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Structural clustering analysis of CVMS-4.26: a 3D seismic velocity model for southern California
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Structural clustering analysis of CVMS-4.26: a 3D seismic velocity model for southern California
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Content
STRUCTURAL CLUSTERING ANALYSIS OF CVMS-4.26:
A 3D SEISMIC VELOCITY MODEL FOR SOUTHERN CALIFORNIA
By
William Karl Eymold
_____________________________________________________________
A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(GEOLOGICAL SCIENCES)
August 2017
Thesis Committee:
Thomas Jordan, Advisor
Charles Sammis
Yehuda Ben-Zion
ii
Table of Contents
Acknowledgements .......................................................................... iv
List of Tables ..................................................................................... v
List of Figures ................................................................................... vi
Abstract .......................................................................................... viii
Chapter 1: Introduction .................................................................... 1
Chapter 2: Model Description .......................................................... 4
2.1: Brief Geologic History ................................................... 4
2.2: Regional Coverage ........................................................ 4
2.3: Cross Sections and Profiles .......................................... 6
2.4: Mantle Depth and Low Velocity Zone .......................... 11
2.5: Validation and Verification ........................................... 13
Chapter 3: Methods ....................................................................... 15
3.1: K-means Clustering .................................................... 15
3.1.1: Example Regionalization Output .......................... 17
3.2: Fuzzy C-Means Clustering ......................................... 19
3.3: Feature Mapping ........................................................ 20
3.4: Correlation Matrices .................................................... 21
3.5: Intracluster Comparisons ............................................. 22
Chapter 4: Results ......................................................................... 23
4.1: Stability of the Analyses .............................................. 23
4.2: K-means++ ................................................................. 26
iii
4.2.1: Target K Value of 7 ............................................... 26
4.2.2: Lower K Values .................................................... 30
4.2.3: Higher K Values .................................................... 36
4.3: Fuzzy C-Means ........................................................... 41
4.3.1: Results from K = 7 ................................................ 42
4.4: Feature Mapping ......................................................... 51
4.4.1: Depth to Mantle .................................................... 51
4.4.2: Low Velocity Zone ................................................ 51
4.4.3: Average Velocity Plots .......................................... 51
4.5: Intracluster Distance .................................................... 54
4.6: Radial Correlation Functions ........................................ 55
Chapter 5: Discussion ................................................................... 60
5.1: K-means Clustering ..................................................... 60
5.2: Fuzzy C-Means Clustering .......................................... 63
5.3: Feature Mapping ......................................................... 64
5.4: Geologic Interpretation of the K = 7 Regionalization ..... 66
Chapter 6: Conclusions and Future Work ....................................... 69
References ..................................................................................... 70
Appendix: Tables and Figures ........................................................ 73
iv
Acknowledgements
I would like to thank everyone who played any part in the past three years of my
academic and personal lives for their support and understanding during the process. My
parents have been great and over the top in their support sometimes but I recognize that
they’re always proud of me and I do appreciate it. It was great living near my best friend
Shane Barkley in San Diego for a while and I’m grateful for the discussions on topics
ranging from the environment to geophysics to baseball. Mostly baseball.
At USC, Marshall Rogers-Martinez has always served to rein me in at times and help
me with a number of complexities regarding computer science and seismology. Again,
extremely frightening that he reined me in. Katie Ardill similarly helped me with geologic
interpretation as well as by accompanying me on multiple coffee runs each week. Jessica
Donovan (now Velasquez) provided ample advice and support even after she completed
her PhD and always at exactly the right times. Cindy Waite has constantly given me
assistance during my infinite pestering sessions and I’m thankful for her putting up with me.
I also want to express gratitude to the ~100 students I got to teach as a TA, especially
Christian Edwards and Iman Khan who have remained in touch years after their GEs were
complete. I truly enjoyed teaching and look forward to continuing.
Special thanks to my committee members Yehuda Ben-Zion and Charlie Sammis for
giving feedback on this document, as well as foolishly permitting me to drive a department
vehicle last fall and to teach Earthquakes this spring, respectively. David Okaya has been
extremely crucial in my development as a computer scientist and I’ve enjoyed all of the past
(and future) projects we’ve worked on together. Of course my advisor Tom Jordan is on a
completely different level of mathematical and seismological understanding and has been a
pinnacle of inspiration. I’m very thankful I got to spend the past few years studying under
him.
Finally (since I waited so long to finish this manuscript) I want to thank my girlfriend
fiancée, Rachel Koons, for her ability to put up with me and offer advice during my perpetual
state of education. Though the last nine months were spent on opposite sides of the
country, I’m glad we’re now back to being within a daily commute from one another and hey
who knows: maybe someday we’ll even be in the same town…
v
List of Tables
Table 1 Average V
s
Belongingness .................................................. 73
Table 2 Average V
p
Belongingness ................................................. 74
Table 3 Average Belongingness for µ = 1.05 – 2.0 ......................... 75
Table 4 Depth to Mantle Velocities ................................................ 76
Table 5 LVZ Behavior for K-Means Outputs .................................. 77
Table 6 Average Upper Crustal V
s
................................................. 78
Table 7 Average Mid-Crustal V
s
.................................................... 79
Table 8 Average Deep-Crustal V
s
.................................................. 80
Table 9 Intracluster Distances ....................................................... 81
vi
List of Figures
Figure 1 S4.26 Surface Velocities .................................................... 5
Figure 2 S4.26 Velocities at 8 km Depth ........................................... 5
Figure 3 V
s
Cross Sections ............................................................... 7
Figure 4 Map Plot of Transects ........................................................ 8
Figure 5 V
s
Profiles for Upper Transects ........................................... 9
Figure 6 V
s
Profiles for Middle Transects ......................................... 10
Figure 7 V
s
Profiles for Lower Transects ......................................... 10
Figure 8 Depth to Mantle Velocities ................................................ 11
Figure 9 Map of Low Velocity Zone ................................................. 13
Figure 10 V
s
K-Means Output for K = 7 ........................................... 17
Figure 11 Distance Reduction Curve .............................................. 23
Figure 12 100 Replicates for V
s
Regionalization ............................. 24
Figure 13 K-Means Output for K = 7 ............................................... 28
Figure 14 Biased K-Means Regionalization .................................... 29
Figure 15 Biased K-Means Centroid Velocity Profiles .................... 29
Figure 16 K-Means Output for K= 3 ................................................ 31
Figure 17 K-Means Output for K= 4 ................................................ 32
Figure 18 K-Means Output for K = 5 ............................................... 34
Figure 19 K-Means Output for K = 6 ............................................... 35
Figure 20 K-Means Output for K = 8 ............................................... 37
Figure 21 K-Means Output for K = 9 ............................................... 38
vii
Figure 22 K-Means Output for K = 10 ............................................. 39
Figure 23 FCM Output for µ = 1.05 ................................................. 44
Figure 24 FCM Output for µ = 1.2 ................................................... 45
Figure 25 FCM Output for µ = 1.5 ................................................... 46
Figure 26 FCM Output for µ = 1.8 ................................................... 47
Figure 27 FCM Output for µ = 2.0 ................................................... 48
Figure 28 Change in Belongingness for 7 Regions .......................... 49
Figure 29 Map of Average Upper Crustal V
s
................................... 52
Figure 30 Map of Average Mid-Crustal V
s
....................................... 52
Figure 31 Map of Average Deep-Crustal V
s
.................................... 53
Figure 32 RCF for Continental Borderlands V
s
............................... 56
Figure 33 RCF for Sierra Nevada Batholith V
s
................................ 56
Figure 34 RCF for Los Angeles Basin V
s
........................................ 57
Figure 35 RCF for Mojave Block V
s
................................................ 57
Figure 36 RCF for Ophiolites V
s
..................................................... 58
Figure 37 RCF for Great Valley V
s
.................................................. 59
Figure 38 RCF for Proterozoic Crust V
s
.......................................... 59
Figure A1 Recovered Checkerboards (Lee et al., 2014a) ............... 82
Figure A2 A Tectonic Map of the U.S. Cordillera (Snoke, 2005) .... 83
viii
Abstract
Over the past 25 years, researchers have produced 3D velocity models which are used in
wave propagation codes to generate synthetic seismograms. While these models are purely
numerical, the behavior of the velocity values will correspond well with known geologic features
if the inversion is performed accurately. We apply k-means clustering analysis to separate the
component V
s
and V
p
velocity profiles in CVMS-4.26, a seismic velocity model for southern
California, into regions without using any a priori identification as to the locations of the
profiles. By grouping the velocity profiles into clusters, we also calculate the centroid velocity
profile for the region and can characterize the distinguishing features which separate the regions
from the model. We also expand on the general k-means clustering by using Fuzzy C-Means
clustering to determine a weighted belongingness of each profile to every cluster in a quasi-
probabilistic manner. Finally, we map several different features in the depths of the velocity
model, including the extent of a pervasive low velocity zone, the general location of the Moho,
and averages for the upper, middle, and lower crustal velocities. These geographic maps are then
compared to the clustering regionalizations to analyze the effects of these features on the
separation of velocity profiles. The results of the k-means clustering demonstrate that the
geologic features embedded in the velocity model can be used in the mathematical routine to
apportion different regions in the state. The region divisions are strongly correlated with the
behavior of the velocity profiles at the surface, the mid-crust, and the Moho depths which present
the greatest amount of heterogeneity in the model. The Fuzzy C-Means analysis expands on
these results by showing the strongest cohesion among regions which do not contain a low
velocity zone in the mid-crust. An in depth statistical analysis is also performed to define the
output regions from a probabilistic standpoint. Finally we show that the algorithm can be applied
successfully to a high resolution crustal velocity model.
1
Chapter 1
Introduction
Over the last 25 years, researchers have developed more realistic 3D seismic velocity models
which have been used in 3D wave propagation codes based on finite difference or finite element
schemes (e.g., Olsen et al., 1995; Graves, 1996; Mozco et al., 2002; Komatitsch et al., 2010;
etc.). At the Southern California Earthquake Center (SCEC), we currently provide two such
community velocity models (CVM) for southern California, namely CVM-S 4.26 and CVM-H
15.1.0 which were released in 2014 and 2015, respectively (Lee et al., 2014a; Shaw et al., 2015).
CVM-S 4.26 (hereon referred to as S4.26) was produced using a technique known as full 3D
tomography (F3DT), in which both the starting velocity model and the Fre ́ chet kernels were
calculated by the full physics of 3D wave propagation (Zhao et al., 2005; Chen et al., 2007b).
The entire procedure required numerical comparisons between observed waveforms of 160 local
earthquake seismograms and their corresponding synthetic seismograms calculated using the
current iteration of the model. After these differences were measured, the model was updated to
accommodate the corresponding changes and the procedure was repeated, as its name implies,
for 26 iterations.
S4.26 contains many interesting features which have been validated by testing the output
waveforms against subsequent earthquake events as well as by comparing with results from
independent geologic and geophysical inquiries (Lee et al., 2014b; Saleeby, 2003; Porter et al.,
2011; etc.). Three of the most dominant and detectable features in the model are the presence of
low velocity sedimentary basin rocks at the surface, the variable depth of the Moho
discontinuity, and the low velocity zone (LVZ) that appears at mid-crustal depth through many
of the velocity profiles. We have designed a series of techniques which can independently
analyze and dissect the 3D model by means of a purely numerical method and showcase the
resulting regions in this paper.
Since its initial development half a century ago, the k-means clustering algorithm has been
applied to a number of very diverse topics as a method to analyze and group large data sets
(MacQueen, 1967). The algorithm operates on the full data set by randomly selecting K starting
centroids, comparing all other data elements to these centroids by calculating a measure of
2
distance, and subsequently assigning each remaining element to a cluster. Following similar
work on global tomography models (Lekic & Romanowicz, 2011 a., b.), we use the built-in
Matlab k-means function to cluster the velocity profiles in S4.26 based on the L
2
norm, also
known as the Euclidean distance. The results from Lekic and Romanowicz (2011a) showed
excellent correlation with tectonic provinces and separated the model into groups representing
such diverse settings as subduction zones, orogenic belts, mid-ocean ridges, and cratons. Their
second study (2011b) parsed the mantle structure into fast and slow regions with potential
mineralogical interpretations. While this clustering approach has been used to separate these
global velocity models which are much sparser in lateral coverage and extend to depths of 350
km or even 2800 km, this technique has never been attempted on a high resolution, 3D crustal
velocity model.
The global data sets show good variation between very distinct geologic terranes, such as
oceanic and continental crust, but the high resolution crustal models used to calculate synthetic
seismograms for local earthquakes pose a much different challenge. Given the tectonic history of
the western margin of North America, it is typical for drastically different geologic formations to
lie juxtaposed in southern California. Additionally, motion along the San Andreas fault (SAF)
over the past 30 Ma has translated contrasting pieces of crust up to 315 km, as constrained by
restoration of the Pinnacles and Neenach volcanics (McQuarrie & Wernicke, 2005). These
juxtapositions allow for contrasts in seismic velocity across a fault zone which will be narrower
than the model’s spacing, and nodes at the surface of S4.26 can be >4000 m/s from their
neighbors (maximum V
s
difference = 2461 m/s; maximum V
p
difference = 4042 m/s).
The maximum depth for S4.26 is 49.5 km and this shallower structure includes fewer areas of
variation, essentially containing only the shallow velocities, potentially a Conrad discontinuity,
and the Moho depth/gradient whereas the global velocity models will have a Moho, an
asthenosphere LVZ, the 410 and 660 km discontinuities, and the core mantle boundary. By
making field measurements, the surface geology can be used as a way to infer behavior at depth
but is significantly limited through observation. In order to make better inferences, the k-means
algorithm will compare the structural behavior at depth across the model (lateral comparison) as
well as coordinating the surface behavior to the extent of the velocity profiles (vertical
comparison). Restriction of the degrees of freedom in the crustal velocity profiles results in
3
allowing these few key characteristics of the different sections of the CVM to dominate the
clustering more than the global models
The input data are the 1D velocity profiles representing 50 km of structure at every node in
the model and after the clustering is complete, each node is assigned to a specific group based on
the minimum L
2
distance from its centroid. In the same way as breaking the global model into
tectonic plates, the results of the clustering regime on the CVM conform well to known
geological terranes as well as surface faults and lithology transitions. As a way to verify the
process, we analyze results across a swath of K values and use the profiles for V
s
, V
p
, and a
combination of these profiles that we refer to as V
s
+V
p
.
In order to expand on this clustering technique, we perform extensive statistical analysis on
each regionalization output. The tightness of each region can be assessed by determining the
mean velocity profile (also known as the centroid velocity profile for the k
th
region) and standard
deviation at all depths and examining where the greatest variations exist for the regions. Radial
correlation functions are also calculated for each of these regions as a way to determine the level
of “communication” between one depth section of the profile to another section throughout the
model (Jordan, et al., 1993). Finally, we apply a variation of the k-means algorithm known as
Fuzzy C-Means Clustering to the same data sets to calculate the strength of correlation of each
output region. This algorithm follows the same L
2
distance calculation for each profile to every k
cluster but rather than assigning the individual profile to a single cluster, the amount of
“belongingness” is measured and represents a quasi-probabilistic association for each profile to
every cluster (Velmurugan, 2014). The final suite of analyses represents a novel technique to
validate and verify future CVMs by associating the different groups with observable geology as
well as to quantify the differences between past or complementary models without the effects of
a priori bias.
4
Chapter 2
Model Description
2.1 Brief Geologic History
The model consists of 992 x 1536 nodes spaced 500 m apart, each containing 100 values for
V
s
, V
p
, and ρ to represent the velocity profile from surface to 49.5 km depth. Due to the
extensive geological history of the western margin of North America, lithology changes very
rapidly across fault lines and the 500 m spacing can result in more gradual changes both laterally
and vertically throughout the CVM when compared to the wider spanning global velocity
models. The past 210 Ma of tectonics have produced an enormous amount of deformation which
has juxtaposed crustal pieces of drastically different composition and age through a series of
subduction, extension, and translation (Godfrey, 1998; Saleeby, 2003; Ducea, 2009; Jacobson,
2011). The geographic range for the study area encompasses all of California from
approximately San Simeon down to the southeast and includes portions of Arizona, Nevada, and
Mexico. The metropolises of Los Angeles and San Diego are located within the model as well as
much of the Great Valley, the Sierra Nevada Mountains, the Mojave Desert, the Coast Ranges,
the Peninsular Ranges, and the Salton Trough. Several major fault traces run throughout the
model space, including but not limited to the San Andreas, Garlock, Banning, Elsinore, San
Jacinto, and San Gabriel faults. While these fault structures are unknown to the model and the
inversion technique, their effects are clearly observed in the small scale velocity contrasts of the
resulting final model (Lee et al., 2014a).
2.2 Regional Coverage
Starting from the surface (Figure 1), several sedimentary basins are contained in the model,
such as those in the Great Valley, Los Angeles Basin (LAB), and the Coachella/Imperial
Valleys. The loosely packed, unconsolidated rock will have low V
s
and V
p
values (warm colors)
and account for a large amount of shaking that is observed during a southern California
earthquake. These structures can extend as far down as 8 km deep (e.g., Great Valley in Figure
2) and contain the lowest V
s
seismic velocities (below 1000 m/s) (Lee & Chen, 2016). By
plotting the slices using the same scale, most of the heterogeneity can be seen near surface and
5
Figure 1. Surface V
p
(left) and V
s
(right) values for S4.26. Low velocities (sedimentary basins)
are plotted in warm colors and high velocities (mountain ranges) in cool colors. Although faults
are unknown to the CVM, the velocities conform well to real world fault traces.
Figure 2. V
p
(left) and V
s
(right) at a depth of 8 km, using the same scale as Figure 1. Although
some parts of the model include lower velocities at this depth (e.g., Great Valley), nearly the
entire model has V
s
> 3000 m/s and exhibits less heterogeneity than the surface values.
6
the majority of the model has V
s
above 3000 m/s by 8 km depth. Note that since the Lamé
parameters were solved for independently, the velocity structures for V
s
and V
p
are similar but
do not simply maintain a constant ratio of √3.
Several mountain belts are located within the study area, including the southern portion of
the Sierra Nevada batholith (SNB) and Peninsular Range batholith (PRB), the Coast Ranges, and
the Transverse Ranges. Each of these formations corresponds to unique tectonic plate
movements over the past 120 Ma and can feature similar but independent seismic velocity
characteristics. The Mojave Block located in the center of the study area is confined by the
Garlock fault to the north and the SAF to the south and extends east across the East California
Shear Zone (ECSZ).
2.3 Cross Sections and Profiles
In order to demonstrate the extent of the model structure at depth as well as its lateral
variation, three cross sections of the model are plotted in Figure 3. These correspond to the Great
Valley and SNB (top), the Mojave Block and ECSZ (center), and the Continental Borderlands,
LAB, and Proterozoic Crust (bottom), respectively. A map view of the lines for these cross
sections as well as the locations used to calculate each average velocity profile are shown in
Figure 4. Furthermore, we show the average seismic velocity profiles from these regions in
Figures 5-7 to highlight the behavior and provide examples of our definition of a LVZ and the
depth to mantle velocities.
The top row of Figure 3 shows a cross section of the upper part of the model featuring the
Coast Ranges, Great Valley, and SNB. Deep structures contain lateral heterogeneity, such as on
either side of the SAF (left side) and the transition from the Great Valley into the SNB (right
side). In the Great Valley, low velocities extend from the surface down to 10 km depth, but
immediately east of this structure are high surface velocities overlying a volume of near-mantle
velocity at only 15 km depth. This represents the Great Valley ophiolites which extend both
north and south of this section. Finally in the right-most portion of the cross section, the SNB
contains the very deep Moho (~35 km) and features a LVZ in the typical 8 to 14 km depth range.
Parts of this region also feature stagnation in the velocity gradient from 15 to 30 km as seen in
the corresponding 1D velocity profile in Figure 5.
7
Figure 3. V
s
cross sections through the Great Valley, Mojave Desert, and Salton Sea. Vertical
dashed lines represent surface fault locations, with the boldest line representing the SAF. Map
view plots are also shown as dashed lines in Figure 4.
A cross section focusing on the Mojave is shown in the middle row of Figure 3 which
demonstrates the extent of the LVZ structure as well as the velocity values and reversals. Note
that on this cross section, the velocity reversal from 8 to 14 km begins on the east side of the San
Gabriel fault, crosses over the SAF into the western Mojave, and extends until approximately
halfway through the ECSZ. While portions of the offshore region (left) and across the Garlock
Fault (right) exhibit velocity reversals, the structures underlying these sections are not nearly as
planar nor coherent as the structure below the western Mojave.
The bottom cross section in Figure 3 runs through the PRB, Salton Sea, and into the area near
the Arizona border. The deep root of the PRB is shown by the abrupt drop in Moho depth from
the offshore structure (left). In the center of the cross section, the Salton Sea has low surface
velocities, thin crust, and a very shallow Moho, as evidenced by the velocities above 4500 m/s
above 20 km in depth. Across the SAF, the planar LVZ reappears for most of the remainder of
the cross section and the Moho stays at just above 30 km depth. All of these structures can be
compared to the maps of the depth to mantle (Figure 8) and LVZ (Figure 9)
8
Figure 4. Map plot for upper, middle, and lower cross sections (dashed black lines) as well as
the central locations for the calculation of average velocity profiles (numbers 1-14). Note that
profile 10 is actually in the center of the ECSZ and profile 12 is actually in the center of the
Salton Sea but their location labels are slightly away from these areas for visual clarity.
Velocity profiles were calculated by averaging 2501 profiles for 14 model areas near the
cross section lines and are plotted in Figures 5-7. The horizontal dashed lines represent divisions
of the crust into the upper crust, mid-crust, and deep crust that are examined throughout this
study. Additionally, the depths to mantle velocities (defined in the next section) are shown as
black asterisks along each profile. In Figure 5, the profiles for regions along the upper cross
section correspond to the Coast Ranges (1, cyan), Tehachapi Anomaly (2, red), Great Valley (3,
orange), Ophiolites (4, magenta), SNB (5, green). At the surface, the lowest velocities (~2000
m/s) are from the Coast Ranges, Great Valley, and Tehachapi Anomaly. In contrast, the higher
velocities for the SNB (2966 m/s) and Ophiolites (3406 m/s) are related to the lithology
differences in these regions. The LVZ in the mid-crust is found in the Coast Ranges, SNB, and
Tehachapi Anomaly and the depth to mantle velocity ranges from 16.5 km (Ophiolites) to 36 km
(SNB). The value for the Ophiolite profile is an example of the mantle velocities at very shallow
depths which is detected by our Moho algorithm, seen in Figure 8 and described in the next
section.
9
Figure 5. V
s
profiles for the upper cross section of Figure 4, profiles 1 – 5.
The middle cross section profiles are for the offshore Continental Borderlands (6, cyan),
LAB (7, red), Western Mojave (8, orange), Basin and Range (9, magenta), and ECSZ (10, green)
and shown in Figure 6. Very low surface velocities are found in the LAB (890 m/s) while the
other four regions have an average of 2502 m/s. Only the LAB, Western Mojave, and Basin and
Range contain a mid-crustal LVZ, and the depth to mantle velocities is shallow for the
Continental Borderlands (19.5 km) and deepest in the ECSZ (31.5 km).
Finally the lower cross section profiles in Figure 7 represent the PRB (11, orange), Salton
Sea (12, red), Eastern Mojave (13, green), and Proterozoic Crust (14, cyan). The Salton Sea has
the lowest surface velocities (1018 m/s) and the highest surface velocities are in the PRB (3250
m/s). These two profiles are also the only ones which include a LVZ and the depths to mantle
vary between 26.5 km (Proterozoic Crust) and 40 km (PRB). All of these features between cross
sections and velocity profiles can be compared to the feature maps for the depth to mantle
(Figure 8) and the LVZ (Figure 9) for the entire model, as well as ultimately to the
regionalization results from the k-means clustering algorithm (Figures 13 and 16-22).
10
Figure 6. V
s
profiles for the middle cross section of Figure 4, profiles 6 – 10.
Figure 7. V
s
profiles for the lower cross section of Figure 4, profiles 11 – 14.
11
2.4 Mantle Depth and Low Velocity Zone
Throughout the structure of S4.26, the depths of two characteristic features, namely the depth
to mantle velocities and the existence of a LVZ, vary across the study area and can be defined in
several different ways. We choose to determine the mantle depth by finding the shallowest point
in each velocity profile where V
s
is above 4200 m/s and the shallowest point in each velocity
profile where V
s
is above 4300 m/s (referred to as Z
4.2
and Z
4.3
, respectively) at or below 15 km
depth, and use the average of these two depths. There were a total of 36 nodes for which the
maximum velocity never reaches 4300 m/s. For these nonconforming profiles, all of which were
located along the bottom left of the model (near Baja California), we assigned “not a number”
(NaN) values. From this same metric, we can calculate the steepness of the velocity gradient in
any portion of the model. Within the model, the depth to mantle velocity ranges from 15 to 47
km, based on lithology and tectonic setting (Figure 8).
Figure 8. Depth to mantle velocities. Shallowest depths are located below the ophiolite structure
in the eastern part of the Great Valley (dark red) and deepest depths occur below parts of the
SNB and PRB (deep blue).
12
While not exactly the definition of the Moho, these elevated velocities correspond to a very
distinct compositional change from the overlaying layers and an alteration into mantle lithology.
Rather than representing the very sharp velocity contrast inferred from geology, this boundary is
gradual between crust and mantle in the CVM. Generally, these depths corroborate with other
Moho studies in the state (e.g., Zhu & Kanamori, 2000; Tape et al., 2012). The shallowest mantle
depths are exhibited as circular features related to the ophiolite structures in the eastern section
of the Great Valley and other locations above 20 km exist in the Continental Borderlands
offshore as well as below the Coachella and Imperial Valleys. The comprehensive characteristics
of the surface, mid-crust, and deep-crust will significantly influence the arrival time and
amplitude of many distinct seismic phases during numerical simulations.
The existence of a LVZ is defined by taking the ratio of the V
s
values at 8 km depth with the
values at 14 km depth if this ratio is above 1. These measurements can be used as diagnostics for
the persistence of a LVZ across the model and for making inferences into boundaries between
thick and thin crust. The resulting plots are also compared to the regionalization to determine
how controlling these features are in the cluster procedure, as a compliment to the specific
analysis of each centroid profile for a given regionalization. These regions are plotted in red in
Figure 9 whereas the lack of an LVZ is shown in blue, based on the ratio of velocities at 8 km
and 14 km for each node.
13
Figure 9. Map of LVZ defined by the ratio of V
s
values at 8 km and 14 km depths. Red regions
correspond to profiles which include a ratio below 1 while blue regions correspond to profiles
with a ratio above 1. The value of this ratio is exhibited by the color intensity.
2.5 Validation and Verification
The performance of the CVM has been extensively verified by comparing the synthetic
seismograms for the observed earthquake data used in the inversion process calculated from
S4.26 with those calculated from CVM-H 15.10.0 (Lee et al., 2014a) as well as validated by
comparing synthetic and observed seismograms for earthquake events that occurred after the
model was released (Lee et al., 2014b). The model is also an improvement to CVM-S 4 and
provides excellent fit to both earthquake seismograms and ambient noise correlagrams up to
frequencies of 0.2 Hz (Lee et al., 2014a). In order to verify the resolution of the model, 3D
checkerboard tests were conducted using different checker sizes. Synthetic seismograms were
calculated from the checkered model and then inverted using an LSQR solver (Lee et al., 2013).
Figure F1 from Lee et al. (2014a) is included in the Appendix as Figure A1 to demonstrate the
performance of these tests at various spacing and depths. All validation and verification checks
were made by Lee et al. prior to the application of k-means to S4.26 and analysis within this
paper.
14
During the course of the inversion process, the “bubble-like features” exist both laterally
and vertically. Examples at the surface are seen in Figure 1 as a circle of low velocity east of the
Arizona border and high velocity between the San Jacinto and Elsinore Faults. Similarly, in the
cross section through the western Mojave, these bubbles occur 25 km below LAB, 30 km below
the SAF, and 9 km below the area just north of the Garlock Fault near the Nevada border. These
are inversion artifacts of the mathematical solution and may not represent real geologic features.
The k-means procedure that we will demonstrate here will similarly contain these features in the
cluster outputs and represent a limit to the resolution for the 3D model.
15
Chapter 3
Methods
3.1 K-means Clustering
We perform cluster analysis on 1,523,712 velocity profiles that compose the S4.26 velocity
model with the built-in k-means function of Matlab that follows the k-means++ algorithm
(Arthur et al., 2006). We define the velocity profiles, called 𝐱 𝑛 , to be vectors of 100 values for
V
s
or V
p
, representing the depths from the base of the 3D model (z = 1, 49.5 km below surface)
up to the surface of the 3D model (z = 100, 0.0 km):
𝐱 n
= [
x
1
⋮
x
100
] , n = 1, … , N (1)
where N is the number of nodes in the model space. At the onset of the clustering analysis, k
velocity profiles are randomly selected from the entirety of the data set to serve as centroids for
the clustering routine, which we call 𝐱̅
(𝑘 )
. We then calculate the distance between every velocity
profile in the model from each 𝐱̅
(𝑘 )
by calculating 𝑑 𝑛 (𝑘 )
:
𝑑 𝑛 (𝑘 )
= ||𝐱 𝑛 – 𝐱̅
(𝑘 )
||
μ
(2)
where μ can be 1 (Manhattan distance) or 2 (Euclidean distance). If we refer to the difference
between 𝐱 𝑛 and 𝐱̅
(𝑘 )
at each depth i as x
i
, we can calculate the norm for μ = 1:
|| x ||
1
= ∑ |x
𝑖 |
𝑍 𝑖 =1
(3)
or for μ = 2:
|| x ||
2
= √∑ x
𝑖 2 𝑍 𝑖 =1
2
, (4)
16
where Z is the length of vector x. After the 𝑑 𝑛 (𝑘 )
is calculated for every profile in the model for
each cluster, the profiles are assigned to the cluster for which they have the smallest value for
𝑑 𝑛 (𝑘 )
, the centroid velocity profile is updated by averaging all assigned profiles:
𝐱̅
(𝑘 )
=
1
𝑁 𝑘 ∑ x
𝑖 (𝑘 ) 𝑁 𝑘 𝑖 =1
, (5)
where N
k
is the number of profiles assigned to cluster k. The procedure then begins again with
these new centroid values as the cluster seeds. This continues for 100 iterations or until
convergence is reached, which is defined to be a new iteration for which the sum of all distances
does not decrease upon submission of the updated clusters.
For the clustering of each of the V
s
, V
p
, and V
s
+V
p
data, we allowed K to range between
3 and 10 random starting centroid velocity profiles and required 10 convergent iterations to run,
accepting the iteration with the minimum total squared L
2
distance as the representative
regionalization for that K value. As a verification exercise for the stability of this approach, we
also ran the analysis on K = 7 for 100 full iterations and calculated the total distance to ensure
selection of a global (rather than local) minimum to serve as the representative regionalization.
Each of these output regionalization assignments, centroid velocity profiles, and sum of distances
for a K value were stored and able to be analyzed statistically, both against other K values and
between different velocity analyses (i.e., V
s
compared to V
p
). The regionalization outputs were
converted to a numerical assignment for a grouped region and plotted in map view to showcase
the behavior of each group along with the conformity to geologic features such as faults,
geologic formations, and coastlines, which were unknown to both the model and the clustering
analysis. The final centroid velocity profiles were systematically examined and analyzed for
behavior at the near surface (< 5 km depth), mid-crust (8-14 km depth), deep crust (21-26 km),
and the location of the depth to mantle.
Finally, in addition to the random clustering k-means analysis, we picked “biased”
velocity profiles to submit as a sample clustering routine. These profiles were determined by
subjectively selecting 7 locations in the model based on geological knowledge which we
believed to represent 7 distinct geological regions. An average of 2501 velocity profiles which
represent the surrounding area of 25 km
2
was assigned as the starting centroid for both V
s
and V
p
17
for the given region. These chosen locations correspond to the offshore Continental Borderlands,
the SNB, LAB, the Western Mojave, the PRB, the Great Valley sedimentary basin, and the
region near the California-Arizona border which we will refer to as “Proterozoic Crust”. During
the routine, the centroid values were updated to reveal their final velocity structure after the
clustering converged and both the starting and ending profiles were stored. The resulting
regionalizations for the biased clustering were compared with the random selection for V
s
and V
p
for K = 7.
3.1.1 Example Regionalization Output
As a demonstration of this clustering procedure, we include the V
s
results for K = 7
regions in Figure 10 here. The left figure shows the output regionalization from the clustering
procedure, color coded based on region. For this specific example, the corresponding
Figure 10. K-means output for K = 7 using V
s
data. (Left) Geographic regionalization of the
clustering routine. (Right) Corresponding color coded centroid velocity profiles.
geologic/geographic regions are referred to as 1) the Continental Borderlands (cyan), 2) the
Sierra Nevada batholith (green), 3) the LA Basin and Coachella/Imperial Valleys (red), 4) the
Mojave Block (yellow), 5) the Ophiolites and Peninsular Range batholith (magenta),
subsequently referred to as the Ophiolites, 6) the Great Valley (orange), and 7) the Proterozoic
Crust (peach). Each of these regions will have a centroid velocity profile, which we plot on the
right using the same color coding. Note that the greatest areas of distinction between the profiles
18
exist near the surface, in the mid-crust, and at the base of the crust where the Moho would occur.
The sedimentary basins (LAB and Great Valley) have the lowest surface velocities but diverge in
the mid-crust, where the LAB contains an LVZ and the Great Valley instead continues to
increase in velocity with depth. The SNB and Mojave Block both include a LVZ as well, but
differ in terms of depth to mantle, where the Mojave Block contains mantle velocities by 30 km
and the deep root of the SNB does not reach mantle until ~35 km depth. Finally, the Ophiolites,
Continental Borderlands, and Proterozoic Crust all exhibit similarly high surface velocities
without a LVZ, but vary in depth to mantle. The Continental Borderlands reach mantle velocities
at the shallowest depth (~ 22 km), while the Proterozoic Crust (~27 km) and Ophiolites (~35 km)
are deeper.
It is important to keep in mind that while the output regionalizations demonstrate 2D
analysis of the map areas, the divisions that result from the algorithm are due to the entwined
relationship between surface expression and crustal seismic velocity behavior at depth. When
features are coherent in multiple dimensions, their extent is more likely to be grouped. These
types of distinctions occur throughout all K values and the different values of K will produce
different geographic divisions of the model. Initially low K values will contain vaguely defined
regions which encompass velocity profiles with a great deal of scatter. As K increases, the
separation of a broad region into two (or more) regions will reveal the distinct characteristic
within the centroid profile that caused the division to occur. We will include the results for each
data setting using each value of K (3-10) below and discuss which regions reveal themselves
along with the distinguishing characteristics for the separation.
19
3.2 Fuzzy C-Means Clustering
While the k-means++ algorithm effectively groups the velocity profiles into K number of
clusters, the results are a discrete assignment for each profile to a “least-distance” centroid
profile and no further quantitative measure of assignment is calculated. Using a similar clustering
analysis referred to as “Fuzzy C-Means Clustering” (fcm), the same style of L
2
distance for every
velocity profile in the model from each centroid velocity profile is calculated and the profiles are
assigned to the centroid with the minimum distance. However, an extra step is taken to calculate
the “belongingness” of each profile to each of the K centroid profiles, which we call 𝑢 𝑛 (𝑘 )
. The
computation for 𝑢 𝑛 (𝑘 )
for a given profile as belonging to the k
th
cluster is:
𝑢 𝑛 (𝑘 )
=
1
∑ (𝑑 𝑛 (𝑘 )
/𝑑 𝑛 (𝑘 ′
)
)
2
𝜇 −1
𝐾 𝑘 ′
=1
, (6)
where K is the total number of clusters, 𝑑 𝑛 (𝑘 )
is the norm distance between 𝐱 𝑛 and 𝐱̅
(𝑘 )
as defined
earlier, k and k’ are clusters in the analysis, and μ is the fuzzy partition matrix exponent for
controlling the degree of fuzzy overlap. The value of μ must always be greater than 1 and in our
analysis, we vary μ from 1.05 to 2.0. Note that for a given velocity profile, the sum of 𝑢 𝑛 (𝑘 )
across
all K values will be 1. The effect is such that profiles which are similar to more than one centroid
will demonstrate that ambiguity rather than giving all members of a cluster equal weighting. The
same style analysis as the regular k-means was performed across K from 3 to 10 regions and on
the V
s
and V
p
profiles in the entire model. Each K value ran for the full 100 iterations, the final
output was stored, and the regionalization (as well as the belongingness for each cluster) was
plotted. The regionalizations for each data set and K value were compared to the k-means
regionalizations based on membership and computational time.
20
3.3 Feature Mapping
Since the key differences in velocity profiles for S4.26 will occur at the surface, mid-crust,
and deep-crust, we determine the average velocities for each profile at these depths as a measure
of behavior within the model. We plot maps of a series of averages for the velocity profiles by
calculating the average velocity from surface to 5 km depth, 8 to 14 km depth, and 21 to 26 km
depth. These ranges will correspond to the upper, middle, and deep portions of the crust and
allow us to examine the extent of similar features at these depths across the study area as well as
to compare the features with the k-means regionalization. Each crustal section will have a
different scale in order to demonstrate the heterogeneity of the model at those depths as well as
more clearly show how different structures conform to faults and other geologic structures.
21
3.4 Correlation Matrices
A radial correlation matrix is a measurement of how sensitive each section of a velocity
profile is to any other section of the velocity profile (Jordan et al., 1993). We are interested here
in how surface behavior is associated with Moho depth and mid-crustal velocity and have
calculated the correlation matrix for the entire S4.26 model as well as each of the regions output
from K = 7. The equation to do so is as follows:
C
k
= ∑
[𝑥 𝑛 (𝑧 ) − 𝑥 ̅ (𝑧 )]∗[𝑥 𝑛 (𝑧 ′
) − 𝑥 ̅ (𝑧 ′
)]
√[𝑥 𝑛 (𝑧 ) − 𝑥 ̅ (𝑧 )]
2
∗[𝑥 𝑛 (𝑧 ′
) − 𝑥 ̅ (𝑧 ′
)]
2
2
𝑁 1
, (7)
where C can correspond to the entire model or a specific region, N is the number of member
profiles considered, n is each profile in N, 𝐱 𝑛 (𝑧 ) is a given velocity profile, and 𝐱̅(𝑧 ) is the
average velocity profile for the model or region. The diagonal elements will by definition always
be 1 and the other elements of the square matrix will range between -1 and 1.
22
3.5 Intracluster Comparisons
One final statistical analysis to perform on the results is to measure how similar the
members of a given region are to each other and then compare that value to how similar the
centroid profiles are between regions. We return to our definition of the distance between every
profile which is a member of a given region and that region's centroid profile, 𝑑 𝑛 (𝑘 )
to calculate
the mean of all 𝑑 𝑛 (𝑘 )
assigned to a cluster and call that 𝑒 𝜇 (𝑘 )
which is calculated for μ = 1 as:
𝑒 1
(𝑘 )
=
1
𝑁 𝑘 ∑ |𝑑 𝑛 (𝑘 )
| (8)
and for μ = 2 as:
𝑒 2
(𝑘 )
= √
1
𝑁 𝑘 ∑ (𝑑 𝑛 (𝑘 )
)
2
𝑁 𝑘 𝑛 2
. (9)
The difference between centroid profiles is defined as:
𝐷 (𝑘 𝑘 ′
)
= || 𝐱̅
(𝑘 )
– 𝐱̅
(𝑘 ′
)
||
μ
, (10)
where 𝐱̅
(𝑘 )
is the centroid profile for the k
th
region and 𝐱̅
(𝑘 ′
)
is the centroid profile for another of
the K regions used. The normalized measure of the cluster separation can therefore be calculated
as:
𝐶 (𝑘 𝑘 ′
)
=
𝐷 (𝑘 𝑘 ′
)
𝑒 𝜇 (𝑘 )
+ 𝑒 𝜇 (𝑘 ′
)
, (11)
which will have a value of 0 along the diagonal. Geometrically, this is a measurement of the
distance between centroid locations in two different regions compared to the sum of the radii of
the corresponding cluster areas. If 𝐶 (𝑘 𝑘 ′
)
is large, this implies a weak overlap in the regional
memberships so that each region is therefore distinct.
23
Chapter 4
Results
4.1 Stability of the Analyses
The procedure worked as predicted in clustering the velocity profiles and took longer
computational time to converge as K increased to 10. The convergence times for V
s
and V
p
using
the same K number were comparable and the combined V
s
+V
p
profiles took slightly longer,
given the double length of the vector submitted for analysis. None of the K runs took more than
an hour to achieve 10 successful convergences. As more clusters were added, the summed
distance for all regions decreased following a K
-1/2
trajectory for all three data sets, as plotted in
Figure 11. This is self-explanatory because each intracluster distance will be reduced if more
regions are separated to subtract the greater distant profiles and form more similar groups for the
new cluster. A sort of inflection point occurs at K = 7, suggesting that the additional
computational time will yield diminishing returns on distance reduction for larger K runs and we
will expand on this regionalization scheme in the discussion section.
Figure 11. Decrease of L
2
Distance as more regions are added. All three datasets reduce
following K
-1/2
. Note that the absolute distances for V
p
and V
s
+V
p
have been scaled by ½ and ⅓,
respectively, to fit on the plot.
24
As verification that the lowest-distanced convergent iterations provide the representative
regionalization, the output from the distances from 100 runs using K = 7 on the V
s
data is shown
in Figure 12. Three different distance results are plotted: the convergent (but not minimum
distanced) iterations in blue, the non-convergent iterations in red, and the convergent and
minimum distanced iterations in green. Note that while some non-convergent iterations have
Figure 12. The sum of squares distances calculated for 100 iterations of K = 7 for the V
s
data
set. The minimum distance occurred in 57 of the 100 iterations and the output regionalization
matches that of our K = 7 representative output.
a reduced distance compared to some convergent iterations, the convergent minimum iterations
are lower than any non-converging iteration. This minimum occurred 57 times in the 100
iterations and matches our output from the V
s
k-means calculations when we used 7 regions and
required 10 convergent iterations. Thus we conclude that our 10 convergent iterations
requirement will satisfactorily determine the representative regionalization for all values of K
using any of the three data sets from S4.26.
The results for each K value show that the routine is both geographically coherent and
stable. While new regions will be revealed as more divisions are permitted, the region
separations were not random and instead followed a distinct difference between the initially
25
general centroid profile and the two resulting centroid profiles. A clear example can be seen in
the change from K = 3 (Figure 16) to K = 4 (Figure 17), where the preliminary grouping of
sedimentary basins features a very generic seismic velocity profile (which has low surface
velocities and increases with depth) is split into a profile featuring a LVZ (LAB) and another
featuring a very shallow peak of velocity followed by a continuous increase with depth (Great
Valley). Until K reaches 8, each output region corresponds to plausible geologic provinces which
can be supported by geologic analysis. When the model is this splintered, the output regions for
the highest K values are generally related to the Coast Ranges, a subset of the ophiolite structure,
the deepest part of the SAF, and bands surrounding the sedimentary basins. However between
the three data sets, most regions are very comparable instead of producing unrelated
regionalizations based on V
s
, V
p
, or V
s
+V
p
which further supports the stability of the technique.
26
4.2 K-means++
When we required each K run to produce 10 convergent iterations, a clear revelation
pattern of different geologic regions developed with limited variation between V
s
, V
p
, and V
s
+V
p
data sets. To highlight the output, we will first expand upon the analysis of the regions based on
our preferred output of K = 7 and then proceed to walk through the other K values systematically
from low to high.
4.2.1 Target K Value of 7
By allowing the code to select 7 starting centroid profiles at random, the model is
separated into regions containing the Continental Borderlands, SNB, LAB, Mojave Block,
Ophiolite/PRB, Great Valley, and Proterozoic Crust. These results are displayed in Figure 13.
Recall that while these regions are not strictly limited to the same areas as their descriptive name
may imply in a geological or geographical sense, these features represent the core identification
for each region and the larger region will be referred to in this manner throughout the analysis.
Differences between the data sets are extremely small and the regions are nearly identical in
regards to geographic extent and seismic velocity profile. The surface velocities are lowest for
LAB (1163 m/s) and highest for the Ophiolites (3093 m/s). Three regions (LAB, Mojave, and
SNB) contain a LVZ in the mid-crustal depth range whereas the Continental Borderlands,
Ophiolites, and Proterozoic Crust all increase with depth for the entire profile. The Great Valley
profile has a very distinct pinch out of 2304 m/s at 2.5 km depth, decreases briefly, then
increases with depth. Finally, the region of the Continental Borderlands contains the shallowest
Moho depth and the SNB has the deepest.
Rather than allowing the k-means++ algorithm to randomly select starting profiles, we
determined a biased set of K profiles which represent unique geologic regions and submitted the
corresponding profiles as starting centroids. The results for V
s
and V
p
show a remarkable match
with the convergent iterations for a K value of 7 (Figures 14 and 15). The starting centroid
locations are marked on the output regionalization map with a black X and were the same
locations for both V
s
and V
p
. It is also interesting to note that when we selected two starting
profiles that were 1 node apart in each of the x and y directions of the model, the same pattern of
27
seven distinct regions was still elucidated by the algorithm, emphasizing the robustness of the
calculation and its ability to parse repeatable patterns.
28
Figure 13. (Top) Results for a K value of 7. (Bottom) Corresponding centroid profiles for
the color coded regions. The left column matches Figure 10 and more detail is included
29
Figure 14. V
s
Biased clustering using 7 centroid location profiles which correspond to LA Basin
(red), the SNB (green), the Mojave block (yellow), Offshore (cyan), the PRB (magenta), the
Great Valley basin (orange), and the Arizona border area (peach). The X is the center of the
biased centroid which was calculated by averaging all profiles in a ±25 nodal area around this
center. Compare this output with that shown in Figure 10.
Figure 15. Starting (Left) and ending (Right) centroid profiles for the biased k-means routine
using the V
s
data set. The starting profiles are exaggerated versions of the more constrained
ending profiles due to the smoothing from more model nodes. Compare the figure on the right
with that in Figure 10.
30
4.2.2 Lower K Values
Under the most basic division (Figure 16), the model is generally separated into the
Continental Borderlands grouped with the Proterozoic Crust (cyan), the Batholiths (green), and
the Sedimentary Basins (red). For consistency, we will always show the new region in the next
color added and attempt to maintain the color scheme of the earlier regions as best as possible.
Each subsequent description of the result will focus on the new region and its profile and only
discuss significant alterations to previously established regions if they occur.
By increasing the K value to 4 (Figure 17), the clustering routine for V
s
now separates the
Great Valley northwest of the Garlock Fault from the other sedimentary basins around Los
Angeles and the Salton Sea. This new velocity profile is very distinct from the original
sedimentary basin as evidenced by its pinch out velocity maximum at 2.5 km depth and lack of a
LVZ in the mid-crust. In contrast to this separation, the V
p
and V
s
+V
p
clustering routines instead
parse the Mojave Block (yellow) from the Batholiths so that the SNB and PRB are now in one
region (green). Both of these centroids feature a LVZ but the Moho is nearly 10 km deeper in the
batholith region than in the Mojave Block.
31
Figure 16. (Top) Regionalizations for K = 3 using V
s
(left), V
p
(center), and combination (right)
profiles. Under this simple clustering, all three data sets produce comparable regions
corresponding to Continental Borderlands and Proterozoic Crust (cyan), Batholiths (green), and
Sedimentary Basins (red). (Bottom) Corresponding centroid profiles for the color coded regions.
32
Figure 17. (Top) Results for a K value of 4. The V
s
analysis separates the Great Valley from the sedimentary
basins whereas the V
p
and V
s
+V
p
analyses separate the Mojave Block from the Sierra Nevada batholiths (both
new regions in yellow). (Bottom) Corresponding centroid profiles for the color coded regions.
33
All three analyses return to match when K = 5 (Figure 18) by regrouping the Great Valley
with the other Sedimentary Basins (red), collectively referred to as LAB, and instead separating
the ophiolite structure on the eastern edge of the Great Valley and the PRB (magenta), which we
will collectively refer to as the Ophiolites, from the eastern SNB (green). Although both of these
regions contain a very deep Moho (below 35 km), the Ophiolites are distinguished by their
higher surface velocities and lack of a mid-crustal LVZ.
The Great Valley (orange) returns as a distinct region in all three data sets when K is
increased to 6 (Figure 19). The corresponding centroid velocity profile features a distinct notch
at 2.5 km where velocity peaks, reduces until 5 km depth, and then proceeds to increase with
depth down to Moho at 29 km. The resulting LAB region (red) has a similar Moho depth but
now includes a very clear LVZ in the mid-crust which was not exhibited when the region was
grouped with the Great Valley (e.g., K = 3, 5).
34
Figure 18. (Top) Results for a K value of 5. All three data sets group the Great Valley with the other sedimentary
basins (red) and the Great Valley ophiolites are grouped with the Peninsular Ranges (magenta), separate from the
Sierra Nevadas (green). (Bottom) Corresponding centroid profiles for the color coded regions.
35
Figure 19. (Top) Results for a K value of 6. The Great Valley (orange) separates from the other
sedimentary basins (red). (Bottom) Corresponding centroid profiles for the color coded regions.
36
4.2.3 Higher K Values
Beyond the target value of K = 7, the clustering begins to diverge based on which data set
is used. The regions also become more geographically ambiguous in definition and feature more
subtle differences in the separated profiles. Each of the data sets reveals a different new region
when K is increased to 8 (Figure 20). For the V
s
analysis, a narrow band between the Ophiolites
and the Great Valley which we call the East Great Valley (EGV) is separated (purple). The EGV
represents an exaggerated version of the Great Valley centroid profile, including the distinct 2.5
km depth velocity notch. Unique to this profile are the very high velocities between 10-20 km
and the deep LVZ between 20 km and the Moho (32 km). The V
p
analysis instead separates the
southern tip of the Sierra Nevadas and the region along the SAF north of the Peninsular Ranges.
We refer to this region as the Deep SAF (purple) due to its Moho depth of 35 km, representing
the deepest centroid for the model. The mid-crust also features lower seismic velocities as well
as a LVZ with higher maximum and minimum velocities compared to the SNB (green). Finally,
the V
s
+V
p
region represents a poorly defined ring surrounding the LAB region, which we will
call the Sedimentary Fringes (purple). These Sedimentary Fringes are not as clear to evaluate for
the combined V
s
+V
p
profile and we will defer to more detail in the next section.
The output for K = 9 (Figure 21) reveals no new regions but instead the previously
defined Sedimentary Fringes are now separated for V
s
and V
p
and the EGV is separated for the
V
s
+V
p
(all plotted in tan). The shallow depths of the Sedimentary Fringes follow the gradient of
the LAB profile but with elevated velocities. They also do not feature a LVZ and contain the
Moho at 30 km.
The final regionalizations at K = 10 (Figure 22) reveal two more distinct regions. In the
V
s
output, the Coast Ranges surrounding Santa Barbara (sea green) are separated from the Great
Valley. This region does not include the 2.5 km velocity notch nor a mid-crustal LVZ, but it does
feature a shallower Moho (27 km) than the Great Valley. The V
p
and V
s
+V
p
outputs instead
separate the Eastern Mojave (sea green) from the Western Mojave (yellow) across the ECSZ.
Though the Eastern and Western Mojave profiles are very similar, their crucial difference is the
lack of a LVZ in the Eastern portion of the block, representing a potential geographic limit of the
regional LVZ by way of the ECSZ.
37
Figure 20. (Top) Results for a K value of 8. Each of the data sets reveal a different new region – the
East Great Valley for V
s
, the Deep SAF for V
p
, and the Sedimentary Fringes for V
s
+V
p
, all of which
are plotted in purple. (Bottom) Corresponding centroid profiles for the color coded regions.
38
Figure 21. (Top) Results for a K value of 9. Both the V
s
and V
p
now separate the Sedimentary
Fringes whereas the V
s
+V
p
separate the East Great Valley (all plotted in tan). (Bottom)
Corresponding centroid profiles for the color coded regions.
39
Figure 22. (Top) Final results for a K value of 10. The V
s
data remove the Coast Ranges from the Great
Valley while the V
p
and V
s
+V
p
instead divide the Mojave Block east and west of the ECSZ. These new regions
are each plotted in sea green. (Bottom) Corresponding centroid profiles for the color coded regions.
40
The geographic coherence of the regionalization supports the argument for stability of the
analysis routine as well as a limit to the value for K. All of the regions for K values below 8 are
nearly identical for all three data sets and differ only in the order of revelation. However,
divergence occurs for the high K values in the regions which are parsed and potentially highlight
a limitation to the clustering technique.
41
4.3 Fuzzy C-Means
If we expand on the general k-means procedure by weighting each region’s membership, we
can demonstrate a somewhat probabilistic association of the velocity profiles with their assigned
cluster. The parameter µ operates as a weighting exponent to control the relative weights of the
squared errors based on the distance between member and centroid velocity profiles. As μ goes
to infinity, the belongingness calculations will blur and defocus towards the fuzziest state. From
previous studies, the range of useful values for µ is between 1 and 30 (with good results typically
confined to 1.5 to 3.0), but there is no theoretical or computational evidence for an “optimal
value” (Bezdek et al., 1984). Our measurements for µ were between 1.05 and 2.0.
At low values of μ, the FCM regionalizations follow a similar output as the k-means whereas
a μ value approaching 2 becomes more blurred, hence the “fuzzy” name. However, the
“belongingness” for each region illustrates the diminishing returns on higher order K values.
Table 1 shows the average belongingness for each cluster of the V
s
data as K increases from 3 to
10 and using the default μ value of 2.0. When K is greater than 5, the average belongingness for
several regions drops off significantly (below 0.3), suggesting a limit to the regional distinctions
for this procedure. Table 2 shows the same data for V
p
instead. The average belongingness for
each K of both V
s
and V
p
are comparable. The output regions were sorted from highest average
𝑢 𝑛 (𝑘 )
(strongest belongingness) to lowest, which effectively demonstrates our certainty for the
membership of each region.
The belongingness results for V
s
with a K value of 7 are shown in Table 3 and the
corresponding map intensity plots are shown in Figures 23 - 27. Throughout all μ values, the
Continental Borderlands region contains the highest average 𝑢 𝑛 (𝑘 )
value among its members,
followed by the Great Valley and the Proterozoic Crust. The LAB and SNB retain strong
membership at low μ values but diminish significantly as μ increases beyond 1.5. The sections of
the model which correspond to the Mojave Block and Great Valley ophiolites are segmented
drastically by a high value of μ, suggesting that these profiles have a very ambiguous character
which can be associated with multiple centroids. Notably, a LVZ is absent from the three most
coherent regions as will be discussed later.
42
4.3.1 Results from K = 7
For μ = 1.05 (Figure 23), the output is very similar to the regions revealed from the
Biased calculation as well as the Random k-means using 7 regions. All regions have an average
𝑢 𝑛 (𝑘 )
above 0.95, suggesting a very strong belongingness for members included. Only slight
differences occur between strengths but the nominal order of strongest to weakest regions are
Continental Borderlands, Great Valley, SNB, Proterozoic Crust, LAB, Ophiolites, and the
Mojave Block. Note that the colors in the top left panel do not remain consistent with the
previously developed scheme nor amongst different values of μ and instead are based on
strongest to weakest membership, identified in the bottom left corner of the subsequent panels.
As μ is increased slightly (up to 1.2), more intermediate colors appear around the core
section of the region (Figure 24). This suggests that the regions are blending into one another or
simply not as “certain” to belong to a specific cluster. There is no longer a strong separation for
the Ophiolites near the Great Valley and the PRB is grouped with the SNB. The order of
belongingness now ranks the Proterozoic Crust third and the Batholiths slide to fifth. This begins
the vagueness of geographical regions which are revealed by the FCM process.
Using μ = 1.5 results in a breakdown of the typical regions we see from k-means (Figure
25). While the strongest four regions still correspond (in order of strength) to Continental
Borderlands, Great Valley, Proterozoic Crust, LAB, and the Batholiths, there is now a blur of the
orange and peach regions which cover much of the Mojave Desert. Again the Ophiolites are no
longer a discrete region in this output and are grouped with the SNB and PRB.
The belongingness calculated for μ = 1.8 shows an even weaker membership for the
seven regions (Figure 26). The Continental Borderlands, Proterozoic Crust, and Great Valley
continue to be the strongest regions, swapping the order between the Proterozoic Crust and Great
Valley regions, but the deep reds there have been significantly weakened compared to Figure 25.
The remaining four regions have maximum belongingness on the order of 0.2 and there is no
clear distinction between the three weakest regions (magenta, orange, peach).
The maximum “fuzziness” calculated 𝑢 𝑛 (𝑘 )
with a μ value of 2.0 and almost completely
blurs the regions (Figure 27). The Continental Borderlands and Proterozoic Crust regions are still
43
distinct (with maximum values now only at 0.575) but the “splotchy” patterns which comprise
the magenta, orange, and peach regions demonstrate a very weak belongingness for much of the
model. On the regionalization panel, it is difficult to identify the magenta region.
44
Figure 23. V
s
output for K = 7 clustering with a μ value of 1.05.
45
Figure 24. V
s
output for K = 7 clustering with a μ value of 1.2.
46
Figure 25. V
s
output for K = 7 clustering with a μ value of 1.5.
47
Figure 26. V
s
output for K = 7 clustering with a μ value of 1.8.
48
Figure 27. V
s
output for K = 7 clustering with a μ value of 2.0.
49
Finally, the reduction in 𝑢 𝑛 (𝑘 )
as a function of μ is plotted in Figure 28. These calculations
were made using the results from the k-means output following equation 6 for the same set of μ
values from 1.05 to 2.0. The average belongingness for each region of assigned profiles
Figure 28. Average belongingness values calculated for the distinct k-means calculations for the
V
s
data. At μ = 1 (theoretical), each region would be assigned a belongingness of 1, against
discretely assigning each velocity profile to one and only one region. From there, the average
for each region drops off at different rates, hence the separation between colored plot lines. For
all regions, note that the average U value never drops below 0.35, unlike the regions developed
by the FCM process.
decreases from 1 at the theoretical μ = 1 calculation to 0.3-0.6 at μ = 2. More telling, however, is
the fact that the Continental Borderlands remain the highest region across all μ and the
Proterozoic Crust remains similarly high. In order of descending average 𝑢 𝑛 (𝑘 )
, the belongingness
curves for the Great Valley, SNB, LAB, the Ophiolites, and the Mojave Block spread apart as μ
increases, again following the pattern mapped in Figure 27, where the latter four regions lose
their unique distinction. For the k-means regions, the minimum average 𝑢 𝑛 (𝑘 )
at μ = 2 is above
50
0.35 which is larger than four of the seven regions calculated by fcm, as seen in Table 3. This
again suggests that these regions are representative of legitimate geological clusters throughout
S4.26.
51
4.4 Feature Mapping
4.4.1 Depth to Mantle
Mapping the depths of features showed conformity between geologic regions and surface
faults across the model. We calculated the deepest mantle depth to be 46.75 km, located in the
PRB east of San Diego. The depth to mantle was shallower than 25 km for 24.25% of the model,
shallower than 30 km for 58.74%, and shallower than 35 km for 92.17%. The shallowest depth
was set to 15 km due to the high surface velocities near the Great Valley ophiolites but mantle
depths above 25 km were very common across the Continental Borderlands and the Imperial
Valley. Along with the PRB, deep mantle depths (below 30 km) exist in regions of the Great
Valley, the SNB, and along the SAF south of the Mojave Block. Drastic differences in depth
occur on either side of faults, such as on the SAF near the Coast Ranges. The mantle depths for
every region in the k-means runs are listed in Table 4 and the map is plotted in Figure 8.
4.4.2 Low Velocity Zone
The regions of the model which contain a LVZ were even better constrained to fault
blocks than the depths to mantle as we showed in the map in Figure 9. There is a clear distinction
between the profiles in the Great Valley basin and the adjacent SNB, where the latter exhibits
LVZ behavior while the former does not. Similarly, the block between the San Gabriel fault and
the SAF is almost solid red, which represents LVZ profiles, and the western Mojave Desert is
separated from north of the Garlock fault, south of the SAF, and west of the ECSZ. As we extend
into the Proterozoic Crust near the Arizona border, the map pattern becomes more mosaic,
demonstrating more crustal heterogeneity as well as less model constraint. Table 5 shows the
existence of a LVZ in each of the regions as K increases to 10.
4.4.3 Average Velocity Plots
The averages calculated for the surface layers (0-5 km depth, Figure 29) clearly show the
sedimentary basin regions with V
s
averages below 3000 m/s and elevated velocities occur in the
regions near the Great Valley ophiolites and Tehachapi anomaly. For all average maps, the scale
is not standardized in order to demonstrate the contrasts in the model for those corresponding
52
depths. The warm colors represent the lowest velocities for that figure and cool colors represent
the highest velocities.
Figure 29. Average V
s
in the upper crustal layers.
Figure 30. Average V
s
in the mid-crustal layers.
53
As we move deeper to the mid-crustal depths (8-14 km, Figure 30), the zones are much
less distinct. The area around the Great Valley ophiolites again stands out with higher velocities
and some fault boundedness occurs across the Garlock Fault and the ECSZ, however most of the
map is ambiguous. Finally the deep crust (21-26 km, Figure 31) shows a distinction between
regions which have already reached the mantle, such as the Continental Borderlands and the
Great Valley, and the deep roots of the SNB and PRB. The averages for these sections in
different K values are shown in Tables 6-8.
Figure 31. Average V
s
in the deep crustal layers.
54
4.5 Intracluster Distance
The intracluster values of C
(kk’)
show that most of the clusters have a high degree of
variance compared to the difference between the region centroids. In fact, the only combination
of regions for which C
(kk’)
is above 1 is the comparison of the Continental Borderlands with the
Sierra Nevada batholith. This generally suggests that the radii for each cluster in the K = 7
analysis span a greater distance than the distances between centroids, implying less distinction
between the regions. The same calculation using the L
1
norm yielded a similar pattern but with
generally smaller ratios. These values are listed in Table 9.
55
4.6 Radial Correlation Functions
The radial correlation functions (RCF) did not provide significant examples of strong
correlation between different depths of the velocity profile for the calculations made using V
s
and V
p
across the entire model. The RCFs for each of the 7 k-means regions were calculated and
plotted next to the corresponding centroid velocity profiles. For nearly all of these regions, the
approximate Moho depth can be seen as a visual change in the RCF plot at the proper depth of
the velocity profile. As an example of the meaning of these functions, if the area of the RCF for a
Z depth of 5 km and Z’ depth of 35 km is blue, that implies the two velocity sections are
positively correlated and an increased velocity at 5 km will indicate an elevated velocity at 35
km. Similarly, if that same area of the RCF were red, an increased velocity at 5 km would imply
a decreased velocity at 35 km. The regions which did not include a LVZ featured a more tightly
constrained band of positive correlation along the diagonal whereas those without the LVZ
contained wider areas of positive correlation. Figures 32-38 show the RCFs and velocity profiles
for each of the regions.
The velocity profile for the Continental Borderlands did not exhibit LVZ behavior and
had the shallowest depth to mantle velocities as discussed in the Feature Mapping section. In the
RCF, the shallow velocities and deep crustal velocities are anti-correlated and there was weak
correlation between the mid-crustal sections and other portions of the velocity profile (Figure
32). On the contrary, the SNB region featured a very deep Moho and includes a LVZ from 8 to
14 km depth (Figure 33). The RCF has positive correlation which is tightly bound along the
diagonal down to the start of the Moho depth at 33 km. Slight areas of negative correlation exist
between the surface and the depths of the LVZ as well as between the mid-crust and the base of
the crust. The LAB has much lower surface velocities than the SNB but also contains a LVZ in
the mid-crust (Figure 34). Only the range between 15 to 30 km contains a wide area of positive
correlation. The velocities above the LVZ at 8 km seem to be negatively correlated down the
base of the crust with the slight exception of the positive correlation between the near surface
and the velocities between 20 to 30 km, where the profile gradient stagnates before increasing at
the Moho. The final region which includes a LVZ is within the Mojave Block (Figure 35).
Similar to the SNB and LAB, the Mojave Block has negative correlation between upper crustal
56
velocities and the velocities immediately above the LVZ and positive correlation between the
surface and the increased velocities between 15 to 30 km.
Figure 32. (Left) Average velocity profile for V
s
of the Continental Borderlands region. (Right)
Radial Correlation Function for the same region.
Figure 33. (Left) Average velocity profile for V
s
of the Sierra Nevada batholith region. (Right)
Radial Correlation Function for the same region.
57
Figure 34. (Left) Average velocity profile for V
s
of the Los Angeles Basin region. (Right) Radial
Correlation Function for the same region.
Figure 35. (Left) Average velocity profile for V
s
of the Mojave Block region. (Right) Radial
Correlation Function for the same region.
The remaining three regions do not exhibit a LVZ and feature distinct velocity profile
characteristics. The Ophiolites have the highest surface velocities among all regions and stagnate
between 10 to 33 km depth, which is expressed in the wide area of positive correlation on the
RCF (Figure 36). Throughout the profile, positive correlation is constrained to near sections,
such as surface velocities with other velocities down to 15 km and the velocities at depths around
33 km with those at 40 km. Outside of these regions, most of the RCF shows negative correlation
58
between depths. In the Great Valley cluster, the velocity profile is more complex than any other
region, which translates to the pattern shown in the RCF (Figure 37). Positive correlation is
wider along the diagonal and can be divided into blocked sections. These blocks represent the
sections above 10 km depth, from 10 to 27 km, and below 32 km. Two main areas of negative
correlation exist between the upper crustal velocities and the mid-crustal velocities as well as
between the velocities of the mid-crust with those around the Moho. These areas correspond to
surface low velocities which overlay high velocities in the mid-deep crust and is likely the result
of the peculiar notch in velocity at 2.5 km depth and the slight decrease in seismic velocity
between 18 to 30 km, respectively. Finally in the Proterozoic Crust, a very simplified seismic
velocity profile increases with depth until Moho around 25 km depth (Figure 38). As for the
RCF, most of the depths are positively correlated with all other depths except for those below 35
km. Compared to Figure 37, the RCF for the Proterozoic Crust has a much thinner visible band
that results from the steep Moho gradient.
Figure 36. (Left) Average velocity profile for V
s
of the Ophiolites region. (Right) Radial
Correlation Function for the same region.
59
Figure 37. (Left) Average velocity profile for V
s
of the Great Valley region. (Right) Radial
Correlation Function for the same region.
Figure 38. (Left) Average velocity profile for V
s
of the Proterozoic Crust region. (Right) Radial
Correlation Function for the same region.
60
Chapter 5
Discussion
The key parts of the procedure are the output grouping for each data set and how that
grouping developed over the course of K values for the general k-means routine or both K and μ
for the FCM analysis. Note that there is a lot of similarity between the three data sets but they
eventually diverge in regards to the output regions as well as for the order of region separation.
Also recall that the regions being discussed follow the general identification for each cluster and
not simply that specific geologic feature. We will discuss the general k-means output, the feature
mapping, and the FCM analysis in detail for the V
s
outputs. Finally we will attempt to interpret
these results in a geological context and provide insight as to further usage of these methods on
local CVMs.
5.1 K-Means Clustering
As part of the Results section, we showcased the fact that the resulting k-means region
outputs reflected a series of mathematically stable and geologically cohesive clusters. As the
value for K is increased, the specific geology became more focused and the centroid velocity
profiles reflected the differences between the broader regions from low K values that led to the
separation. Initially, the three regions corresponded generally to the sedimentary basins,
attenuated continental crust (i.e., thin crust with shallow Moho), and batholithic crust. The
corresponding centroid profiles exemplified generic velocity structures that are typically
observed in these settings. When more regions were permitted, the detailed profiles showed the
heterogeneity of the model and the component geologic regimes that represent California’s
tectonic history.
Distinct differences in lithology and geochemistry allow for the development of new regions
in the higher values of K. An example of this is the parsing of the batholiths from the basic
clustering in K = 3 (Figure 16) into discrete groups in K = 5 (Figure 18). While both the SNB and
PRB are well known batholiths in southern California, their compositions are recognizable and
the result of different styles and timing of subduction. The SNB is the result of arc magmatism
related to subduction of the Farallon plate over the past 85 Ma which was later exhumed,
61
emplacing high velocity structure at the surface (Saleeby, 2003). This same magmatic pulse
produced the PRB south of the Mojave Desert between 110-79 Ma and contains a mostly
granodioritic composition (Ducea, 2009), with a geochemical separation between the eastern and
western portions (Langenheim & Jachens, 2003). This geological contrast is determinant in the
removal of the more mafic western section, which includes the ophiolites, to be grouped together
with the PRB during the k-means analysis. When K is greater than 7, the V
p
data set separates the
southern SNB into a group with the deep portion of the SAF south of the Mojave Block. While it
is interesting that the V
s
and V
s
+V
p
data sets fail to divide this member in the same manner, the
velocity contrasts due to geologic differences produce a strong enough separation to cluster
mathematically.
The sedimentary basins are initially grouped together as well, primarily due to low velocities
in the surface and upper crust. This signal is so strong that it dominates at K values below 6, even
overriding the difference in LVZ inclusion underneath the LAB and lack thereof below the Great
Valley. During the course of the k-means analysis though, the basins are not further partitioned
until the poorly defined “Sedimentary Fringes” are revealed in all data sets between K values of
8 to 10 and the East Great Valley is extracted from the edge of the Great Valley. The separation
of the EGV yields a velocity profile very similar to that of the Great Valley but which includes
generally elevated velocities, especially in the mid-to-deep portion of the crust. This suggests
that the lithology or composition in the mid to lower crust may be more heterogeneous than at
the near surface, potentially exemplifying the structure of the Great Valley ophiolite and
subsurface serpentinization (Godfrey & Klemperer, 1998). The LAB is distinguished from the
rest of the sedimentary basins by its LVZ, which has higher absolute velocities than the LVZ
beneath the Mojave Block and SNB as well as a much more gradual velocity reversal. This
feature will be discussed in more detail in Section 5.3.
Finally the Mojave Block cluster is established early in the k-means routine and remains
during all later K values. The SAF and Garlock Fault serve as a very strong boundary between
the Mojave and other regions, where the Mojave Block is constrained to the North American
Plate side of the SAF south of the SNB and to the Pacific Plate side west of the Great Valley.
Similarly, the Mojave Block is mostly constrained south of the Garlock Fault until it extends into
the Basin and Range province of California, into Nevada. The k-means region is defined
62
primarily by its LVZ and shallower Moho than the nearby SNB and PRB. Interestingly, the
Moho depth beneath the Mojave is comparable to that of the Great Valley despite the otherwise
distinct velocity profiles of these regions.
The geologic and tectonic differences throughout S4.26 are exhibited in the velocity profiles
of the model and different regional divisions result from the number of allowed clusters. At K
values below 7, the output regions group very diverse parts of the crust together and the defining
characteristics of the centroids are blurred. One example of this is the combination of the Great
Valley region with the LAB region despite the very distinct differences in the upper and middle
crust that are typically observed in member profiles. The PRB and SNB regions are another
example, which are not distinct before K = 5 despite the geochemical and tectonic differences.
On the other hand, when K is greater than 7, the additional output regions have minor shifts in
profile behavior and the three data sets do not agree on which of these shifts is most important.
In all three data sets, the region of the Sedimentary Fringes shown in high K values does not
have a clear geologic or tectonic definition. Therefore, based on the distance reduction curve
(Figure 11) as well as geologic intuition, we conclude that the K = 7 output serves as our
preferred clustering for the CVM.
Rather than producing arbitrary and fluctuating regionalizations as the value for K increases,
the procedure maintains a systematic division of the regions as a greater number of clusters are
allowed, and therefore the higher K clustering will more clearly define the regions until K = 8.
New map divisions that appear can be clearly linked to differences in the new centroid velocity
profile that are the result of changes in lithology or tectonic setting. The fact that these regions
are divided mathematically yet still adhere to the rules of geology with no a priori information
regarding geology, fault locations, or any other subjective influences supports the tool as an
objective assessment of the CVM.
It would be interesting to use the same assessment on earlier renditions of the southern
California model (e.g., CVMS-4) to evaluate alterations to the model based on the F3DT
inversion and may serve as a technique to appraise future iterations during model development.
Model resolution can be evaluated by running the k-means algorithm on the different iterations
or models produced by different groups to observe which features and regions remain separated.
In the early stages of our application of k-means to S4.26, we excluded the perimeter 50 nodes
63
from the analysis and still observed matching results to the analysis of the full model, indicating
that the lower resolution areas of the model did not significantly alter the clustering.
Additionally, different depth sections of the model profiles (e.g., in the mid-crust from 8 to 14
km) can be clustered exclusively as a measure of the model resolution at different depths.
5.2 Fuzzy C-Means Clustering
The ability for the fcm analysis to quantify a probabilistic measure of region strength
provides an additional support for the clustering routine. By exclusively focusing on the K = 7
division of the model, the nearly exact same regions revealed by the k-means procedure are
elucidated at the lowest values for µ but quickly dissolve as more fuzziness is permitted. All
regions have an average 𝑢 𝑛 (𝑘 )
above 0.95 at µ = 1.05, but these reduce at different rates as plotted
in Figure 28. By the end of the fcm routine, only the Continental Borderlands, Proterozoic Crust,
Great Valley, and SNB remain intact and the former Mojave Block, Ophiolites, and LAB regions
are either separated into indiscernible mosaic patterns or grouped with previously distinct
regions.
All three of the most cohesive regions (Continental Borderlands, Proterozoic Crust, and
Great Valley) do not exhibit a LVZ whereas the SNB, LAB, and Mojave Block regions do.
While the Ophiolites lack a LVZ, the similarities between other parts of the velocity structure
with the SNB and Continental Borderlands lead it to separate into ill-defined regions throughout
the procedure. During the comparisons of high K values in the k-means analysis, we noted that
the Mojave Block was parsed differently by the V
s
and V
p
data sets. Similarly, there appears to
be an alteration to the region across the ECSZ, but it is very ambiguous as to where that
ambiguity truly exists and how it is represented. This suggests that while the LVZ is crucial in
the division of the profiles in S4.26, the exact definition and extent of this feature remain unclear.
In addition to the lack of a LVZ, the regions that retain an elevated belongingness through all
µ values also tend to include more shallow depths to mantle velocities (e.g., compare Figure 8 to
Figure 27). This may in part highlight the importance of velocity gradients during the clustering,
supported by the fact that the velocities at the surface and in the deep crust seem to be crucial in
determining divisions. Possibly, this is also due to the fact that velocities will not change too
64
much below the Moho depth and could represent a section of the velocity profile which has a
reduced L
2
distance among members.
The Great Valley region poses a complicated pattern during these clustering routines. On the
one hand, its characteristic profile is distinct from any other region based on the shallow notch
structure observed upon its first appearance, exclusively in the V
s
data set, for K = 4 (left
column, Figure 17) until the EGV is parsed when K = 8, (left column, Figure 20). But two
contrasting observations are made between the fcm and k-means results: 1) the Great Valley does
not separate until K = 6 for V
p
and V
s
+V
p
and is actually regrouped for V
s
in the output of K = 5
to instead separate the SNB from the Ophiolites and PRB, and 2) the Great Valley initially has a
higher average 𝑢 𝑛 (𝑘 )
for the K = 7 regions at µ < 1.3 (Figure 28) but reduces faster than the
Proterozoic Crust region at higher µ values, despite the fact that this region is not revealed by k-
means until K = 7. Potential explanations for these phenomena include greater heterogeneity
among the model regions near the Arizona border and the very similar centroid velocity profiles
of the Continental Borderlands and Proterozoic Crust below the 15 to 25 km depths (Figure 10).
The conclusion we reach from the fcm analysis is that certain regions are consistently
internally cohesive and distinct from external regions. In contrast, other regions lose this
singularity quickly as we allow the µ value to increase and permit more blending between the
regions. Ultimately the fcm approach offers additional statistical insight as to the clustering of
data and elucidates greater detail behind the coherence of the regions and the robustness of the
regionalization. Performing independent calculations of 𝑢 𝑛 (𝑘 )
from the crisp k-means output
further quantifies the results and assesses the regions for heterogeneity.
5.3 Feature Mapping
The clustering routines are strongly affected by regions of the greatest amount of
heterogeneity, specifically the surface velocities, the mid-crustal behavior (e.g., presence of a
LVZ), and the deep-crustal velocities (e.g., the depth and gradient of the Moho). The feature
maps provide a visual assessment of the model which can be compared to the regionalizations
output by the k-means algorithm.
65
At the surface, the most distinct signatures are the low velocities in the sedimentary
basins and high velocities in the batholiths (Figure 29). The basins tend to conform to surface
faults and affect the upper crustal portions of the centroid velocity profiles. Elevated velocities
correspond to the ophiolite sequence in the eastern part of the Great Valley as well as in portions
of the PRB, again supported by geochemical and magnetic studies (Godfrey & Klemperer, 1998;
Langenheim & Jachens, 2003). The high velocities in the isolated portion of the Arizona border
may be an example of the bubble-like structures found throughout the model which are simply
inversion artifacts.
In the mid-crust, the ophiolites once again stand out due to their near mantle velocities at
these shallow depths (Figure 30). A distinct division exists on the eastern and western sides of
the ECSZ, suggesting a potential geographic limit to the extent of the LVZ. The portions of the
model with the LVZ are also strongly fault-bound and the k-means divisions closely follow these
contours (Figure 9). The strongest geologic interpretation for this feature is the underplating of
schists related to the Farallon subduction (Luffi et al., 2009). The bulk of the Pelonia-Orocopia-
Rand (POR) schists extend on the continental side of the subduction zone and outcrop at various
locations throughout southern California, including along fault boundaries in the Mojave Block
(Jacobson et al., 2011). Thus, the LVZ as defined here may represent the location of this
underplating below the surface of southern California. The schist solution, while not absolutely
unique, is supported by anisotropy and velocity studies (e.g., Porter et al., 2011). Furthermore,
the shallow subduction during underplating would require the fault to cut through the continental
crust without incorporating upper mantle lithosphere and the missing material would be pulled
deeper during this process (Ducea et al., 2009). This could explain both the presence of the
schists as well as their method of emplacement in the mid-crust.
Finally in the deep part of the crust, there is a clear distinction between model areas offshore
, which are already below Moho depth, and those onshore, with few exceptions such as beneath
the Great Valley and Salton Sea (Figure 31). The complimentary map of the mantle velocity
depths (Figure 8) demonstrates the variations of Moho depth within the model and the division
between thick (e.g., around the SNB and PRB) and thin crustal areas (e.g., attenuated continental
crust such as beneath the Salton Sea). Again, the deep roots beneath mountain belts demonstrate
66
the differences in lithology which offer an explanation as to the centroid velocity profiles
between the mid-crust and upper mantle.
5.4 Geologic Interpretation of the K = 7 Regionalization
Differences in geologic composition dominate the crust of southern California due to the
various stages of tectonic activity over the past 210 Ma. The separations that result from the k-
means analysis correspond to geologic differences throughout the model, which is clear even
from the very basic three region division of the batholiths, sedimentary basins, and attenuated
continental crust (Figure 16). Beyond that though, the regions may contain similar velocity
structures but differ in key crustal sections. These alterations typically reflect the changes in
lithology for the separate geologic provinces.
When the k-means routine parses the sedimentary basins into two separate regions for the
Great Valley and the grouped LAB and Coachella Valley, each new region is on different sides
of the SAF. The Great Valley represents a forearc basin which is bound by the Franciscan
Assemblage to the west and the volcanism of the Sierra Nevadas to the east (Ducea et al., 2009;
Jacobson et al., 2011). Additionally the Great Valley overlies part of the same ophiolite sequence
as the PRB to the south (Jacobson et al., 2011). These unique features explain the separation of
the Great Valley from the other sedimentary basins of the model as well as the grouping of the
Great Valley ophiolite structure with the PRB in K > 4 regionalizations.
The SNB is the result of several pulses of arc magmatism which lasted over 50 Ma from
during the Cretaceous (Saleeby, 2003). The western portion of the SNB is older and more mafic
in composition while the eastern portion becomes felsic and younger toward the east, with a
distinct contrast with the southern portion along the Tehachapi-Rand deformation belt (Saleeby,
2003; Jacobson et al, 2011). Similarly, the PRB formed 120-90 Ma and combines arc rocks with
oceanic crust, giving the formation a very mafic signature (Langenheim & Jachens, 2003). While
originally part of the same massive batholith as the SNB, movement along the SAF over the past
6 Ma has separated the PRB after the plate migrated from offshore to the North American plate
(Porter et al., 2011). The distinct mafic characteristics of the PRB and the western part of the
SNB suggest the separation of the full structures beyond a K value of 4 and also the grouping of
these two portions at higher K values. The geochemistry of southern California offers further
67
support for the legitimacy of the regionalization outputs. Isotopic ratios of strontium (
87
Sr/
86
Sr)
constrain the mantle contribution in crustal rocks. The Sr .706 line (where
87
Sr/
86
Sr = 0.706)
represents the extent of Precambrian rocks in North America and is shown in the Appendix
(Figure A2) (Snoke, 2005). In the output from K = 7 (Figure 13), the boundaries between the
SNB and the Ophiolites as well as the eastern edge of the PRB seem to follow this signature line.
The division beyond composition lies in the inclusion of a LVZ in the SNB that is not exhibited
in the PRB. Both centroid velocity profiles contain comparable velocities in the upper crust but
diverge in the mid-deep crust where the felsic composition of the SNB reduces the velocity to the
lowest values in the model at these depths. Both batholiths also have a deep Moho (> 35 km) but
the boundary in the SNB is much sharper while the PRB contains a gradual change.
The Continental Borderlands represent the majority of the attenuated continental crust on
the Pacific Plate and has the shallowest depth to mantle velocities at 22 km due to its basaltic
composition. The Proterozoic Crust near the Arizona border follows a very similar velocity
profile to the offshore region, essentially only differing between the depths of 18 to 30 km. In
this region, the surface is speckled by Cenozoic volcanics as well as Precambrian basement
rocks, related to the western edge of Pangaea on the North American Plate (Luffi et al., 2009).
The depths to mantle in this region are also exceptionally shallow, typically around 26 km
(Figure 8). The absence of a LVZ in either of these regions would also suggest minimal
deformation due to subduction and limited contribution from underplated schists.
The Mojave Block in the middle of the model represents a very interesting geologic
setting which has experienced widespread tectonic activity over the past 100 Ma. The basement
of the Mojave was layered during the Laramide orogeny and the steepening of the Farallon slab
led to collapse of the region, resulting in its rapid uplift and erosion (Saleeby, 2003; Jacobson et
al., 2011). The lithosphere of the North American Plate was removed and schists were
underplated during subduction, as evidenced by isotopic data (Luffi et al., 2009). During the
Cenozoic, the Mojave experienced two stages of deformation, including normal faulting and the
development of metamorphic core complexes related to the Basin and Range extension.
(McQuarrie & Wernicke, 2005). Although in the k-means analysis the distinction between
eastern and western Mojave only occurs in the K = 10 output of V
p
and V
s
+V
p
, this division may
68
be supported by the lack of continental lithosphere in the western Mojave Desert and its re-
emergence on the eastern side of the ECSZ (Saleeby, 2003).
Finally, many researchers have attempted to recombine the tectonically altered portions
of California by way of a type of section restoration process known as palinspastic restoration
(e.g., McQuarrie & Wernicke, 2005). This is done by producing models of the crust which are
constrained by additional information such as structural knowledge, fault slip rates, geodesy, and
geochemistry. Different blocks can then be effectively run in reverse to suture the previously
joined crustal components and create a tectonic reconstruction of the past. The clustering
algorithm can be run on both current data as well as crustal representations of the past to
compare which sections remained intact and where separations occur over time. An example of
such a study could investigate whether the current thick crustal regions remain grouped in the
past renditions of the crust. Such an endeavor could be complicated by geologic processes such
as extension and compression which would alter the areal and volumetric coverage of the current
regions compared to their original states.
69
Chapter 6
Conclusions and Future Work
The clustering routines available for seismic velocity models show that real segmentation
can occur by mathematically analyzing the data. On a very dense local velocity model, the
results elucidate distinct geologic provinces across southern California which correspond to real
features of the crust. Due to a lack of geologic coherence during the mathematical inversion of
crustal velocities, this routine can serve as a way to assess the model during the inversion
process. First, we envision the k-means algorithm as an independent verification on different
iterations during the CVM calculation. In this manner, it can identify which features have been
updated since the last version and quantify differences and hopefully improvements. It would be
especially interesting to compare iterations calculated by the scattering integral method with
those resulting from the adjoint wavefield method, which were both used to produce S4.26 (Chen
et al., 2007a). Second, this analysis can compare CVMs produced using different inversion
schemes and techniques. One simple example is making a comparison between SCEC’s CVMS-
4.26 and Harvard’s CVMH-15.1. Finally, this technique is efficient and robust enough to run on
other local tomography models, such as those for New Zealand or Taiwan.
Greater ambition could potentially harness this process to segment previously connected
geologic regions in California and analyze palinspastic reconstructions of the crust which
incorporate rotation and translation, but challenges may result when accommodating for
compression or extension. The results of this work as well as future analyses should also verify
the results with external data sets (e.g., gravity anomalies, magnetic studies, structural
assessment) and provide a constraint on the certainties of the results.
70
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Chen, Po, Thomas H. Jordan, and Li Zhao. "Full three-dimensional tomography: a comparison
between the scattering-integral and adjoint-wavefield methods." Geophysical Journal
International 170.1 (2007): 175-181.
Chen, Po, Li Zhao, and Thomas H. Jordan. "Full 3D tomography for the crustal structure of the
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California." Tectonics 17.4 (1998): 558-570.
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Society of America Bulletin 123.3-4 (2011): 485-506.
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Langenheim, V. E., and R. C. Jachens. "Crustal structure of the Peninsular Ranges batholith from
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Lee, En-Jui, et al. "An optimized parallel LSQR algorithm for seismic tomography." Computers
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scattering‐integral and the adjoint‐wavefield methods." Journal of Geophysical Research:
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Lee, En‐Jui, Po Chen, and Thomas H. Jordan. "Testing waveform predictions of 3D velocity
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Full‐3D Seismic Waveform Tomography (F3DT)." Seismological Research
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Lekić, V., and Barbara Romanowicz. "Inferring upper-mantle structure by full waveform
tomography with the spectral element method." Geophysical Journal International 185.2
(2011): 799-831.
Luffi, Peter, et al. "Lithospheric mantle duplex beneath the central Mojave Desert revealed by
xenoliths from Dish Hill, California." Journal of Geophysical Research: Solid
Earth 114.B3 (2009).
McQuarrie, Nadine, and Brian P. Wernicke. "An animated tectonic reconstruction of
southwestern North America since 36 Ma." Geosphere 1.3 (2005): 147-172.
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motion with volume harmonic and arithmetic averaging of elastic moduli and
densities." Bulletin of the Seismological Society of America 92.8 (2002): 3042-3066.
Porter, Ryan, George Zandt, and Nadine McQuarrie. "Pervasive lower-crustal seismic anisotropy
in Southern California: Evidence for underplated schists and active
tectonics." Lithosphere 3.3 (2011): 201-220.
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region." Geological Society of America Bulletin 115.6 (2003): 655-668.
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Tape, Carl, et al. "Estimating a continuous Moho surface for the California unified velocity
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73
Appendix: Tables and Figures
K k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
3 0.625 0.521 0.438 X X X X X X X
4 0.601 0.463 0.414 0.379 X X X X X X
5 0.634 0.469 0.416 0.368 0.334 X X X X X
6 0.605 0.477 0.366 0.363 0.264 0.256 X X X X
7 0.575 0.473 0.393 0.330 0.225 0.216 0.206 X X X
8 0.553 0.469 0.392 0.301 0.189 0.186 0.184 0.176 X X
9 0.545 0.472 0.407 0.287 0.178 0.169 0.168 0.168 0.163 X
10 0.536 0.463 0.406 0.269 0.163 0.162 0.150 0.149 0.147 0.144
Table 1. Average V
s
“belongingness” for the regions as K varies from 3 to 10 with μ = 2.
74
K k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
3 0.668 0.572 0.472 X X X X X X X
4 0.712 0.525 0.489 0.416 X X X X X X
5 0.702 0.542 0.468 0.380 0.370 X X X X X
6 0.677 0.534 0.405 0.339 0.278 0.275 X X X X
7 0.653 0.523 0.368 0.313 0.234 0.234 0.220 X X X
8 0.633 0.517 0.345 0.297 0.216 0.201 0.195 0.188 X X
9 0.622 0.514 0.329 0.301 0.205 0.181 0.176 0.176 0.172 X
10 0.606 0.509 0.403 0.324 0.316 0.198 0.170 0.170 0.162 0.160
Table 2. Average V
p
“belongingness” for the regions as K varies from 3 to 10 with μ = 2.
75
μ k1 k2 k3 k4 k5 k6 k7
1.05 0.9871 0.9811 0.9739 0.9737 0.9698 0.9675 0.9567
1.10 0.9715 0.9693 0.9471 0.9454 0.9360 0.9217 0.9159
1.15 0.9563 0.9327 0.9180 0.9109 0.9098 0.8923 0.8602
1.20 0.9381 0.9090 0.8793 0.8713 0.8643 0.7924 0.7799
1.30 0.8954 0.8493 0.8117 0.8007 0.7682 0.6684 0.6479
1.40 0.8484 0.7859 0.7599 0.7115 0.6792 0.5617 0.5383
1.50 0.8032 0.7259 0.7163 0.6195 0.6081 0.4825 0.4540
1.60 0.7597 0.6743 0.6688 0.5439 0.5319 0.4180 0.3920
1.70 0.7173 0.6290 0.6145 0.4861 0.4515 0.3639 0.3436
1.80 0.6758 0.5790 0.5632 0.4317 0.3635 0.3126 0.3004
1.90 0.6183 0.5165 0.4530 0.3683 0.2407 0.2365 0.2260
2.00 0.5747 0.4733 0.3931 0.3301 0.2249 0.2163 0.2056
Table 3. Average value for 𝑢 𝑛 (𝑘 )
for each region as µ increases to 2.0. Regions have been sorted
so that k1 corresponds to the region with the highest average µ and k7 corresponds to the lowest
average μ.
76
Moho k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
K3 26.0 33.5 29.5 X X X X X X X
K4 26.0 33.5 29.0 31.0 X X X X X X
K5 23.5 34.0 29.5 30.0 35.5 X X X X X
K6 23.5 34.0 28.5 30.0 35.5 31.0 X X X X
K7 20.5 34.5 28.5 30.5 35.5 31.0 27.5 X X X
K8 20.5 34.5 28.5 30.5 36.0 30.5 27.5 24.0 X X
K9 20.5 34.5 28.0 31.0 36.0 30.5 27.0 24.0 30.0 X
K10 20.5 34.5 28.0 31.0 36.0 31.0 27.0 24.0 30.0 28.5
Table 4. Depth to Mantle Velocities (km) for each of the regions in all values of K. The specific
geologic regions that these correspond to are provided in Figures 9, 12-18. These values are
calculated by the average depth between Z
4.2
and Z
4.3
for each centroid profile.
77
LVZ k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
K3 0.97 1.03 0.89 X X X X X X X
K4 0.97 1.03 1.01 0.79 X X X X X X
K5 0.96 1.05 0.87 1.02 0.98 X X X X X
K6 0.96 1.05 1.00 1.01 0.98 0.79 X X X X
K7 0.95 1.05 1.00 1.02 0.97 0.79 0.99 X X X
K8 0.96 1.05 1.00 1.02 0.98 0.80 0.98 0.78 X X
K9 0.96 1.05 0.99 1.02 0.98 0.80 0.98 0.78 1.03 X
K10 0.96 1.05 0.99 1.03 0.98 0.74 0.98 0.79 1.02 0.92
Table 5. LVZ behavior for each of the regions in all values of K. The specific geologic regions
that these correspond to are provided in Figurs 9, 12-18. These values are calculated by the
ratio of V
s
at 8 km depth to the V
s
at 14 km depth, therefore a value above 1 indicates the
presence of a LVZ between these depths and a value below 1 indicates the absence of a LVZ.
78
Surf AVG k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
K3 3146 3077 2264 X X X X X X X
K4 3157 3100 2463 2214 X X X X X X
K5 3158 3029 2218 3012 3294 X X X X X
K6 3157 3026 2340 3047 3289 2204 X X X X
K7 3112 3036 2326 2980 3281 2205 3207 X X X
K8 3114 3032 2322 2983 3293 2190 3206 2475 X X
K9 3114 3030 2192 3077 3300 2197 3214 2485 2748 X
K10 3127 3026 2182 3066 3304 2034 3225 2571 2812 2564
Table 6. Average V
s
(m/s) for each region between surface and 5 km depth.
79
Mid AVG k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
K3 3602 3554 3587 X X X X X X X
K4 3598 3548 3665 3507 X X X X X X
K5 3619 3532 3589 3495 3775 X X X X X
K6 3620 3535 3715 3472 3781 3502 X X X X
K7 3611 3533 3717 3483 3779 3499 3576 X X X
K8 3608 3531 3718 3484 3770 3432 3577 3902 X X
K9 3608 3531 3729 3430 3775 3428 3586 3905 3645 X
K10 3617 3529 3729 3440 3768 3491 3591 3941 3696 3397
Table 7. Average V
s
(m/s) for each region between 8-14 km depth.
80
Low AVG k1 k2 k3 k4 k5 k6 k7 k8 k9 k10
K3 4158 3683 3994 X X X X X X X
K4 4161 3674 3965 3972 X X X X X X
K5 4241 3567 4005 3812 3846 X X X X X
K6 4244 3565 4019 3805 3845 3971 X X X X
K7 4400 3552 4028 3765 3840 3969 3925 X X X
K8 4401 3551 4028 3765 3827 3943 3925 4112 X X
K9 4402 3544 4064 3759 3822 3941 3929 4113 3846 X
K10 4407 3543 4060 3756 3814 3945 3924 4118 3871 3963
Table 8. Average V
s
(m/s) for each region between 21-26 km depth.
81
C
1
(kk')
k1 k2 k3 k4 k5 k6 k7
R1 X 1.271 0.491 0.968 0.574 0.619 0.666
R2 1.271 X 0.696 0.493 0.357 0.685 0.920
R3 0.491 0.696 X 0.497 0.390 0.305 0.412
R4 0.968 0.493 0.497 X 0.407 0.530 0.595
R5 0.574 0.357 0.390 0.407 X 0.416 0.382
R6 0.619 0.685 0.305 0.530 0.416 X 0.567
R7 0.666 0.920 0.412 0.595 0.382 0.567 X
C
2
(kk')
k1 k2 k3 k4 k5 k6 k7
R1 X 1.343 0.841 0.949 0.742 0.914 0.697
R2 1.343 X 0.903 0.554 0.456 0.920 0.985
R3 0.841 0.903 X 0.663 0.739 0.566 0.851
R4 0.949 0.554 0.663 X 0.508 0.776 0.556
R5 0.742 0.456 0.739 0.508 X 0.755 0.543
R6 0.914 0.920 0.566 0.776 0.755 X 0.938
R7 0.697 0.985 0.851 0.556 0.543 0.938 X
Table 9. (Top) C
(kk’)
calculated using the L
1
norms. All values are less than 1 except for the
comparison between k1 (Continental Borderlands) and k2 (Sierra Nevada Batholith). The larger
the number the more tightly constrained the region members are compared to the centroid
velocity profiles for each region; (Bottom) same plot but for C
(kk’)
calculated using the L
2
norm.
Note the generally greater values.
82
Figure A1. Recovered checkerboards at (a–d) 5 km, (e–h) 15 km, and (i–l) 25 km depths
computed using the Jacobian of frequency-dependent GSDF phase delay measured on both
earthquake waveforms (EFP) and ambient-noise correlagrams (AFP) (first column), the
Jacobian of GSDF phase-delay measured only on ambient-noise correlagrams (second column),
the Jacobian of GSDF phase-delay measured only on earthquake waveforms (third column), and
the Jacobian of broadband cross-correlation phase-delay measured on both earthquake
waveforms (EBP) and ambient-noise correlagrams (ABP) (fourth column). The size of the
checker is 10 km by 10 km at 5 km depth and 15 km by 15 km at 15 km and 25 km depths. Figure
and text are direct copy of Figure F1 from (Lee et al., 2014a).
83
Figure A2. A tectonic map of the United States Cordillera, showing selected geological/tectonic
features. Figure and text are direct copy of Figure 2 from (Snoke, 2005).
Abstract (if available)
Abstract
Over the past 25 years, researchers have produced 3D velocity models which are used in wave propagation codes to generate synthetic seismograms. While these models are purely numerical, the behavior of the velocity values will correspond well with known geologic features if the inversion is performed accurately. We apply k-means clustering analysis to separate the component Vs and Vp velocity profiles in CVMS-4.26, a seismic velocity model for southern California, into regions without using any a priori identification as to the locations of the profiles. By grouping the velocity profiles into clusters, we also calculate the centroid velocity profile for the region and can characterize the distinguishing features which separate the regions from the model. We also expand on the general k-means clustering by using Fuzzy C-Means clustering to determine a weighted belongingness of each profile to every cluster in a quasi-probabilistic manner. Finally, we map several different features in the depths of the velocity model, including the extent of a pervasive low velocity zone, the general location of the Moho, and averages for the upper, middle, and lower crustal velocities. These geographic maps are then compared to the clustering regionalizations to analyze the effects of these features on the separation of velocity profiles. The results of the k-means clustering demonstrate that the geologic features embedded in the velocity model can be used in the mathematical routine to apportion different regions in the state. The region divisions are strongly correlated with the behavior of the velocity profiles at the surface, the mid-crust, and the Moho depths which present the greatest amount of heterogeneity in the model. The Fuzzy C-Means analysis expands on these results by showing the strongest cohesion among regions which do not contain a low velocity zone in the mid-crust. An in depth statistical analysis is also performed to define the output regions from a probabilistic standpoint. Finally we show that the algorithm can be applied successfully to a high resolution crustal velocity model.
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Eymold, William Karl
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Structural clustering analysis of CVMS-4.26: a 3D seismic velocity model for southern California
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Geological Sciences
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