelopes Wij(t)—w ith the theoretical prediction— Eq. (8.7). It is well known th at the shape of the coda envelope is very stable. O ur goal is to get Q of the region using this fact, and assum ing th a t the later parts of Wij(t) can be considered as parts of this stable coda en velope. For each frequency band (channel number i) all curves u7,y (t) , j — 1 -f- 4, were plotted in log-log scale on transparent paper, and then all of them were m atched in their “tail” p art by manual shifting of each curve along th e vertical axis. T he am ount of shift of each Wij{t) necessary to m atch all the curves in the channel considered was measured. For the frequency band num ber i and the range of epicentral distances number j > 1 we designate the shift of th e uJ,y(t) curve w ith respect to uJn(t) curve as 5,y. From Eq. (8.4), the am ount of shift along log10 axis should be (<*?; - c ) Sij = ------------------------ log10e, (8.8) where (38 corresponds to shear wave velocity and Q, is the frequency dependent attenuation factor for channels # i. It appeared th a t the dependence S ’,y on dfj is linear, as expected, for channels # 4 and higher. For lower frequency channels, the determ ination of Sii is difficult because coda reaches its stable envelope at lapse tim es which are much longer, than those available from the strong motion 4 1 9 records. To get the coefficient of proportionality, the following regression model was fit to the data, separately for each channel # i: Sij m ij rrijj •logio e. (8.9) = (aidfj + 6,)- Here a,- and 6 ,- are the unknown regression coefficients for the channel considered and m,y are given in Table 8.1. (Note th a t a,- and 6 ,- are in no way related to the sam e notation used in the previous Chapters). The term in square brackets [•] represents the weighting param eter for each range of epicentral distances, ex pressed as the ratio of the num ber of d ata points available for the j — range on the t— channel to the total num ber of d ata points in the i— channel. Having a t - from regression analysis, we can now calculate the shift, resulting from the Eq. (8.9) for the * — frequency band and the j — range of distances as a, (df- — d f ^ , so th a t the shifted curves Wij{t) can be w ritten as logio &ij(t) = log10 w,j(t) - a,- (dfj - d%) -log10 e. (8.10) For each frequency band considered (# 4 and higher), all four shifted curves u/,-y can be plotted together to dem onstrate the quality of the fit (we expect u;,y to coincide in their “tail” portions). The result is presented in Fig. 8.9 (bottom graphs). T he top graphs in this figure show the num ber N(t) of individual envelopes w&(t) (see Eq. (8.7), available for averaging at tim e t at the i— channel and the j — range of epicentral distances. Only portions of u’iy(t) where the corresponding Nij(t) is large enough are well defined and are w orth considering. As it is seen from the figure, the “tails” of u/,-y(t), j = 1 -r 4, can form relatively sm ooth curves for alm ost every frequency band considered. Com paring Eqs. (8.8) and (8.9) we note that - M i ’ f0 = l-l Hz (i=6) 1000 100 10 V'V 18.2 km (j= l) 40.0 km (j=2) 56.6 km (j=3) 99.5 km (j=4) — 10.6 km (j= l) — 32.6 km (j=2) — 46.9 km (j=3) 95.5 km (j=4) 9.8 km (j =1) 32.5 km (j=2) 47.5 km (j=3) 85.0 km (j=4) .01 100 100 100 Time (sec) F ig . 8 .9 a Channels # 4 to # 6 : the shape of the strong m otion coda envelope, as formed by the shifted averaged normalized envelopes u;,y (t), Eq. (8.10), is shown in the bottom of this figure. The reliability of each expressed as the number Nij(t ) of individual envelopes, available for averaging (see Eq. (8.7)) at tim e t in the t— channel and the j — range of epicentral distances, is shown in the top of this figure. The tim e is measured from the “source tim e.” > b . to o 1000 N - ij 100 10 1 I ' ■ I W - 1J .01 f 0=1.7 Hz (i=7) 8.6 km (j= l) 27.4 km (j=2) 43.6 km (j=3) 74.2 km (j=4) - » « - I 1 L I I I I_ _ _ _ _ _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ I I 1 ... I f0=2.5 Hz (i=8) . 7.1 k m ;(j= l) 24.0 km (j=2) 43.9 km (j=3) 70.6 km (j=4) i i I i i_ i t I ' * i I . i i i f0 =4.2 Hz (i=9) 7.0 km (j= l) 23.5 km (j=2) 43.5 km (j=3) 71.1 km (j=4) i I r I i i i i I i i 10 100 1 _J ! i— i I . 10 Time (sec) 100 l 10 100 F ig . 8 .9 b Channels # 7 to # 9 : the shape of the strong motion coda envelope, as formed by the shifted averaged normalized envelopes u;ty (t), Eq. (8.10), is shown in the bottom of this figure. The reliability of each Wij(t), expressed as the number Nij(t ) of individual envelopes, available for averaging (see Eq. (8.7)) at time t in the * — channel and the the j — range of epicentral distances, is shown in the top of this figure. The tim e is measured from the “source tim e.” to f 0 = 7 .2 Hz (i= 1 0 ) 1000 100 ■ V \ \ 6.9 km (j = l) 20.7 km (j=2) 39.9 km (j=3) 65.7 km (j=4) 5.7 km (j= l) 15.2 km (j=2) 33.2 km (j=3) 49.3 km (j=4) 5.4 km (j= l) 14.7 km (j=2) 31.9 km (j=3) 49.2 km (j=4) .01 10 100 100 100 Time (sec) F ig . 8.9c Channels # 1 0 to #12: the shape of the strong motion coda envelope, as formed by the shifted averaged normalized envelopes Wij(t), Eq. (8.10), is shown in the bottom of this figure. The reliability of each Wi} (t), expressed as the num ber Nij{t) of individual envelopes, available for averaging (see Eq. (8.7)) at time t in the t— channel and the the j — range of epicentral distances, is shown in the top of this figure. The time is measured from the “source tim e.” 4 2 3 so th a t Q~1 can now easily be obtained for each t— frequency band. We assum e f}„ = 3.35 km /sec (according to Richter, 1958). The resulting Q~l = Q ~x(f) is shown in Fig. 8.10. This figure also shows the results obtained in California by other authors. Aki and Chouet (1975), Phillips and Aki (1986) and M ayeda et al. (1992) used coda waves to determ ine Q(f). Singh et al. (1982) and N uttli and H erm ann (1981) used spectral ratio m ethod to get Q of S-waves. Trifunac (1993) obtained his Q values analyzing the same d ata base as used in this work b u t using the attenuation equations for Fourier spectral am plitudes (Trifunac and Lee, 1989, 1990). T hus, the “tails” of shifted averaged and normalized “strong m otion coda envelopes,” form more or less sm ooth function of tim e (Fig. 8.9). We next consider th e front portion of uqy(£). In the case of a half space w ith homogeneous characteristics of scatters, Wij(t) should approach the “tail envelope function” from below. (Recall th a t we only consider scattered energy here, and do not take into account the energy of the direct waves). Such behavior of the envelope of scattered energy can be found in Uscinski (1977) for the case of weak scattering. Q ualitatively similar picture also follows from the results of Zeng (1991). The approach of the “tail envelope” from below can be seen on low frequency chan nels (see channel # 4 ). At higher frequencies, the pulses, coming from the source, sta rt “noticing” the low Q zone of sediments. Let us assume th a t the scattering in sedim ents is much more intense than it is in the underlying rock. In this case, when the body waves enter the layer of low Q under the station from below, the scattering becomes more intense, and more energy is being lost by the prim ary waves and (partially) transferred into “strong m otion coda.” As a result, one can see the “overshoot” in Fig. 8.9 (bottom plots). We may assume th a t in the Q - i .0 1 - .001- .0001- Strong m otion data T rifu n ac ( 1 9 9 2 c ) : -------- H orizontal m o tio n V ertical m o tio n E p icen tral d istan ce: • 26 k m ® 51 k m O 76 km Coda analysis □ Mayeda e t al. (1992), C en tral C alifornia E P h illips an d Aki (1986), C entral C alifornia ■ Aki an d C houet (1969), Stone Canyon, C alifornia Spectral ratio m ethod A N u ttli an d H erm ann (1981), C alifornia A Singh e t al. (1982), Im p e ria l F au lt, C alifornia 1 —'—1 — 1 — 10 This study: Frequency (Hz) 100 F ig . 8.10 Our estimates of Q compared w ith other measurements of Q 1 factor in California. 425 early p a rt of the “strong motion coda,” we see the effect of scattering produced by the layer of sediments right under the station (if scattering Q ~ 1 there is much larger th a n in rock). The “overshoot” is more pronounced for higher frequencies and this is natural: shorter waves “recognize” the presence of such sedim ents easier, th a n the longer waves. For very high frequencies the inelastic attenua tion presum ably starts playing its role, and the “overshoot” is less pronounced. Sim ilar effect was discussed already in this work in connection w ith the range of frequencies where the influence of geological and soil site conditions on the duration of strong motion can be noticed. If the above assum ption about the nature of the “overshoot” is tru e, then the shape of the “overshoot” should depend on the depth of sedim ents under the station, h, and, perhaps, on the horizontal characteristic dim ension of th e valley, R. The direct waves are coming to the station not ju st from one direction, b u t from some range of angles (due to the finite source dimension and some disturbances of the direct wave along the propagation path). Hence, the volume of the sedim entary basin (given roughly by h-R) should influence the intensity of the “overshoot.” The bigger volume with high scattering Q -1 (but not strong attenuation) is present under the station, the greater am ount of energy will be “sent back” (transferred from the prim ary waves to the scattered p a rt of the record) and will arrive to the station right after arrival of each direct wave. If the natu re of the form ation of tvy (£) is as described above, we may be able to detect an early local segment of the “strong m otion coda” envelope. T he local segm ent and the global late segments correspond to two different intensities of scattering and intrinsic attenuation. R autian and K halturin (1978) and R autian et al. (1981) describe several branches of the seismological coda, which, they 426 think, m ay correspond to different physical conditions of wave propagation and scattering. Their segments, however, are some stable entities. The shape of these segm ents and the location in the record (with respect to the source tim e) does not depend on the distance to the source. This is natural, because seismologists usually consider “far coda,” which is formed by back scattered waves, so th a t the source and the station can often be considered as one point. O ur “early local branch” of “strong motion coda” cannot be looked a t in this m anner. It reflects (as we think) the scattering characteristics of the sedim entary layer right under the station, and can be seen only between arrivals of the direct waves and right after them . To verify our assum ption about the influence of the geological conditions on the intensity of the “local” branch, the envelopes Wij(t) were constructed for four different cases: (l) d ata w ith h < 0.5 km, (2) h > 0.5 km, (3) R < 15 km and (4) R > 15 km. The same Eq. (8.10) was used to “shift” the averaged envelope Wi.it), w ith a,- taken from the results for all available data and dfj recalculated for each case separately. This means the same “global” Q was assumed for all cases. To com pare the intensity of the scattered portions of the record, Fig. 8.11 was prepared. For each frequency band (# 4 and up), the envelopes for h > 0.5 km and h < 0.5 km are plotted together at the top of the figure, and the envelopes for R > 15 km and jR < 15 km are plotted on the bottom . Small volume of sedim ents (h < 0.5 km and jR < 15 km) corresponds to solid curves, and larger volumes (h > 0.5 km and R > 15 km) are plotted as dotted waves. Notice, th a t each case has its own dfj, and, thus, the envelopes corresponding to h > 0.5 and h < 0.5, as well as R > 15 km and R < 15 km, are slightly shifted w ith h<.5 km .01 R<15 km R>15 km .01 10 Time (sec) 100 F ig . 8 .1 1 a Channel # 4 : com parison of the envelopes of the “strong m o tion coda,” for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km , bottom— R > 15 km). h<.5 km .01 R<15 km R>15 km .01 10 Time (sec) 100 Fig. 8.11b Channel #5: comparison of the envelopes of the “strong mo tion coda,” Wij(t), for various geometries of the sedim entary basin. Solid lines show th e cases w ith “small” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, b o tto m — R > 15 km). 1 W ij .1 h<.5 km h>.5 km .01 l fn=l.l Hz R<15 km R>15 km .01 10 100 Time (sec) F ig . 8 .1 1 c Channel # 6 : comparison of the envelopes of the “strong m o tion coda,” for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sediments (top— depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km , bottom— R > 15 km). Hz h<.5 km h>.5 km .01 W ii R<15 km R>15 km .01 10 Time (sec) 100 F ig . 8 .l i d Channel # 7 : comparison of the envelopes of the “strong mo tion coda,” Wij[t), for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sedim ents (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, bottom— R > 15 km). h<.5 km .01 R<15 km R>15 km .01 10 Time (sec) 100 F ig . 8 .l i e Channel # 8 : comparison of the envelopes of the “strong mo tion coda,” W{j(t), for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, b ottom— R > 15 km). h<.5 km .01 fn=4.2 Hz R<15 km R>15 km .01 10 Time (sec) 100 F ig . 8 .I l f Channel # 9 : comparison of the envelopes of the “strong m otion coda,” for various geometries of the sedim entary basin. Solid lines show the cases w ith “sm all” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, bottom— R > 15 km). h<.5 km .01 R<15 km R>15 km .01 100 Time (sec) F ig . 8 .1 1 g Channel # 10: comparison of the envelopes of the “strong m o tion coda,” Wij(t), for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, bottom— R > 15 km). 1 Wij .1 h<.5 km .01 1 R<15 km R>15 km i . . i J i i i i_L .01 X X 10 100 Time (sec) F ig . 8 .1 1 h Channel #11: comparison of the envelopes of the “strong mo tion coda,” Wij(t), for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, b o tto m — R > 15 km). 4 3 5 f0 = 21 Hz h<.5 km .01 x X X J_ _ L x X x j L R<15 km R>15 km .01 10 Time (sec) 100 F ig . 8 .H i Channel #12: comparison of the envelopes of the “strong mo tion coda,” Wij(t), for various geometries of the sedim entary basin. Solid lines show the cases w ith “small” volume of sediments (top—depth of sedim ents, h < 0.5 km; bottom —small horizontal distance to the nearest rocks, R < 15 km ), dashed lines represent “large” volume of sedim entary deposits (top— h > 0.5 km, bottom— R > 15 km). 4 3 6 respect to each other along the horizontal axis. At low frequencies, no difference is seen in the envelopes describing scattering in large and in small sedim entary volumes. Starting from channel # 6 ( /0 = 1.1 Hz), the dotted curves (h > 0.5 km and R > 15 km) sta rt to position themselves somewhat above the solid curves (h < 0.5 km and R < 15 km ). The effect is visible in the long distance envelopes only. On the channels # 8 and # 9 ( /0 = 2.5 -j- 4.2 Hz) the effect is well pronounced for two pairs of envelopes, corresponding to the longest epicentral distances (dav is about 40 — 70 km). The maximum of the envelope tends to occur later for the dotted curves than for the solid curves, and also the dotted curves “settle” on the “tail” envelope later, than the solid curves do. For very high frequencies, all these effects disappear (however, it should be noted th a t the statistics is not very good there). As we see, the d ata do not contradict our assum ption, th a t for interm ediate frequency, the size of the sedim entary basin (low scattering Q media) contributes to the intensity of the scattered energy in the “strong m otion coda.” At low frequencies, the effect cannot be seen due to the long wave length, and for the high frequencies, strong attenuation may overshadow the effect. There could be other explanations of the “overshoot.” For exam ple, strong scattering m ight occur not in the whole volume of sedim ents, bu t ju st at the boundary between the rock and the sediments. This boundary could be (very roughly) described as a p art of an ellipse w ith the source and the station located at its foci, and the size of this ellipse being determ ined by the distance from the station to the rock, R. (Recall th at R is measured in such a way th a t, when added to the distance from the source to the rock, it gives the shortest possible travel tim e along the path source-rock-station). We concluded earlier (see C hapter 4) 4 3 7 th a t the boundary of a sedim entary basin can reflect a well defined pulse, which can be later observed at the station. It would be logical to assume however, th a t this boundary has a lot of irregularities which give rise to some am ount of irregularly scattered waves, especially at higher frequencies. Moreover, this scattering occurs hot ju st exactly at the location of the boundary as it is seen on the S m ith’s (1964) m ap, bu t in some region of finite w idth along it. In this case, we can expect the “outburst” of the scattered energy on the accelerogram around the tim e, corresponding to the arrival of the scattered waves from this region along the boundary. The further the rock is, the later the energy should come to the station. This may be one of the explanations for the shift of m axim a of the envelopes of the scattered energy towards late t in the case of big R, as com pared w ith the small R (Fig. 8.11). The larger volume where scattering occurs (big R leads to longer boundary between rock and sedim ents) could result in m ore scattered energy (if not overshadowed by attenuation on the way from the rock to the station). As a result, the dotted curves (R > 15 km) in the bottom graphs in Fig. 8.11 could be shifted up with respect to the solid curves (R < 15 km). T he difference in the “overshoots” for the large and the small volume of sedim ents we see in the Fig. 8.11 might also come from the fact th a t sediments tend to amplify the am plitudes of the strong ground m otion in certain frequency bands (Trifunac, 1989a, 1989b). The bigger am plitudes of the prim ary waves, com ing to the station can give rise to higher level of scattered energy observed right after their arrival. The later part of coda—the “tail” envelope—will not be disturbed so much because it describes the average of the properties of the m edium in much larger volume, not ju st directly under the station. 4 3 8 A t the present stage, we are unable to distinguish between these or other possible explanations of the nature of the observed “overshoot.” F urther analysis of this phenom enon should also start from the verification of its very existence, because our m ethod of construction of the envelope of “strong m otion coda” is far from perfect, and the presence of the “overshoot” may be, in p art, attrib u ted to the distortion of the record by various filters, used in the construction of the envelope. We wish to consider here one more elementary example th a t may lead to the estim ate of the attenuation coefficient Q. It is unusual, because we are going to look at the elastic medium, w ithout any scatters and w ithout any absorption. Let us consider a sedim entary layer of the depth h, with the shear m odulus pi and th e density p i, located above a harder halfspace w ith param eters and po (Fig. 8.12). We will use the ray theory, i.e. we assume th a t the wave length is m uch shorter than h. We p u t the receiver at the point O on the free surface. Suppose a sharp pulse of unit am plitude (in the form of a plane wave) approaches the sedim entary layer from below with the angel of accidence &o. After refraction at the point N, the pulse comes to the observation point O. However, this is not the only pulse, observed at point O. O ther pulses, reflected several times between the free surface and the layer boundary, arrive later. The second pulse goes through the p ath shown as M F N O on Fig. 8.12. Due to the reflections from the boundary sediments-rock, each consecutive pulse, coming to the observation point O is be weaker than the previous one, so th a t envelope on the seismogram resembles the typical coda envelope. Thus, certain effective attenuation factor Q can be can be assigned to this process. From Fig. 8.12, the tim e difference between the two consecutive pulses, observed at point O, the first coming through 439 o F ig . 8 .1 2 A pulse with the plane wave front approaches a sedim entary layer w ith rigidity p,i, density pi and thickness h, which rests on top of a halfspace w ith param eters Ho and pq. A sequence of pulses with dim inishing am plitudes is recorded at point O. An apparent quality factor Q can be assigned to the record, obtained at the point O, which is the function of the angle of incidence and the param eters of the media. Note th at both m edia are perfectly elastic. 4 4 0 p ath P N O and the second through path M F N O , is where 2h-cos 0i . At = - ^ 1, (8.13) 0o = V W /» o , /?i = V»i/Pi- (8-14) sint/j pi If the am plitude of the initial pulse (in the rock) is 1, the am plitude of the refracted wave is (Achenbach, 1973): 2p cos 0q £i fl cos 0Q + fli Is - cos $i Pi After th e wave enters the sedim entary layer, reflects back from the free surface (path M FN in Fig. 8.12), and reflects from the boundary rock-sedim ents towards th e surface (path FNO in Fig. 8.12), its am plitude is e -e i , where (Achenbach, 1973) ^ COS 0 1 — f 1 - COS 0 Q e = f ----------- --§9------------------------------------------ (8.15) ^ COS 01 + COS 00 The am plitudes of the consecutive pulses, observed at point 0, can now be w ritten as 2ei, 2eiE, 2eiE2, etc. T hat is, am plitude of the n— pulse is 2eiEu~1. The equation of envelope going through the m axim a and m inim a of the response at point O is y = 2 E iE ^ At\ (8.16) Combining Eqs. (8.13) and (8.16) and making some rearrangem ents, we get y = 2 e i exp < In e 2/i -c o s^ i 1 1 01 and representing Eq. (8.17) as y ~ exp(-7r/t<2_1), we have (8.17) 27T/A-COS 0 1 Q 0 M e - 1) ’ ( -1 } 4 4 1 where e and 0\ can be obtained from Eqs. (8.14)-(8.15). The same equation can be obtained for more realistic case, where the sedi m entary layer is bounded from both sides, i.e. we consider a “box” of sedim ents, w ith the depth h and the horizontal dimension 2i2, em bedded into a harder half space. We assume here th a t R is several times larger than h. Consider the same situation as before: an impulse with the plane wave front approaches the sedi m ents from below, penetrates into the sediments, and the m ultiple reflection takes place. Now, however, the scenario is little bit different. The pulse is “trapped” in th e “box,” reflects from its horizontal and vertical boundaries and passes by the station not once, but several times. Suppose (for simplicity) the station is located in the middle of this rectangular valley, so th a t the distance from the station to each vertical boundary is R. In this case, the pulse, ju st recorded at the station, will reflect from the bottom of the valley approxim ately R/(htaxiOi) tim es before it comes back to this station, and will attenuate eR/(.ht-aa0i) times. T he tim e interval between each pair of consecutive arrivals of the sam e pulse “trap p ed ” in the sedim entary valley is 2h R 2 R At = /? cos0j /itan flj /? isin 0 i The envelope at the station can be obtained now from y „ J \ (8.19) Note, th a t there will be several pulses, travelling in such a m anner inside the sedim entary “box,” because several of them can “enter” the “box” in such a way, th a t they will “hit” the station (see Fig. 8.12). However, the num ber of these pulses is finite (and not large, if R does not exceed h by a large factor), 4 4 2 therefore, the rate of the decay of the am plitudes at the station is represented by Eq. (8.19). After rearrangem ents, Eq. (8.19) gives for the attenuation factor Q the sam e form ula (8.18). T he dependence of Q~l on the angle O q from Eq. (8.18) is shown in Fig. 8.13 for several frequencies and for two groups of param eters: a) h = 3 km , /?i = 2 km /sec, Hi/no = 1/3, pi/po = 3/4; and b) h = 2 km, fli = 1.5 km /sec, P'l/P'O — 1/6, Pi/po = 2/3. The apparent Q~l from Eq. (8.18), as we see, depends on the initial angle of incidence 60, does not change very much for different m edia when physical param eters of sediments and rock are w ithin the range we considered, and is inversely proportional to the frequency of m otion. For the wide range ofthe angle 6q (0-j-60°), Q~x is relatively stable, and the values of Q ~ 1 are close to the upper bound of various Q - 1 , obtained in California (see Fig. 8.10). Hence, “reasonable values” of Q _1 can be obtained from th e record of m otion on the free surface of a medium w ithout any inelastic attenuation and w ithout any “random distribution of random scatters.” We now summarize the conclusions of this Chapter. The strong m otion records can be used to study the scattering properties of the medium. More careful and detailed consideration can provide quantitative results, for example on the dependence Q~l = Q~1(f). Further study of the early portion of the “strong m otion coda” can provide some information on the “local Q at the site.” This inform ation can be used and should be very useful for engineering purposes as well. The knowledge of the shape of the envelope of the scattered energy, coming in between and after strong motion pulses, can also be used in developm ent of the new definition of the strong motion duration. This definition will take 1.7 Hz 2.5 Hz 4.2 Hz 7.2 Hz 13 Hz 21 Hz - 1 \ \ v .001 10 20 30 40 50 60 70 80 90 - 1 .01 .0 0 1 t- 10 20 30 40 50 60 Angle 90 (d egrees) 70 80 90 F ig . 8 .1 3 A pparent Q 1 (as a function of the angle of accidence do and frequency / ) , th a t can be assigned to the process described in Fig. 8.12 for two sets of param eters: a) h = 3 km, = 2 km /sec, = 1/3, p j/p o = 3/4; b) h = ° 1 — /a . — i e i— /--- ' . •j inn, ^ Krn/sec, ni/fi0 — Lf '6, Pi/Po — 3/4; 2 km, /?i = 1.5 km /sec, p,i/p,o = 1/6, P i/po = 2/3. 443 4 4 4 into account the dam ping characteristics of the structure, i.e. the rate of energy dissipation during the stru ctu re’s motion, and compare those w ith the rate of the energy input, generated by the scattered energy coming to the site between and after strong m otion pulses. Then, it will be possible to estim ate the “duration of strong response,” i.e. the cum ulative time, when the response of a certain mode at a certain floor is going to be “significant.” The results for all modes a t all floors can be later combined into the “duration of the total m otion response,” i.e. the expected duration of “strong” response of the structure as a whole. 4 4 5 9. CO N C LU SIO N S In this work, the physical bases and empirical equations for m odelling the duration of earthquake strong ground motion were investigated. We accepted the definition of duration of a function of motion f(t), where f(t) is acceleration, velocity or displacem ent, as the sum of tim e intervals during which th e integral Jq f 2(r)dr gains a significant portion of its final value. All records were band-pass filtered through several narrow filters and duration of strong ground m otion was studied separately in these narrow frequency bands. T he scaling param eters were chosen to be: the m agnitude of the earthquake, the epicentral (or hypocentral) distance, the Modified Mercalli Intensity at the site, the geological and the local soil site conditions, the geom etry of the sedim entary basin, where the recording station is located, and the hypocentral depth (considered in Appendix E). M any models, utilizing some of those param eters, were considered. It was found th a t the duration of strong ground m otion, dur, can be modeled as an exponential function of m agnitude M or approxim ated by a quadratic function of M. The previous investigators (Trifunac and Brady, 1975b; Trifunac and W estermo, 1976a, 1978, 1982; Westermo and Trifunac, 1978), who were using one th ird of our present d ata base, had only a linear term in the function dur — dur(M). O thers, who looked at the exponential dependence of the duration on earthquake m agnitude, did not include the frequency dependent natu re of strong ground m otion in their definition of duration (Kawashima and Aizawa, 1989), and som etim es considered the records obtained in geologically different regions in the sam e regression analysis (Theofanopulos and W atabe, 1989). In all cases, the d a ta base available to other authors, was much less abundant or homogeneous. 4 4 6 T he estim ates of the source dimensions, obtained from our equations, which link the source duration w ith the m agnitude of an earthquake, are in good agree m ent w ith the results of other authors. T he linear dependence of duration on the epicentral distance was confirmed and the corresponding coefficient obtained with good accuracy due to increase in the num ber of the available d ata and by applying careful “cleaning-up” procedure to the initial noisy d ata base. It was shown, th a t the coefficient of proportionality in the dependence of duration on the epicentral distance is different for “soft” and for “h ard ” propagation paths. This observation is in agreement w ith our assum ption about the dispersive nature of the dependence of the strong ground m otion duration on the epicentral distance. T he dependence of duration on the Modified Mercalli Intensity was studied using the different set of equations, than those used previously by Trifunac and W estermo (1976b, 1977) and W estermo and Trifunac (1979). It was shown, th a t duration may increase or decrease w ith the increase of intensity, and th a t the particular behavior depends on the frequency of motion and on the distance to the source. For the first tim e the dependence of duration on the hypocentral depth was dem onstrated (Appendix E). It was also shown th a t the influence of m agnitude and epicentral distance on the duration of strong ground motion are coupled (A ppendix D). The new param eters describing the geometry (the depth and the w idth) of the sedim entary basins surrounding the stations were introduced. The functional form of the dependence of duration on the angular measure of the reflecting capability of the rocks, surrounding the station, was introduced and discussed. 4 4 7 It was shown th a t the duration at a sedim entary site can be prolonged due to the m ultiple reflections from the bottom and the edges of the alluvial basin by several (up to 6 -f- 7) seconds at frequencies near 1 Hz. W hen the detailed description of geological conditions at the site is not avail able, the param eter s can be used instead. The im portance of considering the local soil together w ith the geological site conditions was also dem onstrated. The influence of th e geological and the soil site conditions on the duration of strong ground m otion prevails at different frequencies. D uration can be prolonged by 3.5 sec at the site w ith deep sedim entary layer at frequencies about 0.5 Hz, and by as much as 5 — 6 sec due to the presence of the soft soil underneath the station at frequency of about 1 Hz. T he regression analysis was performed for the great variety of models. Fig. 9.1 provides an overview for choosing the proper model in each particu lar case, depending on w hat earthquake and site param eters are available. Each model is shown in this figure by specifying the set of param eters it considers. The equation num ber of each model is also given for easy reference to the m ain text. There are two recommendations we would like to make regarding the prac tical use of these models, (l) If the station is located on rock, the best model would be the one th a t considers geological param eter s (w ithout considering the local soil conditions param eter sl). In this case, obviously, s should be taken to be equal to 2. (2) The regression coefficients were obtained at some specific set of frequencies only. If the estim ate of the duration of strong m otion at some frequency, not present in this set, is required, we recommend to get it in two steps. First, get the estim ate of duration for two nearby frequencies from our set. Second, interpolate the results linearly and get the estim ate for the initially Yes No dur ( M , M 2A , h , R , h R , R 2h 2 <p) ------------ Eq. ( 4 .1 2 ) _________ _ S f a r ( M , M 2A , h , ^ Eg. ( 4 . 1 5 ) S dur ( M , A , s, sL ) w Eq. (4.17)___ dur ( M , M 2A ) Eq. (4.9)_ _ Which of these three parameters are known? Is the parameter s only available or local soil conditions are also known? available, or site conditions are specified in terms of s Are som e of Is any information about site conditions and/or geology of the region available ? F ig . 9 .1 a The-flow chart for selecting the proper scaling model in term s of the earthquake magnitude. 448 ( ^ m ■ 4 ' ' m m ^ > \ Eg. (5.1) No f Is distance to the source available? J Y es ^ I ^Jf No Is any information about site conditions and/or geology of the region available ? Y es E H No _ ^ f d u r ( l MM) Eq- (5.2L Yes Are som e of R, h or cp available, or site conditions are specified in terms of s and s L ? C t i r ( l m , A ' , l m A ' , h , R , h R , R * h 2 , ------------- Eq. (5.3) _ <P) R, h, c p C * * o m . a \ imm a', R ^ W T - y Eg. (5.6)____________ ^ Eq. (5.8) R, c p R, h, cp s, Which of these three parameters are known? A >'mmA', s ) Eq. (5.10) ('mm ' A ' 'mm A < s > SL ) V * Eq. (5.12) R, h, s, cp SL /" R, h, c p ° ur('MM’ h , R , h R , R ,h , (p) Eq. (5.5) Which of these three parameters are known? , R , R W f \ V Eg. (5.7) dur (Liu. h, h‘) Eq. (5.9) Is the Is the parameter s only available parameter s only available or local soil or local soil conditions are conditions are also known? also known? uur(lm . , s ) Eq. (5.11) ...L IUr('MM- S' SL> Eq. (5.13) F i g . 9 . 1 b T h e - f l o w c h a r t f o r s e l e c t i n g t h e p r o p e r s c a l i n g m o d e l i n t e r m s o f t h e M o d i f i e d M e r c a l l i I n t e n s i t y a t t h e s i t e . 4 4 9 450 r e q u i r e d f r e q u e n c y . W e w o u l d n o t r e c o m m e n d t h e i n t e r p o l a t i o n o f o u r c o e f f i c i e n t s b e t w e e n t h e c e n t r a l f r e q u e n c i e s o f t h e c h a n n e l s , b e c a u s e s o m e c o r r e l a t e d c o e f f i c i e n t s ( s u c h a s c o e f f i c i e n t s f o r M a n d M 2, f o r e x a m p l e ) c a n b e v e r y s e n s i t i v e t o s u c h a p r o c e d u r e . T h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n w a s s h o w n t o b e c o m p o s e d o f s e v e r a l p u l s e s h a v i n g v a r i o u s n a t u r e a n d o r i g i n . I n i t i a l l y , a n u m b e r o f p u l s e s i s e m i t t e d b y t h e s o u r c e . D u r i n g t h e p r o p a g a t i o n t h r o u g h t h e d i s p e r s i v e l a y e r e d m e d i u m , e a c h o f t h e i n i t i a l l y e m i t t e d p u l s e s is s e p a r a t e d i n t o s e v e r a l s u r f a c e w a v e m o d e s . S o m e l a t e a r r i v i n g p u l s e s c a n b e e x p l a i n e d a s r e f l e c t i o n s f r o m t h e v e r t i c a l a n d h o r i z o n t a l b o u n d a r i e s o f a s e d i m e n t a r y b a s i n . T h e d e p e n d e n c e o f t h e n u m b e r o f s t r o n g m o t i o n p u l s e s a n d t h e i r d u r a t i o n o n v a r i o u s p a r a m e t e r s w e r e s t u d i e d t o e s t a b l i s h t h e s c e n a r i o d e s c r i b e d a b o v e . T h e p o s s i b i l i t y o f s t u d y i n g a t t e n u a t i o n w i t h t h e h e l p o f t h e i n c o h e r e n t s c a t t e r e d e n e r g y , r e c o r d e d b y t h e a c c e l e r o m e t e r i n b e t w e e n a n d a f t e r t h e l a s t o f t h e s t r o n g m o t i o n p u l s e s , w a s i n v e s t i g a t e d . T h e s i m p l e e a s y - t o - b e - p r o g r a m m e d m e t h o d o f c o n s t r u c t i n g t h e e n v e l o p e o f t h i s s c a t t e r e d e n e r g y m o t i o n w a s s u g g e s t e d . T h e a t t e n u a t i o n c o e f f i c i e n t Q w a s e s t i m a t e d a n d s h o w n t o a g r e e w i t h t h e r e s u l t s o f s o m e p r e v i o u s s t u d i e s . I t w a s s h o w n t h a t t h e “ s t r o n g m o t i o n c o d a ” h a s s o m e s p e c i f i c f e a t u r e s w h i c h d e p e n d o n t h e v o l u m e o f s e d i m e n t s a t t h e r e c o r d i n g s i t e . T h i s d e p e n d e n c e c o u l d b e u s e d f o r t h e e s t i m a t e o f t h e p h y s i c a l p r o p e r t i e s o f t h e m e d i u m i n t h e l o c a l r e g i o n s u r r o u n d i n g t h e s t a t i o n . T h e r e s u l t s o f t h i s w o r k c a n b e u s e d f o r t h e e s t i m a t e o f d u r a t i o n o f s t r o n g g r o u n d m o t i o n p r o d u c e d b y f u t u r e e a r t h q u a k e s , w h e n t h e p a r a m e t e r s o f t h e s h o c k a n d t h e s i t e a r e k n o w n o r c a n b e e s t i m a t e d . O u r r e g r e s s i o n e q u a t i o n s a r e a l s o u s e f u l f o r c o n s t r u c t i o n o f r e a l i s t i c s y n t h e t i c a c c e l e r o g r a m s , a s t h e d u r a t i o n 451 o f s t r o n g g r o u n d m o t i o n i s o n e o f t h e e s s e n t i a l p a r a m e t e r s n e c e s s a r y f o r t h e g e n e r a t i o n o f a r t i f i c i a l s t r o n g g r o u n d m o t i o n ( T r i f u n a c , 1 9 7 1 b ; W o n g a n d T r i f u n a c , 1 9 7 8 , 1 9 7 9 ) . A n o t h e r f u t u r e a p p l i c a t i o n is r e l a t e d t o t h e e s t i m a t e o f t h e h a z a r d o f e a r t h q u a k e s h a k i n g , w h i c h c a n a l s o b e c h a r a c t e r i z e d b y t h e p r o b a b i l i s t i c m e a s u r e s o f t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n . 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G e n e r a t i o n o f A r t i f i c i a l S t r o n g M o t i o n A c c e l e r o g r a m s , Earthquake Eng. Structural Dynamics, 7 , 5 0 9 - 5 2 7 . W o o d , H . O . a n d F . N e u m a n n ( 1 9 3 1 ) . M o d i f i e d M e r c a l l i I n t e n s i t y S c a le o f 1 9 3 1 , Bull. Seism. Soc. Amer., 2 1 , 2 7 7 - 2 8 3 . W u , R . - S . ( 1 9 8 5 ) . M u l t i p l e S c a t t e r i n g a n d E n e r g y T r a n s f e r o f S e i s m i c W a v e s — S e p a r a t i o n o f S c a t t e r i n g E f f e c t f r o m I n t r i n s i c A t t e n u a t i o n — I . T h e o r e t i c a l M o d e l l i n g , Geophys. J.R. Astr. Soc., 8 2 , 5 7 - 8 0 . 464 W u , R . - S . , a n d K . A k i ( 1 9 8 5 ) . S c a t t e r i n g o f E l a s t i c W a v e s b y a R a n d o m M e d i u m a n d t h e S m a l l S c a le I n h o m o g e n e i t i e s i n t h e L i t h o s p h e r e . , J. Geophys. Res., 9 0 , 1 0 2 6 1 - 1 0 2 7 6 . W u , R . - S . , a n d K . A k i ( 1 9 8 8 ) . 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Res., 9 6 , 6 0 7 - 6 1 9 . 465 A PPEN D IX A Filters Used in the Com putation of Duration and the Limitations They Impose T w o m a j o r s t e p s , w h i c h i n v o l v e d t h e d i g i t a l f i l t e r i n g , a r e r e q u i r e d f o r c a l c u l a t i o n o f t h e f r e q u e n c y d e p e n d e n t d u r a t i o n a s i t i s d e f i n e d i n t h i s w o r k ( s e e E q . ( 2 . 1 ) ) . F i r s t , e a c h c o m p o n e n t o f a c c e l e r a t i o n , v e l o c i t y o r d i s p l a c e m e n t h a s t o b e b a n d - p a s s f i l t e r e d , s o t h a t 1 2 f r e q u e n c y b a n d s a r e o b t a i n e d . N e x t , s e p a r a t e l y f o r e a c h f r e q u e n c y b a n d , t h e s q u a r e o f t h e f u n c t i o n f ( t ) ( w h i c h r e p r e s e n t s b a n d - p a s s f i l t e r e d c o m p o n e n t o f a c c e l e r a t i o n , v e l o c i t y o r d i s p l a c e m e n t ) h a s t o b e i n t e g r a t e d t o g e t / J / 2 ( r ) d r ; t h e r e s u l t s h o u l d b e s m o o t h e d a n d d i f f e r e n t i a t e d : J t { / f2{j)dT I ’ ^ 1 U O J s m o o th e d ! F u n c t i o n ( A . l ) i s u s e d t h e n i n t h e d e f i n i t i o n o f t h e t i m e i n t e r v a l s q u a l i f i e d t o b e c a l l e d t h e “ s t r o n g m o t i o n . ” A s i t is k n o w n ( s e e , f o r e x a m p l e , H a m m i n g , 1 9 8 3 ) t h e s h a r p n e s s o f t h e f i l t e r is r e l a t e d t o t h e l e n g t h o f i t s i m p u l s e r e s p o n s e . I n g e n e r a l , t h e f o l l o w i n g e x p r e s s i o n a p p r o x i m a t e l y h o l d s : ( A / ) - ( r 0) « l , (A. 2 ) w h e r e A / is t h e w i d t h o f t h e f i l t e r ( f o r t h e b a n d p a s s f i l t e r ) a n d T o is t h e r e l a x a t i o n t i m e — t h e l e n g t h o f t h e f i l t e r ’ s i m p u l s e r e s p o n s e . ( T h e r e a d e r s h o u l d n o t m i x A / t h a t d e s i g n a t e s t h e f i l t e r w i d t h a n d / ( f ) t h a t s t a n d s f o r t i m e d e p e n d e n t a c c e l e r a t i o n , v e l o c i t y o r d i s p l a c e m e n t ) . T i m e r = l / A / g i v e s t h e m i n i m u m d u r a t i o n o f a p h y s i c a l p r o c e s s t h a t c a n s t i l l b e r e c o g n i z e d a f t e r b e i n g f i l t e r e d b y t h e b a n d - p a s s f i l t e r w i t h t h e w i d t h A / . 466 O r m s b y f i l t e r s w e r e u s e d i n c a s c a d e t o g e t 1 2 b a n d - p a s s e d v e r s i o n s o f e a c h c o m p o n e n t o f m o t i o n . T h i s p r o c e d u r e is e f f e c t i v e l y e q u i v a l e n t t o t h e a p p l i c a t i o n o f t h e f i l t e r s w i t h t h e c u t - o f f (fi a n d / 4 ) a n d t h e r o l l - o f f ( / 2 a n d / 3 ) f r e q u e n c i e s l i s t e d i n T a b l e A . l ( “ s t e p o n e ” ) f o r e a c h c h a n n e l . T h e c e n t r a l f r e q u e n c y o f e a c h c h a n n e l i s d e f i n e d a s fo = V/Wsi a n d t h e c e n t r a l p e r i o d o f t h e c h a n n e l is To = l / / o - F o r d e f i n i t e n e s s , b u t q u i t e a r b i t r a r i l y , w e d e f i n e t h e w i d t h o f t h e f i l t e r t o b e l o g ( A / ) = { l o g ( / 3 ) - l o g ( / 2 ) } + i { [ l o g ( / 2 ) - l o g ( / i ) ] + [ l o g ( ( / 4 ) - l o g ( / 3 ) ] } , w h i c h s i m p l i f i e s t o : « - w r - T h e s c h e m e w h i c h i l l u s t r a t e s t h e d e f i n i t i o n ( A . 3 ) is s k e t c h e d o n F i g . A . I . T a b l e A . l s h o w s t h e v a l u e s o f A / a n d o f t h e e s t i m a t e d r e l a x a t i o n t i m e r = l / A / f o r e a c h c h a n n e l . T h e v a l u e o f t g i v e s t h e a p p r o x i m a t e l o w e r b o u n d o n t h e d u r a t i o n o f a p u l s e t h a t c a n b e t h e o r e t i c a l l y m e a s u r e i n t h e r e c o r d a f t e r i t h a s b e e n b a n d - p a s s f i l t e r e d . N o t e t h e h i g h v a l u e s o f t f o r t h e l o w f r e q u e n c y c h a n n e l s . R e c a l l i n g t h e r e s u l t s o f t h e r e g r e s s i o n a n a l y s i s p e r f o r m e d o n m a n y m o d e l s d e s c r i b e d i n t h i s w o r k ( s e e , f o r e x a m p l e , T a b l e 4 . 2 , m o d e l E q . ( 4 . 9 ) ) , w e s h o u l d b e a l a r m e d b y t h e f a c t t h a t t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n o n c h a n n e l # 1 i s e q u a l , i n a v e r a g e , t o t h e r e l a x a t i o n t i m e o f t h e b a n d - p a s s f i l t e r o n t h i s c h a n n e l . T h i s m e a n s t h a t t h e b r o a d e n i n g o f t h e f i l t e r u s e d o n t h e l o w e s t f r e q u e n c y b a n d m a y a l l o w u s t o n o t i c e s o m e p r o p e r t i e s o f d u r a t i o n t h a t w e r e m a s k e d b y t o o l o n g r e s p o n s e t i m e o f t h e f i l t e r u s e d h e r e . 467 F o r t h e d e f i n i t o n o f A / t h e r a t i o o f d i s t a n c e s p : q w a s c h o s e d to b e 2 :1 f4 log (f) Af Fig. A . l T h e b a n d - p a s s f i l t e r o f t h e O r m s b y t y p e a n d t h e d e f i n i t i o n o f i t s w i d t h A / . T h e f r e q u e n c i e s fi a n d / 4 r e p r e s e n t t h e c u t - o f f a n d t h e f r e q u e n c i e s / 2 a n d f 3 r e p r e s e n t t h e r o l l - o f f o f t h e O r m s b y f i l t e r a m p l i t u d e s w h i c h f o r m t h i s b a n d - p a s s f i l t e r . F o r t h e d e f i n i t i o n o f t h e w i d t h , t h e r a t i o o f “ d i s t a n c e s ” p:q w a s c h o s e n t o b e 2 :1 ( s o m e w h a t a r b i t r a r i l y ) . T a b l e A . l T h e p r o p e r t i e s o f f i l t e r s u s e d i n t w o - s t e p p r o c e s s o f c a l c u l a t i n g f r e q u e n c y d e p e n d e n t d u r a t i o n o f s t r o n g g r o u n d m o t i o n . _______________________________________________________________________ S t e p o n e : b a n d p a s s f i l t e r i n g S tep tw o: sm oothing o f the t J / 2(T) d t 0 C hannel num ber Central freq. fo Central period T o C u t-off and ro ll-off frequencies o f the bandpass filter, (H z) 1 T T ^ C om er freq. fc « c t ' *0 1 0.075 Hz 13.4 sec . 0 5 - . 0 7 ; . 0 8 - . 1 0 4 2 sec 3.2 0 .0 3 8 H z 2 6 sec 1.9 2 0.1 2 H z 8.2 sec . 0 8 - . 1 0 ; . 1 5 - . 1 7 16 sec 1.9 0 .0 6 H z 16 sec 1.9 3 0.21 Hz 4 .7 sec . 1 5 - . 1 7 ; . 2 7 - . 3 0 8.6 sec 1.8 0.11 H z 9 sec 1.9 4 0 .3 7 Hz 2.7 sec .27 - .30 ; .45 - .50 5 .7 sec 2.1 0 .1 4 H z 7 sec 2.6 5 0.63 H z 1.6 sec .45 - .50 ; .80 - .90 2 .9 sec 1.8 0.1 7 H z 6 sec 3.8 6 1.1 H z 0.85 sec .8 0 - . 9 0 ; 1 .3 0 - 1.50 2.0 sec 2.2 0 .2 0 H z 5 sec 5.9 7 1.7 H z 0.55 sec 1 .3 0 - 1 .5 0 ; 1 .9 0 - 2 .2 0 1.8 sec 3.0 0.23 H z 4 sec 7.3 8 2.5 Hz 0.3 4 sec 1 .9 0 - 2 .2 0 ; 2 .8 0 - 3 .5 0 1.1 sec 2.7 0 .2 6 H z 3.8 sec 11 9 4 .2 H z 0.20 sec 2 .8 0 - 3 .5 0 ; 5 .0 0 - 6 .0 0 0.4 8 sec 2.0 0.2 8 H z 3.6 sec 18 10 7.2 H z 0.12 sec 5 .0 0 - 6 .0 0 ; 8 .7 5 - 10.25 0.2 8 sec 2.0 0 .3 0 H z 3.3 sec 28 11 13 Hz 0 .0 6 9 sec 8.75 - 1 0 .2 5 ; 1 6 .0 0 - 1 8 .0 0 0 .1 4 sec 1.9 0 .3 2 H z 3.1 sec 45 12 21 H z 0 .0 4 5 sec 1 6 .0 0 - 1 8 .0 0 ; 2 5 .0 0 - 2 7 .0 0 0.12 sec 2.5 0.35 H z 2.9 sec 64 4 6 9 O n e o f t h e c o l u m n s i n T a b l e A . l g i v e s t h e r a t i o r / T o f o r a l l f r e q u e n c y b a n d s . T h i s r a t i o s h o w s t h e r e l a t i v e w i d t h o f t h e f i l t e r . Z a p o l s k i ( 1 9 6 0 a , b , 1 9 7 1 ) , u s e d t h e r a t i o t / T q w h i c h w a s c h a n g i n g f r o m 1 f o r fo = 0 . 0 2 2 H z t o 3 f o r fo = 4 0 H z . H i s a n d s i m i l a r d e v i c e s w e r e p r o v e d t o b e v e r y u s e f u l i n s e i s m o l o g y a n d w e r e u s e d b y m a n y s c i e n t i s t s . T h u s , f o l l o w i n g t h i s i n v e s t i g a t o r , w e m a y h a v e t o c o n s i d e r t h e b r o a d e n i n g o f t h e l o w f r e q u e n c y c h a n n e l s t o g e t m o r e i n f o r m a t i o n a b o u t t h e p r o p e r t i e s o f l o n g p e r i o d d u r a t i o n o f s t r o n g g r o u n d m o t i o n . W e w i l l c o n s i d e r n o w t h e s e c o n d s t e p i n t h e p r o c e s s o f c a l c u l a t i n g t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n , i . e . g e t t i n g t h e s m o o t h e d v e r s i o n o f f * f 2[t)(Lt a n d d i f f e r e n t i a t i n g i t . T h e i n t e g r a t i o n w a s c o m p l e t e d w i t h 5 - p o i n t C h e b y s h e v i n t e g r a t o r ( N o v i k o v a a n d T r i f u n a c , 1 9 9 1 ) a n d t h e d i f f e r e n t i a t i o n — w i t h o n e o f t h e s i m p l e s t f i l t e r s d e s c r i b e d b y H a m m i n g ( 1 9 8 3 ) . T h e p r o b l e m s c o m e w h e n t h e d e c i s i o n a b o u t s m o o t h i n g h a s t o b e m a d e . T h e r e a s o n s w h y t h e s m o o t h i n g is n e e d e d w e r e a l r e a d y d i s c u s s e d i n C h a p t e r 2 . H e r e w e w o u l d l i k e j u s t t o e m p h a s i z e , t h a t p e r f o r m i n g s m o o t h i n g ( l o w - p a s s f i l t e r i n g ) w i t h t h e c u t - o f f f r e q u e n c y fc ^ foi w e s e v e r a l l y c u t t h e i n f o r m a t i o n t h a t i n i t i a l l y w a s p r e s e n t i n t h e b a n d p a s s f i l t e r e d r e c o r d . T h e r i g h t h a n d s i d e o f t h e T a b l e A . l ( “ s t e p t w o ” ) g i v e s t h e v a l u e s o f f c a n d t h e e s t i m a t e d r e l a x a t i o n t i m e o f t h e s m o o t h i n g f i l t e r t ' « l / / c f o r a l l 1 2 c h a n n e l s . ( I n t h e c a s e o f t h e l o w p a s s f i l t e r A / i n ( A . 2 ) s t a n d s f o r f c). T h e r a t i o t ' / t o i n t h e r i g h t m o s t c o l u m n o f t h e T a b l e A . l s h o w s b y h o w m a n y t i m e s t h e d u r a t i o n o f t h e s h o r t e s t s i g n a l , w h i c h o n e c a n g e t a f t e r t h e b a n d - p a s s f i l t e r i n g ( “ s t e p o n e ” ) , is p r o l o n g e d , w h e n t h e s a m e s i g n a l i s s m o o t h e d i n t h e l o w - p a s s f i l t e r i n g ( “ s t e p t w o ” ) . W e c a n n o w r e c a l l t h e m o d e l s w h i c h c o n s i d e r t h e l e n g t h o f a p u l s e o f s t r o n g g r o u n d m o t i o n , durpalae ( s e e , f o r e x a m p l e , T a b l e 6 . 3 , m o d e l E q . ( 6 . 1 7 ) ) . H i g h - a n d e v e n i n t e r m e d i a t e - f r e q u e n c y c h a n n e l s s h o w t h a t 470 i n a v e r a g e , d u r p u ls e « t'. T h u s , o n e s h o u l d b e v e r y c a r e f u l i n i n t e r p r e t a t i o n o f t h e r e s u l t s o b t a i n e d u s i n g t h e d u r a t i o n o f a n i n d i v i d u a l p u l s e o f s t r o n g g r o u n d m o t i o n . I f o n e d e s i r e s t o i n v e s t i g a t e t h e f e a t u r e s o f t h e s e p u l s e s m o r e c a r e f u l l y , f o r e x a m p l e f o r t h e p u r p o s e o f b e t t e r u n d e r s t a n d i n g t h e r u p t u r e p r o c e s s a t t h e s o u r c e , t h e n t h e f r e q u e n c y o f t h e s m o o t h i n g f i l t e r f c h a s t o b e i n c r e a s e d . H o w e v e r , s o m e p r a c t i c a l l i m i t o f t h e v a l u e o f fc h a s t o b e i n t r o d u c e d , a s i t h a r d l y m a k e s a n y s e n s e t o t a l k a b o u t 5 0 s t r o n g m o t i o n p u l s e s ( o f f i n i t e d u r a t i o n ) e x c i t i n g a s t r u c t u r e d u r i n g t h e i n t e r v a l o f t i m e i n t e r v a l o f a b o u t , s a y , 3 s e c o n d s . A s a r u l e , e n g i n e e r i n g s t r u c t u r e s h a v e t h e f r e q u e n c y o f t h e f i r s t m o d e n o t e x c e e d i n g s e v e r a l H z , a n d v e r y l i g h t d a m p i n g . H e n c e , s e v e r a l s u c c e s s i v e s h o r t p u l s e s o f s t r o n g m o t i o n , s e p a r a t e d b y s h o r t s i l e n c e g a p s , w i l l b e “ f e l t ” b y t h e s t r u c t u r e a s o n e p r o l o n g e d p u l s e . A s w e s e e , t h e c h o i c e o f t h e f i l t e r s m a y i n f l u e n c e t h e r e s u l t s o f t h e a n a l y s i s p e r f o r m e d o n t h e f i l t e r e d d a t a v e r y s i g n i f i c a n t l y , a n d s o a c o n s i d e r a b l e a t t e n t i o n s h o u l d b e p a i d t o t h e p r o c e s s o f c h o o s i n g f i l t e r s , a n d t h e l i m i t a t i o n t h e y i m p o s e s h o u l d b e t a k e n i n t o a c c o u n t w h e n t h e r e s u l t s a r e i n t e r p r e t e d . 471 A PPEN D IX B A ccepted and Rejected Data: Making the Decision A s i t w a s a l r e a d y m e n t i o n e d i n C h a p t e r 2 , t h e d a t a w e r e “ c l e a n e d - u p ” b e f o r e a n y o f t h e r e g r e s s i o n a n a l y s e s o f t h e m o d e l s d e s c r i b e d i n C h a p t e r s 3 - 8 , a n d i n A p p e n d i c e s D a n d E , w e r e i n i t i a t e d . T h e r e a s o n f o r t h i s w a s t o e x c l u d e a n y “ n o i s e ” o f a n y k i n d t h a t c a n b e p r e s e n t i n t h e d a t a a n d t o g e t a “ c l e a n ” d a t a s e t c a p a b l e t o s e r v e a s t h e b a s i s f o r t h e m e a n i n g f u l a n d r e l i a b l e r e s u l t s . T h e “ c l e a n i n g - u p ” p r o c e d u r e w a s p e r f o r m e d m a n u a l l y . T h e d e c i s i o n o f a c c e p t i n g o r r e j e c t i n g t h e p a r t i c u l a r c a s e w a s m a d e f o r a l l 1 4 8 2 c o m p o n e n t s o f a c c e l e r a t i o n , v e l o c i t y a n d d i s p l a c e m e n t , s e p a r a t e l y f o r e a c h o f t h e 1 2 f r e q u e n c y b a n d s . F o r t h e p u r p o s e o f t h i s a n a l y s i s , 3 * 1 4 8 2 p l o t s l i k e t h e o n e s h o w n o n F i g . 2 . 9 a n d d e s c r i b e d i n d e t a i l i n C h a p t e r 2 , w e r e p r e p a r e d . T h r e e o t h e r s i m i l a r p l o t s a r e s h o w n o n F i g . B . l ; t h e s e a r e t h e v e r t i c a l c o m p o n e n t o f a ) a c c e l e r a t i o n , b ) v e l o c i t y a n d c ) d i s p l a c e m e n t o f t h e S a n F e r n a n d o e a r t h q u a k e , r e c o r d e d a t B e v e r l y H i l l s . W e w i l l p r e s e n t h e r e j u s t a b r i e f d e s c r i p t i o n o f t h e s e p l o t s a n d r e f e r t h e r e a d e r t o C h a p t e r 2 f o r m o r e d e t a i l e d e x p l a n a t i o n s . T h e t o p c u r v e o n t h e l e f t h a n d s i d e o f t h e s e p l o t s d i s p l a y s t h e i n i t i a l u n f i l t e r e d s i g n a l . T h e c u r v e s b e l o w i t r e p r e s e n t t h e b a n d - p a s s f i l t e r e d f u n c t i o n , / ( < ) , f o r a l l c h a n n e l s , s t a r t i n g f r o m t h e o n e w i t h t h e l o w e s t f r e q u e n c y . f( t ) h e r e s t a n d s f o r t h e a c c e l e r a t i o n a(t), v e l o c i t y v(t) or d i s p l a c e m e n t d(t). T h e r i g h t - h a n d s i d e o f t h e f i g u r e g i v e s t h e g r a p h o f / 2 ( r ) d r f o r e a c h c h a n n e l a n d t h e f i n a l n u m e r i c a l v a l u e o f t h i s i n t e g r a l a t t h e e n d o f t h e r e c o r d J 0 <0 / 2 ( r ) d r , w h e r e t o is t h e l e n g t h o f t h e r e c o r d . A f t e r a l l p l o t s a r e p r e p a r e d , t h e n e x t s t e p is t o a s s i g n a s p e c i f i c n u m b e r , t , w h i c h w e w i l l c a l l “ t h e i n d e x o f a c c e p t a n c e , ” t o e a c h o f t h e b a n d - p a s s e d c o m p o n e n t s o f A C C E L E R A T I O N - G/10 472 -0.2 0 0.2 -0.1 0 0.1 -0.1 0+- 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 SAN FERNANDO EA R T H Q U A K E FEB 9, 1971 - 0600 PST 435 NO R TH OfKHURST AVENUE, BA SEM ENT, BE V ER LY H ILLS, C A L . C O M P D O W N .11- 13,25.0-27.0 hz 0.037 G ^a a a A / w v v ^ ^ /V v w v a a .05-.07,.08-.10 hz (Outside passband) _L % .08-10,15-. 17 hz 0.001G # 2 l .15-.17..27-.30 hz J L 0.002 G # 3 .27-.30,.45- 50 hz 0.007 G _L .45-.50,.80-.90 hz 0.007 G # 5 .80-.90,1.30-1.50 hz ‘ — 'W\/\AA/^\/\(y\/w\/LA/vvv\/wvWv '' 1.3-1.5,1.8-2.2 hz 0.007 G # 6 _ _ l ______ 0.013G #■? 1.8-2.2,2.8-3.5 hz _L 0.017G # 8 2.8-3.5,5.0-6.0 hz I ____________I _____ 0.016G # 9 5.0-6.0,8.75-10.25 hz . I ____________I ______ 0.010G #10 8.75— 10.25,16.0— 18.0 hz 0.008G *— — T #11 'm i i ' 16.0-18.0,25.0-27.0 hz 0.004G #12 0 10 20 30 40 TIME - SECONDS 20 1831 H 16.9s 1 . 5 I ' 26.8s 17.2s - 11.Is - 176 I 20.7 s - 78.5 Z | 19.1s 299 16.9s - 253 14.4s 253 10.8s H 14.1s - 92.1 12.0s - 8.0 12.5s H 4 0 F ig . B . l a Original acceleration record (top) and its 12 band-pass filtered chan nels. T he right-m ost column gives the value of the “index of acceptance.” VELOCITY - C M /SE C 4 7 3 SAN FER N A ND O EA R T H Q U A K E FEB 9, 1971 - 0600 PST 435 NO RTH O A K H U R ST AVENUE, BA SEM EN T, BEVERLY H ILLS, C A L . C O M P D O W N .11-. 13,25.0-27.0 hz X cm /’ s 69.0 I 19.1s - .05-.07,.08-.10 hz (Outside passband) J____________I _____ 0 % .08-.10,.15-.17 hz —* X 0.14 cm /s # 2 0.09 I 21.8s - .15-.17,.27-.30 hz L4/ cm /s # 3 14.0 20.4s H 45-.50 hz 3.0 cm /s # 4 26.5 I 13.5s - 00-.90 hz cm/s # 5 _1____ 11.6 19.9 s H .80-.90,1.30-1.50 hz cm/s # 6 20.7 s - 1.3-1.5,1.9-2.2 hz cm /s # 7 0 1 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 1.9-2.2,2.8-3.5 hz cm /s # 8 14.6 s 2.0-3 5,5.0-6.0 hz 0.57 c m /s 5 7 V s # 9 14.2s 5.0-6.0,8.75-10.25 hz 0.20 cm /s #10 0.06 I 14.8s - 8.75-10.25,16.0-18.0 hz X 0.11 cm /s 0.01 16.5s 16.0-18.0,25.0-27.0 hz 0.067 c m /s #12 0.002 I 19.3s 0 10 20 30 40 TIME - SECONDS 2 0 40 F i g . B . l b O r i g i n a l v e l o c i t y r e c o r d ( t o p ) a n d i t s 1 2 b a n d - p a s s f i l t e r e d c h a n n e l s . T h e r i g h t - m o s t c o l u m n g i v e s t h e v a l u e o f t h e “ i n d e x o f a c c e p t a n c e . ” 474 SAN FER N A ND O E A R T H Q U A K E FEB 9, 1971 - 0600 PST 435 NO R TH O A K H U R ST A V EN UE, BA SEM ENT, BEVERLY H ILLS, C A L . C O M P D O W N -2 0 2 -1 0 0 1 -0 .3 0 0.3 g - 0 . 3 w 2 g 0.3 3 - 0 .3 m 0 ® 0 .3 - 0 .3 0 0 .3 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 _L 13,25.0-27.0 hz I 2.3 c m .05-.07,.08-.10 hz (Outside passband) % .08-.10,.15-.17 hz 0 .3 7 c m # 2 15-.17,.27- 30 hz 1.1 c m # 3 .27-.30,.45-.50 hz 1.3 c m # 4 -.90 hz 0 .5 0 c m # 5 .80- 90,1.30-1.50 hz N A W y \ A A / W w \ A w ^ W W 0 .1 6 c m # 6 1.3-1.5,1.9-2.2 hz 0.12 c m # 7 10 20 3 0 4 0 TIME - SECONDS 1.9-2.2,2.8-3.5 hz 0 .0 7 0 c m — f i I i 0 .0 0 5 _ 1 5 .6 s - 1 ~ 1 1 1 #8 1 f i i 2.8-3.5,5.0-6.0 hz 0 .0 4 7 c m - r" 0 .0 0 4 _ 1 1 1 # 9 1 | it).4 : S 1 “ 5.0-6.0,8.75-10.25 hz 0 .0 0 8 c m - 0.000 _ 21.8 s - 1 _ 1 1 1 #10 1 ^ T * ! 1 8.75-10.25,16.0-18.0 hz 0 .0 1 7 c m - * M i ■ 0.000 _ 1 7 .5 s - 1 “ 1 • i p i u p i i #11 I i i 16.0-18.0,25.0-27.0 hz 0.021 c m ~ r" 0.000 _ 1 8 .4 s - l “ 1 1 1 #12 1 i 1 9 .4 s - Ul_ 20 0.6 2 6 .1 s 1 7 .9 s 0 .0 6 20.1s 0 .0 3 21.1 s 4 0 F ig . B .l c Original displacement record (top) and its 12 band-pass filtered channels. The right-m ost column gives the value of the “index of acceptance.” 475 a l l r e c o r d s . A s b e f o r e , w e w i l l c a l l h e r e o n e s u c h c o m p o n e n t a s t h e “ d a t a p o i n t . ” T h e f o l l o w i n g s y s t e m o f i n d i c e s w a s a d o p t e d : i = 0 : t h e f r e q u e n c y b a n d u n d e r c o n s i d e r a t i o n is o u t s i d e t h e n o i s e - f r e e p a r t o f t h e s p e c t r u m , a d o p t e d f o r t h i s c o m p o n e n t d u r i n g d i g i t i z a t i o n a n d d a t a p r o c e s s i n g ( T r i f u n a c a n d L e e , 1 9 7 9 a ; L e e a n d T r i f u n a c , 1 9 9 0 ) . T h i s m e a n s t h e r e a r e n o d a t a i n t h e c h a n n e l t o b e e v e n c o n s i d e r e d . T h e c a s e c o r r e s p o n d s t o t h e e m p t y b o x o n t h e F i g B . l ; i = 3 : t h e b a n d - p a s s f i l t e r e d d a t a e x i s t , b u t p r a c t i c a l l y c o n s i s t o f n o t h i n g b u t n o i s e a n d s h o u l d b e r e j e c t e d ; i = 5 : t h e d u r a t i o n o f s t r o n g m o t i o n s e e m s t o e x c e e d t h e l e n g t h o f t h e r e c o r d a t t h e f r e q u e n c y b a n d c o n s i d e r e d . T h e r e is n o u s e i n a c c e p t i n g t h i s d a t a p o i n t . i = 2 : t h e s t r o n g p a r t o f t h e g r o u n d m o t i o n s e e m s t o b e c o m i n g t o i t s e n d a p p r o x i m a t e l y a t t h e s a m e t i m e w h e n t h e r e c o r d i n g w a s s t o p p e d . T h e d a t a p o i n t c a n b e c o n s i d e r e d i n t h e a n a l y s i s . i — 1 : t h e p e r f e c t d a t a p o i n t , t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n is s i g n i f i c a n t l y le s s t h a n t h e l e n g t h o f t h e r e c o r d , t h e a m p l i t u d e s a r e w e l l a b o v e t h e n o i s e l e v e l . t = 4 : t h e r e c o r d w i t h a n a f t e r s h o c k , c a n b e u s e d i n t h e a n a l y s i s a f t e r b e i n g t r u n c a t e d t o e x c l u d e t h e a f t e r s h o c k . W e w i l l n o w s h o w o n e x a m p l e h o w t h e “ i n d e x o f a c c e p t a n c e ” c a n b e a s s i g n e d a s p e c i f i c v a l u e . F o r t h e p u r p o s e o f t h e d e m o n s t r a t i o n w e p r i n t e d t h e v a l u e o f t , c o r r e s p o n d i n g t o e a c h c h a n n e l , a s t h e r i g h t m o s t c o l u m n i n F i g . B . l . W e c o n s i d e r n o w e a c h c h a n n e l i n F i g . B . l s e p a r a t e l y . F o r t h e l o w e s t f r e q u e n c y b a n d , t h e d e c i s i o n is o b v i o u s : i = 0 o n a l l t h r e e f u n c t i o n s — a c c e l e r a t i o n , 476 v e l o c i t y a n d d i s p l a c e m e n t . C h a n n e l # 2 h a s i = 3 . T h e r e a s o n f o r t h i s i s t h e f o l l o w i n g . A s w e a l r e a d y n o t e d , o u r f r e q u e n c y b a n d s c a n b e c o n s i d e r e d a s r e l a t i v e l y n a r r o w , s o t h a t t h e a c c e l e r a t i o n a(t) , v e l o c i t y v(t) a n d d i s p l a c e m e n t d(t) s h o u l d h a v e v e r y s i m i l a r w a v e s t r u c t u r e , b u t j u s t d i f f e r e n t a m p l i t u d e s . W e e x p e c t , a l s o , t h e i n t e g r a l s a 2 ( r ) d r , v2(t)c I t a n d f* d 2 ( r ) d r t o h a v e s i m i l a r s h a p e s . I f i t i s n o t s o , t h e c h a n n e l u n d e r c o n s i d e r a t i o n m o s t p r o b a b l y h a s n o t h i n g b u t n o i s e . I n s o m e c a s e s , w e a l s o u s e d t h e e s t i m a t e o f d u r a t i o n o f s t r o n g g r o u n d m o t i o n a s o n e c o n t i n u o u s t i m e i n t e r v a l . T h e b e g i n n i n g s a n d t h e e n d s o f t h e s e i n t e r v a l s a r e s h o w n b y d a s h e d l i n e s o n t h e r i g h t - h a n d s i d e i n t h e F i g . B . l , a n d t h e e s t i m a t e o f d u r a t i o n i n s e c o n d s is p r i n t e d f o r e a c h c h a n n e l . I f t h e s i g n a l i n t h e c h a n n e l is w e l l a b o v e t h e n o i s e , t h a n t h e e s t i m a t e o f d u r a t i o n s h o u l d b e s i m i l a r f o r a c c e l e r a t i o n , v e l o c i t y a n d d i s p l a c e m e n t . O n e m o r e c r i t e r i o n is t h e p o s i t i o n o f t h e m a x i m u m a m p l i t u d e s o f a(t), v(t) a n d d(t) , m a r k e d b y a n a s t e r i s k o n F i g . B . l . I f t h e t i m e w h e n t h e m a x i m u m o c c u r s i s t h e v e r y f i r s t o r t h e v e r y l a s t p o i n t o f t h e b a n d - p a s s e d r e c o r d , t h i s c a n b e a n d i n d i c a t o r t h a t t h i s m a x i m u m i s t h e r e s u l t o f t h e f i l t e r i n g p r o c e s s . ( F o r f i l t e r i n g t o b e p e r f o r m e d , t h e r e c o r d is e x t e n d e d b e y o n d t h e f i r s t a n d t h e l a s t p o i n t s b y t h e h a l f - l e n g t h o f t h e f i l t e r , a n d t h i s c a u s e s d i s t u r b a n c e s i n f i l t e r e d o u t p u t w h i c h a r e p r o p o r t i o n a l t o t h e d i s t a n c e f r o m t h e b e g i n n i n g a n d t h e e n d o f t h e r e c o r d ) . T h e l a s t c r i t e r i o n s h o u l d b e u s e d o n l o n g - p e r i o d c h a n n e l s o n l y , b e c a u s e h i g h f r e q u e n c y b a n d s h a v e v e r y s h o r t i m p u l s e r e s p o n s e ( f i l t e r l e n g t h ) a n d t h e d e s c r i b e d e f f e c t c a n n o t b e n o t i c e d . C h a n n e l s # 3 ■ — 6 h a v e t = 5 o n a ( £ ) , v(t) a n d d(t), w h i c h i n d i c a t e s t h a t t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n o n t h e s e f r e q u e n c y b a n d s e x c e e d s t h e l e n g t h o f t h e r e c o r d . I n d e e d , o n e m a y n o t i c e t h a t o n c h a n n e l s # 5 a n d # 6 f* a2(r)dr, fo v2(r)dT a n d d2(r)dT h a v e b i g p o s i t i v e d e r i v a t i v e s a t t — to, w h e r e to is t h e 477 r e c o r d ’ s l e n g t h . T h i s m e a n s s o m e m o r e “ s t r o n g m o t i o n e n e r g y ” i s e x p e c t e d t o c o m e a f t e r t = to a.t t h e s e f r e q u e n c y b a n d s . C o m p a r i n g t h e s t r u c t u r e o f t h e n o n f i l t e r e d m o t i o n , s h o w n o n t h e v e r y t o p o f t h e f i g u r e a n d t h e b a n d - p a s s f i l t e r e d f u n c t i o n s o n c h a n n e l s # 3 - r 6 , o n e m a y e x p e c t a t l e a s t o n e m o r e s t r o n g m o t i o n p u l s e o n c h a n n e l s # 3 a n d # 4 . A l l o f t h e a b o v e f o r c e s u s t o a s s i g n * = 5 t o a l l 4 c h a n n e l s , f r o m # 3 u p t o # 6 . T h e n e x t c h a n n e l t o c o n s i d e r is $ 7 . T h e d e r i v a t i v e o f / 2 ( r ) d r , w h e r e f{t) i s o ( t ) , v(t) o r d(t) i s p r a c t i c a l l y z e r o , s o w e a s s i g n t = 1 t o t h i s c h a n n e l . C h a n n e l # 8 h a s i = 2 f o r a l l f u n c t i o n s o f m o t i o n ( a c c e l e r a t i o n , v e l o c i t y a n d d i s p l a c e m e n t ) . T h e h i g h e s t f r e q u e n c y c h a n n e l , n u m b e r 1 2 , is n o t a c c e p t a b l e a n d h a s t = 3 o n a l l f u n c t i o n s o f m o t i o n . T h e a m p l i t u d e s o f / ( < ) o n t h i s c h a n n e l a r e v e r y l o w , a n d t h e f i n a l v a l u e o f /J / 2(r)dr a t t = to is v e r y s m a l l c o m p a r e d w i t h t h i s v a l u e f o r t h e u n f i l t e r e d r e c o r d ( t h e c o r r e s p o n d i n g n u m b e r s a r e p r i n t e d i n t h e b o x w i t h t h e i n t e g r a l / 2(r)dr a t t h e r i g h t - h a n d s i d e o f t h e f i g u r e s ) . A n o t h e r i n d i c a t i o n o f t h e f a c t t h a t / ( < ) o n a ( h i g h f r e q u e n c y ) c h a n n e l c o n s i s t s o f n o i s e o n l y i s t h e v e r y s m o o t h a n d m o n o t o n i c s h a p e o f t h e i n t e g r a l f 2[r)dr ( t h i s c r i t e r i o n , h o w e v e r , d o e s n o t a l w a y s w o r k ) . T h e “ i n d e x o f a c c e p t a n c e , ” t , h a s d i f f e r e n t v a l u e s f o r a c c e l e r a t i o n , v e l o c i t y a n d d i s p l a c e m e n t a t t h e c h a n n e l s # 9 - h 1 1 . T h e i n t e g r a t i o n f i l t e r s o u t h i g h f r e q u e n c i e s , s o v e l o c i t y h a s w o r s e s i g n a l t o n o i s e r a t i o a t h i g h f r e q u e n c i e s t h a n a c c e l e r a t i o n , a n d d i s p l a c e m e n t h a s w o r s e r a t i o t h a n t h a t f o r t h e v e l o c i t y . T h e c o m p a r i s o n o f t h e s h a p e s o f f(t) a n d f 2(r)dr f o r f(r) — a ( t ) , v(t) a n d d(t) g i v e s t h e f o l l o w i n g v a l u e s f o r t h e i n d e x i: c h a n n e l # 9 , t = 1 , 1 , 3 ; c h a n n e l # 1 0 , t = 1 , 1 , 3 ; c h a n n e l # 1 1 , i = 1 , 3 , 3 . T h e v a l u e s o f i h e r e a r e l i s t e d i n t h e o r d e r : a c c e l e r a t i o n , v e l o c i t y a n d d i s p l a c e m e n t . 4 7 8 W e n e x t s h o w s o m e s t a t i s t i c a l r e s u l t s o f a p p l y i n g t h e a b o v e p r o c e d u r e t o a l l t h e r e c o r d s i n t h e d a t a b a s e . F o r s i m p l i c i t y , w e w i l l c o n s i d e r h e r e t h e d i s t r i b u t i o n s f o r a c c e l e r a t i o n o n l y . F i g . B . 2 s h o w s t h e d i s t r i b u t i o n o f t h e a p p r o v e d c a s e s a s t h e f u n c t i o n o f t h e c e n t r a l f r e q u e n c y o f t h e c h a n n e l , f o r t h e h o r i z o n t a l a n d f o r t h e v e r t i c a l c o m p o n e n t s . A s e x p e c t e d , v e r t i c a l m o t i o n h a s s l i g h t l y m o r e a p p r o v e d h i g h - f r e q u e n c y c a s e s , t h a n t h e h o r i z o n t a l . T h e p e a k o f i — 2 d i s t r i b u t i o n is s u b s t a n t i a l l y s h i f t e d t o w a r d s l o w f r e q u e n c i e s a s c o m p a r e d w i t h t = 1 d i s t r i b u t i o n . T h i s f a c t i s r e l a t e d t o t h e i n c r e a s e o f t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n w i t h t h e i n c r e a s e o f t h e p e r i o d o f v i b r a t i o n . F i g . B . 3 d i s p l a y s t h e d i s t r i b u t i o n o f t h o s e c a s e s t h a t w e r e r e j e c t e d b e c a u s e o f i n s u f f i c i e n t a m p l i t u d e s o f t h e s i g n a l i n t h e c h a n n e l s ( i n d i c e s * = 0 a n d t = 3 ) . T h i s g r a p h s h o w s t h a t t h e r e q u i r e m e n t s f o r t h e s i g n a l t o n o i s e r a t i o , a d o p t e d i n t h i s s t u d y a r e m o r e s e v e r e , t h a n i n t h e o r i g i n a l d a t a p r o c e s s i n g u s e d t o o b t a i n t h e p r e s e n t d a t a b a s e ( T r i f u n a c a n d L e e , 1 9 7 9 a ) . F i g . B . 4 g i v e s t h e d i s t r i b u t i o n o f c a s e s w h e r e t h e d u r a t i o n o f s t r o n g g r o u n d m o t i o n i s a p p r o x i m a t e l y e q u a l t o ( t = 2 ) o r l o n g e r t h a n ( i = 5 ) t h e l e n g t h o f t h e r e c o r d . A s e x p e c t e d , t h e d i s t r i b u t i o n s o f i = 2 a n d i = 5 a r e v e r y s i m i l a r — t h e y j u s t h a v e d i f f e r e n t n o r m a l i z a t i o n . T h e d i f f e r e n c e i n t h e d i s t r i b u t i o n f u n c t i o n f o r t h e s e “ l o n g c a s e s ” b e t w e e n t h e h o r i z o n t a l a n d t h e v e r t i c a l c o m p o n e n t s is d i r e c t l y r e l a t e d t o t h e d i f f e r e n c e b e t w e e n t h e h o r i z o n t a l a n d t h e v e r t i c a l c o n s t a n t c o e f f i c i e n t s (a\h^ a n d a ^ ) i n t h e m o d e l s d is c u s s e d a b o v e ( s e e , f o r e x a m p l e , m o d e l E q . ( 4 . 9 ) , F i g . 4 . 1 ) . A s d i s c u s s e d e a r l i e r , F i g . 2 . 1 1 . d i s p l a y s t h e d i s t r i b u t i o n o f c a s e s , a p p r o v e d f o r t h e a n a l y s i s f o r a l l f u n c t i o n s o f m o t i o n — a c c e l e r a t i o n , v e l o c i t y a n d d i s p l a c e - 4 7 9 goQ Horizontal components of acceleration Approved cases: Y /////A index 1 K S index 2 450 V ertica l c o m p o n e n ts o f a c c e le r a tio n Approved cases: Y /////A index 1 index 2 F req uency (Hz) F ig . B .2 D istribution of cases adopted for the regression analysis. Index i = l — perfect d ata point; index i= 2— acceptable d ata point (the strong m otion seems to end approxim ately at the tim e of the term ination of the recording or of the digitization). o f cases Number o f ca ses 4 8 0 Horizontal components of acceleration 900- Rejected cases: 800- W 2 Z Z A index 3 700- index 0 600- 500- 400- • 300- 2 0 0 - 1 0 0 - w q m v / / ( / a w / f i . .21 .37 .63 1.1 1.7 2.5 4.2 7.2 13 21 .075 .12 V ertica l c o m p o n e n ts o f a c c e le r a tio n 500 450- Rejected cases: 400- Y /////A index 3 350- r^~^l index 0 300- 250- 2 0 0 - 1 0 0 - 50 - .21 .37 .075 .12 .63 1.1 1.7 2.5 4.2 7.2 13 21 F req uency (Hz) F ig . B .3 D istribution of cases, rejected from the analysis, due to low signal to noise ratio. Index i= 0—channel outside initial frequency band of th e record (now signal); index i= 3— low signal to noise ratio. Number o f cases Number o f ca ses Horizontal components of acceleration 3 0 0 y "Long" cases: Y /////A index 5 (rejected) index 2 (approved) W Af/l V ertica l c o m p o n e n ts o f a c c e le r a tio n Long” cases: Y /////A index 5 (rejected) index 2 (approved) 1 0 0 - .075 .12 .21 .63 1.1 1.7 2.5 4.2 F requency (Hz) F ig . B .4 D istribution of cases, where the duration of strong ground m otion is practically equal to (index i=2) or longer than (index i=5) the length of the record. 4 8 2 m ent, and gives an impression of the statistical quality of the d ata base a t each frequency band. The described “cleaning-up” procedure allowed us to get the d a ta base of very good quality. As it has been performed manually, this procedure takes a lot of tim e. U nfortunately, we could not come up w ith an equivalent process which can be performed to be perform ed by a com puter. 483 A PPE N D IX C The Solution o f the Linear Least Square Problem by the Singular Value Decom position and other Related Questions T he general linear regression problem can be form ulated as follows. Given m observations of a function bj, j = 1 -r m, and n param eters for each observation a,y, t = 1 -7 - n, a set of coefficients X{, i = 1 -r n, such th a t n bi = '52aii’xi’ y = 1 -f- m, t= i or, in m atrix form, 6 = A-x, b = {bj}JLt , x = { x i } ? = 1 , A = { a i j } , (C .l) has to be found. The exact solution of (C .l) does not always exist in a general sense. It can be shown th at for m > n a generalized inverse of A exists and the “best possible” solution vector x can be obtained. This x satisfies the least squares fit of the d ata 6 to the model under consideration (Hoel, 1971). A reliable m ethod for com puting the coefficients for general least-squares problem s in based on a m atrix factorization known as the singular value decom position (Press et al., 1986). The m ethod uses the theorem stating th a t for any m atrix A, there exist matrices U, W and V, such th a t A = U-W-Vt , (C. 2) where UT = U~\ VT = V~1 4 8 4 and W is the diagonal m atrix with entries w i,W 2 i-!wm called singular values of the m atrix A . Using Eq. (C.2), the Eq. (C .l) can be solved and the unknown vector x obtained in the form: x = E t=i U {iy b (C.3) where t7(t) and V(t) are the i— column-vectors of the m atrices U and V respec tively. Being a linear fit, the solution (C.3) is approxim ate, and its accuracy should be estim ated. Assuming th a t independent observations bj, j = 1 -f- m, have th e sam e variances c x 2, which can be estim ated as „ / „ N 2 < x 2 = (m — n) 3 = 1 \i= 1 / where m — n is the number of the degrees of freedom, the Eq. (C.3) can be rew ritten as x — E Li=l + ± ~ VW ± . wi Ur (C . 4) S tandard deviations of, of the solution x,, * = 1 -r n, can be estim ated as If some of the singular values w* are zeros, the corresponding param eters ajk, j = 1 -r m , have to be excluded from the model, so th a t the w idth of m atrix A and th e dim ension of the solution vector x are reduced. If some Uk « 0, the m atrix A is “alm ost singular.” In this case, as it is seen from Eqs. (C.4) and (C.5), the variances of all solution coefficients (<Ti)2 become big. This is th e m athem atical form ulation of the good empirical rule “The addition of noisy inform ation makes 4 8 5 the results worse.” In practice, it is useful to assume some threshold level of accuracy, r, and disregard all ay* and x*, which correspond to the singular values th a t do not satisfy oJk > r-max{w,-, * = 1 n ). (C-6) T he reduction of the w idth of the m atrix A practically m eans the change of the linear model considered. However, this is necessary if the result of the regression analysis has to be meaningful. In this work, we consider not ju st an abstract m athem atical problem , b u t a real physical process. Thus, we find it reasonable to assume, for exam ple, th a t the vector of the unknown coefficients x , should change sm oothly w ith frequency. In particular, each x,- should be a reasonably sm ooth function of frequency. We took this into account when the decision was made on which x,- should be left in the model. T he model at each channel was chosen in such a way, th a t all x ,( /) have reasonably sm ooth behavior through all frequencies. To accomplish this, we first obtained the set x,- at each frequency band only using Eq. (C.6) as the criterion of acceptance (if Eq. (C.6) is not satisfied, the corresponding Xi was assumed to be zero). As a second step, we studied each dependence xt'(/"), i — 1 - T - n, and identified those channels and coefficients, where x ,( /) is not reasonably sm ooth (jum ps back and fourth around zero by substantial am plitude, often exceeding the am plitude of variation of £,•(/) in the rest of the frequency scale). Those x ,’ s were assumed to be zero and the calculations were perform ed w ith the new model. These two steps were repeated until all x t( /) had sm ooth behavior. Every attem p t was made to leave as many nonzero coefficients as possible at each frequency band. 4 8 6 In some of our models of the strong ground motion duration, this duration appears to depend on some of the m odel’s param eters in a similar m anner for horizontal and for vertical com ponents, and to posses different dependence on differently oriented com ponents for some other param eters of the model. In these cases, the num ber of coefficients i,- in Eq. (C .l) was bigger, th an the num ber of param eters considered in the model, because two different coefficients were assigned for each param eter, where the behavior of the horizontal and the vertical com ponents was different. As far as the values of a,y are concerned, those were taken to be zero if: a) this is the “horizontal” param eter and a “vertical” d ata point is considered and b) this is the “vertical” d ata param eter and a “horizontal” d a ta point is considered. The rest of o,y (for t ’s, corresponding to param eters, “com m on” for horizontal and vertical components) had their values unchanged and equal to the actual values of the corresponding param eter of the d ata point considered. This m ethod has the advantage when com pared w ith the m ethod of separate fit for the horizontal and the vertical com ponents because it uses more d ata points for obtaining the value of those coefficients, which are “common” for the different orientation. 4 8 7 A PPEN D IX D Coupling of the Effects of the Earthquake M agnitude, M, and the Epicentral Distance, A, on the Duration of Strong Ground M otion As it was shown in this work, the duration of strong ground m otion depends linearly on epicentral distance, A, and exponentially on the m agnitude of an earthquake, M. T h at is to say, we considered only models where the dependence of duration on M and A are uncoupled. However, when the dim ension of the source L is com parable with the epicentral distance A, the uncoupling of dur = dur (M , A) seems unlikely. Indeed, consider the source of length L and a station, located at the epicentral distances A <C L. The duration of strong motion at the station is hardly sensitive to the value of A. The angle subtended at the station by the source is 180°, so the direct waves (originated at the source), come to the recording site from the w ide range of directions. As a result, all the waves cover different distances from the m om ent they were generated until the moment they cam e to the station. The am ount of dispersion, causing the dependence of the duration of strong ground m otion on the distance travelled by the waves, is not the sam e for all those waves. Hence, for a fixed L, duration depends on A linearly only if L < A. T he dependence is very weak in the near range, A < L. The characteristic length of the source, L, decreases with the decrease of m agnitude. T he expected dependence dur — dur(A) for different values of m agnitude is shown on Fig. D .lb. If we still try to approxim ate dur (A) by a straight line, the slope of it should decrease w ith M increasing. 4 8 8 Consider next the case of similar epicentral distances, bu t w ith different source sizes. The expected curves dur = dur(M) for different A are plotted on Fig. D .la . For large M , duration practically does not depend on A . However, the definition of “large” is changing with A. At each A, all the earthquakes w ith L » A should be considered as big, and with L < C A — as small. As a result, we have an asym ptotic exponential curve dur = dur{M), which corresponds to zero epicentral distance. For this distance, all earthquakes should be considered as large. T he curves for positive epicentral distance approach this asym ptote from above. For small M , duration always depends on A; for interm ediate M , some relatively large values of A do influence duration, bu t small values of A do not; for very big M no dependence on A can be noticed, for distances considered in our work. To test the ability of the d ata base to show this coupling, the following model was chosen: dur(f) = consti (/) + const2(/)-M + const3(/)-M 2 + const4(/)-M 3+ (0 .1) + co n sts(/)-A + conste(/)-M A + const7(/)-M A. Here dur(f) is the frequency dependent duration and const,•(/), i = 1 -r 6, are the coefficients to be determ ined from the regression analysis. As it is seen from Eq. (D .l) we chose the linear dependence dur = dur(A) and cubic dur = dur(M). The initially present term consts(f)-M3A was later elim inated as not significant. T he horizontal and the vertical components were considered together, and only th e d a ta for M > 4 and A < 100 km were taken for the regression analysis of Eq. (D .l). The results of this analysis show th at the family of curves sim ilar to w hat is sketched on Fig. D .la can be obtained for the channels # 7 and higher ( / > 1.7 Hz). Fig. D.2 displays this family for the case fo = 13 Hz (channel # 1 1 ). large eq. fo r all A sm all eq. fo r all A a o big A 4 -> c d 2 Q A=0 Magnitude M fa r range fo r all M n ea r range fo r all M o small M ~L (sm all M) ~L (large M) F ig . D . l The coupling of the influence of m agnitude of an earthquake, M, and epicentral distance, A, on the duration of strong ground motion: a) The dependence dur = dur(M) for different m agnitudes. For small earthquake, the source is virtually a point for almost all A, and the duration increases with distance. For large earthquake, the duration practically does not depend on A for each fixed M , because all the sites can be considered to be in the “near range.” b) Dependence dur = dur (A) for different epicentral distances. The switch from “near range” (very weak dependence on A) to “far range” (strong dependence on A) happens when A is of the order of the source dimension L. 489 490 25 A=90 km - A=75 km A=60 km A=45 km A=30 km A= 15 km A=0 km 15 10 M a n i t u d e F ig . D .2 The dependence dur = dur{M, A) according to the results of the regression analysis performed on Eq. (D .l), channel # 1 1 , the central frequency fo = 13 Hz. 4 9 1 M = 4 -r5 M = 5 -6 ! i h i W l i i I J l S i i W ■ * » ! * * ■ ! ■ ; > ■ m h k m ■ M = 6-i-7 a s s s s a s s a s s u iia s iis iis iia s s s s a ! ! i!H !* !! S ifB !l S I* ilSiiB ! ■ ! ■ ! ! HI'!'? ■ » ! ■ ! iffF E iE E ^ iw iirffE S ! ® B * M tN i$ S s S 8 l3 t f iW liiiiiW S M a a s a s s K iii llL M JIl'lU M lllllj l| t|blHJ,l| hit till III ' W * • ! > P W '* * i * < * • * 1 5 * • » > > » n • 5 * ! ■ l S I IS m b W w tti s l i i i s t o s ! ! : ! ! ! ! ! # l l l i l l l i l S I l i l l i f l l l l t i l * • r a IJi] mm M = 7 - h7 .7 [T|i,:|i.liiillii>ili'i f m r r w i m T O T O t r>w: a m s > = : « • v • - * - ■ ’ ■ "I’m . " ' IlHhlm'll.'l I. l.hl,l S S i y ^ M I I ^ S S S S S S iJ i s S S S S B B B B B 3 B jH ! jl j} M iB ! * M & il i 20 30 40 50 60 70 80 90 100 Epicentral distance (km) F ig . D .3 a The dependence dur = dur (A) for various ranges of m agnitude. Shaded area is bounded by the curves corresponding to the highest and the lowest M from each range, as predicted by Eq. (D .l). The actual d ata are shown by asterisks. 492 A=0^-20 km 20 15 10 A=20-i-40 km 20 15 10 0) co n 'T r P T n M n T '7 i* ? n n * M » > ii> i!«*r * 4 li t f l i *!«i* x i a ! i ! ! f ; g ; ? j ! B g ! { f j ! f , i , , T r i i n i ' ■ ~ n ( 0 o • rH + -> 0 A=40-?-60 km 20 15 10 ■ ■ ■ ! h!t '! ji»!! « h s K » l i!hii*i'*|| limn i ■ ■ ~ 1 A=60-rl00 km 20 15 10 Magnutide F ig - D .3 b The dependence dur — dur(M ) for various ranges of epicentral distances. Shaded area is bounded by the curves corresponding to the highest and lowest A from each range, as predicted by Eq. (D .l). The actual d ata are shown by asterisks. 493 T he slope of the linear dependence dur = dur(A) decreases with m agnitude, as expected. Fig. D.3 shows how the actual d ata points can be approxim ated by the curves, obtained from Eq. (D .l) after all constants were determ ined by the linear regression. T he model Eq. (D .l) has to be corrected. Similar to the cases considered earlier in this work (see model Eq. (4.9) and other models w ith sim ilar depen dence dur(M)), the second iteration should be performed. First, the value of m agnitude, M a , where ^ d u r (1)(M , A )= 0 , Should be calculated as function of epicentral distance A. Here d u r ^ ( M , A) is th e duration, estim ated from the first iteration (Eq. (D .l)). Then, th e following model may be considered: ( dur ( /) = like in Eq. (D .l), if M > M a , \d u r (f) = dur(1)(M,A)\M=M^, if M < M A . However, we believe the extra com putations are not necessary. This “coupled” model does not change the prediction of the duration of strong ground m otion sig nificantly when com pared with “uncoupled” model. Moreover, the model w ith the coupled dependence dur = dur(M. A) becomes very unstable if other pa ram eters (geological and soil conditions, for example) are introduced. Thus, we believe, it is sufficient ju st to indicate th a t such effect as coupling of the influ ence of the earthquake m agnitude and the epicentral distance on the duration of strong ground shaking can be extracted from the current d a ta base. 494 A PPE N D IX E D uration of Strong Ground M otion and the H ypocentral D epth In this Appendix, we show how the depth of the hypocenter of an earthquake influences the duration of the strong ground motion. The distribution of d ata available for the analysis is plotted in Fig. 2.3 and was already discussed. Here we ju st em phasize th a t the distribution of the records is far from being uniform w ith respect to the hypocentral depth H, and th a t the accuracy of the depth H is questionable in some cases. Thus, we will not do extensive analysis of the result of th e least-squares-fit of our models, but will ju st describe them . F irst model to consider is the simplest one: dur(f) = ai(f) + a 2(/)-M + a 4(/)-A ' + a is(/)-(2 - s) + an(f)-H. (£.1) Here dur(f) is the frequency dependent duration, M is the m agnitude of an earth quake, s is the geological site condition, H designates the hypocentral depth and A ' = (A 2 -f- H 2) 1/ 2 is the hypocentral distance (with A being the epicentral distance). The value A ' is considered instead of A to exclude the possibility of coupling of the dependence of dur(f) on the epicentral distance, A , and the hypocentral depth, H. Horizontal and vertical com ponents are considered to gether. Fig. E .l and Table E .l show the results of the linear regression analysis of Eq. (E .l). It is seen th a t the coefficients a i ( / ) , a 2(/), 0.4 (f) and ais(f) do not differ from the previous estim ates (see models Eq. (6.4), Fig. 6.3 and Eq. (4.17), Fig. 4.12). The coefficient an(f), scaling the influence of hypocentral depth on d u ration, is significantly different from zero for the frequencies / = 0.2 -f- 1.1 T ab le E . l Results of the regression analysis of Eq. (E .l). Channel number fo (H z) # o f data points N (f) C oefficients a, and their accuracy ("o-interval") 0 dur (sec) durav (sec) ai ± 0 1 a 2 ±02 44 ±04 ais ±015 a n ±017 1 0.075 34 37.8 ±1.9 .0 .0 .0 .0 11.1 37.8 2 0.12 286 17.8 +3.8 .0 .197 ±.019 .0 .02 ±.29 10.4 2 8 .4 3 0.21 811 2.3 ±1.8 .0 .204 +.013 1.39 ±.48 .55 ±.14 7.8 20.7 4 0.37 1091 3.2 ±1.0 .0 .209 +.009 1.37 ±.33 .51 ±.07 7.0 20.3 5 0.63 1496 -2.3 ±2.4 .84 +.44 .199 ±.008 2.74 ±.32 .24 ±.06 7.6 17.9 6 1.1 1952 -5.3 ±1.5 1.64 ±.28 .157 +.006 1.67 ±.23 .09 ±.04 6.6 14.5 7 1.7 2373 -6.3 ±0.9 2.02 ±.18 .132 ±.004 .99 ±.17 -.01 ±.03 5.2 11.9 8 2.5 2663 -4.4 ±0.6 1.54 ±.12 .092 ±.003 .64 ± •1 2 .03 ±.02 3.9 8.9 9 4.2 2269 -6.7 ±0.5 1.88 ±.10 .084 ±.003 .54 ±.10 .03 ±.02 3.2 7.4 10 7.2 2203 -7.6 ±0.4 2.04 ±.08 .077 ±.003 .15 ±.08 .03 ±.02 2.6 6.2 1 1 13 1408 -7.6 ±0.4 1.93 ±.08 .065 ±.004 .1 2 ±.08 .07 ±.02 2.0 5.1 12 21 640 -6.3 ±0.5 1.56 ± .1 1 .080 ±.009 .09 ±.10 .07 ±.03 1.8 4.3 1 M A’ S H C orresponding parameters 495 [const] 2 . 5 4 0 T .20 2.0 ,\ . 1 5 A .10 (Hz) ! ________ L . 0 5 10 f (Hz) -10 .00 [s] 3.0 1 5 2 .5 2.0 1.5 f (Hz) - .2 f (Hz) - . 4 10 F ig . E . l The coefficients at '( /) in Eq. (E .l), plotted versus central frequency of the channels (solid lines). The coefficients are bounded by their “o-intervals” (dashed lines) and by the their estim ated 95% confidence intervals (dotted lines). 4 9 6 4 9 7 Hz. This may be related to the fact th at deeper earthquakes are able to excite m ore surface wave modes in this frequency range. (Recall th a t our d ata base has shallow earthquakes only, and here “deeper” means H « 20 km instead of H & 5 km ). Note, th a t a n (f) ^ 0 approxim ately for the sam e frequency range, where th e influence of the geological conditions on the duration of strong ground m otion is significant. Thus, it is possible th a t the m echanism of this influence is related to the fact th a t the waves are getting “trapped” in and above the layer, where th e m ain portion of energy was released. This should be the case if the upper layers are softer than the lower ones. For higher frequencies, the deeper the source is, the more modes of surface waves it is capable of exiting. For very high frequencies, the attenuation overpowers any such effects. A dditional study shows th a t an(f) significantly differs from zero when d ata w ith H > 8 km only are considered and a-n{f) « 0 for all / if only records w ith H < 8 km are taken into account. The explanation of this may be as follows. Very shallow strike-slip earthquake often breaks the earth ’s surface. The exact position of the hypocenter (the point where the rupture starts) is not im portant here, because it does not characterize the average depth of the fault area which radiates seismic energy. W hen we are dealing w ith com pletely subm erged fault, the hypocentral depth is correlated with the center of energy release. T he results of the regression analysis of the model m(f) = ci (/) + c2(/)-M + c17(f)-H, (E.2) where m ( f) is the num ber of strong motion pulses per one record and c ,(/) are free coefficients to be determ ined from the linear fit, show th a t the increase of d uration due to increase in hypocentral depth can be, at least partially, explained 4 9 8 by the increase of the num ber of strong motion pulses which takes place w hen H increases (Fig. E.2). If the source effects are isolated (the term c2(/)-A f is included in the Eq. (E.2)), this increase in the num ber of pulses can be interpreted as additional modes of surface waves, which were exited due to the lower position of th e source in the layered medium. As the stronger earthquakes will tend to occur at or to be extended to a greater depth (see Fig. E.3), the influence of the m agnitude and the hypocentral dep th on the duration may be coupled. To test this, we tried to fit th e model dur(f) = a1{f)+a2{f}M+a4{f}A+ai5{f){2-s)+ a17{f}H+a18(f).MH (E.3) to the data. The coefficients a t (/) were determ ined from regression analysis and three of them —a 2( /) , a\7(f ) and ais(f )—are shown in Fig. E.4. In the frequency range, where the influence of hypocentral depth on the duratio n of strong ground motion is significant, the correlation term aia(f)-MH is negative. H aving m agnitude as a param eter, the dependence dur — dur(H) is stronger for w eaker earthquakes. The explanation may be as follows. Consider earthquakes of the fixed m agnitude M, bu t of different hypocentral depth H. T he influence of H on duration of strong ground motion can be seen only if the depths H are greater th an some depth H m which is of the order of the w idth of the source. (Recall th a t we are dealing m ostly w ith strike-slip earthquakes in California, and th a t we assum e—as a result of our first model Eq. (E .l)— th a t the source has to be com pletely submerged if the influence of the hypocenter’s depth on the duration of strong ground motion is to be noticed). Obviously, the threshold value of the hypocenter depth, fljvf, increase w ith increasing M, m aking the num ber of the events in the d ata base for which any dependence dur = dur(M) 499 .08 17 .06 — \ .04 .02 .00 10 f (Hz) -.0 2 -.0 4 F ig . E .2 The frequency dependent coefficient cn(f) scaling the num ber of strong m otion pulses as a function of hypocentral depth in Eq. (E.2) (solid line), its “cr-interval” (dashed lines) and estim ated 95% confidence interval (dotted lines). 5 0 0 8 7H CD 6-1 TJ • • • • • • P * # . * j_ 3 ^ 0 0 0 -rH 5 * m w P • • • • • 0 0 tsD c d 0 0 I I 1 ---------------- I -----------------1 -----------------1 0 5 10 15 20 25 30 Hypocentral depth (km) F ig . E .3 The m agnitude M as a function of the hypocentral depth H in our d a ta base. 2.0 3 1 .5 1.0 2 f (Hz) 1 10 A'-V - . 5 - 1.0 1 10 f (Hz) f (Hz) -.1 -.2 F ig . E .4 The frequency dependent coefficients a2(f), a,i7(f ) and aig(f) from Eq. (E.3) (solid line), their u<7-intervals” (dashed lines) and their estim ated 95% confidence intervals (dotted lines). 501 502 can be noticed progressively smaller. As a result, the slope of the linear function dur = dur(M) decreases w ith increasing magnitude.

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