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Integration and validation of deterministic earthquake simulations in probabilistic seismic hazard analysis
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Integration and validation of deterministic earthquake simulations in probabilistic seismic hazard analysis
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Content
INTEGRATION AND VALIDATION OF DETERMINISTIC
EARTHQUAKE SIMULATIONS IN PROBABILISTIC SEISMIC
HAZARD ANALYSIS
by
Feng Wang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(GEOLOGICAL SCIENCES)
December 2013
Copyright 2013 Feng Wang
ii
Epigraph
“If little labor, little are our gains: Man’s fortunes is according to his pains.”
- Robert Herrick
iii
Dedication
To my family.
iv
Acknowledgements
First, I would like to express my deep gratitude to my advisor Professor Thomas H.
Jordan for providing guidance in the last five years of my study and research at USC, for
his generousness to share his ideas with me, for encouraging me to explore unknown
seismological questions, for the time he has put into developing my research, oral
presentation and writing skills, and for all those research opportunities he has given me in
the Southern California Earthquake Center.
Secondly, I would like to thank Dr. Geoffrey Ely for kindly providing me advices and
help on my skills of programming and software development, and for generously sharing
with me his research experiences. I would like to also thank Philip Maechling, Kevin
Milner, Scott Callaghan, Fabio Silva, and SCEC Community Modeling Environment
(CME) Collaboration for their help on downloading and manipulating CyberShake
dataset and performing the probabilistic seismic hazard calculations using OpenSHA
(www.opensha.org).
Thirdly, special thanks are given to John McRaney for kindly helping me handle
problems in payroll and fellowships. I also want to thank Cindy Waite for patiently
helping me handle problems in registration, degree process, and all the paperwork, and
sincere appreciations are given to John Yu for patiently helping me with technical
problems of computing resources.
v
At last, I am grateful to have constant support and encouragements from my girlfriend
Jianghao Wang and all my friends here at USC, who have made my life at USC
enjoyable and unforgettable, and I would like to thank my parents and my younger
brother in China. I hope my efforts and achievements here at USC could make them
happy and proud.
CyberShake database is generated and partially supported by University of Southern
California Center for High Performance Computing and Communications
(http://www.usc.edu/hpcc). MySQL (http://www.mysql.com) is used to extract necessary
peak amplitudes of simulated ground motions from CyberShake dataset. Dataset for
developing NGA GMPEs were provided and supported by Pacific Earthquake
Engineering Research (PEER) center (http://peer.berkeley.edu). A Package of MATLAB-
language functions and scripts for evaluating the Spudich and Chiou (Earthquake Spectra
2008) directivity correction terms for NGA GMPEs used in the research is downloaded
from http://earthquake.usgs.gov/research/software/. Other data analysis, calculations,
visualizations were performed using Python (http://www.python.org), NumPy
(http://www.numpy.org), and Matplotlib (http://www.matplotlib.org). Geographic figures
were made using the Generic Mapping Tools (GMT, http://gmt.soest.hawaii.edu/).
vi
Table of Contents
Epigraph ii
Dedication iii
Acknowledgements iv
List of Tables x
List of Figures xi
Abstract xxiii
Chapter 1 Introduction 1
1.1
Probabilistic Seismic Hazard Analysis ................................................................... 1
1.2
Ground Motion Prediction Models ......................................................................... 5
1.2.1
Ground Motion Prediction Equations .............................................................. 5
1.2.2
Physics-Based Ground Motion Simulations .................................................... 8
1.3
Summary ............................................................................................................... 11
Chapter 2 Averaging-Based Factorization of PSHA Models 15
2.1
Abstract ................................................................................................................. 15
2.2
Introduction .......................................................................................................... 16
2.3
Generalized PSHA ................................................................................................ 18
2.3.1
Earthquake Rupture Forecast ......................................................................... 18
2.3.2
Time-Independent Hazard Model .................................................................. 21
2.4
Ground Motion Prediction Equations (GMPEs) .................................................. 22
2.4.1
Model-based factorization ............................................................................. 22
2.4.2
Empirical Source Directivity Models ............................................................ 25
2.4.3
Hazard Calculation based on GMPEs ........................................................... 29
2.5
Simulation-Based Hazard Models ........................................................................ 30
2.5.1
Physics-Based Earthquake Simulations ......................................................... 31
vii
2.5.2
Simulation-Based Seismic Hazard Model: CyberShake ............................... 34
2.5.3
Discussion ...................................................................................................... 37
2.6
Averaging-based Factorization ............................................................................. 42
2.6.1
Excitation Functional ..................................................................................... 42
2.6.2
ABF Representation of Simulation-Based Excitation Functionals ............... 44
2.6.3
ABF Representation of GMPE-Based Excitation Functionals ...................... 46
2.6.4
ABF Representation of Excitation Differences ............................................. 48
2.7
Conclusions .......................................................................................................... 50
Chapter 3 Comparisons of NGA GMPEs and CyberShake Model Using
Averaging-Based Factorization 52
3.1
Abstract ................................................................................................................. 52
3.2
Introduction .......................................................................................................... 53
3.3
Application of ABF .............................................................................................. 55
3.3.1
Conditional hypocenter and slip distribution ................................................. 56
3.3.2
Reduction of the Source Set K by Disaggregation ........................................ 57
3.4
Validation of ABF by NGA Intercomparisons ..................................................... 61
3.4.1
Map Interpolation Methodology .................................................................... 63
3.4.2
A-Factor: Regional Excitation Level ............................................................. 64
3.4.3
B-Map: Site-Specific Effects ......................................................................... 65
3.4.4
C-Map: Path Effects ...................................................................................... 68
3.4.5
D-Map: Directivity Effects ............................................................................ 73
3.5
ABF Comparisons of CyberShake and NGA Models .......................................... 75
3.5.1
Regional Excitation Levels ............................................................................ 75
3.5.2
Basin Effects .................................................................................................. 76
3.5.3
Path Effects .................................................................................................... 78
3.5.4
Directivity Effects .......................................................................................... 81
viii
3.5.5
Directivity-Basin Coupling ............................................................................ 84
3.6
Discussion ............................................................................................................. 88
3.7
Conclusions .......................................................................................................... 91
Chapter 4 Rupture Directivity Forecasting of Large Earthquakes 92
4.1
Abstract ................................................................................................................. 92
4.2
Introduction .......................................................................................................... 93
4.3
Effects of Rupture Complexity on Source Directivity ......................................... 96
4.3.1
Earthquake Slip Models ................................................................................. 96
4.3.2
Directivity Effects ........................................................................................ 100
4.3.3
Distance Attenuation and Basin Effects ...................................................... 106
4.4
Conditional Hypocenter Distributions for Hazard Calculations ......................... 109
4.4.1
Conditional Hypocenter Distributions ......................................................... 109
4.4.2
Dependence of Directivity Effects on CHDs .............................................. 112
4.4.3
Effects of CHDs on CyberShake Seismic Hazard Calculation ................... 114
4.5
Discussion ........................................................................................................... 122
4.6
Conclusions ........................................................................................................ 124
Chapter 5 Basin Amplification in 3D Velocity Structures 125
5.1
Abstract ............................................................................................................... 125
5.2
Introduction ........................................................................................................ 126
5.3
Basin Effects for Different 3D Velocity Models ................................................ 129
5.3.1
CVM-S4 and CVM-Harvard ....................................................................... 129
5.3.2
Regional Excitation Levels .......................................................................... 131
5.3.3
Basin Effect Maps ....................................................................................... 132
5.4
Basin-Excited Seismic Waves ............................................................................ 143
5.4.1
Response Spectra at Basin Sites .................................................................. 143
5.4.2
Wavelet Analysis of Basin Responses ......................................................... 148
ix
5.5
Directivity-Basin Coupling ................................................................................. 156
5.6
Conclusions ........................................................................................................ 161
Chapter 6 Summary 164
6.1
ABF Variance Analysis ...................................................................................... 164
6.2
Achievements ..................................................................................................... 169
6.3
Prospective .......................................................................................................... 172
Bibliography 175
x
List of Tables
Table 2.1 Model parameters used by four NGA GMPEs. ................................................. 24
Table 2.2 Velocity models and rupture model generators used in CyberShake models. .. 39
Table 3.1 Fault sources identified using disaggregation given hazard level y = 0.3g for SA
at 3.0 s ................................................................................................................................ 60
Table 3.2 Correlation coefficients between residual b and basin depth for different target
models using BA08 as the reference model. ..................................................................... 67
Table 4.1 Averaged standard deviation σ
d
(r,k) of D(r,k,x) of CyberShake (GenSlip v2.1
and v3.2) over r and k for SA at 2.0, 3.0, 5.0, and 10.0. ................................................. 105
Table 4.2 Total standard deviation σ
d
for SA at 3.0, 5.0, and 10.0 s using different CHDs
and different target and reference models in ABF analysis ............................................. 113
Table 5.1 Cross-correlation of residual maps b(r) for SA at 2.0, 3.0, 5.0, and 10.0s with
different target models (see text for the abbreviation of each target model). The
directivity-corrected BA08 is used as the reference model. ............................................ 140
Table 5.2 Averaged standard deviation σ
d
(r,k) of D-maps for CyberShake models (CVM-
S4 and CVM-H) and directivity-corrected BA08 over r and k for SA at 2.0, 3.0, 5.0, and
10.0s. ................................................................................................................................ 161
xi
List of Figures
Figure 1.1 Basic components of PSHA. .............................................................................. 3
Figure 1.2 Occurrence probability of fault ruptures in California for different magnitude
thresholds during the next 30 years (original figures are from Field et al., 2009). ............. 3
Figure 1.3 Illustration of hazard curve quantifying the probability of exceedance for PGA
in 50 years at a specific site r. ............................................................................................. 4
Figure 1.4 The predicted mean (solid red line) and standard deviation (red dash line) of
PGA for given magnitude 6.0 and 7.5 earthquakes as a function of distance from a source
to a site. Red solid dots are observations. The normal distribution is also shown as blue
patches. The GMPE used is developed by Boore et al. (Boore et al., 1997; Field et al.,
2009). ................................................................................................................................... 7
Figure 1.5 Illustration of ground motion simulations for 1992 Landers, California
earthquake (M
w
=7.3, shown as red dot) at a site Barstow (black triangle). Rupture model
for this earthquake shows three fault segments along which rupture propagates, and the
final slip distribution (cross-section view) from the finite source modeling given by Wald
and Heaton (1994). Observed (black) and synthetic (red) ground motion velocities are
shown for two horizontal components (fault parallel and fault normal), and the
geometrical mean of spectral accelerations for those components in gravity unit are
computed with 5% damping for a range of periods. Simulations were performed for 1D
velocity model using broadband platform developed in SCEC (Graves and Pitarka, 2010).
........................................................................................................................................... 10
Figure 2.1 Radiation pattern for (a) double-couple point source and (b) finite source. e
R
,
e
ih
, and e
ph
are radial, vertical, and transverse directions. (rake=0; strike=0; dip=90) ...... 25
xii
Figure 2.2 Rupture and site geometry for computing isochrone directivity predictor. ..... 28
Figure 2.3 Examples of fD and IDP to illustrate the directivity effects considered in
empirical models. Soild thick black line is the fault trace on the surface, and black dot
shows the epicenter location. fD has the natural logarithmic scale showing the correction
to SA at 3.0s. Strikes for both faults are 0
o
. ....................................................................... 29
Figure 2.4 Illustration of kinematic rupture models developed by Graves and Pitarka
(2010). (a) Panels from top to bottom are final slip with rupture front, rise time (duration
of slip time function), and rake direction, respectively. (b) Slip velocity function
generated by using two triangle functions. ........................................................................ 32
Figure 2.5 Fence diagram of P-wave velocity in and around the Los Angeles region
basins. Cross section locations shown as red lines in lower left panel. Modified from
Magistrale et al. (2000). ..................................................................................................... 33
Figure 2.6 Map of the sampling region for CyberShake in Southern California. Black
squares indicate grid of sites where CyberShake hazard curves have been computed. Blue
lines indicate selected fault ruptures from UCERF2.0. Red stars represent possible
epicenral locations. White letters indicate the major sedimentary basins in the Southern
California. (OP=Oxnard Plain, SFV=San Fernando Valley, SGB=San Gabriel Basin,
LAB=Los Angeles Basin, CB=Chino Basin, SBB=San Bernardino Basin) ..................... 35
Figure 2.7 The comparison of ground motion intensity (3 s spectral acceleration)
distribution for one rupture k at one site r obtained by different GMPMs. The histogram
shows the variability of simulated lnY. Each colored solid line indicates the probability
density that is generated using mean and sigma from each NGA GMPE. ........................ 39
xiii
Figure 2.8 Hazard maps of 3s SA with 2% probability of exceedance in 50 years
calculated using (a) AS08, (b) BA08, (c) CB08, (d) CY08, and (e) CyberShake (Using
OpenSHA). ........................................................................................................................ 40
Figure 3.1 Method of source selection using hazard disaggregation. Sources are identified
using CyberShake SourceIDs, and sites are represented by CyberShake SiteIDs. Hazard
level for the calculation is SA-3s at y
0
= 0.3 g. (a) Dots are scaled and colored to the
fractional contribution p(k | r) of each source to the hazard at each site in the CyberShake
model. (b) Contribution of each source to the hazard averaged over all sites. Solid dots
represent the 20 sources with the largest average contributions, which we selected as our
source set. .......................................................................................................................... 59
Figure 3.2 Locations of sources chosen in the analysis. Letters and numbers indicate the
fault segments, which correspond to those in Table 3.1. ................................................... 61
Figure 3.3 The A factors of CyberShake and NGA models (AS08, BA08, CB08, and
CY08) at four periods. Scales are logarithmic. ................................................................. 64
Figure 3.4 Maps of residual factor b(r) for SA at 3.0 seconds using target models (a)
BA08, (b) CB08, (c) CY08, and (d) AS08. For (a), BA08 model calculated for reference
site condition (no site effect) is used as the reference model, and for (b), (c), and (d),
BA08 model (with site effect) is used as the reference model. Contours in (a) are V
S30
with 0.1 km/s interval, which are obtained from Wills and Clahan (2006) for southern
California. Contours in (b) are basin depth to shear wave speed equal to 2.5 km/s (Z
2.5
with 0.53 km interval), and in (c), and (d) are basin depth to shear wave speed equal to
1.0 km/s (Z
1.0
with 0.11 km interval). Basin depth model is from CVM-SCEC. The good
correlation shows that ABF recovers the V
s30
and basin effects. ....................................... 66
xiv
Figure 3.5 Comparisons of model-based basin amplifications for SA-3s (blue open
circles) with the residuals b(r) from the ABF representation (red open circles) for (a)
CB08 vs. Z
2.5
, (b) CY08 vs. Z
1.0
, (c) AS08 vs. Z
1.0
, and (d) AS08 vs. V
S30
. In all cases,
BA08 is used as the reference model. ................................................................................ 68
Figure 3.6 C-maps at SA-3s for (a) Source 86 (San Andreas fault) and (b) Source 273
(Sierra Madre fault). Heavy black lines are the fault surface traces. Left maps in both
cases show C(r,k) of the target model, CB08; middle maps show C(r,k) of the reference
model, BA08; and right maps show the target-reference residual, c(r,k). The dipole
pattern in the lower right panel reflects the larger hanging-wall effect of the BA08
reference model (cf. Figure 3.10). ..................................................................................... 69
Figure 3.7 The maps of c(r,k) for SA at 3.0 seconds using target model AS08 and
reference model BA08. All sources as listed in Table 3.1 are shown here and solid lines
indicate fault surface traces. .............................................................................................. 70
Figure 3.8 The maps of c(r,k) for SA at 3.0 seconds using target model CB08 and
reference model BA08. Same sources as Figure 3.7 are shown. ....................................... 71
Figure 3.9 The maps of c(r,k) for SA at 3.0 seconds using target model CY08 and
reference model BA08. Same sources as Figure 3.7 are shown. ....................................... 72
Figure 3.10 Comparison of hanging wall and footwall effects on of SA at 3.0 s computed
by four NGA GMPEs for the source 273 at V
S30
= 760 m/s. ............................................. 73
Figure 3.11 Maps of residual factor d(r,k,x) at SA-3s for four epicenters (black dots) of
Source 255 computed using directivity-corrected BA08 as the target model and BA08 as
the reference model. Heavy black lines are the fault trace. Contour lines are the SC08
directivity correction terms normalized to zero mean over the hypocenters, plotted at 0.15
intervals. ............................................................................................................................ 74
xv
Figure 3.12 The residual factor b(r) for SA at 3.0 second using Cybershake as target
model and directivity-corrected BA08 as reference model shown (a) as map-based, and as
function of (b) Z
1.0
and (c) Z
2.5.
......................................................................................... 76
Figure 3.13 Maps of residual factors b(r) using (a) CyberShake and (b) AS08 as the target
model and directivity-corrected BA08 as the reference models for spectral accelerations
at T = 2.0, 3.0, 5.0, and 10.0 seconds (as shown on the top row). Color scales are the
same as. The SA-3s maps in (a) and (b) are the same as Figure 3.12(a) and Figure 3.4(d),
respectively. ....................................................................................................................... 78
Figure 3.14 C-maps of SA-3s for (a) Source 86 and (b) Source 273. Left maps in both
cases show C(r,k) for the CyberShake target model; middle maps show C(r,k) for the
BA08 reference model; and right maps show the target-reference residual, c(r,k). Heavy
black lines are the fault surface trace. ................................................................................ 80
Figure 3.15 Maps of residual factors c(r,k) at 3.0 seconds for the target model
CyberShake using directivity-corrected BA08 model as the reference model. ................. 81
Figure 3.16 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake as the
target model and directivity-corrected BA08 as the reference model. The hypocenters of
source 112 are shown as black dots in each subplot. ......................................................... 82
Figure 3.17 The maps of standard deviations of d(r,k,x) using CyberShake as the target
model and directivity-corrected BA08 as the reference model for all sources used in the
analysis. These d-maps assume a uniform conditional hypocenter distribution. .............. 84
Figure 3.18 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake as the
target model and directivity-corrected BA08 as the reference model. The hypocenters of
source 93 are shown as black dots in each subplot. ........................................................... 86
xvi
Figure 3.19 Maps of residual factor d(r,k,x) for target model CyberShake and reference
model BA08 with directivity correction from SC08 model. Black line shows the wall-to-
wall rupture (from Parkfield to Bamboy Beach section of south San Andreas Fault). Each
of black squares shows different epicenter locations along the fault. Number in each map
of d(r,k,x) indicates the location of the corresponding hypocenter. .................................. 87
Figure 4.1 Peak spectral acceleration at 3.0 calculated by TeraShake simulations. (a)
Simulation using scenario that starts at the southeastern end of San Andreas fault
rupturing toward the northwest (SE-NW); (b) Simulation using scenario that starts
northwestern end of San Andreas fault rupturing toward the southeast (NW-SE).
Modified from (Olsen et al., 2006). ................................................................................... 95
Figure 4.2 Illustration of kinematic rupture models developed by Graves and Pitarka
(2007, 2010). (a) GenSlip v2.1: More simple rupture description with constant rupture
velocity and smooth rupture front; (b) GenSlip v3.2: More complex rupture model with
variable rupture velocities for different depth and rough rupture front. Each row shows
final slip with rupture front, rise time (duration of source time function), and rake
direction, respectively. ....................................................................................................... 97
Figure 4.3 Relationship between the IDP (black dots) and normalized isochrone velocity
(red dots) and the ratio between rupture velocity and shear-wave velocity. ..................... 99
Figure 4.4 Rupture surface trace and possible epicenter locations for source 93. .......... 101
Figure 4.5 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake with
GenSlip v2.1 rupture models as the target model and BA08 with SC08 directivity
correction as reference model. The source 93 with all hypocenters is shown. The number
in each subplot indicates the order of hypocenter shown in Figure 4.4 (0:southern end; 11:
northern end). Color scale is the same as Figure 3.16. .................................................... 102
xvii
Figure 4.6 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake with
GenSlip v3.2 rupture models as the target and reference model, respectively. Source and
hypocenters are the same as in Figure 4.5. ...................................................................... 103
Figure 4.7 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (for GenSlip v3.2) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. ...................................................... 105
Figure 4.8 Map of residual b(r) for SA at 3.0 second using CyberShake with GenSlip
v3.2 and GenSlip v2.1 rupture models as the target and reference model, respectively.
Color scale is changed in order to show the small differences between models. ............ 107
Figure 4.9 The residual map c(r,k) for SA at 3.0 seconds using CyberShake with GenSlip
v3.2 and GenSlip v2.1 rupture models as the target and reference model, respectively. 108
Figure 4.10 Distribution of hypocenter position (along-strike and up-dip) for (a) all
events, (b) strike-slip events, and (c) non-strike-slip events used in empirical directivity
modeling. ......................................................................................................................... 110
Figure 4.11 Schematic diagram showing different conditional hypocenter distributions
given a fault plane. The parameters for Beta distribution are shown along with each
probability distribution (upper panel), and corresponding randomization of those
hypocenter distributions in 2D fault plane (lower panel). (Credit to: Thomas Jordan) ... 111
Figure 4.12 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (with GenSlip v3.2) as the target model and directivity-corrected BA08 as
the reference model for all sources used in the analysis. The CHD used in calculating
d(r,k,x) is the centroid-biased hypocenter distribution illustrated in Figure 4.11. .......... 113
xviii
Figure 4.13 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (for GenSlip v3.2) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. The CHD used in calculating d(r,k,x)
is the periphery-biased hypocenter distribution illustrated in Figure 4.11. ..................... 114
Figure 4.14 (a) Particle velocities at a given time for rupture along a material interface
(thin horizontal line). The slipping region is marked by the thick segment on the fault and
is propagating to the right (right-lateral strike-slip fault). (b) Enlarged view of the white
box in (a) showing asymmetric particle velocities around the rupture pulse. Retrieved
from Dor et al (2006). ...................................................................................................... 116
Figure 4.15 Faults of the southern San Andreas system included in the current study and
region that includes investigation sites (two small box). After Dor et al (2006). The site
STNI at which the hazard curves are calculated is shown (black triangle), and
CyberShake region is also shown. ................................................................................... 117
Figure 4.16 Asymmetric hypocenter distributions (probability density) with different set
of parameters. Each set of parameters is assigned a name and x from 0 to 1 means south
to north in this case. ......................................................................................................... 118
Figure 4.17 Hazard curves for the station STNI using two sources [93,112] as shown in
Figure 4.15 and different CHDs are considered during the hazard calculations. ............ 119
Figure 4.18 Probability gain factors calculated from the probability of exceedance given
hazard level based on Figure 4.17. The black line is the reference line. ......................... 120
Figure 4.19 Maps of the probability gain factors of hypocenter distribution (a) A1-N, (b)
A1-N, (c) A1-S, and (d) A2-S with respect to uniform distribution given hazard level SA
at 3.0 s equals to 0.3g. ..................................................................................................... 121
xix
Figure 5.1 (a) Map view of shear wave velocity at depth 1000m for CVM-S4. (b) Map
view of shear wave velocity at depth 1000m for CVM-H11.9.1. (c) 1D shear wave profile
underneath the station LGB shown as black triangle in (a) and (b). ............................... 130
Figure 5.2 The A factors of CyberShake and NGA models (AS08, BA08, CB08, and
CY08) at four periods. Black groups use CVM-H, and red groups use CVM-S4. Scales
are logarithmic. ................................................................................................................ 132
Figure 5.3 Maps of residual factors b(r) using CyberShake model with (a) GP and CVM-
S4, (b) GP and CVM-H, (c) AWP-ODC and CVM-S4, and (d) AWP-ODC and CVM-
Harvard as the target model and directivity-corrected BA08 as the reference model for
spectral accelerations at T = 3.0s. .................................................................................... 133
Figure 5.4 Maps of residual factors b(r) using CyberShake model with GP and CVM-S4
as the target model, and CyberShake model with GP and CVM-H as the reference model
for spectral accelerations at T = 3.0s. .............................................................................. 134
Figure 5.5 Maps of residual factors b(r) using directivity-corrected BA08 as the reference
model, and CB08, CY08, AS08, and CyberShake with GP and CVM-S4 as the target
models (each column) for spectral accelerations at T = 3.0, 5.0, and 10.0 s (each row).
Color scales are the same as Figure 5.3. The basin depth parameters used in CB08, CY08,
and AS08 are extracted from CVM-S4. .......................................................................... 135
Figure 5.6 Maps of residual factors b(r) using directivity-corrected BA08 as the reference
model, and CB08, CY08, AS08, and CyberShake with GP and CVM-H as the target
models (each column) for spectral accelerations at T = 3.0, 5.0, and 10.0 s (each row).
Color scales are the same as Figure 5.3. The basin depth parameters used in CB08, CY08,
and AS08 are extracted from CVM-H. ............................................................................ 136
xx
Figure 5.7 Maps for basin depth parameters Z
1.0
and Z
2.5
(each column) extracted from
CVM-S4 and CVM-Harvard (each row as indicated with the texts). ............................. 138
Figure 5.8 Comparison of median SA for M
w
=7 strike-slip earthquakes at an R
JB
distance
of 10km for different site conditions: soil sites (V
S30
=270m/s) with average soil depth
(Z
1.0
=0.5 km, Z
2.5
=2.3 km), shallow soil depth (Z
1.0
=0.1 km, Z
2.5
=0.9 km), and deep soil
depth (Z
1.0
=1.2 km, Z
2.5
=4.8 km) depths and rock sites (V
S30
=760 m/s). ........................ 139
Figure 5.9 Maps of residual factors b(r) using CyberShake with CVM-S4 as the target
model and directivity-corrected BA08 as the reference models for spectral accelerations
at T = 3.0, 5.0, and 10.0 s (as shown on the top row). Color scales are the same as Figure
5.3. Panel (a) The source set used in the ABF analysis is different from sources listed in
Table 3.1, and the CHD is uniform distribution. Panel (b) The source set used in the ABF
analysis is the same as sources listed in Table 3.1, and the CHD is periphery-biased
distribution. ...................................................................................................................... 142
Figure 5.10 Selected basin sites located within CyberShake region. Names of sites are
shown beside the location of those sites marked as black triangles. ............................... 144
Figure 5.11 Relative accelerations for input synthetic seismograms (black traces on the
first row) at three natural period of the SDOF system (second row: 3.0s, third row: 5.0s,
and forth row: 10.0s). Each column indicates the component of the synthetic seismogram.
Each black dot indicates the position where the corresponding PSA is located and its
value. ................................................................................................................................ 146
Figure 5.12 Impulse response function (response acceleration) for three periods. Dots
show the maximum absolute values (PSA), indicating the group delay for different
periods. ............................................................................................................................ 146
xxi
Figure 5.13 Averaged t
max
for peak spectral acceleration at 3.0, 5.0, and 10.0s. The names
of basin sites are also shown. ........................................................................................... 147
Figure 5.14 Illustration of wavelet analysis for a given source and site. Two-components
seismograms (velocity in m/s) are show on the top and the continuous wavelet transforms
are show on the bottom (colored), in which white triangles show the maximum values of
the powers for each frequency. ........................................................................................ 150
Figure 5.15 The variability of averaged maximum power values (top row) and corrected
time shifts where the maximum power values are located (bottom row). The black dashed
lines show the averaged S-wave arrival time. Red dots: CVM-S4; Blue dots: CVM-H. 151
Figure 5.16 Maps of spectral value and peak time at period 3.0s calculated for the
component N35E. The velocity model is CVM-S4. ........................................................ 152
Figure 5.17 Maps of spectral value and peak time at period 3.0s calculated for the
component N35E. The velocity model is CVM-H. ......................................................... 153
Figure 5.18 Maps of spectral values at period 2.0, 3.0, 5.0, 7.0 and 10.0s (each column)
calculated for the component N35E and E35S (each row). The velocity model is CVM-
S4. .................................................................................................................................... 154
Figure 5.19 Maps of spectral values at period 2.0, 3.0, 5.0, 7.0 and 10.0s (each column)
calculated for the component N35E and E35S (each row). The velocity model is CVM-H.
......................................................................................................................................... 154
Figure 5.20 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake (CVM-S4)
as the target model and directivity-corrected BA08 as the reference model. The source 93
with all hypocenters is shown. The number in each subplot indicates the order of
hypocenter as shown in Figure 4.15 (0:southern end; 11: northern end). ....................... 157
xxii
Figure 5.21 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake (CVM-H)
as the target model and directivity-corrected BA08 as the reference model. The source 93
with all hypocenters is shown. The number in each subplot indicates the order of
hypocenter as shown in Figure 4.15 (0:southern end; 11: northern end). ....................... 158
Figure 5.22 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (CVM-S4) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. The CHD is uniform distributed. . 159
Figure 5.23 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (CVM-H) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. The CHD is uniform distributed. . 160
Figure 6.1. Budget of the residual variance (σ
G
2
), as given by Equation (6.1), for the
NGA mean and CS13.2 (the left and the right one at each period, respectively). Each of
the ABF terms is represented by a different color. For NGA mean, σ
F
2
is the total aleatory
variance; For CS13.2, it is the excitation variance from the source-complexity effect at
constant magnitude. ......................................................................................................... 167
Figure 6.2. Budget of the residual variance (σ
g
2
), as given by Equation (6.1), for residual
CS11-NGAmean and CS13.2-NGAmean (the left and the right one at each period,
respectively). Each of the ABF terms is represented by a different color. Black dashed
line shows the total regression variance of the directivity-corrected NGA meanσ
T
2
. ..... 169
xxiii
Abstract
Seismic hazard models based on empirical ground motion prediction equations
(GMPEs) employ a model-based factorization to account for source, propagation, and
path effects. An alternative is to physically simulate these effects using earthquake source
models combined with three-dimensional (3D) models of Earth structure. We generalized
the implementation of those hazard models in probabilistic seismic hazard analysis from
the seismological perspectives, and developed an averaging-based factorization (ABF)
scheme to facilitate the geographically explicit comparison of these two types of seismic
hazard models. Through a sequence of averaging and normalization operations over
various model components, such as slip distribution, magnitudes, hypocenter locations,
we uniquely factorize model residuals into several factors. These residual factors
characterize differences in basin effects, distance attenuation, and effects of source
directivity and slip variability. We illustrate the ABF scheme by comparing CyberShake
model for the Los Angeles region with the Next Generation Attenuation (NGA) GMPEs.
Relative to CyberShake, all NGA models underestimate the basin effects. Using the
GMEPs with directivity corrections, we quantify the extent to which the empirical
directivity model capture the source directivity effects demonstrated by physics-based
ground motion prediction model. In particular, empirical directivity corrections for NGA
models underestimate source directivity effects in CyberShake, and do not account for
xxiv
the coupling between source directivity and basin excitation that substantially enhance
the low-frequency seismic hazards in the sedimentary basins of the Los Angeles region.
We then investigate seismologically to what extent the complex rupture processes and
conditional hypocenter distributions affect the ground motion predictions and seismic
hazard assessment. At last, considering two 3D velocity models for Southern California
used in simulations, we use different CyberShake studies to physically understand the
basin excitations and directivity-basin coupling effects. To our knowledge, this is the first
systematical and quantitative integration and validation of deterministic earthquake
simulations in probabilistic seismic hazard analysis.
1
Chapter 1
Introduction
1.1 Probabilistic Seismic Hazard Analysis
Natural earthquakes are the major threats to lives and properties of people living close
to the active tectonic regions. For example, California’s 35 million people live among
some of the most active earthquake faults in the United States. Earthquakes have major
economic and societal consequences as can be seen from the aftermath of the large
earthquakes worldwide. Public safety needs reliable assessments of the earthquake hazard
to maintain appropriate building codes for safe construction and earthquake insurance for
loss protection. In order to assess risk to a structure from the ground shaking induced by
earthquakes, we must first estimate the ground-motion intensities from an earthquake and
evaluate the facility of interest under that intensity. One possible way is that we could
identify a “worst-case” earthquake that induces the “worst-case” ground motion, but
conceptual problems arise quickly and are difficult to overcome. First, the choice of a
“worst-case” earthquake can be difficult and subjective due to the physics and
randomness of earthquake occurrences. Secondly, the “worst-case” ground motion
intensity associated with an existing “worst-case” earthquake is practically difficult to
accurately predict. Fortunately, the problems identified in the deterministic thinking of
hazard and risk assessment do provide a useful motivation for thinking about probability-
based alternatives.
2
Instead of searching for the “worst-case” ground motion, probabilistic-based
alternatives consider the uncertainties in location, size, and rate of future earthquake
provided by the earthquake rupture forecast (ERF) model and resulting ground-motions
predicted by ground-motion prediction model (GMPM). The mathematical approach for
combining ERF and ground motion predictions is known as probabilistic seismic hazard
analysis, or PSHA (Cornell, 1968; McGuire, 2004; Power et al., 2008).
Figure 1.1 presents the basic components in PSHA. As the starting point of PSHA, a
standard ERF model specifies the spatial geometries, sizes, and the likelihoods of
potential earthquake ruptures in a time span. The collection of all possible earthquake
sources (faults) in the ERF is represented as a set K, and each member (sample) in the set
is labeled by an index k (i.e.
k∈K ), indicating the k
th
fault surface. For a given source, a
set M(k) is used to assemble possible ruptures (such as potential magnitudes and rupture
length) for the source, and m labels each member in the set, i.e.
m∈ M(k). The ERF
models provide the probability P(k,m|T) of occurrence of earthquakes for each given
rupture m of a source k in a time span T within a region of interest. For example, the
Uniform California Earthquake Rupture Forecast, Version 2 (UCERF2) identifies fault
spatial geometry of larger, active faults; fault slip rates for seismic-moment release
estimation, rate of all possible damaging earthquakes throughout the region, and the
probability that each earthquake in the given earthquake rate model will occur during a
3
specified time span (Field et al., 2009). Figure 1.2 shows the maps for probability of
occurrence at three magnitude thresholds given by the UCERF2 model.
Figure 1.1 Basic components of PSHA.
Figure 1.2 Occurrence probability of fault ruptures in California for different magnitude
thresholds during the next 30 years (original figures are from Field et al., 2009).
4
With a ground motion prediction model (GMPM), the conditional probability
P(Y>y|r,k,m) can be calculated for an earthquake rupture given an ERF model, and the
resulting P(Y>y|r,T) provides the probability of ground shaking intensity Y at a site r
induced by potential earthquakes exceeding a particular intensity level y. Ground motion
intensity measures are typically used to quantify the earthquake-induced ground motions,
such as peak ground velocity (PGV), peak ground acceleration (PGA), and 5%-damped
and period-dependent spectral acceleration (SA). As shown in Figure 1.3, the probability
of exceedence at a series of ground motion levels (e.g. PGA) within a time interval (e.g.
50 years) for a specific site gives so-called hazard curve. Furthermore, choosing a given
probability of exceedance (e.g. 10%), a corresponding ground motion level is found for
the site, or fixing a ground motion-level, a corresponding exceedance probability is found
for the site, resulting in a hazard map of either ground motion or exceedance probability
for many sites in a region R (i.e.
r∈R ).
Figure 1.3 Illustration of hazard curve quantifying the probability of exceedance for PGA
in 50 years at a specific site r.
5
1.2 Ground Motion Prediction Models
There are two types of GMPMs used in PSHA, including empirical ground-motion
prediction equations (GMPEs) and physics-based ground-motion simulations. Seismic
waves radiated from a particular earthquake rupture propagate through three-dimensional
(3D) structures of the earth subsurface and lead to ground shaking at a specific location.
For example, sites located in the rupture propagation directions on average experience
stronger ground motions at long periods due to forward rupture directivity (Ben-
Menahem, 1961; Zhao et al., 2006; Graves et al., 2011b) and sites with different local
conditions (e.g. soil or rock) experience different shaking levels due to site effects (King
and Tucker, 1984; Choi and Stewart, 2005; Olsen et al., 2008; Bielak et al., 2010; Graves
et al., 2011a). The goal of GMPMs is to capture different physical effects, such as source
directivity, path scattering, and site amplifications.
1.2.1 Ground Motion Prediction Equations
The empirical GMPEs or attenuation relations, commonly used in PSHA (Petersen et
al., 2008; Graves et al., 2011b) with standard ERF model, are developed using statistical
regression on observed ground motions from both global and regional networks. A
typical GMPE has the form
lnY = lnY m;θ ( )+εσ
T
(1.1)
where lnY is the natural logarithm of the ground motion intensity (e.g. PGA, PGV, and
SA), ln
Y represents the mean prediction of variability in ground motions, σ
T
is the
aleatory standard deviation, and ε is the number of standard deviations which quantifies
6
the fraction of lnY from ln
Y , and follows standard normal distribution, quantify the
randomness of ground motion predictions. The GMPEs predict the mean ground motion
at a given location as a function of m with statistically determined coefficients θ, where
m represents the model parameters considered in the GMPEs, which relate to the
earthquake sources (such as magnitude, focal mechanism, and fault geometry),
propagation path (such as the closest distance from a site to fault surface), and site local
conditions (such as average shear wave velocity, depth to bedrock).
Those relations assume that the measured variability of ground motions at many sites
for past earthquakes accurately represents the variability of ground motions expected at a
single site for future earthquakes (Anderson and Brune, 1999; 2008), which is so-called
ergodic assumption. The ground motion distribution for a site is characterized by mean
prediction and standard deviation, assuming that ground motions induced by an
earthquake rupture (k,m) follows a log-normal distribution, and P(Y>y|r,k,m) can be
calculated using the cumulative log-normal distribution. Figure 1.4 presents the ground
motion predictions using a GMPE developed by Boore et al. (1997) for rock sites with
different distances from two earthquakes with different magnitudes. The red solid lines
indicate the mean prediction and red dashed lines represent the first standard deviation.
First, lots of scatters relative to the mean can be seen from the observed ground motions
(red dots) for a given magnitude and distance. Secondly, limited data are found to
constrain the predictions of mean and standard deviation for larger earthquakes and close
distances. Both observations show large uncertainties for ground motion predictions
using GMPEs.
7
Figure 1.4 The predicted mean (solid red line) and standard deviation (red dash line) of
PGA for given magnitude 6.0 and 7.5 earthquakes as a function of distance from a source
to a site. Red solid dots are observations. The normal distribution is also shown as blue
patches. The GMPE used is developed by Boore et al. (Boore et al., 1997; Field et al.,
2009).
Uncertainties generally represent that the different parameter values, models or
outcomes have different probability (Field et al., 2003; Graves et al., 2011b). Aleatory
uncertainties represent those for which, given enough time or realizations, all possibilities
will eventually be sampled. Epistemic uncertainty, on the other hand, represents different
possibilities for an entity that has only one true, but currently unknown, value. In this
case, the mean value and standard deviation has epistemic uncertainty due to different
model parameterizations in Equation (1.1) by different modelers. The distribution that
indicates the probability for each possible ground motion value shows the aleatory
8
uncertainty. Due to the ergodic assumption, the variability in ground motion at a single
site-source combination is the same as the variability in ground motion observed in a
more global dataset that has relatively large aleatory variability. To reduce the aleatory
variability, people usually identify those components of ground motion variability at a
single sites that are repeatable rather than purely random, so that these may be removed
from the aleatory standard deviation and merged into the quantification of the epistemic
uncertainty, i.e., changing the mean prediction and reduce the total standard deviation,
resulting in more reliable estimation of conditional probability of exceedance (Atik et al.,
2010; Lin et al., 2011; Graves et al., 2011b). However, the approach strongly depends on
the amount of recorded ground motions at a single station from multiple occurrences of
earthquake in the same region, and leads to large epistemic uncertainty due to lack of
knowledge about those repeatable effects.
1.2.2 Physics-Based Ground Motion Simulations
Earthquake-induced ground motions at a geographical location are complex
excitations resulted from a dynamic fault failure, seismic wave propagation, and site local
conditions. Considering this fact, the deterministic earthquake simulations require the
three-dimension (3D) subsurface structures, the detailed earthquake source description
(such as hypocenter location and slip spatial and temporal heterogeneity across the fault
rupture surface), and numerical calculation of wave propagation and attenuation with
efficient wave propagation codes. Simulating the large-scale ground-motion time series
for historical and future earthquakes has become more feasible with advances in the study
of fault rupture processes, wave propagation effects, and site response, and significant
9
growth in computational power and efficiency (Mai and Beroza, 2002; Olsen et al., 2008;
Bielak et al., 2010; Graves et al., 2011a).
Figure 1.5 illustrates a broadband ground-motion simulation and comparisons with
observed ground shaking intensities for the 1992 Landers, California earthquake
(M
w
=7.3) at a station (Barstow) located in southern California. The kinematic fault
rupture model provides hypocenter location, seismic moment or magnitude, and slip
spatial and temporal heterogeneity across those segments (Wald and Heaton, 1994;
Anderson and Brune, 1999). The deterministic low-frequency (<1Hz) and semi-stochastic
high-frequency (>1Hz) ground-motion simulations are performed using a hybrid
approach (2008; Graves and Pitarka, 2010), and the resulting ground motions are
comparable with observations.
The implementation of the physics-based simulations in PSHA has been investigated
within the Southern California Earthquake Center (SCEC) CyberShake project (Bernard
and Madariaga, 1984; Spudich and Frazer, 1984; Graves et al., 2011b) for southern
California, a suitable natural laboratory with detailed representation of subsurface
structures and plate-boundary fault system. The CyberShake hazard model implements
seismic reciprocity for a suite of “pseudo-dynamic” rupture variations (i.e. possible
earthquake scenarios) that sample the aleatory variability in kinematic source models, and
provides over 415,000 simulated ground-motion time series for sites in Los Angeles
region of southern California, large enough to sample the probability distribution of
ground-motion intensities at those sites. In addition, several CyberShake studies have
been generated using different 3D velocity models (CVM-SCEC and CVM-Harvard) for
10
southern California based on full-waveform inversion (Chen et al., 2007; Press et al.,
2007; Tape et al., 2009) and different kinematical rupture descriptions based on studies of
rupture process of natural earthquakes (Mai and Beroza, 2002; Guatteri et al., 2004;
Graves et al., 2011b) so that the epistemic uncertainty of ground motion predictions is
considered in simulation-based models.
Figure 1.5 Illustration of ground motion simulations for 1992 Landers, California
earthquake (M
w
=7.3, shown as red dot) at a site Barstow (black triangle). Rupture model
for this earthquake shows three fault segments along which rupture propagates, and the
final slip distribution (cross-section view) from the finite source modeling given by Wald
and Heaton (1994). Observed (black) and synthetic (red) ground motion velocities are
shown for two horizontal components (fault parallel and fault normal), and the
geometrical mean of spectral accelerations for those components in gravity unit are
computed with 5% damping for a range of periods. Simulations were performed for 1D
velocity model using broadband platform developed in SCEC (Graves and Pitarka, 2010).
11
1.3 Summary
In this dissertation, our goal is to provide a general framework in which we
systematically study the integration of deterministic earthquake simulations in PSHA, in
particular CyberShake for the southern California region, and quantitatively compare
empirical GMPEs and simulation-based ground motion prediction models in the
framework, establishing confidence on the efficacy of advantages of simulation-based
models in PSHA, especially the consideration of 3D velocity models and pseudo-
dynamic source descriptions. Moreover, from seismological point of view, we put our
emphasis on quantifying the physical effects of source complexity and velocity structure
on the ground motion predictions and resulting hazard curves at particular sites.
In Chapter 2, we generalize the description for PSHA to incorporate physics-based
simulations in place of GMPEs, and we specialized this description to two
implementations of ground motion prediction models in the hazard assessment, NGA
GMPEs and CyberShake for southern California. We define our notation with the help of
the kinematically complete ERF model that explicitly includes the rupture variations,
including hypocenter locations and slip distributions. Following the notation, we present
two different types of hazard models, empirical ground motion prediction equations and
simulation-based ground motion prediction models. Comparing the variability considered
in both models and the differences in hazard calculations, we would like to compare
those models in the general framework of PSHA. The model-based factorization provides
a decomposition in which model factors represent distinguishable physical effects, such
as source, path, and site effects. This factorization allows us to compare NGA GMPEs in
12
a systematical way. However, simulation-based models incorporate complex source
information and wave propagation, which has no model factors from the model-
factorization. In order to compare these two types of models, we develop a scheme that is
based on simple mathematical averaging operations following a certain hierarchical
sequence. Through a sequence of averaging operations over various model parameters,
we uniquely factorize model residuals into several terms, which then characterize
differences in various physical effects between models, such as basin excitation, distance
attenuation, and source directivity, allowing us to directly compare different types of
models.
In Chapter 3, we apply the averaging-based factorization (ABF) scheme that
conforms to the implementation of ground motion predictions in seismic hazard analysis,
and facilitates the geographically explicit comparisons between empirical GMPEs and
simulation-based model. We present the numerical results for comparisons between
CyberShake and the NGA GMPEs. For spectral acceleration at long periods (>2s), NGA
models underestimate the basin-excitation effects relative to CyberShake; Moreover,
basin excitations in the physics-based model are not a simple function of basin depth as
parameterized in the NGA models. Using the directivity relations of Spudich and Chiou
(2008), we quantify to what extent the empirical directivity models capture the source
directivity effects predicted by physics-based model. We found that the empirical
directivity corrections for NGA models underestimate source directivity effects in
CyberShake, and they do not account for the directivity-basin coupling that substantially
13
enhances the low-frequency seismic hazards in the sedimentary basins of the Los Angeles
region.
In Chapter 4, we focus on the effects of rupture directivity forecasting on ground
motion predictions and hazard calculations. The ability to forecast rupture directivity can
substantially improve the long-term forecasting of strong ground motions and its
application in PSHA. We consider three types of information relevant to directivity
forecasting obtained from the study of finite fault models for regional and global
moderate and large earthquakes, and crustal fault zones, including small-scale rupture
complexity, conditional hypocenter distribution (CHD), and directivity bias inferred from
damage asymmetry or material contrast across faults. CyberShake, a simulation-based
seismic hazard model, provides the physical framework for investigating the problem,
and we apply the ABF scheme as presented in Chapter 2 to quantify how information
related with directivity effects affects ground motion prediction and seismic hazard
calculations. The results show that directivity effects considered in CyberShake could be
lowered by increasing in small-scale rupture complexity and centroid bias of the CHDs.
Moreover, we did experiments of using different CHD parameterized by Beta distribution,
and found the asymmetric CHDs that are consistent with bimaterial ruptures bias can
substantially alter seismic hazard assessments.
In Chapter 5, we focus on the effects of 3D geological structures, in particular,
sedimentary basins, on ground motion predictions. Two CyberShake models with
different 3D velocity models are used to examine the basin amplifications for different
14
geological structures, including CVM-SCEC version 4 (CVM-S4) and CVM-Harvard.
We investigate the frequency-dependence of basin effects by using ABF scheme and
calculating the B factor for spectral acceleration as excitation functionals at various
periods from those CyberShake models. Meanwhile, we compute the basin amplification
using NGA GMPEs with different basin depth models from those velocity models, and
we found that, CyberShake models with both velocity models have larger basin effects
than NGA models that consider basin amplification. Using CVM-Harvard model, AS08
model has the similar frequency-dependence as simulation-based model, CyberShake
with a correlation around 0.9 in the frequency band 0.1-0.3Hz. By calculating the
corresponding times of maximum power spectra values and pick times where the spectral
acceleration values are chosen, we find that the cause of large basin effects is due to the
surface waves and basin excited resonances. In addition, we compare the directivity-basin
coupling effect that depends on velocity structures, and found that, for a ShakeOut-type
rupture, the directivity-basin coupling effect is much lower in CVM-Harvard than CVM-
S4, while the directivity effects have relatively weaker dependence on velocity structure.
At last, we summarize with the variance analysis to take full advantage of ABF’s
intrinsic structure, and compare with the regression (aleatory) variance from NGA08
models to quantify the changes in ground-motion variability by using more complex slip
models and velocity structures in simulation-based model, CyberShake, relative to NGA
08 models.
15
Chapter 2
Averaging-Based Factorization of PSHA Models
2.1 Abstract
We generalized the description for PSHA to incorporate physics-based simulations in
place of GMPEs, and we specialized this description to two implementations of ground
motion prediction models in the hazard assessment, NGA GMPEs and CyberShake for
southern California. First, kinematically complete ERF model explicitly includes the
rupture variations, including hypocenter locations and slip distributions, and is enough to
include the standard ERF model for PSHA. Secondly, GMPEs use a model-based
factorization, in which model factors represent distinguishable physical effects, such as
source, path, and site effects, while CyberShake model incorporates complex source
information and wave propagation, which has no model factors from the model-based
factorization. We compare the hazard maps produced by both types of models, and found
that CyberShake predicts higher ground motion in the sedimentary basins where most
people live, and large discrepancies are found near the San Andreas Fault. In order to
interpret those differences found in hazard calculations, we developed a technique call
averaging-based factorization for the comparison of PHSA models that does not depend
on any model-based factorization but instead relies on a hierarchical sequence of
averaging operations.
16
2.2 Introduction
Probabilistic Seismic Hazard Analysis (PSHA) combines the probabilities of all
earthquake scenarios with predictions of resulting ground motions, and estimates the
seismic hazard at a site (Cornell, 1968; Kohler et al., 2003; McGuire, 2004). It requires
an earthquake rupture forecast that provides the likelihood of possible earthquake
ruptures, and prediction models for specifying earthquake-induced ground motions at a
specific location.
The ground motion prediction equations (GMPEs) currently used in PSHA, such as
the Next Generation Attenuation (NGA) relations (Wills and Clahan, 2006; Power et al.,
2008), are regression models that fit standardized sets of shaking intensities derived
primarily from strong-motion recordings of historical earthquakes (Abrahamson et al.,
2008; Chiou et al., 2008; Ancheta et al., 2013). These empirical attenuation relations
predict shaking intensity as a random variable with an expectation expressed as the
product of factors accounting for various effects, including earthquake size, faulting
orientation, distance to site, and site conditions. This type of GMPE representation will
here be called a model-based factorization. The GMPE factors are increasing in number
and complexity as better data and theory allow model developers to capture additional
predictable earthquake phenomena, such as directivity, basin, and hanging-wall effects
(Day et al., 2008; Spudich and Chiou, 2008; Akkar and Bommer, 2010; Star et al., 2011;
Bozorgnia et al., 2012; Spudich et al., 2012; Boore et al., 2013).
17
An alternative method for predicting ground motions is numerical simulation. This
physics-based approach to PSHA is exemplified by the CyberShake simulation-based
hazard models that have been developed by the Southern California Earthquake Center
(SCEC) using an efficient computational method based on seismic reciprocity (Mai et al.,
2005; Zhao et al., 2006; Graves et al., 2011b). In CyberShake, the rupture probabilities
are taken from an ERF, and the rupture description is augmented with enough
information to fully determine a kinematic seismic source, including a hypocenter and a
slip distribution over the fault surface, as illustrated in Figure 1. This information is
contained in two probability distribution functions (pdfs) that are conditioned on the
ERF-specified rupture geometry: the conditional hypocenter distribution and the
conditional slip distribution. A discrete (though very large) set of source models are
constructed by sampling the ERF and these auxiliary distributions. The sample sources
are input to an wave propagation code that uses a 3D anelastic model of geologic
structure to simulate the ground motions (Ben-Zion, 2001; Dor et al., 2006; Olsen et al.,
2008; Bielak et al., 2010; Graves et al., 2011a), from which the intensity measures are
calculated. The intensity measures are then recombined with the source probabilities to
compute hazard curves at specific sites and regional hazard maps from spatial
distributions of sites (Boore et al., 2006; Graves et al., 2011b).
In this chapter, we developed a technique for the comparison of PSHA models that
does not depend on any model-based factorization but instead relies on a hierarchical
sequence of averaging operations applied to large ensembles of intensity-field
realizations drawn from the probabilistic models. These operations lead to a unique
18
averaging-based factorization (ABF) of the intensity field for each source realization. We
begin by recasting PSHA in a generalized seismological form that can accommodate both
GMPE-based and simulation-based hazard models. We introduce a five-level ABF
representation consistent with this discretized formulation, and we discuss the physical
interpretation of its five elementary factors.
2.3 Generalized PSHA
PSHA combines an ERF with a ground motion model to estimate P(Y > y | r, T), the
probability that the ground shaking intensity Y at a site r will exceed some hazard level y
during a future interval T. The locus of P versus y is the seismic hazard curve at r, and a
plot of y(r) for fixed P is a seismic hazard map for a region R, which we sample at a
discrete set of sites, r ∈ R. Common measures of the intensity Y are peak ground
acceleration (PGA), peak ground velocity (PGV), and SA(f), the response spectral
acceleration at seismic frequency f.
2.3.1 Earthquake Rupture Forecast
A standard fault-based ERF specifies the spatial geometries, magnitudes, and
likelihoods of potential earthquake ruptures in the interval T with fault discretization
adequate to represent the seismic hazard. The example used here is the Uniform
California Earthquake Rupture Forecast, Version 2 (UCERF2), which provides the
rupture probabilities of almost 8,000 discrete fault sources that support ruptures with
moment magnitudes M
> 5 in the California region (Field et al., 2009). UCERF2 is a
19
time-dependent (renewal) probability model, but we restrict our attention to its time-
independent (Poisson) component, which has been used to represent California seismicity
in the 2008 revisions of the National Seismic Hazard Maps (Petersen et al., 2008).
UCERF2 also accounts for off-fault earthquakes through a background seismicity model
(Field et al., 2009), but this background seismicity has not yet been included in the
CyberShake modeling and is therefore excluded from our analysis. In UCERF2,
epistemic uncertainties are represented by a set of ERFs organized into a logic tree
comprising 480 branches. In the CyberShake implementation of UCERF2, ruptures have
only been sampled from a single branch representing the mean hazard (Graves et al.,
2011b).
Each discrete fault source is labeled with an integer k, which is a member of an index
set K(r) that represents all sources relevant to the site r. Relevance of a source is usually
based on the hazard level of interest; e.g., for large values of y, very small sources or
sources very far away will contribute insignificantly to the hazard and can therefore be
safely excluded from K(r). For notational simplicity, the dependency of the source set on
site locations will not be explicitly expressed until it is needed.
For the k
th
source, the fault area is determined, but the ruptures can have variable
magnitudes corresponding to different average stress drops. Values are drawn from a
conditional magnitude distribution p(m | k). For UCERF2, this pdf is a truncated
Gaussian with a mean value given by a magnitude-area relationship and a constant
standard deviation 0.12. In the CyberShake implementation, the magnitude-area
relationship has been adjusted to conform to a self-similar scaling consistent with
m(k)
20
recorded ground motions (Graves et al., 2011b). We represent the magnitude distribution
by a discrete sample set, m ∈ M(k).
To obtain kinematical completeness, we must augment the ERF with a conditional
hypocenter distribution p(x | k) and a conditional slip distribution p(s | k, x). We represent
the former by a discrete sample set on the k
th
fault surface, x ∈ X(k). The simplest
conditional hypocenter distribution is a uniform model where all hypocenter locations are
equally likely, which is the default model for CyberShake (Graves et al., 2011b) and has
some observational support (McGuire et al., 2002).
A sample from p(s | k, x) specifies a vector-valued slip function s(ξ, t ; k, x) at all
points ξ on the k
th
fault surface and at all times t after an (arbitrary) origin time.
CyberShake’s current implementation employs the pseudo-dynamic rupture models of
Graves and Pitarka (2004; 2010), in which s(ξ, t ; k, x) depends only on the hypocenter x
and the total local slip vector . The latter is taken as a sample of a self-
affine Gaussian random field, independent of x, with an isotropic spatial spectrum that
decays as the square of the wavenumber, which is a model that approximates our current
understanding of rupture dynamics (Mai and Beroza, 2002; Guatteri et al., 2004). The
moment magnitude of a rupture (Hanks and Kanamori, 1979) is a logarithmic functional
of the seismic potency, defined to be the length of the total integrated slip vector,
m s ( )~ log s ξ,t;k,x ( )dtdξ
−∞
+∞
∫
Σ
k
∫
(2.1)
s(ξ,t;k, x)dt
−∞
+∞
∫
21
where Σ
k
is the area of the fault rupture surface. Therefore, the conditional magnitude
distribution given by the ERF can always be subsumed into p(s | k, x) via this magnitude
functional. We represent p(s | k, x) by a discrete sample set, s ∈ S(k, x).
2.3.2 Time-Independent Hazard Model
From a kinematically complete ERF discretized in the way presented above, the
probability of exceeding (PoE) for the intensity level y at site r can be computed as
P Y > y |r,T ( )=1− 1−P k |T ( ) p x |k ( ) p s |k,x ( )P Y > y |r,k,x,s ( )
s∈S k,x ( )
∑
x∈X k ( )
∑
⎛
⎝
⎜
⎞
⎠
⎟
k∈K
∏
(2.2)
where P(k | T) is the probability of an event at the k
th
source in time T, and
P(Y > y | r, k, x, s) is the conditional probability of exceedance for a particular earthquake
rupture. This expression assumes the discretized conditional probabilities have been
normalized to unity:
p(x |k)
x∈X(k)
∑
= p(s |k,x)
s∈S(k,x)
∑
= 1 (2.3)
For a time-independent ERF, the conditional event probabilities depend only on
occurrence rate v
k
of events of the k
th
source:
P(k |T) = 1 − e
−ν
k
T
(2.4)
Consequently, the total rate of exceedance per unit time (e.g. one year) is given by
22
λ(Y > y |r) = ν
k
P(Y > y |r,k)
k∈K
∑
= ν
k
p(x |k) p s |k,x ( )P Y > y |r,k,x,s ( )
s∈S(x,k)
∑
x∈X(k)
∑
k∈K
∑
(2.5)
and the exceedance probability reduces to a Poisson form,
P Y > y |r,T ( ) = 1 − e
−λ Y>y|r ( )T
(2.6)
2.4 Ground Motion Prediction Equations (GMPEs)
2.4.1 Model-based factorization
Empirical GMPEs have been developed for estimating exceedance probabilities from
a standard ERF assuming ergodicity; i.e., that the variability of ground motions at many
sites for past earthquakes accurately represents the variability of ground motions
expected at a single site for future earthquakes (Anderson and Brune, 1999). The
predicted ground-motion intensity at a site r is represented as a log-normal random
variable
(2.7)
where ε is the fraction of logarithmic standard deviations
σ
T
by which the logarithmic
ground motion intensity lnY deviates from the mean ln
Y , and is assumed to be a
normally distributed random variable with zero mean and unity standard deviation,
quantifying the randomness of ground motions at a site r for a specific earthquake
rupture. In this form, the mean intensity depends on the slip function only through the
lnY r,k,x,m;ε ( )= lnY r,k,x,m ( )+εσ
T
Y
23
magnitude functional m(s), and all other aleatory variability in s is subsumed into the
second term of Equation (2.7).
The recent GMPEs, in particular, the Next Generation Attenuation (NGA) relations,
have been developed to determine the mean intensity and standard deviation σ
T
by
statistical regression of recored ground-motions (Abrahamson and Youngs, 1992; Joyner
and Boore, 1993), including Abrahamson and Silva (2008), Boore and Atkinson (2008),
Campbell and Bozorgnia (2008), and Chiou and Youngs (2008) which we refer to
subsequently as AS08, BA08, CB08, and CY08, respectively. The fifth NAG relation,
Idriss (2008) is not used in this dissertation because it applies only to rock sites. The
mean intensity can be factorized into four terms,
lnY(r,k,x,m) = F
1
(r) + F
2
(r,k) + F
3
(r,k,x) + F
4
(k,m) (2.8)
where F
1
, F
2
, F
3
, and F
4
are the logarithms of factors that represent the average site, path,
rupture directivity, and source-size effects, respectively. In the originally published forms
of the NGA relations, the model-based factorization excluded an explicit dependence on
the hypocenter x (F
3
= 0), but a correction term expressing this dependence was
developed by Spudich and Chiou (2008) based on an isochrone representation (Bernard
and Madariaga, 1984), and we include it here. In Equation (2.7), we use indices to
represent model parameters m in Equation (1.1), and the coefficients determined from
regression are not explicitly written for simplicity. A summary of the model parameters
used in NGA GMPEs is presented in Table 2.1. In our index notation, r relates with site
parameters, and k indicates the geometry and faulting mechanism of a fault with
Y
24
hypocenter location x. The effect of slip spatial-temporal perturbations is not explicitly
considered as a model factor. Hence, s represents only the moment magnitude, i.e., s
reduces to m, and S(k,x) reduces to M(k).
Table 2.1 Model parameters used by four NGA GMPEs.
NGA GMPEs AS08 BA08 CB08 CY08
Source
parameters
M
W
Rake and dip
angles
Z
TOR
, W, and
aftershock flag
M
W
and rake
angle
M
W
Rake and dip
angles
Z
TOR
M
W
Rake and dip
angles
Z
TOR
, W, and
aftershock flag
Path
parameters
R
JB
, R
rup
, and
R
X
Hanging
wall flag
R
JB
R
JB
and R
rup
R
JB
, R
rup
, and
R
X
Hanging wall
flag
Site
parameters
V
S30
and Z
1.0
V
S30
V
S30
and Z
2.5
V
S30
and Z
1.0
M
W
: Moment magnitude.
R
JB
:
Shortest distance from the recording site to the surface projection of the rupture.
R
rup
: Shortest distance from the recording site to the rupture.
R
X
: Site coordinate measured perpendicular to the fault strike from the surface projection of the
up dip edge of the rupture, with the down-dip direction being positive;
V
S30
: Shear wave velocity averaged over the top 30 m.
W : Fault down-dip width
Z
TOR :
Depth to the top of the rupture.
Z
1.0
: Depth to the 1.0 km/s shear-wave velocity horizon.
Z
2.5
: Depth to the 2.5 km/s shear-wave velocity horizon.
In this study, we use a python package called pynga developed by the author of this
dissertation to empirically predict ground motions. This software package has been tested
and applied in broadband platform for validate simulation models.
25
2.4.2 Empirical Source Directivity Models
Spudich and Chiou (2008), referred as to SC08, developed a model correction that
accounts for the source-directivity effect on the mean ground-motion intensities estimated
by NGA GMPEs for different hypocenter positions on the fault, and provided a way to
estimate F
3
in Equation (2.8). The source directivity effects have been studied for a finite
source that gives different radiation patterns relative to the point source.
Figure 2.1 Radiation pattern for (a) double-couple point source and (b) finite source. e
R
,
e
ih
, and e
ph
are radial, vertical, and transverse directions. (rake=0; strike=0; dip=90)
26
Figure 2.1 shows the radiation pattern for a point source and a finite source (with
rupture velocity = 0.8*shear wave velocity) in a whole space. The factor that changes the
point radiation pattern to finite source radiation pattern is 1−
v
r
β
cos ψ ( )
⎡
⎣
⎢
⎤
⎦
⎥
−1
, where v
r
and
β are the rupture and shear wave velocity, respectively, and ψ is the azimuth from source
strike to source-site direction. For the simple strike-slip focal mechanism, the transverse
component (shear wave) shows larger radiation in the direction of rupture propagation
(black arrow, ψ=0
o
) relative to the opposite direction (ψ=180
o
).
Spudich and Chiou (2008) developed physically-based predictor variables by using
isochrone theory (Bernard and Madariaga, 1984; Spudich and Frazer, 1984) to calculate
the directivity correction, i.e. the term F
3
in Equation (2.8)
for each NGA GMPEs except
Idriss’s model. Figure 2.2 shows the source-site geometry used in developing the
directivity parameter. Isochrones are the locus of points on the fault that radiate elastic
waves all of which arrive at a given station at the same time (isochrones time). Each
station has a different isochrones distribution, that is, each station sees different parts of
the fault at given time. Hence, isochrones can be used to extract that part of the rupture
that produces a peak in the ground motion for a given station (Schmedes and Archuleta,
2008). Defining a hypocenter location as x and an arbitrary point on fault as ξ, the
isochrones time is defined as
τ ξ,r;x ( )=T ξ,r ( )−T x,r ( )+t
*
ξ;x ( ) (2.9)
27
where T(.) shows the travel time of seismic wave (P or S wave) from a point on fault to a
station r, and t
*
is the rupture propagation time from one point on the fault to another
point on the fault. The isochrones velocity (different from the rupture velocity), which
relates both hypocenter and site, is defined as
c = ∇
ξ
τ ξ,r;x ( )
−1
(2.10)
For simple geometry like in Figure 2.2, we can derive the approximate isochrones
velocity along with (2.9) for shear wave propagation as
c= T x,r ( )−T x
0
,r ( )+ x−x
0
( )/v
r
⎡
⎣
⎤
⎦
/ x−x
0
( )
⎡
⎣
⎤
⎦
−1 −1
=
D
R
rup
/β −R
hypo
/β +D /v
r
=
β
β /v
r
− R
hypo
−R
rup
( )
/D
(2.11)
The directivity parameter developed in SC08 is the isochrones directivity predictor
(IDP), which consists of three parts
IDP=C
r
S
r
R
ri
(2.12)
C
r
=
min( c',c
0
)−v
r
/β
c
0
−v
r
/β
(2.13)
S
r
= ln min d
0
,max s,h ( ) ( )
⎡
⎣
⎤
⎦
(2.14)
where R
ri
is the geometrical mean of radiation pattern for the two horizontal components
of a point source, C
r
represents the azimuthal dependence due to directivity as shown in
Figure 2.1, and S
r
represents the effective rupture length that affect the directivity at the
28
site. c
0
and d
0
are constant parameters determined subjectively during regression, and in
SC08 model, the ratio between rupture and shear wave velocity is assumed to be 0.8
(constant). The most important parameter in IDP that controls the modification on the
point source radiation pattern to form the radiation for finite source is
c ' which is the
normalized isochrone velocity as in Equation (2.11) by shear wave velocity as
c'= c /β .
In Figure 2.3, we show the example for the F
3
(fD shown in the figure) and IDP using a
MATLAB package developed by Spudich and Chiou (2008) for strike slip (SS) and
reverse (RV) events. Warm color in IDP shows large effects that appear in the direction
of rupture propagation.
Figure 2.2 Rupture and site geometry for computing isochrone directivity predictor.
29
Figure 2.3 Examples of fD and IDP to illustrate the directivity effects considered in
empirical models. Soild thick black line is the fault trace on the surface, and black dot
shows the epicenter location. fD has the natural logarithmic scale showing the correction
to SA at 3.0s. Strikes for both faults are 0
o
.
2.4.3 Hazard Calculation based on GMPEs
The exceedance probability in Equation (2.5) can be expressed in our index notation
as
λ Y > y |r ( )= v
k
p x |k ( ) p m |k ( )P Y > y |r,k,x,m ( )
s∈S k,x ( )
∑
x∈X k ( )
∑
k∈K
∑
(2.15)
The conditional magnitude distribution p(m | k) is assumed to be normalized to unity over
its sample set M(k). The conditional exceedance probability at hazard level y is given by
an integral over ε,
P(Y > y |r,k,x,m) = f (ε) H[lnY(r,k,x,m;ε)− lny]dε
−∞
+∞
∫
(2.16)
−50 0 50
−150
−100
−50
0
50
100
150
East [km]
North [km]
SS; SA 3 s, BA08
fD
−0.5
0
0.5
−50 0 50
−150
−100
−50
0
50
100
150
East [km]
North [km]
IDP
1
2
3
4
−50 0 50
−150
−100
−50
0
50
100
150
East [km]
North [km]
RV; SA 3 s, BA08
fD
−0.5
0
0.5
−50 0 50
−150
−100
−50
0
50
100
150
East [km]
North [km]
IDP
0.5
1
1.5
2
2.5
30
where f(ε) is the normal pdf and H is the Heaviside function. which is zero if lnY is less
than lny, and 1 otherwise. The summation over the sample sets in Equation (2.15) are
Monte Carlo integrations that can be made arbitrarily accurate by choosing large enough
samples (Press et al., 2007). The first is a necessary discretization, because it is intrinsic
to the ERF, but the second two summations are not, since the simple forms usually
adopted for the conditional distributions (e.g., uniform or truncated Gaussian,
respectively) can be calculated analytically or by simple numerical quadrature. However,
Equation (2.15) allows for a more direct comparison of the NGA models with simulation-
based hazard models, which are necessarily discretized.
2.5 Simulation-Based Hazard Models
The empirical GMPEs capture the average effects of source, wave propagation, and
site conditions, but do not represent some other important deterministic aspects of source
directivity, basin amplification, and directivity-basin coupling, as well as small-scale
variations caused by rupture-process complexity and 3D geological structure, which have
potential influences on the ground motion predictions and hazard calculations. In
addition, lack of ground motion data for constraining the form of mean predictions cause
limitation of NGA GMPEs and the implementation in PSHA under the ergodic
assumption. All those raised issues motivate the development of ground motion
prediction using deterministic earthquake simulations, and further application of
simulation-based ground motion prediction in seismic hazard calculations. The three-
dimensional (3D) physics-based earthquake simulations incorporate the understanding of
31
fault rupture process, wave propagation physics, and site response characterization,
providing a more comprehensive description and robust estimates of the ground shaking
at a site expected from future earthquakes.
2.5.1 Physics-Based Earthquake Simulations
Recent advances both in seismology and in cyber-infrastructure have made possible
for large scale, high-resolution, physics-based predictive earthquake simulations. The
elements essential for accurate ground motion predictions include earthquake description,
3D structural model of earth, calculation of wave propagation and attenuation, and high-
performance computational facilities. First, we need detailed representation of fault
geometry (length, width, strike, dip, etc.) with source model that capture the complexities
of dynamic fault failure. Such source model could be pseudo-dynamic, meaning that the
description of rupture process is kinematic but the property is consistent with the
dynamic physics during earthquakes (Mai and Beroza, 2002). For example, Figure 2.4
shows one example of kinematic source descriptions for earthquake rupture. The effect of
different slip models on the ground motion prediction using simulation-based model will
be discussed in Chapter 4. Secondly, we need earth models with good resolution of near
surface (depth < 30km) seismic velocity structure and attenuation details, including
geotechnical layer and sedimentary basins, in which radiated seismic waves from a
source propagate towards a site. For example, the community velocity model developed
by researches through the SCEC (CVM-S) gives detailed seismic velocity structure for
the southern California as shown in Figure 2.5. The effects of different 3D velocity
models on the ground-motion prediction will be discussed in Chapter 5. Thirdly, with
32
detailed source and velocity description, we need efficient numerical methods for
calculation of wave propagation and attenuation (scattering and anelastic attenuation),
that is, the computer program that numerically solves the 3D elastic/anelastic wave
equations (Aki and Richards, 2002) by discretizing the whole calculation region into
finite grids. The way to discretize and solve the wave equation, for example, includes the
finite differences and finite elements (Olsen et al., 1995; Graves, 1996; Bielak et al.,
2005). The major discrepancy among those numerical methods is the differences between
mesh and grid representation of the same material model. However, even the largest
differences in the synthetic seismograms are small (Bielak et al., 2010). Furthermore, the
significant growth in computational power and efficiency have made the large-scale
simulations for future earthquakes much more feasible (Cui et al., 2007).
(a) (b)
Figure 2.4 Illustration of kinematic rupture models developed by Graves and Pitarka
(2010). (a) Panels from top to bottom are final slip with rupture front, rise time (duration
of slip time function), and rake direction, respectively. (b) Slip velocity function
generated by using two triangle functions.
33
Figure 2.5 Fence diagram of P-wave velocity in and around the Los Angeles region
basins. Cross section locations shown as red lines in lower left panel. Modified from
Magistrale et al. (2000).
The current deterministic earthquake simulation could reach up to 4 Hz using a finite
element method with a velocity model that has minimum shear wave velocity down to
200 m/s (Taborda and Bielak, 2013). Undergoing research incorporate high-frequency
characteristics in the source representation and the structural heterogeneity of the seismic
velocity models into current simulation methods by considering aspects such as
geometrical complexity of the faults (Shi and Day, 2013), the random distributions of slip,
rupture velocity, and rise time, and the stochastic characteristics of the material properties
of near-surface layers (Frankel and Clayton, 1986). Meanwhile, hybrid methods combine
the low frequency (<1 Hz) deterministic simulation with high frequency (up to 10 Hz)
stochastic simulation (Boore, 1983; Beresnev and Atkinson, 1997), producing broadband
earthquake ground motion predictions up to 10 Hz. One limitation of simulations is the
34
modeling of site effects due to the local site condition (this is included in the observed
ground motion responses), and the solution currently is to use the empirical site
amplification factors to correct the simulated results (Graves and Pitarka, 2010). One, or
more, historic earthquakes in Southern California have been selected as target events, and
the results of high-frequency ground motion simulations (both hybrid and deterministic)
have been compared with observed data from these earthquakes for validating physics-
based simulations. Furthermore, deterministic earthquake simulations have been used as
supplementary “data” for improving the GMPEs, such as the consideration of basin
amplification and source directivities (Day et al., 2008; Spudich and Chiou, 2008).
2.5.2 Simulation-Based Seismic Hazard Model: CyberShake
The probabilistic seismic hazard analysis using the physics-based ground-motion
simulations has been investigated in Southern California by SCEC CyberShake project.
Figure 2.6 shows the sampling region of CyberShake with sites of interest (total 235
shown) at which deterministic simulations are performed for each earthquake rupture
given by the kinematically complete ERF model generated from the UCERF2 model.
With a lot of active faults, high probability of large earthquakes occurring, and
sedimentary basins accommodating large ground motions, the region, where a large
number of people live, provide a natural laboratory to apply the simulation-based seismic
hazard analysis.
35
Figure 2.6 Map of the sampling region for CyberShake in Southern California. Black
squares indicate grid of sites where CyberShake hazard curves have been computed. Blue
lines indicate selected fault ruptures from UCERF2.0. Red stars represent possible
epicenral locations. White letters indicate the major sedimentary basins in the Southern
California. (OP=Oxnard Plain, SFV=San Fernando Valley, SGB=San Gabriel Basin,
LAB=Los Angeles Basin, CB=Chino Basin, SBB=San Bernardino Basin)
For each fault surface k in the UCERF2 model, hypocenters are spaced evenly along-
strike about every 20 km at a constant focal depth, resulting in an epicenter sampling set
X(k). A number of realizations of s(ξ, t | k, x) are generated for each epicenter using the
pseudo-dynamic rupture model of Graves and Pitarka (2004), resulting in a sample set
S(k, x). The total number of rupture variations—the number of elements in the union of
S(k, x) over all k and x—is approximately 415,000.
For each site, a strain Green tensor (SGT) was calculated for horizontal system of
force couples using the SCEC Community Velocity Model, Version 4 (CVM-S4, Kohler
et al., 2004). Seismic reciprocity then allowed the horizontal-component seismograms for
119˚ 118˚ 117˚ 116˚
34˚
35˚
0 100
km
0 100
km
SBB SGB
CB
LAB
SFV OP
36
each rupture variation to be computed by a (numerically cheap) quadrature of the SGT
over each rupture variation. We represent each of those rupture variations as F(ξ,t|k,x,s),
where ξ represents the spatial location on the rupture surface. Secondly, instead of
forward simulating wave propagation from each rupture variation at a site, the seismic
reciprocity is used to compute ground motions for each site from each of those rupture
variations with the form (Aki and Richards, 2002; Zhao et al., 2006)
(2.17)
where H represents the strain Green tensors (SGTs) that are calculated only once for each
site given the three-dimensional (3D) community velocity models (Magistrale et al.,
2000; Suss and Shahi, 2003), but saved at each fault surface Σ. With calculated SGT for
each site, the ground motion (simulation gives waveforms) is just the time convolution
between SGT and rupture variation. Because the number of sites was more than three
order of magnitude smaller than the number of rupture variations, the use of reciprocity
required about 1000 times less computer time than direct simulation of the wavefields for
each rupture variation. This efficiency makes feasible a full rendering of the hazard curve
for each site. The CyberShake hazard map for Los Angeles was thus derived from an
ensemble of almost 200 million seismograms.
In simulation-based hazard models, Y (r,k,x,s) is fully determined by the site
location r, source index k, hypocenter location x, and slip function s, and the conditional
exceedance probability in equation (2.5) is given simply by
u r,t |k,x,s ( )= dτ
−∞
∞
∫
dΣ ξ ( )
Σ
∫
H ξ,τ;r,t |k,x,s ( ) :F ξ,t−τ |k,x,s ( )
37
P Y > y |r,k,x,s ( )= H lnY r,k,x,s ( )− lny ⎡
⎣
⎤
⎦
(2.18)
The Heaviside function expresses the fact that the deterministic functional Y (r,k,x,s)
either exceeds the hazard level y or it does not—the probability is either one or zero.
Thus, the aleatory variability represented by p(m | k) and σ
!
in the GMPEs is replaced by
the conditional slip distribution p(s | k, x), because other sources of variability, in
particular path and site effects, are assumed to be modeled. No ergodic assumption need
be made, and any modeling errors (e.g., in the seismic velocity model) are represented as
epistemic uncertainties.
2.5.3 Discussion
By analyzing observed ground motions, NGA GMPEs estimate the mean and
standard deviation to quantify the aleatory variability in ground motion. Different from
implementation of NGA GMPEs shown in Equation, simulation-based hazard model,
CyberShake, deterministically calculates the ground-motion intensities at a specific site
for each specific rupture variation in a large sampling set that considers the aleatory
variability in kinematical source models, generating a distribution for ground motions.
Figure 2.7 shows the distribution of 3s SA (in natural logarithmic unit) for one
specific rupture in UCERF2.0 and one site r in CyberShake sampling region calculated
by different GMPMs. The variability of simulation-based ground motion prediction
lnY(r,k,x,s) for different hypocenter locations and slip distributions is indicated by the
histogram. The probability distributions of ground motions predicted by NGA GMPEs
are shown as lines. Comparing the distribution for CyberShake with those for GMPEs,
38
we found that mean intensities are comparable, but the standard deviations are different
between CyberShake and NGA GMPEs. The ergodic assumption necessary for
implementing empirical GMPEs in PSHA has potentials to have an upward bias in the
estimated hazard level (Anderson and Brune, 1999), and the repeatable source, site, and
path effects found in observations have been found to significantly reduce the single-site
standard deviation by up to 47% with respect to the total aleatory uncertainties given by
GMPEs (Atik et al., 2010; Lin et al., 2011). On the other hand, CyberShake is a explicitly
simulation-based model for each site without the ergodic assumption for the probabilistic
hazard estimates, and considers those repeatable effects for each source-site pair by
including the deterministic earthquake rupture and wave propagation effects in predicting
the ground motion response, reducing the aleatory variability due to earthquake
occurrences. As depicted in Figure 2.7, different NGA GMPEs provide different values
of mean and standard deviation, representing the epistemic uncertainty in empirical
model developments. It is also important to notice that, due to the lack of knowledge, 3D
velocity models and kinematic descriptions used in CyberShake simulations are not
perfectly representing the nature. These epistemic uncertainties could significantly affect
the resulting estimations of ground-motion intensities and hazard assessment. There are
several CyberShake studies that use available 3D velocity models (CVM-SCEC and
CVM-Harvard), kinematic source models (Graves and Pitarka 2007, 2010), and two
numerical codes (for calculating SGT) so that the epistemic uncertainty is considered in
simulation-based PSHA as logic trees shown in Table 2.2.
39
Table 2.2 Velocity models and rupture model generators used in CyberShake models.
Logic trees
Velocity models CVM-SCEC4 CVM-Harvard v11.9
Rupture models
GenSlip v2.1 (Graves and
Pitarka, 2004)
GenSlip v3.2 (Graves and Pitarka,
2010)
Numerical codes Graves and Pitarka (2007)
Anelastic wave propagation (AWP-
ODC)
Figure 2.7 The comparison of ground motion intensity (3 s spectral acceleration)
distribution for one rupture k at one site r obtained by different GMPMs. The histogram
shows the variability of simulated lnY. Each colored solid line indicates the probability
density that is generated using mean and sigma from each NGA GMPE.
40
Figure 2.8 Hazard maps of 3s SA with 2% probability of exceedance in 50 years
calculated using (a) AS08, (b) BA08, (c) CB08, (d) CY08, and (e) CyberShake (Using
OpenSHA).
Using the same earthquake rupture model based on UCERF2 and CVM-S as the
velocity model from which basin depths (Z
1.0
and Z
2.5
) are extracted for NGA GMPEs,
significant discrepancies have been found among hazard maps produced by CyberShake
and NGA GMPEs. Figure 2.8 present the hazard maps of 3s SA for 2% probability of
41
exceedance in 50 years within the CyberShake sampling region calculated by NGA
GMPEs and the original CyberShake (using CVM-S and rupture model by Graves and
Pitarka, 2004). First, we can see the discrepancies between NGA GMPEs due to the
epistemic uncertainties (different means and standard deviations, or different ground
motion distributions), for example, BA08 gives larger hazard along the San Andreas
Fault and CB08 model gives larger basin effects. Secondly, hazard map produced by
CyberShake shows similar patterns as NGA GMPEs, except higher hazard is found in
sedimentary basins (refer to Figure 2.6) relative to NGA GMPEs (Graves et al., 2011b).
We would like to quantify and understand those similarities and differences between
the two types of models. The NGA models are easy to compare with one another,
because all four have similar model-based factorizations. For example, they differ
because the basin-effect factor of CB08 is a function of Z
2.5
(the depth to the 2.5 km/s
surface in seismic shear velocity v
s
), those for CY08 and AS08 are functions of Z
1.0
(with
different functional dependencies and parameters), and that for BA08 is unity (i.e., no
basin effect is included). In the CyberShake model, however, the basin effects involve
complex wavefield interactions within the entire 3D seismic velocity structure; therefore,
they are not simple functions of a single basin-depth parameter. Moreover, the basin
effects can be substantially enhanced by source directivity effects (e.g., Olsen et al.,
2006, 2008; Graves et al., 2008), a phenomenon we will call directivity-basin coupling.
42
2.6 Averaging-based Factorization
We have developed a technique for the comparison of PSHA models that does not
depend on any model-based factorization but instead relies on a hierarchical sequence of
averaging operations applied to large ensembles of intensity-field realizations drawn from
the probabilistic models. These operations lead to a unique averaging-based factorization
(ABF) of the intensity field for each source realization. The elementary factors of the
ABF are themselves spatial fields that can be mapped and thus easily compared. The
ratios of intensity fields from different models can be also be uniquely factored, allowing
the direct display of model differences.
2.6.1 Excitation Functional
ABF applies a series of averaging operations to the excitation functional, defined as
the logarithmic intensity of the ground motions sampled at r ∈ R from a rupture of source
k ∈ K(r) with hypocenter x ∈ X(r, k) and slip function s ∈ S(r, k, x),
G(r,k,x,s) ≡ lnY(r,k,x,s) (2.19)
The dependences among the sample sets imply a five-level seismological hierarchy,
which we represent as the graph
A←
R
B ←
K
C ←
X
D ←
S
E (2.20)
The lowest level E comprises the unaveraged excitation fields {G(r,k,x,s) }, and each
arrow indicates averaging over the subscripted sample sets; e.g., the elements at level E
43
are averaged over S to obtain the elements at level D. The ABF recipe entails four levels
of averaging:
Level D. Compute the expectation over the slip functions s for each r, k, and x,
G(r,k,x,s)
S
= p(s |k,x) G(r,k,x,s)
s∈S(k,x)
∑
(2.21)
Level C. Compute the expectation over the hypocenters x for each r and k,
G r,k,x,s ( )
S,X
= p x |k ( ) G r,k,x,s ( )
S
x∈X k ( )
∑
(2.22)
Level B. Compute the expectation over the sources k for each r,
G r,k,x,s ( )
S,X,K
= p k |r ( ) G r,k,x,s ( )
S,X
k∈K
∑
(2.23)
Level A. Compute the expectation over all sites r,
G r,k,x,s ( )
S,X,K ,R
= p r ( ) G r,k,x,s ( )
S,X,K
r∈R
∑
(2.24)
The first two levels of the seismological hierarchy comprise ensembles of excitation
fields on R, the third is a single excitation field on R, and the fourth is a scalar.
In Equations (2.21) and (2.22), the expectations are taken over the conditional slip
and hypocenter pdfs, which have been previously defined. Equations (2.23) and (2.24)
introduce two new pdfs, a conditional probability for the sources, p(k | r) and an absolute
probability for the receivers, p(r). The latter expresses how sites are weighted in the
interpretation of the hazard level (2.24). We could plausibly weight the sites inversely
with their areal density on a seismic hazard map or, in different applications, by
44
population or some other hazard-exposure measure. Since the CyberShake site density is
fairly uniform across the Los Angeles region (Figure 2.6), we adopt simple uniform
weighting: p(r) = 1/N
R
, where N
R
= 235 is the total number of sites.
Appropriate specification of the conditional source probability p(k | r) will also be
application-specific. Many studies focus on the threat level set by some intensity
threshold, y
0
. In the next Chapter, we describe how source probabilities can be
constructed by disaggregation of the site-specific hazard curves at y
0
when applying ABF
in comparing PSHA models.
2.6.2 ABF Representation of Simulation-Based Excitation Functionals
Having ordered the expectations (2.21)-(2.24) according to the seismological
hierarchy (2.20), we can exactly and uniquely represent each excitation functional as the
sum of five terms,
G r,k,x,s ( )= A+B r ( )+C r,k ( )+D r,k,x ( )+E r,k,x,s ( ) (2.25)
The first is the regional excitation level (2.24),
A = G r,k,x,s ( )
R,K ,X,S
(2.26)
and the other four are the differential excitation fields; i.e., differences among successive
pairs in the hierarchy:
B r ( )= G r,k,x,s ( )
S,X,K
− G r,k,x,s ( )
S,X,K ,R
(2.27)
45
C r,k ( )= G r,k,x,s ( )
S,X
− G r,k,x,s ( )
S,X,K
(2.28)
D r,k,x ( )= G r,k,x,s ( )
S
− G r,k,x,s ( )
S,X
(2.29)
E r,k,x,s ( )=G r,k,x,s ( )− G r,k,x,s ( )
S
(2.30)
Equations (2.27)-(2.30) constitute the ABF representation of the excitation functional.
The differential excitation fields can be interpreted in terms of distinctive
seismological phenomena. In Chapter 3, we will provide numerical examples from the
averaging-based factorizations of NGA and CyberShake models; here, we make a few
theoretical observations. The differential excitation fields are defined such that the
expectation of each with respect to its trailing variable is exactly zero:
B r ( )
R
= C r,k ( )
K
= D r,k,x ( )
X
= E r,k,x,s ( )
S
= 0 (2.31)
Therefore, the variations in E are entirely due to the aleatory variability within the sample
set of slip functions S(r, k, x), while the expected excitation over this set, which is
proportional to the mean magnitude m , is promoted to level D. We therefore call E the
source-complexity effect. Likewise, the variations in D are entirely due to the aleatory
variability within the sample set of hypocenters X(r, k), and we therefore refer to D as the
directivity effect. We note that the average directivity effect is promoted to level C, which
is consistent with GMPE model-based factorization in Equation (2.8). The variations in C
come primarily from excitation differences caused by wave propagation from source k to
site r, which are dominated by distance attenuation, so we call C the path effect. Finally,
46
the differential response of sites to the average excitation by all sources is measured by B,
so it represents the site effect.
Maps can be made to display the differential excitation fields. For example, B(r) is a
map of the mean site effect (referred to as the B-map), and C(r, k) is a map of the mean
path effect for the k
th
source (referred to as C-maps). The maps proliferate at the lower
averaging levels, so it is often convenient to map the dispersion rather than their values.
For example, we can define the sample variances
σ
D
2
r,k ( )= D
2
r,k,x ( )
X
, r,k fixed (2.32)
σ
E
2
r,k,x ( )= E
2
r,k,x,s ( )
S
, r, k and x fixed (2.33)
The σ
D
-map displays the site-specific dispersion in excitation due to directivity, whereas
the σ
E
-map displays the dispersion due to source complexity. In the current CyberShake
implementation, the σ
E
-map does not depend on the hypocenter location x and only
weakly depends on the source k.
2.6.3 ABF Representation of GMPE-Based Excitation Functionals
The seismological hierarchy for the GMPEs must slightly modified to account for the
form of the conditional exceedance probability given by Equation (2.16). We represent
the integral over aleatory variability in Equation (2.16) as a summation over a sample set
drawn from a normal distribution, ε ∈ N. Then we can express the ABF hierarchy in
graphical form similar to Equation (2.20):
47
A←
R
B ←
K
C ←
X
D ←
M∪N
E (2.34)
In other words, we take expectations over both the magnitude sample set M(k) and the
normal sample set N as the first level of averaging. The former promotes the mean
magnitude m to the D level, whereas the latter delivers zero.
The averaging scheme in Equation (2.34) preserves the ABF representation of
Equation (2.25). Moreover, it makes clear the correspondence between the averaging-
based factors in Equation (2.25) and the NGA model-based factors in Equation (2.8):
B r ( )~F
1
r ( ) (2.35)
C r,k ( )~F
2
r,k ( ) (2.36)
D r,k,x ( )~F
3
r,k,x ( ) (2.37)
E r,k,x,s ( )~F
4
k,m ( )+σ
T
ε (2.38)
This correspondence will be quantified and numerically validated in Chapter 3.
The E maps for GMPEs are uninformative, because they contain only aleatory
variability that is either a constant for all sites (from the dispersion of m) or spatially
uncorrelated noise (from the dispersion of ε). In the limit of large sample sizes, the σ
E
-
map defined by Equation (2.33) is just a constant (independent of r, k, and x) equal to the
root-mean-square of the magnitude dispersion and aleatory variability,
σ
E
= σ
M
2
+σ
T
2
(2.39)
48
where σ
M
represents the variability of ground motions due to magnitude dispersion
given fault rupture area. Such magnitude dispersion is provided in ERF models
independent of any ground motion prediction models. The total uncertainty σ
T
here has
been reduced by adding directivity corrections in the NGA08 GMPEs, and it includes the
inter-event variability that represents the scatter of ground motion intensities excited by
different seismic events and the intra-event variability that indicates the variability of
ground motion intensities excited by a particular seismic event at different sites.
For simulation-based models, the σ
E
defined in Equation (2.33) could be explicitly
represented as
σ
E
= σ
M
2
+σ
U
2
(2.40)
where U is the sample set of magnitude-normalized rupture variations, such as variations
in slip distributions given a magnitude.
2.6.4 ABF Representation of Excitation Differences
Our main motivation for developing ABF is the comparison problem: how can
differences between two hazard models be characterized in terms of physical effects
when they do not share a common model-based factorization? We solve this problem by
applying ABF to the excitation differences (logarithms of the intensity ratios) sampled at
common sites from common sources. The residual excitation is defined to be the
49
difference between the excitation of a target model and that of a reference model.
Variables of the latter are superscripted with a tilde:
g(r,k,x,s) = G(r,k,x,s) −
G(r,k,x,s)
residual = target − reference
(2.41)
With the convention, the reference model is chosen to be a simplified standard, then any
physical effects in the target model missing from the standard will have their appropriate
sign. We average the residual in Equation (2.41) over common elements of the sample
set; e.g. over sitesr∈ R∩
R and sources k∈K∩
K . We assume these intersections are
large enough to sample the pdfs in Equations (2.21)-(2.24). In our numerical examples,
we make them identical. Then, the ABF representation of the residual excitation
functionals takes the same form as Equation (2.25):
g r,k,x,s ( )= a+b r ( )+c r,k ( )+d r,k,x ( )+e r,k,x,s ( ) (2.42)
The lower-case letters denote differences in corresponding terms of the ABF
representations of G(r,k,x,s) and
G(r,k,x,s) ; e.g., the regional excitation residual is
a ≡A−
A . By definition, the expectation of each term with respect to its trailing variable
is exactly zero:
b r ( )
R
= c r,k ( )
K
= d r,k,x ( )
X
= e r,k,x,s ( )
S
= 0 (2.43)
Maps of the residual differential excitation fields are the main tools we use for model
comparisons. For example, the b-map shows the mean site-effect residual of the target
model relative to the reference model, and c-maps display the mean path-effect residual
50
for the k
th
source. We can also compute σ
d
-maps and σ
e
-maps from the analogs of
Equation (2.32) and (2.33). For comparisons involving GMPEs as either target or
reference models, the sample set S(k) = M(k)∪N produces e-maps that only contain the
dispersion information analogous to (2.39).
2.7 Conclusions
We generalized the description for PSHA to incorporate simulated ground motions in
place of GMPEs, and we specialized this description to two implementations of ground
motion prediction models in the hazard assessment, NGA GMPEs and CyberShake for
southern California. The key of implementing simulation-based ground motion
predictions is to use kinematically complete ERF model, which is general enough to
include the standard ERF for PSHA using empirical GMPEs. The implementation of
simulation-based ground motion prediction shows conceptual differences relative to the
case where empirical GMPEs are used as the ground motion prediction models.
Simulations use the extended ERF, in particular, the kinematically complete ERF model
to produce the ground motion distributions deterministically, while NGA GMPEs use the
ergodic assumption and lognormal distribution for ground motions. Our general
description of the framework incorporates both types of ground motion prediction
models.
In order to understand the differences in hazard maps calculated from NGA GMPEs
and CyberShake, it is necessary to decompose the predicted ground motions into
51
separated components and quantify the differences in the contribution of each component
by NGA GMPEs and physics-based CyberShake calculations to ground-motion
intensities. We developed a technique that does not depend on any model-based
factorization but relies on a hierarchical sequence of averaging operations. These
operations lead to a unique averaging-based factorization (ABF) of the intensity field for
each source realization. The elementary factors of the ABF are themselves spatial fields
that can be mapped and thus easily compared. The ratios of intensity fields from different
models can be also be uniquely factored, allowing the direct display of model differences.
Detailed numerical examples of applying ABF to compare empirical GMPEs and
simulation-based model, CyberShake will be presented in the next chapter.
52
Chapter 3
Comparisons of NGA GMPEs and CyberShake Model
Using Averaging-Based Factorization
3.1 Abstract
We apply an averaging-based factorization (ABF) scheme that conforms to the
implementation of ground motion predictions in seismic hazard analysis, and facilitates
the geographically explicit comparisons between empirical GMPEs and simulation-based
model. Through a sequence of averaging operations over various model parameters, we
uniquely factorize model residuals into several terms, which then characterize differences
in various physical effects between models, such as basin excitation, distance attenuation,
and source directivity. We compare CyberShake with the Next Generation Attenuation
(NGA) GMPEs. For spectral acceleration at long periods (>2s), NGA models
underestimate the basin-excitation effects by up to a factor of 3 relative to CyberShake;
Moreover, basin excitations in the physics-based model are not a simple function of basin
depth as parameterized in the NGA models. Using the directivity relations of Spudich and
Chiou (2008), we quantify the extent to which the empirical directivity models capture
the source directivity effects predicted by physics-based model. We found that the
empirical directivity corrections for NGA models underestimate source directivity effects
in CyberShake, and they do not account for the directivity-basin coupling that
substantially enhances the low-frequency seismic hazards in the sedimentary basins of
the Los Angeles region.
53
3.2 Introduction
Earthquake ground motions evaluated for engineering applications by empirical
GMPEs incorporate the average effects of earthquake source, wave propagation, and
local site conditions. However, for large earthquakes and close site-source distance (e.g.
M
w
>7.5 and distance < 20km as discussed in Chapter 1), the ground motion prediction is
not well constrained. Simulated ground motions, which capture complex source features,
3D wave propagation, and site effects, have the potential to provide a valuable
supplement to empirical methods for the cases where observations are sparse. The
verification among earthquake simulations using different numerical methods has been
investigated (Bielak et al., 2010). However, such simulation-based ground motion
prediction technique has not found significant practical applications to date because of a
general sense among engineers that the simulated motions have not been adequately
validated. This, then, raises the issue of how simulated motions should be validated (Star
et al., 2011). With the help of kinematically complete ERF model, the implementation of
deterministic earthquake simulations in PSHA, CyberShake, has been investigated in the
southern California. As discussion in Chapter 2, the hazard map by CyberShake shows
significant discrepancies comparing with those calculated by NGA GMPEs, and we need
an approach to compare these types of models.
SCEC’s broadband platform provides an environment for validating simulated
waveforms with recorded waveforms of historical earthquakes, and validating simulated
ground motions of scenario events with empirical predictions by NGA GMPEs.
Waveform comparison provide direct validation, but are not an entirely sufficient
54
validation from an seismological perspective because they omit the factors that contribute
to the final differences between models, and they do not adequately demonstrate the
correlations with important independent variables such as magnitudes, distances, and
local basin structures. Star et al. (2011) compared ground-motion intensities obtained
from simulations of several earthquake scenarios with those calculated using NGA
GMPEs. They statistically investigated the variability of residuals with various model
parameters that characterize different physical effects as described in Equation. Such
comparisons are important in further engineering application with physics-based
simulations.
In this Chapter, we apply the averaging-based factorization (ABF) methodology
developed in Chapter 2 following the generalized framework of PSHA, which uniquely
and exactly decompose the excitation functionals of a particular model into different
map-based factors that effectively represent various physical effects. Comparing
corresponding map-based terms of CyberShake and NGA GMPEs, we quantify the
differences and provide seismological interpretations for those differences. We evaluate
specific attributes critical to ground motion hazard analysis, including site-to-site
variability, distance attenuation, and directivity effects. Such comparison is a necessary
first step in establishing confidence in the efficacy of simulated motions for engineering
application. The CyberShake model considered in this Chapter uses the CVM-SCEC4
and rupture model by Graves and Pitarka (2004), and NGA GMPEs are as presented in
Chapter 2. We represent as CS11 and NGA08 GEMPEs.
55
3.3 Application of ABF
Following the ABF method introduced as in Equation (2.25), the excitation
functionals of a given hazard model could be decomposed into distinct factors, the
physical meaning of which we will illustrate in the following section. In this dissertation,
the excitation functional is calculated from a component of the acceleration field as the
maximum amplitudes of response spectrum of a 5%-damped linear oscillator with
characteristic frequency f, i.e. SA(f), and comparisons are restricted to the relatively low
frequencies (f < 0.5 Hz) currently simulated in the CyberShake model. We will employ
the geometrical mean of SA(f) for two orthogonal components (north-south and east-
west) to measure the horizontal shaking intensity, although the standard NGA08 median
measure of the azimuthal variation, rotD50 (Boore et al., 2006; 2010), is available from
the CyberShake seismograms and could be easily substituted. These residual factors (in
natural logarithmic units) characterize the decomposed differences between the target and
reference models associated with each parameter. The choice of both the target and the
reference model is arbitrary, depending on the application of ABF for comparing models.
Specifically, one could use mean predictions by NGA08 GMPEs as the target and
reference models and connect the factors from ABF with those in MBF. Treating CS11 as
the target model and each of those NGA08 GMPEs as the reference model, we could
quantify the differences between the two types of models, and illustrate them
geographically because all those residual factors (except a) are map-based.
56
3.3.1 Conditional hypocenter and slip distribution
According to Equations (2.26)-(2.30), in order to do the decomposition using ABF for
a given hazard model, one need to calculate the levels defined in Equations (2.21)-(2.24)
with knowing those weighting functions, including p(s | k, x), p(x | k), p(k | r), and p(r),
where p(r) has been discussed in Chapter 2 as 1/N
R
(N
R
=235). In this section, we will
focus on the discussion of conditional hypocenter and slip distribution, and the next
section discuss the specification of p(k | r).
According to the specification of the kinametically complete ERF model, for a given
source k, the fault area is determined, but the ruptures can vary due to variability
corresponding to different hypocenter locations (rupture initiation location) and
difference average stress drop (magnitude). First, we represent the former by a discrete
sample set on the k
th
fault surface. The simplest conditional hypocenter distribution is a
uniform model where all hypocenter locations are equally likely, which is the default
model for CS11 and has some observational support (McGuire et al., 2002). The specific
weight is determined by one over the total number of hypocenters N
X
(k) for a given
source k. Secondly, possible magnitudes are sampled from the conditional magnitude
distribution p(m | k), which is a truncated Gaussian with mean value determined by
magnitude-area relationship and a constant standard deviation σ
M
=0.12, resulting in total
number of magnitudes N
M
(k). For a given magnitude m, a sample set of vector-valued slip
function s(ξ, t ; k, x) at all points ξ on the k
th
fault surface is generated, and each member
in this sample set has equal weight that is calculated by one over the total number of
members in the set, i.e. N
F
(m,k,x). Therefore, the conditional slip distribution p(s | k, x) is
57
the combination of conditional magnitude distribution and the uniform distribution for
vector-valued slip functions given those magnitudes.
The total numbers of hypocenters and vector-valued slip functions vary from source
to source, and are specified in the kinematically complete ERF model used in CS11. We
will show them when necessary in the following section after obtaining a specific source
set. Moreover, the effects of different conditional hypocenter distributions on the
resulting factors from ABF will be discussed in Chapter 4.
3.3.2 Reduction of the Source Set K by Disaggregation
In the prototype CyberShake implementation of Graves et al. (2011), sources with
nearest points more than 200 km from a site are excluded as insignificant to the hazard.
Nevertheless, there are still approximately 400 UCERF2 sources for which excitations
are computed at each site. To reduce the size of the source set, we ranked the sources
using a disaggregation procedure similar to Bazzurro and Cornell (1999). At an intensity
level y
0
, the exceedance rate for the k
th
source is
λ(y
0
|r,k) = ν
k
P(Y > y
0
|r,k) (3.1)
Ordering these rates such that λ
k
≥λ
k+1
and truncating the set at size N
K
yields the
conditional source probability:
p(k |r) = λ(y
0
|r,k)/ λ(y
0
|r,k)
k=1
N
K
∑
(3.2)
58
Note that the notation has been simplified by suppressing the dependence of these
probabilities on y
0
.
We applied this disaggregation procedure to the CyberShake sources by fixing the 3-s
spectral acceleration (SA-3s) at y
0
= 0.3 g. The high level restricted the sources to large
faults and reduced their number to 280. We normalized the source rates λ(y
0
|r,k) by the
sum over this set to obtain p(k | r). Owing to the size of the Los Angeles region and the
variability of its 3D crustal structure, this procedure resulted in conditional source
probabilities that were very heterogeneous in r, as shown by the scaled dots in Figure
3.1(a). To simplify the interpretation of the B-map (the only differential excitation field
that depends on the source-set weighting), we averaged p(k | r) over R to obtain the mean
probabilities p(k) in Figure 3.1(b); we then reduced the K set by selecting sources with
the 20 largest p(k) (filled circles) and normalized the sum of these unconditional
probabilities to unity (Table 2). The reduced set, which includes both strike-slip and
reverse-slip faults, comprises about 60% of the total source probability. Figure 3.2 shows
the fault traces of these sources, and Table 3.1 lists the source names and fault
parameters. Numerical experiments showed that increasing N
K
above 20 had a negligible
effect on the B-maps used in subsequent sections of this paper. One measure of this low
sensitivity is the mean excitation level for the CyberShake model, which varied from A =
–2.89 for N
K
= 20 to A = –2.94 for N
K
= 30.
59
Figure 3.1 Method of source selection using hazard disaggregation. Sources are identified
using CyberShake SourceIDs, and sites are represented by CyberShake SiteIDs. Hazard
level for the calculation is SA-3s at y
0
= 0.3 g. (a) Dots are scaled and colored to the
fractional contribution p(k | r) of each source to the hazard at each site in the CyberShake
model. (b) Contribution of each source to the hazard averaged over all sites. Solid dots
represent the 20 sources with the largest average contributions, which we selected as our
source set.
60
Table 3.1 Fault sources identified using disaggregation given hazard level y = 0.3g for SA
at 3.0 s
IDs Source Name Average
rake/dip (°)
(%)
8
*
Elsinore;GI+T 180/90 2.0
10 Elsinore;GI+T+J+CM 180/85 5.7
15 Elsinore;T+J+CM 180/85 2.0
64 S. San Andreas;CH+CC+BB+NM+SM 180/90 5.5
85 S. San Andreas;PK+CH+CC+BB+NM+SM 180/90 19.3
86 S. San Andreas;PK+CH+CC+BB+NM+SM+NSB 180/90 13.6
87 S. San Andreas;
PK+CH+CC+BB+NM+SM+NSB+SSB
180/90
3.1
88 S. San Andreas;
PK+CH+CC+BB+NM+SM+NSB+SSB+BG
180/86
1.9
89 San Andreas;
PK+CH+CC+BB+NM+SM+NSB+SSB+BG+CO
180/86
2.0
93 S. San Andreas;SM+NSB+SSB+BG 180/81 1.9
112 San Jacinto;SBV+SJV+A+C 180/90 7.6
218 Newport Inglewood Connected alt 1 180/89 3.2
219 Newport Inglewood Connected alt 2 180/90 3.2
231 Palos Verdes 180/90 5.0
232 Palos Verdes Connected 180/90 6.2
254 San Cayetano 90/42 5.4
255 San Gabriel 180/61 2.6
267 Santa Susana, alt 1 90/67 2.3
271 Sierra Madre 90/53 3.4
273 Sierra Madre Connected 90/51 4.0
*
Abbreviations in source names can be found in UCERF2.0 database of Field et al (2009)
61
Figure 3.2 Locations of sources chosen in the analysis. Letters and numbers indicate the
fault segments, which correspond to those in Table 3.1.
3.4 Validation of ABF by NGA Intercomparisons
The ABF methodology can be validated by intercomparisons among the NGA08
GMPEs, for which the model-based factorizations are known. According to equations
(2.7) and (2.8), the NGA08 excitation functionals can be written in the form
G r,k,x,m;ε ( )=F
1
r ( )+F
2
r,k ( )+F
3
r,k,x ( )+F
4
k,m ( )+σ
T
ε (3.3)
The averaging scheme represented in the graph (2.34) yields:
62
E k,m ( )=F
4
k,m ( )+σ
T
ε− F
4
k,m ( )
M
(3.4)
D r,k,m ( )=F
3
r,k,x ( )− F
3
r,k,x ( )
X
(3.5)
C(r,k) = F
2
(r,k) − F
2
(r,k)
K
+ F
3
(r,k,x)
X
− F
3
(r,k,x)
X,K
⎡
⎣
⎤
⎦
+ F
4
(k,m)
M
− F
4
(k,m)
M,K
⎡
⎣
⎤
⎦
(3.6)
B(r) = F
1
(r) − F
1
(r)
R
+ F
2
(r,k)
K
− F
2
(r,k)
K,R
⎡
⎣
⎤
⎦
+ F
3
(r,k,x)
X,K
− F
3
(r,k,x)
X,K,R
⎡
⎣
⎤
⎦
(3.7)
A(r) = F
1
(r)
R
+ F
2
(r,k)
K,R
+ F
3
(r,k,x)
X,K,R
+ F
4
(k,m)
M,K
(3.8)
Equations (3.4)-(3.7) express the averaging-based factors in terms of the NGA08 model-
based factors, detailing the proportionalities noted in equations (2.35)-(2.38). The first
term in square brackets in Equation (3.6) is the hypocenter-averaged directivity effect,
which is small but varies systematically with source-site orientation, and the second is the
magnitude-averaged source-site effect, which shift the overall level of each site and each
source. The square brackets in Equation (3.7) comprise the source-averaged path and
directivity effects, which will be discussed below for the NGA08 models.
The NGA08 excitation functions were computed for the same sample sets R, K, and X
as the CS11. In particular, R contains the 235 CyberShake sites in Figure 2.6, and K
contains the 20 UCERF2 sources in Figure 3.2. We averaged over R and X assuming
uniform probabilities and over K assuming the source probabilities p(k) listed in Table
3.1.
63
3.4.1 Map Interpolation Methodology
The elementary factors of the ABF are themselves spatial fields that can be mapped
and thus easily compared. However, due to the limited number of sites (as shown in
Figure 2.6), we use an interpolation method for visually showing those map-based
factors. The method for interpolating the values at all those sites to map-based image
follows Creager and Jordan (1984). Known values
for points (j=1,2,3,…,J), the
unknown values u(r
i
) for points r
i
(i=1,2,3,…,I) could be determined by solving the linear
equation
(3.9)
where the kernal matrix L
ji
is constructed as . The solution is obtained by
the stochastic inversion (H and Franklin, 1971) as
(3.10)
where is the covariance matrix for values of unknown values to smooth the inversion
results, and is the noise autocorrelation matrix to account the noise in the known
values. One specific example each covariance matrix is
(3.11)
and
(3.12)
In this dissertation for map interpolations, we use κ and η as 0.01 and 0.001, respectively.
u s
j
( )
r
j
u
j
= L
ji
u
i
L
ji
=δ r
j
− r
i
( )
u=C
uu
L
T
LC
uu
L
T
+C
nn
( )
−1
u
C
uu
C
nn
C
uu
r
i
,r
i'
( )= exp −κ |r
i
−r
i'
| ( ),i,i'=1,2,3,4...I
C
nn
r
j
, r
j'
( )
=η
2
δ r
j
− r
j'
( )
, j, j'=1,2,3,...,J
64
3.4.2 A-Factor: Regional Excitation Level
The A-factor measures the regional excitation level in natural logarithmic unites of
gravitational acceleration g, obtained by averaging over all sites and all rupture variations
for all sources (Equation 3.8). At the long periods considered here (2-10 s), the regional
excitation levels for the NGA08 models decrease approximately inversely with period
(Figure 3.3). The maximum difference between them is only about 12% at periods of 2 s
but grows to almost a factor of two at 10 s, reflecting the increase in epistemic
uncertainty owing to less data at very long periods (Chiou et al., 2008). BA08 predicts the
highest excitation at 2-5 s; though it is exceeded by CB08 at 10 s. CY08 consistently
predicts the lowest excitations.
Figure 3.3 The A factors of CyberShake and NGA models (AS08, BA08, CB08, and
CY08) at four periods. Scales are logarithmic.
65
3.4.3 B-Map: Site-Specific Effects
The differential field B(r) measures the inter-site variability in the excitation after
averaging over all rupture variations for all sources. The NGA08 models describe two
types of site effects (Table 2.1): a dependence of the excitation on V
S30
, the average shear
velocity in the upper 30 m (V
S30
or soil effect), and a dependence on the local thickness of
sediments, parameterized in terms of the iso-velocity depths Z
1.0
for AS08 and CY08 and
Z
2.5
for CB08 (basin effect). BA08 includes a V
S30
factor but no explicit basin effect. The
site effects were computed from the V
S30
model
(Wills and Clahan, 2006) and the iso-
velocity surfaces from the CVM-S4 structure (Kohler et al., 2003).
To demonstrate that the averaging over ruptures can recover site effects, we first
chose the target model to be BA08 computed using site-dependent values of V
S30
and the
reference model to be BA08 computed using a constant value of V
S30
= 760 m/s; i.e., with
no site effect. ABF returns the V
S30
effect, as shown in Figure 3.4 by the high correlation
between the residual b-map, displayed in colors, and the Wills-Clahan V
S30
model,
displayed as contours.
We then selected AS08, CB08, and CY08, including both V
S30
and basin effects, as
the target models, and we computed b-maps using BA08 with the V
S30
effect as the
reference model. The averaging recovers basin effects that correlate well with the Z
1.0
and
Z
2.5
contours (Figure 3.4). For AS08, the V
S30
and basin effects were not independently
factorized, as they were for CB08 and CV08 models; instead, a factor distinguishing soil
effects for deep and shallow basins was calibrated by 1D site-effect simulations
(Abrahamson and Silva, 2008). Consequently, in areas where Z
1.0
is less than 200 m but
66
V
S30
is low (shallow soft soil sites; e.g., on the Mojave block northeast of the San
Andreas fault), AS08 model gives a large reduction in excitation relative to the reference
model, as depicted by colder colors outside the basins in Figure 3.4d. The CB08 and
CY08 models do not include this reduction.
Figure 3.4 Maps of residual factor b(r) for SA at 3.0 seconds using target models (a)
BA08, (b) CB08, (c) CY08, and (d) AS08. For (a), BA08 model calculated for reference
site condition (no site effect) is used as the reference model, and for (b), (c), and (d),
BA08 model (with site effect) is used as the reference model. Contours in (a) are V
S30
with 0.1 km/s interval, which are obtained from Wills and Clahan (2006) for southern
California. Contours in (b) are basin depth to shear wave speed equal to 2.5 km/s (Z
2.5
with 0.53 km interval), and in (c), and (d) are basin depth to shear wave speed equal to
1.0 km/s (Z
1.0
with 0.11 km interval). Basin depth model is from CVM-SCEC. The good
correlation shows that ABF recovers the V
s30
and basin effects.
119˚ 118˚ 117˚
34˚
35˚
210 1 2
b(r)
0.3
0.4
0.5
0.7
0.7
(a)
119˚ 118˚ 117˚
34˚
35˚
0.3
2.58
3.15
3.72
(b)
0.34
0.45
0.56
0.67
(c)
0.34
0.45
0.56
0.67
(d)
67
In addition to the contour plot of basin depths on top of residual b factors shown in
Figure 3.4, we compute the (linear) correlation coefficients between b for SA at four
periods (2.0, 3.0, 5.0, and 10.0s) and corresponding basin depths depending on the target
model. Those correlation coefficients are listed in Table 3.2. It can be seen that the
residual b of CB08 has the largest correlation with basin depth with Z
2.5
, which is used
for basin amplification parameterization in CB08 model.
Table 3.2 Correlation coefficients between residual b and basin depth for different target
models using BA08 as the reference model.
Period (s)
Target Models
AS08 CB08 CY08
2.0 0.766 0.956 0.789
3.0 0.802 0.954 0.697
5.0 0.838 0.950 0.500
10.0 0.882 0.944 0.352
According to Equation (3.7), the B factor is dominated by the variations in F
1
(r). In
Figure 3.5, we compare the basin terms F
1
r ( )− F
1
r ( )
r
for AS08, CB08, and CY08 with
the corresponding b residuals, plotted as functions of the basin-depth parameters. For
CB08 and CY08, these residuals closely match. In both cases, the small but systematic
positive shifts in b relative to the basin terms are due the source-averaged path and
directivity effects given by the square brackets in Equation (3.7). A similar, somewhat
larger shift can also be seen for AS08 (Figure 3.5c), although the pattern is complicated
by the dependence of the AS08 basin amplification term on V
S30
, as plotted in Figure
3.5d.
68
Figure 3.5 Comparisons of model-based basin amplifications for SA-3s (blue open
circles) with the residuals b(r) from the ABF representation (red open circles) for (a)
CB08 vs. Z
2.5
, (b) CY08 vs. Z
1.0
, (c) AS08 vs. Z
1.0
, and (d) AS08 vs. V
S30
. In all cases,
BA08 is used as the reference model.
3.4.4 C-Map: Path Effects
The differential field C(r, k) shows the site-dependent variability after averaging over
the rupture variations for each source k, and it is dominated by a magnitude-dependent
attenuation of the excitation with the Joyner-Boore distance, r
JB
. Representative maps are
displayed for two sources (86 and 273) in Figure 3.6. In both cases, C(r,k) is larger near
the source (warmer colors) and decays with distance. The distance decay is less for the
larger magnitude source (Source 86), consistent with the magnitude dependence of the
NGA distance attenuation factors.
69
(a)
(b)
Figure 3.6 C-maps at SA-3s for (a) Source 86 (San Andreas fault) and (b) Source 273
(Sierra Madre fault). Heavy black lines are the fault surface traces. Left maps in both
cases show C(r,k) of the target model, CB08; middle maps show C(r,k) of the reference
model, BA08; and right maps show the target-reference residual, c(r,k). The dipole
pattern in the lower right panel reflects the larger hanging-wall effect of the BA08
reference model (cf. Figure 3.10).
Figure 3.6 also shows the residual maps c(r,k) calculated by subtracting C(r,k) of the
reference model (BA08) from C(r,k) of the target model (CB08). The c-maps for all 20
sources: AS08-BA08, CB08-BA08, and CY08-BA08 are shown in Figure 3.7-Figure 3.9.
For the strike-slip source, the distance attenuation in CB08 is very similar to BA08, as
70
confirmed by the very small c-residual. For the reverse fault source, the residual is
positive on the footwall and negative on the hanging wall. Similar dipole patterns are
observed for reverse faults when AS08 and CY08 are the target models and BA08 is the
reference; they occur because BA08 has the largest hanging-wall effect among the
NGA08 models, as illustrated for Source 273 at reference site condition (V
S30
=760 m/s)
in Figure 3.10.
Figure 3.7 The maps of c(r,k) for SA at 3.0 seconds using target model AS08 and
reference model BA08. All sources as listed in Table 3.1 are shown here and solid lines
indicate fault surface traces.
119˚ 118˚ 117˚
34˚
35˚
210 1 2
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
71
Figure 3.8 The maps of c(r,k) for SA at 3.0 seconds using target model CB08 and
reference model BA08. Same sources as Figure 3.7 are shown.
−119˚ −118˚ −117˚
34˚
35˚
−2 −1 0 1 2
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
72
Figure 3.9 The maps of c(r,k) for SA at 3.0 seconds using target model CY08 and
reference model BA08. Same sources as Figure 3.7 are shown.
73
Figure 3.10 Comparison of hanging wall and footwall effects on of SA at 3.0 s computed
by four NGA GMPEs for the source 273 at V
S30
= 760 m/s.
3.4.5 D-Map: Directivity Effects
The differential field D(r, k, x) measures, in dimensionless logarithmic units, the site-
dependent variability obtained averaging over the slip variations for each source k and
hypocenter x as in Equation (3.5). In the NGA08 models considered here, this directivity
effect is determined by Spudich & Chiou’s (2008) azimuth-dependent correction to the
74
excitation determined by the relative location of a site and a hypocenter. In Figure 3.11,
we display residual maps d(r,k,x) for four hypocenters (black dots) of Source 255 using
the BA08 model with the SC08 directivity correction as the target model and BA08 with
no directivity correction as the reference model. These maps correlate almost exactly
with the contours of the directivity term F
3
(r,k,x) – .
Figure 3.11 Maps of residual factor d(r,k,x) at SA-3s for four epicenters (black dots) of
Source 255 computed using directivity-corrected BA08 as the target model and BA08 as
the reference model. Heavy black lines are the fault trace. Contour lines are the SC08
directivity correction terms normalized to zero mean over the hypocenters, plotted at 0.15
intervals.
.
F
3
r,k,x ( )
x
119˚ 118˚ 117˚
34˚
35˚
0.4 0.2 0.0 0.2 0.4
0.2
0.2
0.05
0.1
0.25
0.35
0.1
0.5
0.1
75
3.5 ABF Comparisons of CyberShake and NGA Models
The comparisons in the previous section validate the ability of averaging-based
factorization to recover the model-based factorizations of the NGA models. Here we
compare the NGA GMPEs with the CyberShake prototype, CS11, which has no explicit
model-based factorization. In our calculations, we chose CS11 as the target model and
one of the GMPEs as the reference model. We then decomposed the residual excitation
fields into their differential components according to Equation (2.42).
3.5.1 Regional Excitation Levels
At periods of 5 s or greater, the A-factor for CS11 is higher than the regional
excitation levels of the NGA GMPEs, exceeding their average by about a factor of two at
10 s (Figure 3.3). In this long-period range, the CS11 level decreases approximately
inversely with period, similar to the NGA mean. At shorter periods, however, the CS11
level does not maintain this frequency scaling, falling about 50% below the NGA level at
2 s. The apparent roll-off can be attributed to the low-pass filtering of the CS11
seismograms near the nominal cutoff frequency of the simulations, which was 0.5 Hz
(Graves et al., 2011b). However, the influence of this filtering on the differential
excitation maps presented below is negligible, because it was applied equally to all
seismograms and did not significantly alter the relative amplitudes of the basin, path, and
directivity effects.
76
3.5.2 Basin Effects
To compare the CyberShake basin effects with those predicted by the NGA08
GMPEs, we chose CS11 as the target model and the directivity-corrected BA08 model,
which has a V
S30
effect but no basin effect, as the reference model. Figure 3.12 displays
the residual map b(r) for SA-3s. The warm colors appearing in the sedimentary basins
indicate high basin amplifications from the CVM-S4 structure used in the CS11
simulations. The cold colors are confined to regions with small basin depths but low
values of V
S30
(< 360 m/s), where the BA08 reference model predicts larger site
amplifications than CS11. Figure 3.12b and Figure 3.12c plot the b-residuals as functions
of Z
1.0
and Z
2.5
; the scatter indicates that the basin effects from the CyberShake
simulations are not simple functions of either Z
1.0
or Z
2.5
(cf. Figure 3.5c).
Figure 3.12 The residual factor b(r) for SA at 3.0 second using Cybershake as target
model and directivity-corrected BA08 as reference model shown (a) as map-based, and as
function of (b) Z
1.0
and (c) Z
2.5.
119˚ 118˚ 117˚
34˚
35˚
210 1 2
b(r)
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
b(r)
0.10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Z
1.0
(km)
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
b(r)
01 234 56 7
Z
2.5
(km)
77
Comparing the CS11 b-map at SA-3s with similar maps for AS08, CB08, and CY08
(Figure 3.4), we find that the basin amplifications from CS11 are higher than any of the
NGA GMPEs (red regions on the b-maps). Among the latter, the closest is AS08, which
shows a larger basin effect than the other NGA models and which also best approximates
the CS11 amplitudes at sites of soft but shallow soils (blue regions on the b-maps). This
relative agreement is not too surprising, because AS08 was the sole NGA model for
which the site-effect factors were calibrated using synthetic seismograms (Abrahamson
and Silva, 2008).
We expand this comparison in Figure 3.13, where we plot the b-maps for CS11 and
AS08 at four periods. The CS11 amplifications are consistently higher, reflecting the
strong basin resonances and basin-directivity coupling observed in earthquake
simulations using the CVM-S4 structure (Olsen et al., 2006; Graves et al., 2008; Taborda
& Bielak, 2013). In CS11, the basin effect increases with period, reaching its highest
amplification at 5 s, and then decreases from this peak to 10 s. In AS08, the amplification
increases uniformly with period.
78
(a)
(b)
Figure 3.13 Maps of residual factors b(r) using (a) CyberShake and (b) AS08 as the target
model and directivity-corrected BA08 as the reference models for spectral accelerations
at T = 2.0, 3.0, 5.0, and 10.0 seconds (as shown on the top row). Color scales are the
same as. The SA-3s maps in (a) and (b) are the same as Figure 3.12(a) and Figure 3.4(d),
respectively.
3.5.3 Path Effects
The CyberShake C-maps include the path effects explicitly modeled by the GMPEs
(e.g., distance attenuation, hanging-wall effects, and hypocenter-averaged directivity) but
also path effects specific to the source-site geometry that arise from the 3D wave
propagation. In Figure 3.14, we compare the CS11 and BA08 C-maps and plot the
residual c-maps. Figure 3.15 displays the c-maps for all 20 sources.
period = 2.00 s
119˚ 118˚ 117˚
34˚
35˚
3.00 s 5.00 s 10.00 s
period = 2.00 s
119˚ 118˚ 117˚
34˚
35˚
3.00 s 5.00 s 10.00 s
79
Much of the variability in Figure 3.15 can be attributed to 3D path effects, but other
types of systematic differences are also apparent. Perpendicular to the fault strike, the
residual amplitudes are lower, on average, at sites away from the sources for CS11 than
for BA08, indicating that the overall attenuation rate with distance is slightly greater for
CS11 than for BA08. This observation agrees with the differences between simulations
and GMPEs calculated by Star et al. (2011) for a Southern California earthquake
scenario. The magnitude-dependence of the distance attenuation for CS11 is about the
same as for BA08.
For strike-slip sources (first three rows plus Source 255 of Figure 3.15), sites located
along the extension of the fault strike generally show more positive residuals than those
at intermediate angles (30º-60º) to the strike, indicating that the hypocenter-averaged
directivity effect for CS11 is larger than for BA08. This difference is consistent with the
D-maps, which show that the CS11 directivity effects are generally larger than those of
the NGA models.
For reverse-slip sources (fourth row except Source 255), the footwall sites are more
positive than the hanging-wall sites, which can also be explained as an average directivity
effect. In this case, however, the discrepancy is partly due to the single focal depth
assumed for the CS11 hypocenters given a source. The average CS11 directivity for the
reverse faults would be lowered if the hypocenters sampled the focal depth more
uniformly.
80
(a)
(b)
Figure 3.14 C-maps of SA-3s for (a) Source 86 and (b) Source 273. Left maps in both
cases show C(r,k) for the CyberShake target model; middle maps show C(r,k) for the
BA08 reference model; and right maps show the target-reference residual, c(r,k). Heavy
black lines are the fault surface trace.
81
Figure 3.15 Maps of residual factors c(r,k) at 3.0 seconds for the target model
CyberShake using directivity-corrected BA08 model as the reference model.
3.5.4 Directivity Effects
The classical directivity effect results from the constructive interference of seismic
waves propagating in the same direction as the rupture (Ben-Menahem, 1961; Spudich &
Chiou, 2008). More generally, directivity effects describe the dependence of the
excitation field on the direction of rupture propagation. They can show up at every level
of the averaging hierarchy as in Equation (2.20); e.g., directivity effects averaged over all
82
rupture variations of a source k contribute to C(r, k), as we have just described. However,
the rupture parameter most expressive of directivity is the conditional hypocenter, x given
k, and our main tool for assessing directivity effects is therefore D(r, k, x).
Figure 3.16 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake as the
target model and directivity-corrected BA08 as the reference model. The hypocenters of
source 112 are shown as black dots in each subplot.
We compared the directivity effects of CS11 with those NGA08 by choosing the
former as the target model and BA08 with the SC08 directivity corrections as the NGA08
reference model. The resulting d-maps for each hypocenter of Source 112 are shown in
83
Figure 3.16. Because only one source depth (20km) was simulated, epicenters (black
dots) uniquely identify the hypocenters. By definition, the average of the residual
d(r, k, x) over x for each source k and site r is exactly zero. The most positive values (red)
are at sites in a fan northwest of the source for the hypocenter furthest away; i.e. for a
rupture that propagates towards the sites. The most negative values (blue) are in the same
region for the closest hypocenter; i.e., for a rupture that propagates away from the sites.
This pattern is a clear indicator that the CS11 directivity effects are larger than those
predicted by the SC08 directivity correction to BA08.
In Figure 3.17, we display the σ
d
-maps for all 20 sources. Each map plots the
standard deviation of d(r, k, x) for a fixed source k. For Source 112, this map summarizes
the high variability of the CS11-BA08 residuals at sites northwest of the rupture, as
observed in Figure 3.16. In general, the strike-slip sources show high standard deviations
in fans beyond the fault terminations, and examination of the corresponding d-maps
confirms that this variability also results from CS11 directivity effects that exceed those
of BA08.
The source radiation pattern modulates the directivity effect, so that the along-strike
directivity for reverse-slip sources tends to be smaller than their up-dip directivity.
Consequently, owing to the single focal depth used in the simulations, the standard
deviations are relatively smaller for most of the reverse-slip faults in our source set.
Diagnostic investigations of directivity effects for dip-slip faults in the CyberShake
simulations will require the consideration of conditional hypocenter distributions p(k, x)
in which the dependence on x is fully two-dimensional.
84
Figure 3.17 The maps of standard deviations of d(r,k,x) using CyberShake as the target
model and directivity-corrected BA08 as the reference model for all sources used in the
analysis. These d-maps assume a uniform conditional hypocenter distribution.
3.5.5 Directivity-Basin Coupling
Many of the small-scale differences between the CS11 and NGA08 directivity
amplifications seen in Figure 3.17 can be attributed to 3D wave-propagation effects. The
most important is directivity-basin coupling. A good example can be observed in the σ
d
-
85
map for Source 93, which shows high variability in the Los Angeles basin. The d-maps
for this source are displayed in Figure 3.18. High directivity amplifications are evident
for sites beyond the fault terminus in the northwestern part of the CyberShake region, as
discussed previously, but we can also see the anomalous excitations in the Los Angeles
basin caused by the coupling of northwest-propagating directivity pulses into basin-
guided surface waves, similar to the coupling described by Olsen et al. (2006, 2008),
Graves et al. (2008), and Day et al. (2012). Such effects are absent in the NGA08 models
(except to the extent that they increase the aleatory variability modeled by the σ
T
ε term
in Equation (2.7)).
A second, even better example is Source 89, which is a “wall-to-wall” rupture of the
entire southern San Andreas fault from Bombay Beach to Parkfield (Figure 3.19).
Positive residuals are seen in the Los Angeles basin for hypocenters located near the
southeastern terminus of the fault (e.g., hypocenters 0-3), again due to the coupling of
northwest-propagating directivity pulses into basin-guided waves. These anomalies
persist up to hypocenter 7, where they then decrease dramatically. This behavior
identifies a “coupling point” in the vicinity of hypocenters 7-8, where the coupling of the
directivity pulse into the basin surface waves is strongest (Day et al., 2012). At
hypocenters beyond 19, positive anomalies form again but now in the Ventura-Santa
Clara-Simi Valley basins northwest of Los Angeles. This behavior identifies a second
coupling point at hypocenters 18-19, in this case near the Big Bend in the San Andreas
south of the Carrizo Plain.
86
Figure 3.18 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake as the
target model and directivity-corrected BA08 as the reference model. The hypocenters of
source 93 are shown as black dots in each subplot.
87
Figure 3.19 Maps of residual factor d(r,k,x) for target model CyberShake and reference
model BA08 with directivity correction from SC08 model. Black line shows the wall-to-
wall rupture (from Parkfield to Bamboy Beach section of south San Andreas Fault). Each
of black squares shows different epicenter locations along the fault. Number in each map
of d(r,k,x) indicates the location of the corresponding hypocenter.
88
3.6 Discussion
The ABF method has been validated by confirming its ability to recover factors of
NGA GMPEs, and it has been applied to compare seismic hazard models. We found ABF
can effectively represent the site-specific, wave propagation path, and source directivity
effects of the specific models, both GMPE-based and simulation-based models.
Applying ABF methodology to compare the simulation-based CyberShake model,
CS11, and NGA GMPEs for SA at low frequencies (0.1-0.5 hz), we found that basin
effects in CyberShake are up to an order of magnitude larger than those in the NGA08
models through the residual factor b(r), and basin excitation is not a simple function of
basin dept. Moreover, the basin excitation in simulation-based model with 3D velocity
structure shows stronger frequency-dependence than GMPE-based models. The residual
factor c(r,k) indicates that the distance attenuations in CyberShake are faster than those
considered in NGA GMPEs, and sites located down-strike or up-dip (foot wall) direction
of a source (fault) would on average experience strong ground motion shaking due to the
constructive interference of seismic energy radiated from hypocenters. Using SC08
model to represent the directivity effects for NGA GMPEs, we found that CyberShake
predicts relatively larger directivity effects than NGA GMPEs for SA at 3.0s, in
particular, directivity-basin coupling effects could significantly enhance the ground
motions in sedimentary basins. Such effects are not considered in empirical modeling of
ground motions using GMPEs. The directivity-basin coupling effects strongly depend on
the source-site geometry and the location of basins, and are not only included in the D
89
factor but are also distributed into C, B, and A factors by the averaging operations. For
example, the larger A factors obtained for CS11 relative to NGA08 (Figure 3.3) are in
part due to directivity-basin coupling.
The CS11-NGA08 discrepancy seen in the σ
d
-maps for strike-slip sources can be
reduced by increasing the source complexity described by the conditional slip distribution
p(s | k, x). Slip complexity reduces the coherence of the directivity pulse and thereby
lowers the overall amplitude of the directivity effects. This possibility is being
investigated by CyberShake calculations that use the modified pseudo-dynamic rupture
model of Graves & Pitarka (2010), which features stronger local variations in rupture
speed, rise time, and rake angle than the Graves & Pitarka (2004) model employed in
CS11. Preliminary results show that more complex rupture model reduces CyberShake-
NGA directivity-effects discrepancy by a factor of two. Another way to reduce the
discrepancy is to modify p(x | k) such that hypocenters near the ends of the faults become
less likely. Whether the conditional hypocenter distribution should be “centroid-biased”
rather than uniform is an observational question being addressed by current research
(Donovan and Jordan, 2012). We will present the reductions by those modifications using
ABF methodology in Chapter 4.
The basin excitation strongly depends on the shape and composition (seismic wave
velocity structures) of sedimentary basins, and at different frequencies, the amplifications
due to basin structures significantly vary. CyberShake calculations with two different 3D
velocity models, CVM-SECE, version 4 (CVM-S4) and CVM-Harvard (CVM-H),
represent the epistemic uncertainties in the subsurface structures. Preliminary
90
experiments show that the basin excitations calculated from CVM-H are smaller than
from CVM-S4, and they show a stronger frequency dependence. Detailed comparison
and physical implication will be presented in Chapter 5.
Comparing Equations (2.39) and (2.40), the aleatory variability is considered
differently in NGA GMPEs and CyberShake. Simulation-based model does not require
the ergodic assumption, and consider the aleatory randomness in rupture process by
specifying the CyberShake CHD and CSD (noting that uncertainties in these conditional
distributions is a form of epistemic uncertainty). Given the same magnitude dispersion
from ERF model, numerical experiments show that σ
M
, which weakly depends on
sources, is larger in CS11 than that in BA08 by a factor of 2 for SA-3s (similar
differences for SA at 5 and 10s). Considering the directivity corrections using SC08 for
BA08, σ
T
for SA-3s is about 0.6, while σ
U
for SA-3s in CS11 is about 0.2. Without the
ergodic assumption, simulation-based model CS11 significantly reduces the aleatory
variability relative to BA08.
The effects of the epistemic uncertainties in ERF models on seismic hazard are
common to both GMPE-based and simulation-based hazard models. The simulation-
based methods may simplify the implementation of new, time-dependent ERFs, such as
UCERF3 (Field et al., 2013) and RSQsim (Richards-Dinger and Dieterich, 2012)
(Richard-Dinger and Dieterich, 2012). We have adopted a time-independent model,
UCERF2, in this Chapter, but the basic formulation of ABF can easily be extended to
time-dependent problems.
91
3.7 Conclusions
In this chapter, we applied the ABF methodology to compare simulation-based hazard
model, CyberShake, and empirical NGA GMPEs. Based on five level of averaging
operations, ABF allows us to partition different physical effects considered in different
seismic hazard models into factors. The comparison between simulation-based and
GMPE-based models using ABF can both serve as a guide to future attenuation-
relationship developments and provide an interim model for use in southern California.
Treating CyberShake as the representative of the virtual world for this region, we could
modify the NGA GMPEs based on those residual factors, and reduce the epistemic
uncertainties in the empirical models due to the lack of observations for constraining
some of physical effects that are considered by simulation-based models. One limitation
of the current comparison is that the excitation function calculated by NGA GMPEs are
rotation-independent horizontal geometrical mean (GMRotI50) of a specific ground
motion intensity measure (Boore et al., 2006), while the current CyberShake models only
have the traditional geometrical mean calculated. Boore et al. (2006) statistically analyze
the ratio between GMRotI50 and traditional geometrical mean of as-simulated ground
motions, then found that the mean ratio is 1.02 with standard deviation 1.11 for SA at
3.0s. Moreover, it is possible to obtain the same rotation-independent ground motion
intensity measures by processing synthetic seismograms from CyberShake database.
92
Chapter 4
Rupture Directivity Forecasting of Large Earthquakes
4.1 Abstract
The ability to forecast rupture directivity can substantially improve the long-term
forecasting of strong ground motions and its application in PSHA. In this chapter, we
consider three types of information relevant to directivity forecasting obtained from the
study of crustal fault zones, including small-scale rupture complexity, conditional
hypocenter distribution (CHD), and directivity bias inferred from damage asymmetry or
material contrast across faults. We apply the ABF scheme as presented in Chapter 2, and
compare CyberShake models that use two different slip models as in Graves and Pitarka
(2004, 2010). The CyberShake directivity effects are generally larger than predicted by
Spudich and Chiou (2008) NGA directivity factor, but those calculated from the Graves
and Pitarka (2010) sources are smaller than those of Graves and Pitarka (2004), owing to
the greater incoherence of the wavefields from the more complex rupture models. We
found that CyberShake directivity effects are lowered by increasing in small-scale rupture
complexity by a factor of two, from 36% to 18%. We also did experiments of using
different CHD parameterized by Beta distribution, and found the asymmetric CHDs that
are consistent with biomaterial rupture bias can substantially alter seismic hazard
assessments.
93
4.2 Introduction
Rupture directivity effects cause spatial variations in ground motion amplitude around
faults. Sites located in the rupture propagation directions on average experience stronger
ground motions due to forward rupture directivity (Ben-Menahem, 1961), as shown in
Figure 2.1. These variations become significant at longer periods according to large
historical earthquakes (Somerville et al., 1997). Large-scale earthquake simulations for
ground motions expected from earthquakes shows the constructive interference (focusing)
due to rupture propagation along finite faults, resulting generally higher amplitudes to the
forward rupture direction, and relatively smaller amplitude to the backward rupture
direction. For example, Figure 4.1 shows the TeraShake simulations for two scenarios
with magnitude M
w
7.7 along the 200km section of the San Andreas fault between Cajon
Creek and Bombay Beach at the Salton (Olsen et al., 2006). For different rupture
directions (SE-NW and NW-SE), the ground motion responses are different due to
directivity effects.
Now that the rupture directivity effect, a physical effect, exists in natural, both
simulation-based model and empirical GMPEs consider it in ground motion predictions
as well as PSHA (As discussed in Chapter 2). However, the rupture directivity
forecasting in seismic hazard analysis has many questions with few answers. For example,
is rupture directivity random or can it be statistically predicted, what are the effects of
kinematic rupture process and distribution of possible hypocenter locations along the
fault on ground motion predictions and PSHA calculations, and what conditional
hypocenter distribution should be used in simulation-based hazard models? In order to
94
answer those problems, we consider information relevant to directivity forecasting
obtained from the study of crustal fault zones, including small-scale rupture complexity,
conditional hypocenter distribution (CHD), and directivity bias inferred from damage
asymmetry or material contrast across faults.
Small-scale heterogeneities in kinematic rupture propagation determine the seismic
wave energy radiation and interactions within fault surface, including source directivity
effects. Finite-fault source inversions reveal the spatial complexity of earthquake slip
over the fault surface. Mai and Beroza (2002) developed a stochastic characterization of
earthquake slip complexity, based on published finite-source rupture models, in which
the distribution of slip as a spatial random field constrained to fit certain wave number
property (wavenumber
-2
) with specific spatial correlation length associated with the
magnitude of an earthquake. This background distribution is then used to determine an
initial estimate of the rupture front arrival time at each subfault with given rupture
velocity. The other important parameter controls the directivity effects is the distribution
of hypocenter locations along a given fault plane. McGuire et al. (2002) found that most
earthquakes appearing to be predominantly unilateral, but this likely is unrelated to
material contrast across a fault. The material contrast may produce a statistically
preferred direction of earthquake ruptures, that is, rupture likely propagates to the
direction as the slip direction in the softer material on one side of the fault surface (Ben-
Zion, 2001; Dor et al., 2006). Additional parameters, such as stress distribution of
earthquakes (critical in dynamic rupture process) could affect the preferred rupture
95
direction seen from observations and should be considered for studying the directivity
effects (Harris and Day, 2005).
Figure 4.1 Peak spectral acceleration at 3.0 calculated by TeraShake simulations. (a)
Simulation using scenario that starts at the southeastern end of San Andreas fault
rupturing toward the northwest (SE-NW); (b) Simulation using scenario that starts
northwestern end of San Andreas fault rupturing toward the southeast (NW-SE).
Modified from (Olsen et al., 2006).
In this chapter, we use CyberShake as the framework to answer those questions
associated with forecasting rupture directivity. The implementation of simulation-based
model allows us to test different forecasting models for directivity, and the ABF scheme
enables us to compare directivity forecasting in empirical and simulation-based models.
Using various conditional hypocenter distributions parameterized by Beta distribution,
we quantify the effects on source directivity and hazard calculations.
96
4.3 Effects of Rupture Complexity on Source Directivity
Considering that the rupture complexity controls the seismic wave energy radiation
from sources and constructive interferences during the rupture process, i.e., the azimuthal
variations in the amplitudes and waveforms directly result from spatially varying slip on
the fault, spatially varying radiation pattern over the fault, and the magnitude and
direction of the rupture velocity, we first present two types of rupture models GenSlip
v2.1 and GenSlip v3.2 presented by Graves and Pitarka (2004; 2010), and then apply
ABF method to compare CyberShake models that use these two rupture models,
quantifying the effects of rupture complexity on source directivity in ground motion
predictions, as well as directivity-basin coupling effects.
4.3.1 Earthquake Slip Models
The generation of a full kinematic rupture prescription requires specification of the
spatial variable dislocation (slip) time function (as shown in Figure 2.4) across the entire
rupture surface. The necessary input parameters for this process are fault location and
geometry (strike, dip etc.), seismic moment or magnitude, rupture initiation point
(hypocenter), and slip direction (rake). Figure 4.2 shows the two types of kinematic
rupture models generated using a software generated following Grave and Pitarka (2007,
2010): GenSlip v2.1 and GenSlip v3.2. As depicted in Figure 4.2, the GenSlip v2.1 shows
smoother rupture front and simple rise time distribution across the fault surface than
GenSlip v3.2. In both models, the slip distribution is assumed to be random (stochastic)
with a roughly wavenumber-squared spectral decay (Herrero and Bernard, 1994;
97
Somerville et al., 1999; Mai and Beroza, 2002), which is based on stochastic realization
of spatial distribution of slip along the fault.
Figure 4.2 Illustration of kinematic rupture models developed by Graves and Pitarka
(2007, 2010). (a) GenSlip v2.1: More simple rupture description with constant rupture
velocity and smooth rupture front; (b) GenSlip v3.2: More complex rupture model with
variable rupture velocities for different depth and rough rupture front. Each row shows
final slip with rupture front, rise time (duration of source time function), and rake
direction, respectively.
Given a hypocenter, rupture initiation time T
r
(i)
(gives the rupture front) for each
subfault i is the determined by first specifying a background distribution rupture velocity
v
r
and then applying a timing perturbation δt that scales with the local slip A
r
(i)
with the
form
98
T
r
(i)
= R
r
(i)
/v
r
−δt A
r
(i)
( )
(4.1)
where R
r
(i)
is the rupture distance from hypocenter to each subfault, and the time
perturbation δt is defined by
δt A
s
(i)
( )
=Δt
log A
s
i ( )
( )
−log A
a
( )
log A
m
( )−log A
a
( )
(4.2)
where A
m
and A
a
is the maximum and average slip across the fault, respectively. For
GenSlip v2.1, Δt is set as 0.5, constant over the fault, and for GenSlip v3.2, Δt is scaled
with magnitude, resulting in large time perturbation or rough rupture front as in Figure
4.2. For GenSlip v2.1, the rupture velocity is assumed to be 0.8 times of local shear-wave
velocity near fault, while in GenSlip v3.2, the rupture velocity is given by
v
r
=
0.56×β z< 5km
0.8×β z> 8km
⎧
⎨
⎪
⎩
⎪
(4.3)
where β is the local shear-wave velocity, z is the depth from top of the fault, and a linear
transition is applied between dpeth 5 and 8 km. The slip direction is homogeneous in
GenSlip v2.1, but vary across the fault with a standard deviation 15
o
about a prescribed
mean value (same value used in GenSlip v2.1) in GenSlip v3.2. The random distribution
of rake angle follows a von Karma correlation function (Mai and Beroza, 2002), shown
as interpolated colorred image along with the direction of rakes in Figure 4.2. The rise
times (or duration of slip rate function) in both model have some dependence on
magnitude, and GenSlip v3.2 also consider the correlation between the rise time and the
local slip. More randomness shown in the GenSlip v3.2 indicates more radiation patches
99
with rough rupture fronts from the source than the GenSlip v2.1, which has more smooth
and coherent propagation process. It leads to less constructive interference during rupture
propagation, and less source directivity effects. CyberShake models that use these two
slip descriptions allow us to quantify the changes in directivity effects on ground motions
due to source complexity.
Figure 4.3 Relationship between the IDP (black dots) and normalized isochrone velocity
(red dots) and the ratio between rupture velocity and shear-wave velocity.
The empirical model presented in Chapter 2 (SC08) considers the source complexity
by using the ratio between rupture velocity and local shear-wave velocity, and following
Equation (2.11), we could calculate the isochrone velocity that controls the azimuth
variations of source directivity. We use the same strike-slip event used in Figure 2.3 and
visually present the normalized isochrone velocities and IDP for different ratios of
rupture velocity and shear-wave velocity in Figure 4.3. We can see the rapid change in
100
normalized isochrone velocities as the ratio increases. As seen from Equation (2.13),
instead of using
c
'
as the parameter to control the directivity distribution, SC08 model
use a scaled (lying in the range [0,1]) parameter C
r
with c
0
equal to 2.45. This is
determined by correlating the residual between NGA predicted mean and simulation
results with
c
'
(Spudich and Chiou, 2008). This is the reason why IDP reaches a plateau
when
c
'
>2.45 seen from Figure 4.3. Because IDP will be used in modeling f
D
for
directivity corrections of NGA GMPEs, increasing velocity ratio will not affect the
directivity correction. However, simulation-based model could help us to quantify the
effects of different rupture velocity profiles (included in the two slip models as in Figure
4.2) on ground predictions and hazard calculations.
4.3.2 Directivity Effects
To illustrate the differences in directivity effects using different slip models, we apply
ABF method introduced in Chapter 2, and calculate the residual map d(r,k,x) using
CyberShake models with GenSlip v2.1 and v3.2 as the target and reference model,
respectively. We first choose one source (SourceID=93 as in Table 3.1) that is similar to
the TeraShake and ShakeOut simulations (e.g. Figure 4.1) but has more scenarios
(multiple hypocenters and rupture descriptions). The fault surface trace and possible
hypocenter locations are shown in Figure 4.4, and the rupture starts from the Bombay
Beach and stops at south Mojave. The map of d(r,k,x) for each hypocenter (indices) is
shown in Figure 4.6. We change the color scale (-1.0 to 1.0 in natural logarithmic scale)
that is different from (-2.31 to 2.31 in natural logarithmic scale) in order to exaggerate the
differences in directivity effects given by the target and reference models. As depicted in
101
Figure 4.6, for hypocenters located in the south part of the fault, rupture propagates from
southeast to northwest, and positive (larger) residuals in d(r,k,x) can be seen in the
forward direction of the rupture. It shows that CyberShake with GenSlip v2.1, or simple
rupture process, gives larger directivity. For hypocenter located in the north part of the
fault, negative (smaller) residuals can be found in the forward direction of the rupture.
We also found that larger directivity effects in CyberShake with GenSlip v2.1 are
enhanced by presence of sedimentary basin, resulting in larger coupling effects found in
basins (e.g. Los Angeles basins) than CyberShake with GenSlip v3.2, consistent with the
positive (large) values in the major basins. The reason is that the heterogeneities along
the rupture plane reduce the coherency of surface waves propagating towards a site, i.e.,
affect the constructive and destructive interferences between rupture pulses radiated from
the source, decreasing the directivity effects at the site.
Figure 4.4 Rupture surface trace and possible epicenter locations for source 93.
119˚ 118˚ 117˚ 116˚
33˚
34˚
35˚
0
1
2
3
4
5
6
7
8
9
10
11
102
Figure 4.5 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake with
GenSlip v2.1 rupture models as the target model and BA08 with SC08 directivity
correction as reference model. The source 93 with all hypocenters is shown. The number
in each subplot indicates the order of hypocenter shown in Figure 4.4 (0:southern end; 11:
northern end). Color scale is the same as Figure 3.16.
103
Figure 4.6 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake with
GenSlip v3.2 rupture models as the target and reference model, respectively. Source and
hypocenters are the same as in Figure 4.5.
Recalling the maps of σ
d
shown in Figure 3.17, where we use CyberShake with
GenSlip v2.1 as the target model and directivity-corrected BA08 as the reference model,
we use CyberShake with GenSlip v3.2 as the target model and BA08 with directivity
correction as the reference model, and Figure 4.7 shows the maps of σ
d
for all sources as
104
listed in Table 3.1. Comparing the two maps in Figure 3.17 and Figure 4.7, we found that
the variability of d(r,k,x) over hypocenters is reduced by using more complex kinematic
source description (rougher rupture front). From the definition of σ
d
, the bigger it is, the
larger directivity (since the mean over d factor is zero) variability is.
In order to get a quantitative sense how the changes in σ
d
are, we further calculate the
averaged σ of σ
d
(r,k) over all sites and sources, resulting a number for SA at a given
period. We take excitation functions as SA at three periods (3.0, 5.0, and 10.0 s). Table
4.1 shows the averaged σ of σ
D
(r,k) for excitation functions of two CyberShake models
with GenSlip v2.1 and v3.2 and for corresponding residual excitation function of the two
models using directivity corrected BA08 as the reference model. First, for the same
period, we see that using the empirical directivity model as the reference model, σ is
reduced by a factor of ~1.5. Secondly, without consideration of reference model, the
CyberShake with GenSlip v3.2 reduces the σ by a factor of 1.7 for shorter period (3s),
but for longer period (10s), σ are similar for both CyberShake models. This is because
for longer period, the characteristic lengths of the rupture heterogeneities in both slip
models are similar. According to the trend of changes of σ
over periods found for
GenSlip v3.2, for shorter period, the characteristic lengths of the rupture heterogeneities
is smaller, i.e., more incoherence in rupture propagation, resulting in smaller directivity
effects relative to longer period (less incoherence).
105
Figure 4.7 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (for GenSlip v3.2) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis.
Table 4.1 Averaged standard deviation σ
d
(r,k) of D(r,k,x) of CyberShake (GenSlip v2.1
and v3.2) over r and k for SA at 2.0, 3.0, 5.0, and 10.0.
Period (s)
Models
GenSlip v2.1 GenSlip v2.1-BA08 GenSlip v3.2 GenSlip v3.2-BA08
2.0 0.426 0.346 0.242 0.170
3.0 0.407 0.307 0.263 0.172
5.0 0.389 0.273 0.293 0.177
10.0 0.349 0.262 0.325 0.216
119˚ 118˚ 117˚
34˚
35˚
0.0 0.2 0.4 0.6 0.8
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
106
4.3.3 Distance Attenuation and Basin Effects
It is seen that the rougher rupture process decrease the directivity effects in the last
section. We then choose CyberShake with GenSlip v3.2 and v2.1 as the target and
reference model, respectively, and compute the residual b(r) and c(r,k) for SA at 3.0 s.
The residual basin effect map in Figure 4.8 shows two major features. First, positive
values (larger effects) are in sedimentary basins. The reason might be more basin
excitation occurring due to the burst pattern of increased peak values radiated out from
those ruptures. From the comparison of DBC effect in Figure 4.6, we see that
CyberShake with GenSlip v3.2 has smaller directivity-basin coupling because of less
directivity effect that is enhanced by basins for each hypocenter. Here the excitation of
basin waves means the general excitation of incident waves radiated from the source and
scattered during the wave propagation. Due to the more pockets of waves radiated from
the source and scattering in the propagation, the basin amplification on average is larger
for CyberShake with GenSlip v3.2. Secondly, smaller values (residuals) are found along
San Andreas Fault area. This might be due to the averaging operation over all 20 sources,
and according to Table 3.1, the weights given to sources located along San Andreas Fault
are generally high, and those sources have average larger near fault effects for more
smoother rupture description.
107
Figure 4.8 Map of residual b(r) for SA at 3.0 second using CyberShake with GenSlip
v3.2 and GenSlip v2.1 rupture models as the target and reference model, respectively.
Color scale is changed in order to show the small differences between models.
The residual maps c(r,k) are shown in Figure 4.9 for all sources (Table 3.1). For those
rupture along the San Andreas Fault, CyberShake with GenSlip v3.2 shows smaller
distance attenuation (positive values) than that with GenSlip v2.1. Olsen et al. (2008)
performed TeraShake-type simulation using dynamic source model, and found that a star
burst pattern of increased peak values radiated out from the fault. These rays of elevated
peak ground motions are generated in areas of the fault where the dynamic rupture pulse
119˚ 118˚ 117˚
34˚
35˚
10 1
b(r)
108
changes abruptly in speed, direction, or shape (corresponding to the rupture heterogeneity
considered in GenSlip v3.2). For this reason, using rougher kinematic source description
that is based on the study of dynamic ruptures, the bursts of elevated ground motion
could be expected. For other ruptures where CyberShake region cover the forward
direction of rupture propagation, sites located in the along-strike direction and footwall
side of ruptures on average experience larger directivity effects as expected.
Figure 4.9 The residual map c(r,k) for SA at 3.0 seconds using CyberShake with GenSlip
v3.2 and GenSlip v2.1 rupture models as the target and reference model, respectively.
119˚ 118˚ 117˚
34˚
35˚
210 1 2
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
109
4.4 Conditional Hypocenter Distributions for Hazard Calculations
In the above section, we investigate the effects of rupture heterogeneity on ground
motion predictions using simulation-based model with ABF scheme, and when doing
averaging-based operation over hypocenters, we use uniformly distributed hypocenters
p(x|k)=1/N
h
, where N
h
is the number of possible hypocenter locations for a given source
k, which varies from source to source. However, for PSHA, the use of the conditional
hypocenter distribution (CHD) p(x|k) indicates the consideration of aleatory uncertainty
in earthquake occurrence (just like considering a magnitude distribution), and CHD itself
has epistemic uncertainties, i.e., what form of CHD should we use in PSHA? In this
section, we put the emphasis on various conditional hypocenter distributions and their
effects on directivity forecasting and hazard calculations, and we explore the PSHA
implications of the directivity data by modifying the uniform CHD of CyberShake model
with CVM-SCEC and GenSlip v2.1.
4.4.1 Conditional Hypocenter Distributions
Mai et al (2005) used a database of more than 80 finite-source rupture models from
more than 50 earthquakes (M
w
4.1-8.1) with different faulting styles occurring in both
tectonic and subduction environments to analyze the location of the hypocenter with
respect to the overall fault dimension. They found that a Gamma distribution adequately
characterizes the observed distribution for which the hypothesis of uniformity (evenly
distributed hypocenter locations along the fault) can be rejected with a high confidence
level. Besides, according to the correlation between hypocenter locations and slip spatial
distributions, they found that earthquake ruptures tend to (preferentially) initiate in zones
110
that show significant slip on the fault plane. Moreover, developing empirical directivity
models (Somerville et al., 1997; Spudich and Chiou, 2008; Rowshandel, 2010), modelers
use information related with hypocenter locations (along-strike and up-dip) in the NGA
flatfile (peer.berkeley.edu/nga/NGA_Flatfile.xls, last accessed June 14, 2013). Figure
4.10 shows the distribution of hypocenter location relative to the overall fault dimension
(normalized to [0,1]), including along-strike (hypocenter location relative to the fault
length) and up-dip (hypocenter location relative to the fault width). We found that the up-
dip distribution of hypocenter locations indicates that rupture tends to initiate at deeper
part of the fault surface, while along-strike distribution of hypocenter locations for all
events shows has no clear bias, although hypocenters tend to appear in the middle of the
rupture surface.
Figure 4.10 Distribution of hypocenter position (along-strike and up-dip) for (a) all
events, (b) strike-slip events, and (c) non-strike-slip events used in empirical directivity
modeling.
111
Considering the statistical features found in observations for hypocenter location
distributions, we decide to use a more general distribution other than Gamma distribution.
The Beta distribution has the form for the probability density function as
pdf x;α,β ( )=
x
α−1
1− x ( )
β−1
t
α−1
1−t ( )
β−1
dt
0
1
∫
(4.4)
where α and β controls the shapes of the distribution, and x is the hypocenter. As an
example, Figure 4.11 shows Beta distribution for three different sets of parameters and
correponding hypocenter locations. For each distribution, we define a name, like
centroid-biased, uniform, and periphery-biased (or unilateral) for each set of parameters.
Figure 4.11 Schematic diagram showing different conditional hypocenter distributions
given a fault plane. The parameters for Beta distribution are shown along with each
probability distribution (upper panel), and corresponding randomization of those
hypocenter distributions in 2D fault plane (lower panel). (Credit to: Thomas Jordan)
112
4.4.2 Dependence of Directivity Effects on CHDs
We extend the CyberShake models to include multiple CHDs parameterized as Beta
distributions as discussed in the above section, and compute the average standard
deviation σ
d
of directivity factor D(r,k,x) for SA at various periods shown in Table 4.2.
We found that centroid-biased hypocenter distribution leads to smaller directivity
variability than the uniform distribution; and periphery-biased hypocenter distribution
shows larger directivity variability than the other two distributions. In order to see more
detailed variability of D factor, we present in Figure 4.12 and Figure 4.13 the standard
deviation maps of D for SA at 3.0s of each source listed in Table 3.1. Comparing Figure
4.12 and Figure 4.13 with Figure 3.17, we can see that using centroid-biased CHD
reduces and using periphery-biased CHD increases the directivity effects. The reason is
that when using periphery-biased CHD, the most contribution to the variability of
directivity effects is those hypocenters located near the end of a fault, leading to larger
effective rupture length that increases the constructive interferences. While using
centroid-biased CHD, the most contribution to the variability of directivity effects is from
those hypocenters located in the middle of a fault, leading to smaller effective rupture
length that decreases the constructive interferences. For the same CyberShake model,
using different CHDs, the C and B factors do not vary because they mainly indicate the
distance attenuation and basin effects.
113
Table 4.2 Total standard deviation σ
d
for SA at 3.0, 5.0, and 10.0 s using different CHDs
and different target and reference models in ABF analysis
Period Model
Conditional Hypocenter Distribution
centroid-biased uniform periphery-biased
3.0
GenSlip v2.1 0.340 0.407 0.442
GenSlip v2.1–SC08 0.257 0.308 0.335
GenSlip v3.2 0.227 0.263 0.282
GenSlip v3.2–SC08 0.151 0.172 0.183
5.0
GenSlip v2.1 0.326 0.389 0.421
GenSlip v2.1–SC08 0.233 0.273 0.294
GenSlip v3.2 0.252 0.292 0.313
GenSlip v3.2–SC08 0.156 0.177 0.188
10.0
GenSlip v2.1 0.307 0.349 0.369
GenSlip v2.1–SC08 0.245 0.262 0.269
GenSlip v3.2 0.284 0.325 0.345
GenSlip v3.2–SC08 0.199 0.216 0.223
Figure 4.12 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (with GenSlip v3.2) as the target model and directivity-corrected BA08 as
the reference model for all sources used in the analysis. The CHD used in calculating
d(r,k,x) is the centroid-biased hypocenter distribution illustrated in Figure 4.11.
119˚ 118˚ 117˚
34˚
35˚
0.0 0.2 0.4 0.6 0.8
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
114
Figure 4.13 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (for GenSlip v3.2) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. The CHD used in calculating d(r,k,x)
is the periphery-biased hypocenter distribution illustrated in Figure 4.11.
4.4.3 Effects of CHDs on CyberShake Seismic Hazard Calculation
The changes in CHD lead to variability in directivity effects, and could affect the
hazard calculations. In this section, we focus on quantifying the variability in hazard
calculations with several asymmetric hypocenter distributions which represent rupture
119˚ 118˚ 117˚
34˚
35˚
0.0 0.2 0.4 0.6 0.8
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
115
bias associated with elasticity contrasts across crustal fault zones, such as those described
by Dor et al (2006).
Large fault zones that accommodated significant slip have interfaces that separate
different media (Ben-Zion and Sammis, 2003) due to the production of damaged fault
zone material and the juxtaposition of different rock bodies across the fault. Ruptures on
faults that separate different materials have a preferred propagation direction, which
correlates with the velocity structure predicted by theory. In Figure 4.14, we show the
model II (shear) rupture on a fault that separates different media (Andrews and Ben-Zion,
1997) tends to propagate preferentially in the direction of slip on the compliant side of
the fault. The slower velocity block is above the fault as manifested by the fronts of the
radiated seismic waves. The existence of a material contrast across the fault produces an
asymmetric motion in the different media that is especially prominent near the fault
(Figure 4.14a). As shown in Figure 4.14b, particle velocities in the more compliant
material (y>0) are larger than in the stiffer medium (y<0). Various studies (Rubin and
Gillard, 2000; Henry and Das, 2001; Wang and Rubin, 2011) have found that both small
and large earthquakes show directional asymmetry that is compatible with a preferred
propagation direction associated with the local velocity structure. McGuire et al (2002)
analyzed rupture properties of large global earthquakes and found that most are
predominantly unidirectional.
116
Figure 4.14 (a) Particle velocities at a given time for rupture along a material interface
(thin horizontal line). The slipping region is marked by the thick segment on the fault and
is propagating to the right (right-lateral strike-slip fault). (b) Enlarged view of the white
box in (a) showing asymmetric particle velocities around the rupture pulse. Retrieved
from Dor et al (2006).
Considering the case where ruptures have preferred propagation direction, the rupture
directivity forecasting should incorporate this information. In KCERF, given a fault,
hypocenters are evenly distributed along the fault, and the conditional hypocenter
distribution (CHD) is used to consider the aleatory variability of hypocenter locations. As
seen in Figure 4.11, Beta distribution is able to generate symmetric CHD beside the
uniform distribution. With different sets of parameters for Beta distribution, we could
sample the asymmetric hypocenter distribution to better forecast the rupture directivity
and its effect on seismic hazard calculations. In order to do so, we consider two sources
(93 and 112 as listed in Table 3.1), along sections of which Dor et al (2006) observed
systematic asymmetry in the pattern of damage expressed by fault zone rocks. The two
117
sources are shown in Figure 4.15, where black squares indicate the location of epicenters
and red arrow associated with each fault indicates the preferred rupture direction, and two
small boxes are the study area of Dor et al (2006). Overall, ruptures along the two faults
as shown are southeast to northwest, and we choose four sets of parameters for Beta
distribution and the resulting probability density is shown in Figure 4.16. For example,
Asymmetry1-South (A1-S) means higher weights are assigned to hypocenters located at
the south part of the given fault. For the same asymmetric direction, we also distinguish
the extent of asymmetric distribution by A1-S and A2-S, corresponding to centroid-
biased asymmetry and periphery-biased asymmetry, respectively.
Figure 4.15 Faults of the southern San Andreas system included in the current study and
region that includes investigation sites (two small box). After Dor et al (2006). The site
STNI at which the hazard curves are calculated is shown (black triangle), and
CyberShake region is also shown.
118
Figure 4.16 Asymmetric hypocenter distributions (probability density) with different set
of parameters. Each set of parameters is assigned a name and x from 0 to 1 means south
to north in this case.
Based on Equation (2.2) using CyberShake model, we calculate the probability of
exceedance (PoE) in 50 years given the earthquake occurrence for the two sources at a
site STNI shown in Figure 4.15. Figure 4.17 shows the hazard curves for different CHDs
which represent different asymmetric hypocenter distributions p(x|k). We also use
symmetric hypocenter distributions (uniform, centroid-biased, and periphery-biased) as
references. As depicted in Figure 4.17, we found that for hazard level (SA at 3.0s)
smaller than 0.1g, each case provides similar hazard estimations, and for hazard level
119
larger than 0.1g, hazard curves calculated using different CHDs start to deviate from each
other.
Figure 4.17 Hazard curves for the station STNI using two sources [93,112] as shown in
Figure 4.15 and different CHDs are considered during the hazard calculations.
To better quantify the differences between them, we calculate the probability gain
factor (PGF) by
pgf =
PoE
CHD
PoE
uniform
(4.5)
120
where the denominator is the PoE calculated using the uniform hypocenter distribution,
and the numerator could be PoE calculated from other CHDs. The results are shown in
Figure 4.18. We found that the gain factor of A2-S (periphery-biased asymmetric
hypocenter distribution) is up to a factor of 4 larger than the A1-N (centroid-biased
asymmetric hypocenter distribution), the reason is that due to the bias that rupture tends
to initiate from the south part of both sources, larger directivity pulse is generated and
propagates towards the site which is located within the sedimentary basin (Los Angeles
basin), and the directivity-basin coupling further enhances directivity effect, resulting in
large ground motion and high hazard. In addition, we found that the centroid-biased
hypocenter distribution lower the hazard relative to the uniform distribution, which is
consistent with what we see in the last section.
Figure 4.18 Probability gain factors calculated from the probability of exceedance given
hazard level based on Figure 4.17. The black line is the reference line.
121
Beside the hazard calculation at a single site, we also calculate the hazard map given
a CHD. Since we are interested in the ratio between hazard maps with different CHDs,
we compute the probability gain factor for each site given a hazard level, and for all sites
considered in CyberShake, we present the map-based probability gain factor in Figure
4.19. We found that for A1-S (Figure 4.19d), the hazard level in the CyberShake region is
significantly increased due to the large directivity effect and directivity-basin coupling.
Figure 4.19 Maps of the probability gain factors of hypocenter distribution (a) A1-N, (b)
A1-N, (c) A1-S, and (d) A2-S with respect to uniform distribution given hazard level SA
at 3.0 s equals to 0.3g.
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4.5 Discussion
The rupture directivity forecasting needs the information related with rupture
propagation modeling and location of hypocenters, which together determine the seismic
wave energy radiation and interferences from an earthquake source. With simulation-
based ground motion prediction model, we could test the effects of different slip models
and different hypocenter locations. ABF analysis allows us to quantify those differences
between models. In addition to compare different CyberShake models, ABF method can
be used to compare current existing NGA models with updated NGA models (e.g. NGA-
west 2 project), and to compare current existing empirical directivity models with
updated models (Spudich et al., 2012).
Considering the empirical directivity model as the observations, we found that the
simulation-based model CyberShake shows larger directivity effects than SC08
directivity corrections to the NGA GMPEs, and directivity-basin coupling enhances the
directivity effects. Increasing in small-scale rupture complexity (e.g. GenSlip v3.2) could
significantly decrease the source directivity considered in CyberShake models. The
changes in directivity effects due to rupture complexity would be expected since the
coherence of rupture propagation would produce constructive seismic wave energy that
would lead to large directivity pulses, while less coherence in rupture propagation would
increase the chance of destructive interference, leading to less directivity effects. In
addition to pseudo-dynamic rupture models, CyberShake simulations with pure dynamic
rupture models have been performed, and the ABF method can be used to compare those
123
models and quantify the changes in directivity effects due to more complex and
sophisticated rupture description.
The frequency-dependent feature in source directivity indicates that at longer periods,
directivity effects considered in both slip models become close due to the similar
coherent rupture propagation process. Treating NGA GMPEs as the observations, the
changes in source models and CHDs could make the directivity in simulation-based
model closer to NGA GMPEs (e.g., in Table 4.2, the total standard deviation of d factor
is 0.151 using centroid-biased hypocenter distribution).
The aleatory variability of hypocenter locations for a given fault is considered by the
CHD p(x|k) in seismic hazard calculations. However, what distribution we should use
indicates the epistemic uncertainty. Using a sample set of different CHDs parameterized
with Beta probability function, we apply both the ABF analysis and probabilistic seismic
hazard analysis to quantify the effects of various hypocenter distributions. First, the
centroid-biased hypocenter distribution lowers the directivity effects in CyberShake
(Table 4.2) because of less chance for hypocenters that locate at the further end of a fault
from a site. Secondly, the asymmetric CHDs consistent with material contrasts, i.e.
rupture bias, can substantially alter seismic hazard estimates, especially in sedimentary
basins from our study. However, the form of the CHD is not well constrained by existing
data, and we need to consider multiple CHDs in logic tree for probability hazard analysis.
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4.6 Conclusions
In this chapter, we consider the information relevant to rupture directivity forecasting,
including small-scale rupture complexity, conditional hypocenter distribution (CHD). We
apply ABF method to decompose the excitation functions of CyberShake models with
two different type of slip models, GenSlip v2.1 and v3.2, and found that rupture
propagation process that depends on the local slip heterogeneity significantly affect the
source directivity and the directivity-basin coupling. Due to more widely distributed
seismic wave energy radiated from the source, the scattering and basin excitation effects
are increased using the more complex slip model. We also extend the CyberShake models
to include multiple CHDs parameterized as Beta distributions, which can be used to
represent rupture bias associated with elasticity contrasts across crustal fault zones, such
as those described by Dor et al. (2006). At present, the degree of CHD asymmetry is
difficult to estimate, because it depends on the dynamical modeling of ruptures in fault
zones described by asymmetric elastic and inelastic properties, which are poorly
constrained. However, the ABF analysis of the CyberShake models using a sample set of
CHDs allows us to quantify the changes in rupture directivity and directivity-forecasting
probability gains that could be achieved by plausible improvements to CHD models of
fault zones in Southern California. We show that these gains are sufficiently large to
motivate more systematic approaches to CHD quantification.
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Chapter 5
Basin Amplification in 3D Velocity Structures
5.1 Abstract
Three-dimensional geological structure could significantly affect the ground motion
predictions, in particular the effects due to sedimentary basins. We use two CyberShake
models with different 3D velocity models to examine the basin amplifications for
different geological structures, including CVM-SCEC version 4 (CVM-S4) and CVM-
Harvard. Using ABF method, we calculate the B maps at various periods from those
CyberShake models, and investigate the frequency-dependence of basin effects in both
simulation-based models and NGA GMPEs. The basin excitations calculated from CVM-
H are smaller than those from CVM-S, and they show a stronger frequency dependence,
primarily because the shear velocities in the deeper parts of the basins are, on average,
lower in CVM-H. Owing to this difference, the substitution of CVM-H for CVM-S
reduces the CyberShake-NGA basin-effect discrepancy. Among the NGA models, that of
Abrahamson & Silva (2008) is the most consistent with the CyberShake CVM-H
calculations, with a basin-effect correlation factor greater than 0.9 across the frequency
band 0.1-0.3 Hz. In addition to the basin effects, we compare the directivity-basin
coupling (DBC) effect for two velocity models. The basin excitations of the incident
directivity pulse at different periods are also presented.
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5.2 Introduction
It is well known that seismic waves are amplified in sediment-filled basins, relative to
the surrounding bedrock (Bard and Bouchon, 1980). The basin amplification has been
seen to cause large damages and devastation of buildings. For example, the 1985
Michoacan earthquake (M
w
8.1) produced 1 to 2 second reverberation in the shallow lake-
bed zone of Mexico City, which selectively matched resonances of most buildings (10-20
floors), although the earthquake is 360 km away (Esteva, 1988). Another example is the
1995 Kobe earthquake in Japan (M
w
6.9). This moderate earthquake located at the edge of
a basin resulted in extremely destructive ground motions within basins, not only because
of the amplification by sedimentary basins but also from the generation of surface waves
along the sharp basin edges (Pitarka et al., 1996; Frankel et al., 2009). It is imperative to
understand the causes of ground motion amplification in sedimentary basins in order to
incorporate this amplification effect in PSHA and mitigate the loss of life and property
during major earthquakes.
Previous studies have investigated the physical factors that contribute to the ground
motion amplification in the simple geometries (2D and 3D) of basins (Bard and Bouchon,
1985; Rial, 1989; Olsen and Schuster, 1995). First, the surface waves generated due to
the sharp basin edges are usually characterized by large amplitudes that significantly
influence the amplification of low velocity basins (Bard and Bouchon, 1980; Graves et
al., 1998). Secondly, the impedance effects due to contrast of sediment and bedrock
boundary increases the amplitude of ground motions, partially explaining site
amplification in basins (Williams et al., 1993). Thirdly, resonance from the vertically
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interfering waves controls the low-frequency elastic responses of 2D and 3D simplified
basin models (Rial et al., 1992). For a layered medium, seismic resonance is the
constructive reinforcement of multiply reflected waves where the layer thickness is tuned
to the seismic wavelength. For simple basin models, the resonance frequency can be
satisfactorily estimated using simple formula which depending on the basin geometry and
seismic wave velocity (Bard and Bouchon, 1985). However, due to the complexity in
analyzing the resonance property for 3D basin structure (Rial et al., 1991), a number of
studies have simulated 3D seismic wave propagation in regional geological models that
include basin structure (Wald and Graves, 1998; Komatitsch et al., 2004; Lee et al., 2008;
Frankel et al., 2009). Those simulations show that underlying structure influences both
the amplitude and duration of ground motions, and the spectral ratios are comparable
with observations for ground motions for stations located within basins with respect to
rock sites.
Empirical basin amplification models have been developed and incorporated in
empirical GMPEs for prediction ground motions, and applied in PSHA (Frankel et al.,
2007). Those models use the basin depth to specific shear wave velocities (e.g. Z
2.5
and
Z
1.0
as in Table 2.1) as the parameterization (1D sense). The form of those models for
deeper basin depth (Z
1.0
>200m and Z
2.5
>3.0km) uses the basin terms developed by Day
et al (2008) from theoretical 3-D ground motion simulations for the Los Angeles, San
Gabriel, and San Fernando basins in southern California. The overall amplitude of this
basin amplification term was calibrated empirically from the regression analysis against
observations. However, according to results from the 3D simulations of wave
128
propagation (Olsen et al., 2006; Graves et al., 2011a), 3D basin structure play an
important role in basin amplification. Furthermore, significant progress has been made
toward the goal of developing comprehensive 3D velocity structure models of the Los
Angeles region, and two high-resolution velocity models for this region, CVM-SCEC
CVM-Havard (Magistrale et al., 2000; Kohler et al., 2003; Suss and Shahi, 2003), has
increase the epistemic uncertainties in basin amplification found in simulations using
these two models.
CyberShake, simulation-based hazard model, which has a large number of earthquake
simulations using 3D velocity models, CVM-SCEC and CVM-Harvard, enables us to
perform analysis for basin amplification given 3D velocity models that incorporate basin
structure. As seen from Chapter 3, the ABF method provides a framework to decompose
the ground motions into to different factors in which the B factor quantifies the basin
amplification and allows us to compare different models, such as between CyberShake
model and NGA GMPEs, or between CyberShake models with different slip models
(seen from Chapter 4). In this chapter, we first compare CyberShake models with
different basin structures and quantify the frequency-dependent of basin effects, and then
we perform analysis of the resonance feature of both 3D basin structures. The slip model
used in this chapter is GenSlip v3.2.
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5.3 Basin Effects for Different 3D Velocity Models
Probabilistic seismic hazard analysis using simulation-based model, CyberShake,
whish uses two available velocity models, CVM-SCEC and CVM-Harvard, provide
different hazard assessments in the sedimentary basins. Those differences could be
mainly caused by differences in the velocity models used in the deterministic simulations.
In this section, we will focus on the southern California region, which is a natural
laboratory for studying the basin effects. We first introduce the development of both
velocity models and their differences briefly, and then we use ABF method to compare
the CyberShake models with these two velocity models, in particular compare the
residual b map at various periods.
5.3.1 CVM-S4 and CVM-Harvard
Motivated by the factor that Southern California is one region where the seismic
hazard is high and the sedimentary basin effects are known to substantially influence
strong ground motions, the development of 3D crustal models that incorporate basin
information from well bores and active seismic survey (e.g. oil exploration), as well as
large-scale structural tomography (Hauksson, 2000). Example of such 3D velocity
models are the SCEC CVM (Magistrale et al., 2000) and the Harvard model (Suss and
Shahi, 2003). The purpose of the 3D Community Velocity Model for southern California
is to provide a unified reference model for the several areas of research that depend of the
subsurface velocity structure in the analysis. These include strong motion modeling,
seismicity location, and tomographic velocity modeling. Those models describe the
130
seismic P- and S-wave velocities and densities, and are comprised of basin structures
embedded in crust over a variable depth Moho. The basin structures in Harvard model are
also compatible with the locations and displacements of major faults represented in the
SCEC Community Fault Model (CFM) (Plesch et al., 2007).
Figure 5.1 (a) Map view of shear wave velocity at depth 1000m for CVM-S4. (b) Map
view of shear wave velocity at depth 1000m for CVM-H11.9.1. (c) 1D shear wave profile
underneath the station LGB shown as black triangle in (a) and (b).
131
Those velocity models have been used in simulations for historical and future
earthquakes with some success, and full-waveform tomography has been developed to
improve those models for predicting regional high-resolution strong ground motion maps
that could lead to effective and reliable earthquake-hazard analysis (Chen et al., 2007;
Tape et al., 2009). In this chapter, the most up to date velocity models are CVM-SCEC
version 4.0 (CVM-S4) and CVM-H11.9.1 (hereinafter we use CVM-H to represent this
version). Figure 5.1 shows the map view of two velocity models at depth 1000m and the
1D shear-wave velocity profile beneath the station LGB that is located within Los
Angeles basin. The basin structures are different between these two models. Simulation-
based hazard model, CyberShake, uses these two velocity models to produce hazard
curves and maps, which show significant discrepancies in those basin areas. In the
following sections, we are going to compare those models in detail using the method we
developed in Chapter 2.
5.3.2 Regional Excitation Levels
Following Equation (2.26), we calculate the regional excitation levels for CyberShake
and NGA08 models with two velocity models, shown in Figure 5.2. The red lines show
the case where CVM-S4 is used as the 3D velocity model in the simulations and basin
depth parameters are extracted from this model, while the black lines show the case
where CVM-H is used as the 3D velocity model in CyberShake simulation and basin
depth parameters are extracted from this model. First, comparing with Figure 3.3 where
132
CVM-S4 and GenSlip v2.1 are used, as expected, the excitation levels for NGA08
models do not change. Second, for the two velocity models, the excitation levels for
BA08 are the same since this NGA model does not explicitly consider basin effects using
basin depth parameter. Thirdly, CVM-H shows larger excitation level than CVM-S4,
probably due to basin effects that we will discuss in the following section.
Figure 5.2 The A factors of CyberShake and NGA models (AS08, BA08, CB08, and
CY08) at four periods. Black groups use CVM-H, and red groups use CVM-S4. Scales
are logarithmic.
5.3.3 Basin Effect Maps
Using the ABF method, we calculate the residual maps for b(r) between CyberShake
models and empirical GMPEs. The excitation functions are taken as spectral acceleration
at low frequencies (at 3.0, 5.0, and 10.0s). We use those residual maps to evaluate the
133
frequency dependence of the basin amplification effects. We also put emphasis on the
correlations of two velocity models with different basin structures with basin
amplification effects in simulation-based models. The source model used in this chapter
is GenSlip v3.2, i.e. more complex slip model (less directivity and directivity-basin
coupling as seen from Chapter 4). Currently, the numerical methods used for simulations
in CyberShake include Graves and Pitarka (Graves and Pitarka, 2010) and anelastic wave
propagation (AWP) code by Olsen (Olsen et al., 1995), hereinafter referred as GP and
AWP-ODC, respectively.
Figure 5.3 Maps of residual factors b(r) using CyberShake model with (a) GP and CVM-
S4, (b) GP and CVM-H, (c) AWP-ODC and CVM-S4, and (d) AWP-ODC and CVM-
Harvard as the target model and directivity-corrected BA08 as the reference model for
spectral accelerations at T = 3.0s.
134
Figure 5.3 shows residual factors b(r) using different CyberShake models as the
target model and directivity-corrected BA08 as the reference model for spectral
accelerations at T = 3.0s. For the same velocity model (e.g. CVM-S), target models with
two different numerical methods (GP and AWP-ODC) have the similar basin
amplification effects as seen from Figure 5.3(a) and (c), and the linear cross-correlation
between the two maps is 0.993. This indicates that the basin effects do not dependent on
numerical modeling of wave propagations, verifying the two numerical models. Using
the same numerical method (e.g. GP), target models with two different velocity models
(CVM-S and CVM-H) have different basin amplification effects within sedimentary
basin as seen from Figure 5.3(a) and (b), and the linear cross-correlation between the two
maps is 0.756. In Figure 5.4, the differential map between Figure 5.3(a) and (b) is shown,
indicating that two velocity models have different basin structures that can affect basin
amplification significantly.
Figure 5.4 Maps of residual factors b(r) using CyberShake model with GP and CVM-S4
as the target model, and CyberShake model with GP and CVM-H as the reference model
for spectral accelerations at T = 3.0s.
119˚ 118˚ 117˚
34˚
35˚
210 1 2
b(r)
135
Figure 5.5 Maps of residual factors b(r) using directivity-corrected BA08 as the reference
model, and CB08, CY08, AS08, and CyberShake with GP and CVM-S4 as the target
models (each column) for spectral accelerations at T = 3.0, 5.0, and 10.0 s (each row).
Color scales are the same as Figure 5.3. The basin depth parameters used in CB08, CY08,
and AS08 are extracted from CVM-S4.
136
Figure 5.6 Maps of residual factors b(r) using directivity-corrected BA08 as the reference
model, and CB08, CY08, AS08, and CyberShake with GP and CVM-H as the target
models (each column) for spectral accelerations at T = 3.0, 5.0, and 10.0 s (each row).
Color scales are the same as Figure 5.3. The basin depth parameters used in CB08, CY08,
and AS08 are extracted from CVM-H.
We further calculate the residual factors b(r) using different CyberShake models as
the target models, which take two velocity models and one wave propagation modeling
code (GP). The directivity-corrected BA08 is used as the reference model, and the
excitation functions are spectral Accelerations at T = 3.0, 5.0, and 10.0s. Figure 5.5
shows the case where the velocity model is CVM-S4, from which the basin depth
parameters (Z
2.5
and Z
1.0
) used in CB08, CY08, and AS08 are extracted, and Figure 5.6
137
shows the case where the velocity model is CVM-H, from which the basin depth
parameters (Z
2.5
and Z
1.0
) are extracted. Although Figure 5.5 is the case where slip model
is GenSlip v3.2, the frequency-dependent feature in basin effects is the same as the case
where GenSlip v2.1 was used (Figure 3.13a), indicating the stability of B(r) as a manifest
of the site-specific effect.
The basin effects are different among CB08, CY08, and AS08 for all periods due to
the different scaling relations with basin depth parameters as shown in Figure 5.7 for
CVM-S4 and CVM-H. In order to understand the difference among NGA models, we did
the similar comparison between NGA models as described in Abrahamson et al (2008),
where the basin depth scaling for M
w
7 strike-slip earthquakes at a distance of 10 km is
compared in Figure 5.8. We show the ratio of median predictions from AS08, CB08, and
CY08 with respect to that from BA08. For shallow basin depths (Z
1.0
=0.1 km, Z
2.5
=0.9
km), the AS08 model has a large reduction in the long-period (T>1s) ground motion, but
the other two models do not have an effect on the long-period ground motion for shallow
soil/sediment sites, which is consistent with those cooler colors in b-map of AS08 seen
from both Figure 5.5 and Figure 5.6. The AS08 shallow soil/sediment scaling is stronger
due to the use of 1-D analytical site response results to constrain the model. For the deep
soil/sediment sites (Z
1.0
=1.2 km, Z
2.5
=4.8 km), the three models (AS08, CB08, and CY08)
all show a large increase in the long-period motion as compared to the BA08 model that
does not include soil/sediment depth scaling, consistent with warmer colors in the b-map
of those models. At T=10 sec period, the AS08 and CB08 models show the strongest
scaling relative to BA08 model due to the use of the 3-D analytical basin response results
138
to constrain their models, while the basin effects in CY08 at this period relative to BA08
is smaller than the other models as seen from both Figure 5.5 and Figure 5.6.
Figure 5.7 Maps for basin depth parameters Z
1.0
and Z
2.5
(each column) extracted from
CVM-S4 and CVM-Harvard (each row as indicated with the texts).
139
Figure 5.8 Comparison of median SA for M
w
=7 strike-slip earthquakes at an R
JB
distance
of 10km for different site conditions: soil sites (V
S30
=270m/s) with average soil depth
(Z
1.0
=0.5 km, Z
2.5
=2.3 km), shallow soil depth (Z
1.0
=0.1 km, Z
2.5
=0.9 km), and deep soil
depth (Z
1.0
=1.2 km, Z
2.5
=4.8 km) depths and rock sites (V
S30
=760 m/s).
For all periods, the CyberShake basin effects in both velocity models are generally
larger than those from the three NGA models that provide basin-effect factors (AS08,
140
CB08, and CY08), and CyberShake model with CVM-H shows similar frequency-
dependent basin effects with AS08 model. Table 5.1 shows the cross-correlation between
basin effect map of CyberShake models and AS08, where we denote the CybeShake
model with CVM-S4 and CVM-H as CS1 and CS2, respectively, and denote the AS08
with basin depth Z
1.0
extracted from CVM-S4 and CVM-H as AS1 and AS2, respectively.
First, the correlation between basin effects of CyberShake with CVM-H and AS08 is
larger than that between CyberShake with CVM-S4 and AS08, which can be seen in
Figure 5.5 and Figure 5.6. Secondly, the correlation between basin effects of CyberShake
models with CVM-H and CVM-S4 is relatively small, indicating the significant
difference in basin amplifications associated with those velocity models. For example, at
3.0s, basin effects is larger in CVM-S4 than CVM-H, while at 10s, basin effects is larger
in CVM-H than CVM-S4.
Table 5.1 Cross-correlation of residual maps b(r) for SA at 2.0, 3.0, 5.0, and 10.0s with
different target models (see text for the abbreviation of each target model). The
directivity-corrected BA08 is used as the reference model.
Period (s)
CS1 CS2 CS1
AS1 AS2 CS2
2.0 0.847 0.813 0.682
3.0 0.882 0.915 0.756
5.0 0.897 0.960 0.783
10.0 0.917 0.964 0.887
Now we focus on the differences between CyberShake models. For the same velocity
models, the basin amplification in both CyberShake studies increases systematically with
periods. This is a 3D effect, that is to say, higher-mode resonances present in 1D case are
smoothed out by lateral scattering, so that the long-period resonances dominate (Day et
141
al., 2008). We also found CVM-H provides larger basin amplification at 3.0s than CVM-
H, but smaller basin amplification at 10.0 than CVM-H. Short-period waves are subject
to short wavelength variations due to local focusing and interference effects. Very long-
period waves are influenced principally by large-scale averages of the seismic velocity
structure. CVM-S4 has relatively deeper basins than CVM-H but the lateral changes (i.e.,
smaller scale variations) of the basin depth are much larger than CVM-H, resulting in
larger basin amplification at shorter-period (e.g. 3.0s) but smaller basin amplification at
longer-period (e.g. 10s).
At last, we confirm that the frequency-dependent basin effects considered in
simulation-based model do not depend on slip model used in numerical simulations,
source set K, and weighting functions we used in the ABF analysis. Comparing Figure
3.13 (GenSlip v2.1) and Figure 5.5 (GenSlip v3.2), the basin effects due to different slip
models are similar. We calculate the residual factor b(r) using CyberShake with CVM-S4
as the target model and directivity-corrected BA08 model as the reference model. Figure
5.9(a) shows maps of residual b(r) using a source set that is different from what listed in
Table 3.1, but the same CHD (uniform) for the ABF analysis. Comparing with Figure 5.5,
the basin effects due to different source set K for averaging operations are similar.
Furthermore, Figure 5.9(b) shows maps of residual b(r) using the same a source set listed
in Table 3.1, but different CHD (periphery-biased) for the ABF analysis. Comparing with
Figure 5.5, the basin effects due to different conditional hypocenter distribution for
averaging operations are similar. Comparing Figure 5.9 with b(r) of CyberShake in
Figure 5.5, the basin effects due to different source set K and conditional hypocenter
142
distribution p(x|k) are similar, in particular the shape of larger basin amplification
correlates with the basin structure as shown in Figure 5.7. The stability of b-map
indicates that basin effects due to wave excitation within basins is intrinsic property
related only with basin structure, independent with slip models, sources, and hypocenter
locations.
Figure 5.9 Maps of residual factors b(r) using CyberShake with CVM-S4 as the target
model and directivity-corrected BA08 as the reference models for spectral accelerations
at T = 3.0, 5.0, and 10.0 s (as shown on the top row). Color scales are the same as Figure
5.3. Panel (a) The source set used in the ABF analysis is different from sources listed in
Table 3.1, and the CHD is uniform distribution. Panel (b) The source set used in the ABF
analysis is the same as sources listed in Table 3.1, and the CHD is periphery-biased
distribution.
143
5.4 Basin-Excited Seismic Waves
At each period, basin effects of CyberShake models show discrepancies in amplitudes
(Figure 5.5 and Figure 5.6). We would like to investigate the physical properties of
seismic waves excited within sedimentary basins specified by different 3D velocity
models, and try to answer the following two questions: Do the waves from seismic
resonance in basins cause maximum values in the response spectra for different
oscillation frequencies? What are the effects of basin structures on the resonance of basin
excitations? First, we focus on spectral accelerations of basin responses considered in
CyberShake simulations, and we then put our emphasis on the seismic resonance
phenomena of a basin structure by performing the time-frequency analysis using wavelet
to extract the resonance frequencies and spectral properties for CVM-S4 and CVM-H.
5.4.1 Response Spectra at Basin Sites
We select a number of sites located within the sedimentary basins, as shown in Figure
5.10. The basin depths to an iso-velocity (shear wave) surface underneath those selected
sites are relatively high (Z
2.5
>3.0km and Z
1.0
>0.2km based on CVM-S4), indicating they
are located within deep basins and would experience larger basin effects (Abrahamson
and Silva, 2008; Campbell and Bozorgnia, 2008). We use sources as listed in Table 3.1.
For a given source, instead of using all rupture variations, we use the mean magnitude
and a hypocenter that locates within the middle of the fault trace. For each rupture-site
pair, we extract the synthetic seismograms of two horizontal components for a given
144
CyberShake model. One of which is 35 degrees from north to east (N35E) and the other
is 35 degrees from east to southern (E35S). There are about 700 seismograms extracted
from CyberShake database for each velocity model used in CyberShake.
Figure 5.10 Selected basin sites located within CyberShake region. Names of sites are
shown beside the location of those sites marked as black triangles.
We calculate the acceleration of a single-degree-of-freedom (SDOF) oscillator with
specific natural period ω
n
and damping ς (Nigam and Jennings, 1969). In terms of the
relative displacement, x(t), the oscillator has the equation of motion
x+2ω
n
ζ x+ω
n
2
x =− u
g
t ( ) (5.1)
119˚ 118˚ 117˚
34˚
35˚
0 50
km
s022
s024
s026
s070
s072
s115
s159
s307
s345
s347
s387
s389
s431
s433
s470
s472
s474
s512
s514
145
where
u
g
t ( ) is the recorded or simulated acceleration at a specific site, and the solution
to Equation (5.1) by assuming x(0)=0 is
x t ( )=
−1
ω
n
1
2
−ζ
2
u
g
τ ( )e
−ω
n
ζ t−τ ( )
sin ω
n
1
2
−ζ
2
t −τ ( )
⎡
⎣
⎤
⎦
dτ
0
t
∫
(5.2)
which could be solved analytically and numerically. The peak spectral acceleration (PSA)
is the maximum absolute value of the relative acceleration
x t ( ) , and the time point
where we take the maximum absolute value is denoted as t
max
. Figure 5.11 shows the
relative acceleration at three natural periods (3.0, 5.0, and 10.0s). The black dots show the
coordinate (t
max
, PSA) for each period, from which we can see the change of the t
max
.
The information of t
max
somehow indicates that the wave group in the input
acceleration
u
g
t ( ) that contributes the most to the PSA by
PSA ω
n
,ζ ( )= x t
max
( )=
d
2
dt
2
−1
ω
n
1
2
−ζ
2
u
g
t −τ ( )e
−ω
n
ζτ
sin ω
n
1
2
−ζ
2
τ
⎡
⎣
⎤
⎦
dτ
0
t
∫
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
t=t
max
(5.3)
where t
max
is also a function of ω
n
and ς. Therefore, we calculate the t
max
for each
seismogram given a site and a source, and average over all sources for the site. To
consider the group delay time for each ω
n
due to the calculation shown in Equation (5.3),
we calculate the impulse response of the SDOF shown in Figure 5.12. We also correct the
calculated t
max
by the average S-wave arrival time (assuming V
s
=2.5 km/s) to remove the
bias due to distance between a source and a site.
146
Figure 5.11 Relative accelerations for input synthetic seismograms (black traces on the
first row) at three natural period of the SDOF system (second row: 3.0s, third row: 5.0s,
and forth row: 10.0s). Each column indicates the component of the synthetic seismogram.
Each black dot indicates the position where the corresponding PSA is located and its
value.
Figure 5.12 Impulse response function (response acceleration) for three periods. Dots
show the maximum absolute values (PSA), indicating the group delay for different
periods.
147
Figure 5.13 shows the resulting t
max
for two horizontal components at all sites. Three
rows show the t
max
for SA at 3.0, 5.0, and 10.0s, respectively. Again, due to two different
velocity models, we have two sets of t
max
for each component, shown as red (CVM-S4)
and blue dots (CVM-H), respectively. The measurement of t
max
allows us to determine
frequency-dependent feature of the wave groups that contribute the most to the PSA that
is then used in the ABF analysis and PSHA. Those wave groups are mainly surface
waves and basin-excited waves due to resonance of incident seismic waves, and we will
focus on the resonance in a basin.
Figure 5.13 Averaged t
max
for peak spectral acceleration at 3.0, 5.0, and 10.0s. The names
of basin sites are also shown.
148
5.4.2 Wavelet Analysis of Basin Responses
For each seismogram, we did the continuous wavelet transform to get time-frequency
spectra. Generally, the continuous wavelet transform of a signal u(t) is defined by
CWT
α,β ( )
u t ( ) { }
=
1
α
u τ ( )ψ
*
τ −β
α
⎛
⎝
⎜
⎞
⎠
⎟
dτ
−∞
+∞
∫
(5.4)
where t is the time, α is the scale parameter, β is the translational parameter, and ψ is the
analyzing wavelet (function basis or mother wavelet). The analyzing wavelet used in the
dissertation is the Morlet wavelet (Kristekova et al., 2006), which is defined as
ψ t ( )=π
−1/4
exp iω
0
t ( )exp −t
2
/2
( )
(5.5)
where ω
0
defines the instrinct of the wavelet, allowing trade between time and frequency
resolutions (tradeoff). Conventionally, the restriction ω
0
>5 is used to avoid problems
with the Morlet wavelet at low ω
0
(high temporal resolution). Such wavelet has a
spectrum with the zero amplitude at negative frequencies, which is called the progressive
wavelet. Specially, β=t, α=ω
0
/2πf, corresponding to
W t, f ( )=CWT
α ,β ( )
u t ( ) { }
(5.6)
W(t, f) is the time-frequency representation of the input signal u(t). Time shift t will be
the translational parameter (β), and frequency f relates with the scale parameter (α). We
use the same selection of sources in section 5.5.1, but extract the two-components
synthetic seismograms for all sites given those sources.
149
Figure 5.14 illustrates the continuous wavelet transforms of selected two-components
seismograms calculated at site ‘s387’ located within Los Angeles basin as in Figure 5.10
from the source 87 as in Table 3.1. The CyberShake model used in this illustration is the
CVM-S4. From the time-frequency representation of an input seismogram, we can see
the seismic energy distribution. In terms of frequency components of a spectrum, for this
particular example, wave energy is concentrated at low frequencies and time shift after
the direct S wave arrival. The site is located ~112 km away from the hypocenter,
indicating that surface waves and waves generated due to vertical constructive
interferences are dominant. In addition to the main wave group, we also observe a ‘V’
shape time-frequency spectra for time window 80~150s and frequency band 0.2~0.35Hz,
indicating the reverberation of waves in the basin.
The relationship between frequencies and maximum power values for those basin
sites allows us to investigate the basin-trapped wave energy, and the relationship between
frequencies and peak times for those basin sites enables us to identify waves that
contribute the most to the maximum ground motions as indicated in Figure 5.11.
We could show similar figure as Figure 5.14 for each source at each site, but we find
that the averaged values over sources are enough to integrate available information. First,
for a specific site, we calculate the average maximum power values and average time
shifts for each frequency over all sources. Secondly, in order to remove effect of
distances between each source and the site, we correct the averaged time shift by the
150
average S-wave arrival time (assuming V
s
=2.5 km/s), then the corrected time shifts
indicates the relative delay to the direct S wave arrivals.
Figure 5.14 Illustration of wavelet analysis for a given source and site. Two-components
seismograms (velocity in m/s) are show on the top and the continuous wavelet transforms
are show on the bottom (colored), in which white triangles show the maximum values of
the powers for each frequency.
Figure 5.15 shows an example of averaged maximum power values (top row) and
corrected time shifts (bottom row) at the site ‘s387’ averaged over sources. Because there
are two velocity models used in CyberShake calculations, we get two sets of averaged
maximum power values and time shifts for each site at each frequency, which are
indicated by red (CVM-S4) and blue (CVM-H), respectively. Because we do the average
151
over all sources, the variability of maximum power values along with frequencies
approximately indicate the intrinsic response at the site (located within the basin). We
found, for this particular site, CVM-H tends to have longer-period (<0.2 Hz) waves that
arrival 25 seconds after the direct S-wave, and CVM-S4 tends to have shorter-period
(>0.25Hz) waves that arrival 25 seconds after the direct S-wave. This is consistent with
the differences between those velocity models, especially in the basins.
Figure 5.15 The variability of averaged maximum power values (top row) and corrected
time shifts where the maximum power values are located (bottom row). The black dashed
lines show the averaged S-wave arrival time. Red dots: CVM-S4; Blue dots: CVM-H.
152
In Figure 5.15, given a frequency, we could obtain a spectral value, as well as a S-
arrival corrected peak time for each of all sites in the CyberShake region (Figure 2.6),
and we denote as S
p
(r) and P
t
(r), respectively. These quantities can be presented as map-
based images as in Figure 5.16. Comparing with residual b-map at 3.0s using CyberShake
(with CVM-S4) as the target model and directivity-corrected BA08 as the reference
model as shown in Figure 5.5, we found that there are significant correlation between b(r)
and S
p
(r), and the peak times in basins are about 20s after the average S-wave arrival. We
could calculate S
p
(r) and P
t
(r) of the same component for CyberShake model with CVM-
H. As shown in Figure 5.17, the maps for averaged spectra and peak times are different
from the case where the velocity model is CMV-S4 because of the differences in velocity
models.
Figure 5.16 Maps of spectral value and peak time at period 3.0s calculated for the
component N35E. The velocity model is CVM-S4.
0.0 0.1 0.2 0.3
Power[m/s]
2
10 0 10 20 30 40 50
PeakTime[s]
153
Figure 5.17 Maps of spectral value and peak time at period 3.0s calculated for the
component N35E. The velocity model is CVM-H.
We normalize S
p
(r) to have zero mean over r, i.e. S
p
r ( )− S
p
r ( )
r
in order to obtain
relative spectral value at each site for each period, and show the normalized spectral
maps of two components at five periods (2.0, 3.0, 5.0, 7.0, and 10.0s) for CVM-S4 and
CVM-H in Figure 5.18 and Figure 5.19, respectively. For CVM-S4, the spectral value
increases with period, reaching its highest amplification approximately at 3.0s, and then
decreases from this peak to 10s, while for CVM-H, the spectral value increases with
period, reaching its highest amplification approximately at 5.0s, and then decreases from
this peak to 10s.
0.0 0.1 0.2 0.3
Power[m/s]
2
10 0 10 20 30 40 50
PeakTime[s]
154
Figure 5.18 Maps of spectral values at period 2.0, 3.0, 5.0, 7.0 and 10.0s (each column)
calculated for the component N35E and E35S (each row). The velocity model is CVM-
S4.
Figure 5.19 Maps of spectral values at period 2.0, 3.0, 5.0, 7.0 and 10.0s (each column)
calculated for the component N35E and E35S (each row). The velocity model is CVM-H.
Based on Rayleigh’s principle in which the total energy of the elastic system is
conserved, Paolucci (1999) developed a method which allows a quick calculation of the
resonance frequencies of valleys filled with stratified sediments. For a simple 3D
N35E
2.00 3.00 5.00 7.00 10.00
E35S
210 1 2
N35E
2.00 3.00 5.00 7.00 10.00
E35S
210 1 2
155
homogeneous alluvial valley whose shape is characterized by two parameters, the deepest
depth of the basin h and the equivalent valley half width a, the ratio of the fundamental
resonance frequency f
0
of such valley and the normalization frequency f
h
= v
s
/(4H)
increases as the ratio h/a becomes larger, where H is the depth of a homogeneous layer
and v
s
is the shear wave velocity.
We treat the basin depth Z
2.5
as the lower boundary of sedimentary basins, and
represent the horizontal distance of non-zero Z
2.5
and the largest Z
2.5
of a basin as l
h
and
l
v
, respectively. The ratio l
v
/l
h
is much larger in CVM-H than that in CVM-S4, such as in
Los Angeles basin and Ventura-Santa Clara-Simi Valley, resulting in smaller resonance
frequency in CVM-H than CVM-S4. For CVM-S4, l
v
is approximately 6.4km, and for
CVM-H, l
v
is approximately 5.0km. After simple calculations, the approximate resonance
frequency is around 0.3Hz for CVM-S4 and is about 0.2Hz for CVM-H, which is
consistent with the results from wavelet analysis.
The wavelet analysis presented here is basically to calculate the spectra of the
maximum response in time domain for each site. Such time-frequency spectra are the
result of source spectra, path scattering, and site response, but here we treat the maximum
response in frequency domain as the resonance due to basin structures. The maximum
value in the spectra indicates the periodic response in time domain, which is different
from the response spectra (e.g. PSA). This can be seen from the frequency-dependent of
basin amplification revealed in B(r) (Figure 5.5 and Figure 5.6) and basin response
(Figure 5.18 and Figure 5.19). The traditional seismic resonance analysis is based on the
156
locked-mode assumption, that is to say, no seismic energy leak out the simple system
(e.g. 1D layer and 2D homogeneous cosine-shape valley). In order to systematically
analyze the resonance property of 3D basins, the modal analysis considering the leaking
modes (transmission from low-velocity sediments to surrounding medium) is necessary.
5.5 Directivity-Basin Coupling
In this section, we use ABF to calculate the residual d map for target models
CyberShake with different velocity models and illustrate the effects of basin structures on
the directivity-basin coupling effects that has been discussed in Chapter 3. Such coupling
effect mainly appears within sedimentary basin where the incident large directivity pulse
is amplified, substantially increasing the ground motion intensities and further affect the
PSHA.
Using directivity-corrected BA08 as the reference model, Figure 5.20 and Figure 5.21
show the residual maps d(r,k,x) for SA at 3.0 seconds using CyberShake models with
CVM-S4 and CVM-H as the target models, respectively. The source is source 93 as in
Figure 4.15, and the slip model is GenSlip v3.2. Referring to the basin depth models
shown in Figure 5.7, we found that the shapes of basins control the directivity-basin
coupling effect. For those hypocenters located in the south portion of the rupture surface,
both models show larger effect (warm color) in places where the basin depth to a given S-
wave velocity (1.0 or 2.5 m/s) is deep. CVM-S4 shows larger directivity-basin coupling
157
than that of CVM-H, in particular in Los Angeles basin. The coupling points are
identified similarly in both models (between hypocenter 5 and 6).
Figure 5.20 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake (CVM-S4)
as the target model and directivity-corrected BA08 as the reference model. The source 93
with all hypocenters is shown. The number in each subplot indicates the order of
hypocenter as shown in Figure 4.15 (0:southern end; 11: northern end).
119˚ 118˚ 117˚
34˚
35˚
10 1
0 1 2 3
4 5 6 7
8 9 10 11
158
Figure 5.21 The residual map d(r,k,x) for SA at 3.0 seconds using CyberShake (CVM-H)
as the target model and directivity-corrected BA08 as the reference model. The source 93
with all hypocenters is shown. The number in each subplot indicates the order of
hypocenter as shown in Figure 4.15 (0:southern end; 11: northern end).
We also calculate the residual d-maps using SA at 3.0 s as the excitation functionals
for all other sources listed in Table 3.1, but only show in Figure 5.22 and Figure 5.23 the
standard deviations of d(r,k,x) over hypocenters of each source by using the averaging
operation defined in Equation (2.32). We found several sources that have the condition
(the relative locations of hypocenter and the basins) to initiate the directivity-basin
coupling show relatively larger variability in CVM-S4 than in CVM-H, indicating that
119˚ 118˚ 117˚
34˚
35˚
10 1
0 1 2 3
4 5 6 7
8 9 10 11
159
the waves at 3.0 s excited in CVM-S4 by the incident directivity pulse have larger
amplitudes than those waves excited in CVM-H.
Figure 5.22 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (CVM-S4) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. The CHD is uniform distributed.
119˚ 118˚ 117˚
34˚
35˚
0.0 0.2 0.4 0.6 0.8
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
160
Figure 5.23 The maps of standard deviations of d(r,k,x) for SA at 3.0 seconds using
CyberShake (CVM-H) as the target model and directivity-corrected BA08 as the
reference model for all sources used in the analysis. The CHD is uniform distributed.
Furthermore, we calculate the averaged of σ
d
(r,k) over all sites and sources,
resulting a number for SA at a given period. We take excitation functionals as SA at three
periods (3.0, 5.0, and 10.0 s). Table 5.2 shows the averaged of σ
d
(r,k) for excitation
functions of two CyberShake models with CVM-S4 and CVM-H and for corresponding
residual excitation functionals of the two models using directivity corrected BA08 as the
reference model. For shorter period (3s), CyberShake model with CVM-S4 (0.263) shows
119˚ 118˚ 117˚
34˚
35˚
0.0 0.2 0.4 0.6 0.8
8 10 15 64 85
86 87 88 89 93
112 218 219 231 232
254 255 267 271 273
σ
σ
161
relatively larger average standard deviations than model with CVM-H (0.256), while at
longer period (10s), CyberShake model with CVM-H (0.337) shows relatively larger
average standard deviations than model with CVM-S4 (0.325), which is consistent with
the observations about the basin effects in longer and shorter periods for those velocity
models. Directivity basin coupling is larger for 3.0s in CVM-S4 due to the small-scale
shallower basin depth with lower velocity, while DBC is larger for 10.0 in CVM-H due
to the deeper basin depth with lower velocity.
Table 5.2 Averaged standard deviation σ
d
(r,k) of D-maps for CyberShake models (CVM-
S4 and CVM-H) and directivity-corrected BA08 over r and k for SA at 2.0, 3.0, 5.0, and
10.0s.
Period (s)
Models
CVM-S4 CVM-S4 - BA08 CVM-H CVM-H - BA08
2.0 0.242 0.170 0.235 0.165
3.0 0.263 0.172 0.256 0.168
5.0 0.293 0.177 0.281 0.173
10.0 0.325 0.216 0.337 0.224
5.6 Conclusions
In this chapter, we apply ABF method to quantify the effects of different 3D velocity
models on the basin excitation and directivity-based coupling over a frequency band 0.1-
0.3 Hz.
Low-velocity sedimentary basins could trap the seismic wave energy and amplify the
ground motions. Three-dimensional velocity model used in simulations has epistemic
162
uncertainty due to ways of obtaining those models, such as full-waveform tomography.
The initial model, misfit measurements, ray path coverage, and inversion methods in the
tomography would affect the inversion results. Multiple velocity models (like logic tree)
are used to consider the epistemic uncertainty, including CVM-S4 and CVM-H. We
found at all periods (3.0, 5.0, and 10.0s), CyberShake basin effects are larger than those
considered in NGA GMPEs. Among the NGA models that consider the basin effects, that
of Abrahamson & Silva (2008) is the most consistent with the CyberShake CVM-H
calculations, with a basin-effect correlation factor greater than 0.9 across the frequency
band 0.1-0.3 Hz.
Among CyberShake models with two different velocity models, basin effect of CVM-
S4 tends to be larger at shorter period (e.g. 3s), but to be smaller at longer period (10s)
than that of CVM-H. This is because CVM-H has large-scale laterally low-velocity
sediments. Although CVM-S4 has deeper basin (e.g. Los Angeles basin) vertically, the
lateral variations of basin depth are larger, i.e. more small-scale variations, resulting
smaller basin effects at longer period than CVM-H.
By presenting the peak times where the SA values are picked, we confirm that the
wave groups that contribute the most to the basin amplification is the surface waves and
waves due to the seismic resonance at a natural frequency determined by the basin
structure. At different frequencies, the tuning of basin resonance waves would cause the
frequency-dependent basin effects, which can be seen from the residual b(r). Using the
closing system assumption (no wave energy leak out a basin), one can calculate the
163
resonance modes given a basin structure (Paolucci, 1999). However, in reality, there are
leaky modes that carry part of seismic wave energy out the basin. This motivates detailed
modal analysis of frequency-dependent basin excitation in 3D structures. Simple
experiments using wavelet analysis indicates that the CVM-S4 has resonance frequency
around 3.0s and the CVM-H has resonance frequency around 5.0s. More detailed modal
analysis of the basin resonance is necessary to fully understand the behavior of waves
excited in 3D sedimentary basins.
By calculating the d(r,k,x) using directivity-corrected BA08 as the reference model,
we found that the DBC first depends on the velocity structure (presence of basins) and
secondly the changes in DBC due to different velocity models have relatively small
effects on the overall directivity effects by showing the changes in total standard
deviations , approximately 0.004.
σ
164
Chapter 6
Summary
6.1 ABF Variance Analysis
The map-based comparisons presented in this dissertation demonstrate that ABF can
be a powerful tool for understanding the structural features of hazard models, especially
simulation-based models like CyberShake that incorporate the complexities of 3D wave
propagation. However, owing to the richness of the ABF hierarchy, there are simply too
many maps for us to explore individually, so a variance analysis can be usefully
employed to draw conclusions from a map ensemble (e.g., Figure 3.17). In this chapter,
we extend our variance analysis to take full advantage of ABF’s intrinsic structure, and
we show how it allows a compact quantification of the excitation differences among the
CyberShake studies and NGA08 models.
From Equations (2.26)-(2.30), it is easy to prove that the total expected variance of
the excitation functional (2.25) can be expressed as the sum of four terms, each
corresponding to a different averaging level:
σ
G
2
≡ [G(r,k,x,s)−A]
2
S,X,K,R
= σ
B
2
+ σ
C
2
(r)
R
+ σ
D
2
(r,k)
K,R
+ σ
E
2
(r,k,x)
X,K,R
≡ σ
B
2
+ σ
C
2
+ σ
D
2
+ σ
E
2
(6.1)
165
The dimensionless variances of the directivity effect and source-complexity effect, σ
D
2
and σ
E
2
, are defined in Equations (2.32) and (2.33), and σ
B
2
and σ
C
2
are the analogous
variances for the site and path effects:
σ
B
2
= B
2
(r)
R
(6.2)
σ
C
2
(r) = C
2
(r,k)
K
(6.3)
In all cases, the expectations involve summations over the sample sets weighted by their
corresponding probabilities.
According to (6.1), the different terms in the ABF representation (2.25) are
uncorrelated, which is a consequence of their zero-mean properties (Equation 2.31), and
the variances from terms at the same averaging level contribute to the total variance
through their expectations over the remaining independent variables. For example, the
contribution to the total expected variance σ
G
2
from the dispersion expressed in the σ
D
-
maps for the various sources k is the expectation of σ
D
2
over both k and r, which we
denote by σ
D
2
.
As previously noted (Equation 2.39), the E-level variances for the NGA08 models are
expressed as the sum of the magnitude-effect excitation variance σ
M
2
(k) , specified by the
ERF, and the aleatory variance σ
T
2
obtained from the regression residuals (in this case,
with respect to the mean excitation values that include the SC08 directivity corrections).
For the CyberShake model, the magnitude-effect variance also depends only on the
166
UCERF2 source index k, and it is augmented with the variance in excitation arising from
the source complexity at constant magnitude, denoted σ
F
2
, which depends on all source
indices:
σ
E
2
(k,x,s) = σ
M
2
(k) + σ
F
2
(k,x,s) (6.4)
Because these two variables are independently distributed, σ
M
2
+ σ
F
2
can be substituted
for σ
E
2
in the ABF variance representation (6.1), where the overbar again indicates the
variance expectation over all independent variables.
How the “σ
G
2
-budget” of the NGA08 models differs from that of CS13.2 is plotted in
Figure 6.1, from which we note that for all those periods, the variances σ
B
2
and σ
D
2
from
CS13.2 are larger than those from NGA08 models, indicating larger basin and directivity
effects in CS13.2 as presented previously. The total expected variance from CS13.2 and
NGA08 models increases with period. The variance σ
M
2
for both models has no
dependence on periods, but σ
M
for CyberShake is about three times the NGA values for
all those periods. This is consistent with CyberShake ground motions scaling with
moment and NGA ground motions scaling with moment^1/3 or one third of magnitude
(as discussed in Graves et al., 2011b). The variance σ
D
2
for CS13.2 increases with
periods, and the same changes are found in σ
B
2
. This indicates that the directivity-basin
coupling is larger at SA-10s because the basin-guided waves are well excited at longer
period. The attenuation rate with distance for CS13.2 becomes smaller as period
increases, resulting in reduction in σ
C
2
, while the attenuation rate with distance for NGA
167
models is reversed with respect to CS13.2 (consistent with C-maps). This observation is
consistent with the differences in SA at 10s between simulations and GMPEs calculated
by Star et al. (2011) for a Southern California earthquake scenario. It might be due to the
effect of 3D wave propagation.
Figure 6.1. Budget of the residual variance (σ
G
2
), as given by Equation (6.1), for the
NGA mean and CS13.2 (the left and the right one at each period, respectively). Each of
the ABF terms is represented by a different color. For NGA mean, σ
F
2
is the total aleatory
variance; For CS13.2, it is the excitation variance from the source-complexity effect at
constant magnitude.
We could also show the total variance structure of CS11 that uses GenSlip v2.1 (more
coherent source) and CVM-S4, but in order to compare the changes in ground-motion
variability due to different slip and velocity models, we show the total variance of a
168
residual field g. Such variance is obtained directly from Equation (6.1) by replacing the
upper-case dependent variables with their lower-case equivalents, as in Equation (2.42).
This defines a “σ
g
2
-budget” for the various models. The “σ
g
2
-budget” of CS11-
NGAmean and CS13.2-NGAmean for SA at 3, 5, and 10 s is plotted in Figure 6.2. Each
of the ABF residual terms shown by a different color quantifies the variance of a residual
effect. From the comparison of σ
g
2
-budgets of CS11 and CS13.2 relative to NGA mean,
we found the residual directivity effects between simulation-based model and empirical
GMPEs are reduced using more complex slip model GenSlip v3.2, and the residual basin
effects are reduced using CVM-Harvard. However, at 10s, σ
b
2
of CS13.2-NGAmean is
larger than that of CS11-NGAmean. This is because at this period, the basin effects in
BA08 and CY08 are not well correlated with those in CS13.2 as shown in Figure 5.6.
Treating CyberShake at the representation of a virtual world, the comparison between
the residual variance and the total aleatory variance that is obtained from the regression
analysis of observed ground-motion intensities allow us to quantify the changes in ground
motion variability using simulation-based models in PSHA. Seen from Figure 6.2, using
more complex source model and CVM-H, simulation-based model reduces the ground
motion variability.
169
Figure 6.2. Budget of the residual variance (σ
g
2
), as given by Equation (6.1), for residual
CS11-NGAmean and CS13.2-NGAmean (the left and the right one at each period,
respectively). Each of the ABF terms is represented by a different color. Black dashed
line shows the total regression variance of the directivity-corrected NGA meanσ
T
2
.
6.2 Achievements
In this dissertation, motivated by substantial differences in probabilistic seismic
hazard estimations using different ground motion prediction models, I integrate
information related with earthquake rupture forecast and ground motion predictions with
a new perspective to PSHA, and generalized the description for PSHA to incorporate
physics-based simulations in place of ground motion prediction equations. Introducing
the kinematically complete ERF model, I provide a general framework for describing
170
different ground motion prediction models, in particular, the NGA GMPEs and physics-
based earthquake simulations, CyberShake.
In order to compare the two types of seismic hazard models, I developed a
decomposition method following the generalized framework of PSHA. The averaging-
based factorization (ABF) does not depend on any model-based factorization used in
GMPEs, but instead relies on a hierarchical sequence of averaging operations. Validating
ABF method by NGA intercomparisons, we found that, although the ABF is a simple
mathematical operation, decomposed factors relates with physical effects considered in
ground motion predictions, where A factor shows the average excitation level, B factor
indicates the site effects (V
s30
and basin effects), C factors reveal the path effects, and D
factors represent the source directivity effects.
We then apply ABF methodology to compare CyberShake and NGA GMPEs for SA
at low frequencies (0.1-0.5 hz). According to the residual factor b(r), basin effects in
CyberShake are larger than those in the NGA models, and basin excitation is not a simple
function of basin depth, especially at places where the basin edge effects are significant.
Moreover, the excitation of basin-guide waves in 3D velocity structure leads to stronger
frequency-dependence of basin excitation considered in the simulation-based model than
empirical relations. The residual factor c(r,k) indicates that the distance attenuations in
CyberShake are faster than those considered in NGA GMPEs, and sites located down-
strike or up-dip direction of a source (fault) would on average experience strong ground
motion shaking due to the constructive interference of seismic energy radiated from
171
hypocenters. Using SC08 model to consider the directivity effects for NGA GMPEs, we
found that CyberShake have produced relatively larger directivity effects than NGA
GMPEs for SA at 3.0s, in particular, the directivity-basin coupling effect could
significantly enhance the ground motions in sedimentary basins. Such effect is not
considered in empirical modeling of ground motions using GMPEs. The effects on
ground motion due to variability of average slips shift the overall level for all sites and
the differences of such shift between CyberShake and NGA GMPEs have standard
deviation smaller than 0.3.
CyberShake simulations consider the epistemic uncertainties in velocity models and
rupture descriptions, resulting different CyberShake studies and hazard calculations. We
apply ABF method to compare different studies and quantify the effects of velocity
models and kinematic descriptions of fault rupture processes on ground motion
predictions. Increasing small-scale variations in rupture process, including rupture
velocity, slip spatial distribution, rupture initiation time for each subfault, could
substantially decrease the source directivity effects and further change the seismic hazard
assessment. Differences in 3D velocity models used in CyberShake can affect the basin
effects and directivity-basin coupling that are due to 3D wave propagations excluded in
GMPEs. Using both velocity models, CyberShake shows larger basin effects than NGA
models at frequency band 0.1-0.3hz, and CyberShake basin effects using CVM-Harvard
show a stronger correlation with AS08 model, indicating that this model may provide a
good estimation of the details in sedimentary basins, in particular, the Los Angeles basin.
172
Besides the epistemic uncertainty, the ABF method allows us to incorporate CHD and
CSD that reveal the aleatory variability in rupture process. The forecasting of hypocenter
locations used in the kinematically complete ERF is very important in the PSHA,
regardless of NGA GMPEs and CyberShake, and we quantitatively show the effects of
different conditional hypocenter distributions on the ground motion prediction and
seismic hazard calculations. We extended the CyberShake models to include multiple
CHDs parameterized as Beta distributions, which can be used to represent rupture bias
associated with elasticity contrasts across crustal fault zones, such as those described by
Dor et al. (2006). At present, the degree of CHD asymmetry is difficult to estimate,
because it depends on the dynamical modeling of ruptures in fault zones described by
asymmetric elastic and inelastic properties, which are poorly constrained. However, the
ABF analysis of the CyberShake models using a sample set of CHDs allows us to
quantify the changes in rupture directivity and directivity-forecasting probability gains
that could be achieved by plausible improvements to CHD models of fault zones in
Southern California. We show that these gains are sufficiently large to motivate more
systematic approaches to CHD quantification.
6.3 Prospective
On one hand, simulation-based seismic hazard model incorporates sophisticated 3D
velocity models and pseudo-dynamic rupture description. The results from ABF of
comparing simulation-based models with current available attenuation relations (e.g.
173
NGA-West) could serve as a guide to future attenuation-relationship developments (e.g
NGA-West 2) and provide an interim model for use in Southern California. Treating
CyberShake as the representative of the virtual world for this region, we could modify the
NGA GMPEs based on those residual factors, and reduce the epistemic uncertainties in
the empirical models due to the lack of observations for constraining some of physical
effects that are considered by simulation-based models.
On the other hand, ground motion predictions and hazard calculations using
attenuation relations are easy and quick. With the increasing data and understanding of
earthquake rupture, propagation, and site amplifications, attenuation relations become
more and more complex and reduces both the aleatory and epistemic uncertainties
appearing in PSHA, while simulation-based model also uses updated velocity models and
rupture descriptions. We could apply the ABF method to validate the simulation-based
models against new attenuation relations. For example, basin effects in CyberShake with
CVM-Harvard correlate closer with those in AS08 model than CyberShake with CVM-
S4, suggesting that the tomography based on CVM-S4 should aim to converge with
CVM-Harvard. In addition, ABF method allow use to testify the physical effects of
different CHD and CSD on directivity effects and to provide the recommendation of
those distributions for directivity forecasting in further PSHA.
At last, my PhD research builds a bridge between seismological science, deterministic
earthquake simulations, and earthquake engineering. The new perspective to PSHA
would help one side to understand the other side. As more broadband earthquake
174
simulations are performed and validated with historical earthquakes, the engineers’
demands of using deterministic earthquake simulations in developing attenuation
relations and seismic hazard calculations would be satisfied, and the method developed
here could be used to quantitatively validate between each of those models.
175
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Abstract (if available)
Abstract
Seismic hazard models based on empirical ground motion prediction equations (GMPEs) employ a model-based factorization to account for source, propagation, and path effects. An alternative is to physically simulate these effects using earthquake source models combined with three-dimensional (3D) models of Earth structure. We generalized the implementation of those hazard models in probabilistic seismic hazard analysis from the seismological perspectives, and developed an averaging-based factorization (ABF) scheme to facilitate the geographically explicit comparison of these two types of seismic hazard models. Through a sequence of averaging and normalization operations over various model components, such as slip distribution, magnitudes, hypocenter locations, we uniquely factorize model residuals into several factors. These residual factors characterize differences in basin effects, distance attenuation, and effects of source directivity and slip variability. We illustrate the ABF scheme by comparing CyberShake model for the Los Angeles region with the Next Generation Attenuation (NGA) GMPEs. Relative to CyberShake, all NGA models underestimate the basin effects. Using the GMEPs with directivity corrections, we quantify the extent to which the empirical directivity model capture the source directivity effects demonstrated by physics-based ground motion prediction model. In particular, empirical directivity corrections for NGA models underestimate source directivity effects in CyberShake, and do not account for the coupling between source directivity and basin excitation that substantially enhance the low-frequency seismic hazards in the sedimentary basins of the Los Angeles region. We then investigate seismologically to what extent the complex rupture processes and conditional hypocenter distributions affect the ground motion predictions and seismic hazard assessment. At last, considering two 3D velocity models for Southern California used in simulations, we use different CyberShake studies to physically understand the basin excitations and directivity-basin coupling effects. To our knowledge, this is the first systematical and quantitative integration and validation of deterministic earthquake simulations in probabilistic seismic hazard analysis.
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Wang, Feng
(author)
Core Title
Integration and validation of deterministic earthquake simulations in probabilistic seismic hazard analysis
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Geological Sciences
Publication Date
09/24/2013
Defense Date
08/27/2013
Publisher
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earthquake rupture forecast,ground motion prediction equations,numerical simulations,OAI-PMH Harvest,probabilistic seismic hazard analysis,seismology
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Jordan, Thomas H. (
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), Abrahamson, Norm (
committee member
), Ben-Zion, Yehuda (
committee member
), Miller, Meghan S. (
committee member
), Wilson, John P. (
committee member
)
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fengo.win@gmail.com,srwin_d@hotmail.com
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Tags
earthquake rupture forecast
ground motion prediction equations
numerical simulations
probabilistic seismic hazard analysis
seismology