Stress-strain characterization of complex seismic sources

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STRESS-STRAIN CHARACTERIZATION OF COMPLEX SEISMIC SOURCES

by

Alan Juárez Zúñiga

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(GEOLOGICAL SCIENCES)

December 2022

Copyright 2022 Alan Juárez Zúñiga

ii
Table of Contents
List of Tables .................................................................................................................................................................................................... iv
List of Figures ................................................................................................................................................................................................... v
Abstract ............................................................................................................................................................................................................ xiv
Chapter 1: Stress-Strain Characterization of Complex Seismic Sequences Using Probabilistic
Hierarchical Models…………… .................................................................................................................................................................... 1
Abstract ......................................................................................................................................................................................................... 1
1.1 Introduction ................................................................................................................................................................................ 2
1.1.1 Mechanism Complexity ..................................................................................................................................................... 6
1.1.2 Stress-Strain Characterization (SSC) Model ......................................................................................................... 12
1.2 Earthquake Mechanism Datasets .................................................................................................................................... 20
1.2.1 Fault Zone Seismicity Catalogs ................................................................................................................................... 23
1.2.2 Cumulative Offsets of California Fault Segments ................................................................................................ 29
1.2.3 Ridgecrest Datasets ......................................................................................................................................................... 31
1.3 Inversion of Focal-Mechanism Catalogs for SSC Model Parameters ............................................................... 34
1.3.1 Probabilistic Estimation of the SSC Model Parameters ................................................................................... 36
1.3.2 Observational Earthquake Mechanism Uncertainties ...................................................................................... 41
1.3.3 Probability Distributions of Focal-Mechanism Orientation ........................................................................... 43
1.4 SSC+Noise Model .................................................................................................................................................................... 53
1.4.1 Monte Carlo Simulation of SSC+Noise Mechanism Catalogs .......................................................................... 54
1.4.2 Numerical Integration of SSC+Noise PDFs on the Sphere Surface .............................................................. 58
1.5 Earthquake Catalog Biases ................................................................................................................................................. 66
1.5.1 Orientation Biases ............................................................................................................................................................ 66
1.5.2 Biases From Mechanism Weighting Scheme ........................................................................................................ 70
1.5.3 Sensitivity to Noise Levels ............................................................................................................................................ 74
1.6 The SSC-Lab: Application to Seismic Sequences in California ............................................................................ 76
1.6.1 Seismicity of the San Jacinto Fault Zone ................................................................................................................. 76
1.6.2 Seismicity in the Parkfield Segment of the San Andreas Fault ..................................................................... 79
iii
1.6.3 The 2019 Ridgecrest Sequence .................................................................................................................................. 82
1.6.4 SSC Model of Southern California .............................................................................................................................. 86
1.6.5 SSC Model of Northern California .............................................................................................................................. 89
1.7 Temporal Monitoring of Stress Fields .......................................................................................................................... 92
1.8 Mechanism Complexity and Cumulative Offsets ...................................................................................................... 95
1.8.1 Tectonic Anisotropy ...................................................................................................................................................... 101
1.8.2 𝜅′ 𝑣𝑠.𝜅 .................................................................................................................................................................................. 107
1.8.3 Estimation of the Fault Orientation Concentration Parameter .................................................................. 108
1.9 Discussion and Conclusions ............................................................................................................................................ 112
Acknowledgements ............................................................................................................................................................................. 118
Chapter 2: Effects of Shallow Velocity Perturbations on Three-Dimensional Propagation of Seismic
Waves……………….. ..................................................................................................................................................................................... 119
Abstract .................................................................................................................................................................................................... 119
2.1 Introduction ........................................................................................................................................................................... 119
2.2 Material and Methods ........................................................................................................................................................ 123
2.2.1 Numerical Modeling ...................................................................................................................................................... 123
2.2.2 Spectral Lag-Time Measurements ........................................................................................................................... 125
2.3 Effects of Shallow Velocity Perturbations on Seismic Waves ........................................................................... 128
2.4 Discussion and Conclusions ............................................................................................................................................ 133
Data and Resources ............................................................................................................................................................................. 136
Acknowledgments ............................................................................................................................................................................... 137
References .................................................................................................................................................................................................... 138
Appendix A. Supplemental Material to Chapter 1 ................................................................................................................. 156

iv
List of Tables
Table 1.1. Fault zone information. The codes of each fault zone correspond to Figure 1.4. LON1
and LAT1 are the geographical coordinates (longitude and latitude) of the northern
extreme of the fault segment and LON2 and LAT2 the coordinates of the southern extreme.
Length in kilometers is the distance between points (LON1,LAT1) and (LON2,LAT2). .................. 28
Table 1.2. California faults zones and cumulative offsets. Some faults are divided into sections,
as shown in Figure 1.4. We report the minimum, median, and maximum of all reported
values in the literature. Some references are compilations from other works. Therefore, we
cite the compilation only. ............................................................................................................................................ 31
Table 1.3. Baseline values of the total-moment fractions 𝐹𝛼 for selected values of 𝜎, 𝜅, and for
𝑅 =0.5. Figure 1.18 shows the residuals between these values and inverted. .................................. 75
Table 1.4. Fault zone inversion results. The columns for 𝑅, 𝜅, and 𝐹0 show the MLE value and
the 67% confidence interval. 𝐹0 is the estimation corrected for noise, and 𝛿𝐹𝛼 =
𝐹𝛼(𝑑𝑎𝑡𝑎)−𝐹𝛼𝑚𝑜𝑑𝑒𝑙,𝛼 =0,…,4, are the residuals of data minus model. 𝐹𝛼𝑚𝑜𝑑𝑒𝑙 are
calculated from the SSC+Noise model. .................................................................................................................. 99
Table 2.1. Earthquake locations and source parameters. ................................................................................... 124

v
List of Figures
Figure 1.1. Stress-oriented basis sets {M𝛼:𝛼 =0,…,4} for selected values of the differential
stress ratio 𝑅. The inset plot shows the coordinate system axes. s0 is the direction of
maximum shear traction, n0 the vector normal to the fault plane, and r𝛼 the principal stress
directions. ........................................................................................................................................................................... 18
Figure 1.2. Aki moment fraction 𝐹0 and total moment fractions 𝐹𝛼 as functions of the
differential stress ratio 𝑅 and the stress sensitivity parameter 𝜅. (a) Aki moment fraction
𝐹0. (b) Total moment fraction of zeroth degree (𝐹0). Solid lines are the limits between low
(𝐹+=0.2), and high (𝐹+=0.4) complexity. Dashed lines are intervals of 0.1 𝐹+. (c)-(f)
Moment fractions of higher-degree (𝐹1−𝐹4). ................................................................................................. 19
Figure 1.3. Seismicity distribution along faults in California. (a) Locations of earthquakes in
Northern California. Dots are colored by earthquake hypocenter depth and sized by
earthquake magnitude. Red stars show the epicenters of large earthquakes since 1981.
Black solid lines are major mapped fault traces. The inset map shows the location of the
area of the study. (b) Similar to (a), but for Southern California. SAF is the San Andreas Fault,
and SAF-C is its creeping section in Northern California. ............................................................................. 23
Figure 1.4. California faults and fault zones analyzed. The colored dotted boxes delimit fault
zones, and the solid colored line indicates their traces. Gray dots are epicenters of
earthquakes shown in Figure 1.3. The codes and names of the fault zones are in Table 1. ......... 29
Figure 1.5. Source mechanism uncertainties of the Hauksson, Cheng, and Lin catalogs of the
Ridgecrest sequence. (a) Map showing the mainshock, aftershock, and seismicity epicenters
in the Ridgecrest area. Red stars show the epicenter location of the Mw 6.4 and 7.2
earthquakes of July 2019. Their mechanisms are also shown as focal spheres. Gray dots are
epicenters of earthquakes from 1981 to 2021, and colored dots are the aftershocks used in
this analysis. (b) Earthquake mechanism uncertainty 𝜎 . Histograms are empirical
vi
distributions of 𝜎 estimated from cos𝜎 =cosPU1cosPU2, where PU1 and PU2 are reported
nodal plane uncertainties in the catalogs. Solid lines are their fit by two-parameter gamma
distributions. (c), (d), and (e) show the event counts in logarithmic scale as a function of
magnitude Mw and uncertainty 𝜎 for the Hauksson, Cheng, and Lin catalogs, respectively
(white areas = no events). ........................................................................................................................................... 34
Figure 1.6. SSC model parameter estimation. Input data are a source mechanism catalog 𝒞, from
which ℱ,𝐸0, and 𝔈0, are measured, and together with 𝜎, their empirical distributions are
fitted: 𝛤, ℒ𝒩, and 𝛣 are the Gamma, Logistic Normal, and Beta distributions, respectively.
The likelihood function is proportional to the probability of the data given the SSC+Noise
model parameters. The calculations in the SSC+Noise box correspond to the forward
modeling of SSC+noise moment tensor densities, and more details are shown in Figure 1.11.
................................................................................................................................................................................................. 41
Figure 1.7. Aki moment fraction 𝐹0, zeroth-degree moment fraction 𝐹0, and complexity factor
𝐹+ expectations from a moment tensor density with uncertainties. (a) No complexity 𝜅 →
∞. Uncertainties are one-dimensional only. (b) 𝜅 =20, 𝑅 =0.5, and 𝜎 is the total standard
deviation of four-dimensional noise. (c) 𝜅 =0, 𝑅 =0.5, and 𝜎 is the total standard
deviation of five-dimensional noise. The dashed grey lines divide the low, moderate, and
high complexity regions in all panels. .................................................................................................................... 43
Figure 1.8. Total moment fraction and Aki moment fraction expectations for 3D isotropic
uncertainties on the orientation of double-couple mechanism as functions of the variance
angle. (a) Aki moment fraction 𝐹0, zeroth-degree moment fraction 𝐹0, and complexity
factor 𝐹+=1− 𝐹0 for different values of angular standard deviation 𝜎, such as the von
Mises concentration parameter is 𝜏 =1/𝜎2. (b) Higher-degree total moment fraction 𝐹𝛼 ≥
1. In both panels, dashed lines are computed from truncated distributions with 𝜑𝛼 ≤30𝑜. ...... 47
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Figure 1.9. Visualization of the marginal probability distributions of the normal, shear, and null
vectors (n,s,b) of a double couple mechanism for different values of the von Mises
concentration parameter 𝜏 =1/𝜎2, (a) 𝜎 =5𝑜, (b) 𝜎 =10𝑜, and (c) 𝜎 =15𝑜. ................................. 50
Figure 1.10. Comparison of parametrizations of mechanism orientation uncertainty from
numerical simulations: A mechanism m0 is rotated by a random Rodrigues rotation matrix
R𝜑,u, whose rotation axis u is uniformly distribted in the unit sphere, and rotation angle 𝜑
follows a von Mises distribution with zero mean and concentration parameter 𝜏 =1𝜎2. (a)
Standard deviation of the angles between a rotated mechanism m and the SOR basis set
whose leading term is the mean mechanism m0. (b) Standard deviations of the angles
between the normal, null, and slip directions of the rotated mechanism and the mean
mechanism. (c) Standard deviations of the rotated mechanism’s strike, dip, and rake
directions. ........................................................................................................................................................................... 52
Figure 1.11. Modeling of SSC+noise mechanisms and estimation of synthetic moment fractions.
Input data are values of 𝑅 and 𝜅, and the distribution of mechanism errors. The SSC and
rotation-matrix distributions are sampled to generate a synthetic moment tensor catalog.
From the catalog, empirical distributions of ℱ and 𝐸0 are estimated. .................................................... 54
Figure 1.12. Total moment fractions (𝐹𝛼(𝑅,𝜅,𝜎)) calculated from Monte Carlo simulations of
the SSC and noise distributions. The top panels (a-d) show 𝐹𝛼 for 𝑅 =0.1, and the bottom
panels (e-h) for 𝑅 =0.5. In each panel, colored lines are for selected values of 𝜅. In (c) and
(g) solid lines are 𝐹2 and dashed lines 𝐹3. .......................................................................................................... 56
Figure 1.13. Synthetic Aki moment fraction 𝐹0 and total moment fraction 𝐹𝛼 as functions of 𝑅,
𝜅 , and 𝜎. The gray contour line in (a) and (b) corresponds to 𝐹0=0.6,0.8 and 𝐹0=
0.6,0.8, respectively. In (c)-(f) is 𝐹𝛼 =0.1. ......................................................................................................... 58
Figure 1.14. Synthetic total moment fraction 𝐹𝛼 as functions of 𝑅, 𝜅 , and 𝜎. ............................................. 64
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Figure 1.15. Synthetic total moment fraction 𝐹𝛼 as functions of 𝜅 , and 𝜎, for two selected
values of 𝑅. Top panels are 𝑅 =0.1, and bottom panes for 𝑅 =0.5. ....................................................... 65
Figure 1.16. Projections and angles between the focal mechanism of Hauksson and those of
Cheng and Lin. (a) Empirical distribution of the angle 𝜂𝐻,𝐶 between the focal mechanisms
of Hauksson and Cheng. 𝜇 and 𝜎 are the mean and standard deviations, respectively. (b)
Distribution of 𝑥0𝐻 =M0:m𝑛(𝐻) vs. 𝑥0𝐶 =M0:m𝑛(𝐶). The solid stair-like lines are the
scaled kernel density estimate (histograms) (c) and (e) are similar to (a), but for the pairs
Hauksson-Lin and Cheng-Lin. (d) and (f) are similar to (b). ........................................................................ 70
Figure 1.17. Sensitivity of 𝐸0 and ℱ to mechanism weighting scheme using the Ridgecrest
dataset of Hauksson. We bin earthquakes in magnitude intervals between 1.5 and the upper
cutoff 𝑀𝑤, which ranges between 2-8 with one unit increment. (a) Empirical distributions
of the stress differential ratio as a function of the upper cutoff magnitude. (b)- (g)
Distributions of total moment fractions measured from different weighting schemes of the
mechanisms m𝑛: purple stars and error bars are the average and one standard deviation
confidence interval from the distributions for magnitude weighted mechanism, m𝑛 =
𝑀𝑤,𝑛m𝑛. In blue, the average and confidence interval from moment weighted mechanisms,
m𝑛 =𝑀𝑤,𝑛m𝑛, and in green from unit tensors, m𝑛. ................................................................................... 73
Figure 1.18. Sensitivity of the likelihood inversion scheme to catalog uncertainty calculated
from synthetic catalogs with 𝜅 =5,10,15, 𝑅 =0.5, and 𝜎 =5,20,25,…,50. (a) Estimated
strain sensitivity factor, 𝜅 for different levels of simulated noise uncertainty 𝜎. (b)
Estimated differential stress ratio 𝑅 for different levels of simulated noise uncertainty 𝜎.
(c)-(f) Difference between expected moment fractions from synthetic simulations and the
inverted from the likelihood function. ................................................................................................................... 75
Figure 1.19. SSC modeling of the seismicity in the San Jacinto fault from the dataset in JJ21. (a)
Observed (red) and modeled (blue) distributions of the stress differential ratio (𝐸0). (b)-
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(f) show the observed (red) and modeled (blue) distributions of the total moment fractions.
The red dot is the MLE of total moment fractions corrected for noise. (g) Distribution of
observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. For this dataset,
Γ(𝜎|𝛼 =8.75,𝛽 =2.48), which gives mean𝜎 =21.75𝑜. (h) Distribution of measured stress
principal axis 𝔈0. (i) Likelihood function of the SSC parameters 𝑅,𝜅 estimated from the
observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set
for the San Jacinto fault seismicity. ......................................................................................................................... 78
Figure 1.20. SSC modeling of the seismicity in the Parkfield segment of the San Andreas Fault
as shown in Figure 1.4. (a) Observed (red) and modeled (blue) distributions of the stress
differential ratio (𝐸0). (b)-(f) show the observed (red) and modeled (blue) distributions of
the total moment fractions. The red dot is the MLE of total moment fractions corrected for
noise. (g) Distribution of observed mechanism uncertainty 𝜎, and its fit by a gamma
distribution. For this dataset, Γ(𝜎|𝛼 =34.3,𝛽 =0.82), which gives mean𝜎 =28.4𝑜. (h)
Distribution of measured stress principal axis 𝔈0. (i) Likelihood function of the SSC
parameters 𝑅,𝜅 estimated from the observed shape factor and total moment fractions in
panes (a)-(f). (j) and (k) show the Pearson correlation matrices of observed and modeled
moment fractions. (l) SOR basis set for the San Jacinto fault seismicity. ............................................... 81
Figure 1.21. Stress-oriented representation (SOR) basis set 𝔅M0=M𝛼,𝛼 =0,…,4 from the
Hauksson dataset ( ℋ ). Moment tensors are represented as focal spheres with
compressional quadrants in color and tensional in white. ........................................................................... 84
Figure 1.22. Empirical distributions of the total stress differential ratio (𝐸0), the total moment
fractions (𝐹𝛼), and stress principal directions calculated from the Ridgecrest catalog of
Hauksson. (a) Distributions of 𝐸0. The red histogram is calculated from measured data. The
blue histograms is the empirical distribution of 𝐸0 estimated from a synthetic catalog
x
whose input parameters are the maximum likelihood estimations: R=0.42, 𝜅 =5.2, and
Γ(𝜎|𝛼 =12.6,𝛽 =1.9) where 𝛼 and 𝛽 are estimated from the data uncertainties. The red
dot is the estimation corrected for noise. Panels (b)-(f) show the histograms are empirical
distributions of 𝐹𝛼,𝛼 =0,…,4. The blue histogram is the distribution of 𝐹𝛼 computed from
the maximum-likelihood estimate catalog. The red dot is the estimation corrected for noise
(g) Pearson correlation matrix of the observed 𝐹𝛼. (h) is the Pearson correlation matrix
from modeled 𝐹𝛼. (i) Distributions of the orientation of the stress principal axis 𝔈0. .................... 85
Figure 1.23. Likelihood function 𝐿𝐹𝐸0,ℱ,𝔈0; Σ,𝜅 of the data given the SSC model parameters
computed from three Ridgecrest datasets: (a) Cheng, (b) Hauksson, and (c) Lin. ............................. 86
Figure 1.24. Stress and strain characterization of seismicity in Southern California. (a) Stress
differential ratio 𝑅. (b) Strain sensitivity parameter 𝜅. (c) Stress tensor plotted as source
mechanism and colored by differential stress ratio. Moment tensors of large earthquakes
(Mw>6) from 1974-2019 are indicated by the black and white source mechanisms. ..................... 88
Figure 1.25. Stress and strain characterization of seismicity in Northern California. (a) Stress
differential ratio 𝑅. (b) Strain sensitivity parameter 𝜅. (c) Stress tensor plotted as source
mechanism and colored by differential stress ratio. Moment tensors of large earthquakes
(Mw>6) from 1983-2014 are indicated by the black mechanisms. .......................................................... 91
Figure 1.26. Temporal monitoring of the stress field in the Ridgecrest fault zone using the
mechanism in the catalog of Hauksson. (a) Evolution of the strain sensitivity parameter 𝜅
between the years 2002-2022. (b) Change of the strain sensitivity parameter 𝜅 after the
Mw6.4 foreshock on July 2, 2019. (c) and (d) show the evolution of the differential stress
ratio. (e) and (f) is the azimuth of the principal compressive stress Az(𝜎1). (g) and (h) show
the depth distributions of earthquakes. Dots are the epicenters in time and depth, colored
and sized by earthquake magnitude. ..................................................................................................................... 94
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Figure 1.27. SSC inversion results for the fault zones in Table 1. Colored markers are the
maximum likelihood estimation of the SSC parameters 𝑅 and 𝜅, and gray error bars are
their 67% confidence interval. Gray segmented lines are the limits between low, moderate,
and high complexity intervals of the complexity scale. .................................................................................. 99
Figure 1.28. Correlation between fault cumulative offsets and measures of mechanism
complexity for different fault zones in California. (a) Cumulative slip against strain
sensitivity factor. (b) Cumulative slip against zeroth-degree moment fraction. Horizontal
gray bars indicate the minimum and maximum reported offsets. Vertical bars are the 67%
confidence interval on the estimated 𝜅 and 𝐹0 corrected for noise. 𝑟 is the Spearman
correlation coefficient. ................................................................................................................................................ 100
Figure 1.29. Scatter plot showing the correlations between residuals of total moment fractions.
Residuals are 𝛿𝐹𝛼 =𝐹𝛼(𝑑𝑎𝑡𝑎)−𝐹𝛼𝑚𝑜𝑑𝑒𝑙,𝛼 =0,…,4. 𝑟 is the Pearson’s correlation
coefficient. ........................................................................................................................................................................ 101
Figure 1.30. Examples of spherical distributions of observed and modeled fault orientation
vectors. We present data only for the EGM, SAP, and IMP fault segments. 𝑝n is the empirical
distribution of observed vectors and 𝑝𝑠n;𝜅,𝑠 is the SSC probability distribution from the
best fit of parameters 𝜅, and 𝑅. Spherical distributions are plotted in the orthogonal
projection perpendicular to r1. .............................................................................................................................. 105
Figure 1.31. Probability density function 𝑝n,𝜅,𝜖 ∝exp𝜅cos2𝜃0n+𝜖cos2𝜃1n, for selected
values of 𝜖 and 𝜅, and R = ½. The projection is orthogonal to r1 ⁠. The PDF is scaled to a
maximum value of unity. ........................................................................................................................................... 106
Figure 1.32. Total moment fractions 𝐹𝛼 as functions of the stress sensitivity parameter 𝜅 and
fault concentration parameter 𝜖. We set R = ½. .............................................................................................. 106
Figure 1.33. Effect of 𝑝n∝e𝜅𝑠n and ∝e𝜅′cos2𝜃0n in the estimation of the strain sensitivity
factor and the zeroth-degree moment fraction corrected for noise. In the inversions for 𝜅′
xii
we set 𝜖 =0. (a) 𝜅 against 𝜅′. (b) 𝐹0′ against 𝐹0. The gray dashed line is the fit by a line to
the estimations. ............................................................................................................................................................. 108
Figure 1.34. Anisotropic SSC inversion results for the fault zones in Table 1. Colored markers
are the maximum likelihood estimation of the SSC parameters 𝑅 and 𝜅, and 𝜖. Gray error
bars are their 67% confidence interval. (a) Cumulative slip against fault orientation
concentration parameter. (b) Estimations of stress-differential ratio and strain sensitivity
factor. The low-moderate complexity boundary corresponds to 𝐹0=0.8 and the
moderate-high to 𝐹0=0.6 computed for 𝜖 =0 . (c) Cumulative slip against strain
sensitivity factor. (b) Cumulative slip against zeroth-degree moment fraction corrected for
noise. 𝑟 is the Pearson’s correlation coefficient. ............................................................................................. 110
Figure 1.35. Correlations between residuals of total moment fractions. Residuals are 𝛿𝐹𝛼 =
𝐹𝛼(𝑑𝑎𝑡𝑎)−𝐹𝛼𝑚𝑜𝑑𝑒𝑙,𝛼 =0,…,4. 𝑟 is the Pearson’s correlation coefficient. 𝐹𝛼𝑚𝑜𝑑𝑒𝑙 are
estimated from numerical simulations using the MLE estimations of the SSC parameters 𝑅,
𝜅, and 𝜖. ............................................................................................................................................................................. 111
Figure 2.1. Community Velocity Model (CVM-S4.26) of Southern California and earthquake
locations in the simulation region. (a) Shear wave velocity (Vs) at the surface of the
simulation region. The source locations and mechanisms are indicated by the focal
mechanisms (black-compression, white-extension). The shear-wave velocity is reduced by
30% on the surface within the Los Angeles basin. The star in the LA basin indicates the
location of the profiles in (b). Gray-thin lines are faults in Southern California. (b) Velocity
and density profiles on the top 5 km of the crust on the LA basin region. The shear wave
velocity is reduced by 30% on the top 0.5 km, The P-wave speed and density are not
modified. ........................................................................................................................................................................... 124
Figure 2.2. Time-lag measurements using the phase cross-correlation spectrum. (a) Synthetic
seismograms computed with the reference velocity model and the modified model. (b)
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Wavelet phase cross-spectrum 𝜑𝑔,ℎ𝜔,𝑡. The dashed line is the cone of influence. (c) The
relative time-shift calculated from the cross-spectrum in (b). Red and blue colors indicate
positive and negative time lags, respectively. We taper the time-frequency dependent time
shifts outside the cone of influence. The solid line indicates the profile at 10 s period used
in (d). (d) Timeshift at 10 s period. The orange dots show the sections that the algorithm
considers for the averaged measurement. ........................................................................................................ 127
Figure 2.3. Validation test of the measuring technique. (a) Time-frequency dependent velocity
change model. The 𝛿𝑣𝑣 model is constant in time and in frequency varies as 0.1sin10𝜔. (b)
Theoretical phase cross-spectrum. (c) Expected time-frequency dependent time-shifts the
synthetic model of 𝛿𝑣𝑣. (d) Reference and perturbed signals. Both signals have the power
spectra of red noise. (e) Calculated phase cross-spectrum. (f) Reconstructed time-shifts. ....... 128
Figure 2.4. Period-dependent distributions of velocity perturbations and spectral amplitude
reduction times. The top row shows the estimated relative velocity perturbation (𝛿𝑣𝑣) at
(a) 5 s, (b) 10 s, and (c) 20 s. The bottom row shows the measured amplitude reduction
times (−𝑙𝑜𝑔𝐴1𝐴2𝜔) at (a) 5 s, (b) 10 s, and (c) 20 s. In all the subplots, the red triangle
indicates the location of the record in Figure 2.2. The thick red line is the border of the LA
basin, where the velocity was reduced by 30%. The focal mechanism indicates the
earthquake location and source mechanisms. ................................................................................................. 132
Figure 2.5. Average estimations of relative velocity perturbations and amplitude reduction
times from the ten pairs of simulations. The top row is the average velocity perturbations
measured at (a) 5s, (b) 10 s, and (c) 20 s. The bottom row is the average amplitude
anomalies at (d) 5 s, (c) 10 s, and (f) 20 s. The thick red line is the border of the LA basin,
where the velocity was reduced by 30%. The focal mechanism indicates the earthquake
location and source mechanisms. .......................................................................................................................... 133

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Abstract
Earthquakes are complex physical systems that involve dynamic stress changes, fracture and damage
generation, and slip. Furthermore, they release accumulated strain energy in the Earth's crust as heat
and seismic waves. Understanding the physical processes that initiate, propagate, and terminate
earthquake ruptures, as well as the interaction of seismic waves with complex geological structures,
is essential for modeling seismic hazards and improving earthquake forecasting systems.
In this work, I investigate two aspects of the earthquake problem: (1) characterization of complex
earthquakes and seismic sequences and the tectonic stress fields that drive them, and (2)
understanding the effects of changes in rock properties on the propagation of seismic waves.
Chapter 1 is devoted to characterizing the mechanism complexity observed in earthquakes and
seismic sequences. Section 1.1 summarizes our mathematical theory for representing complex
seismic sources using orthogonal moment tensor fields (Jordan & Juarez, 2019; Jordan & Juarez,
2020). Moreover, we introduce the concept of earthquake mechanism complexity and the total
moment fractions that measure the seismic moment partitioning into a basis set of up to six-moment
tensors. The total moment fractions are a measure of mechanism complexity.
In a subsequent paper (Jordan & Juarez, 2021), we developed a physics-based probabilistic model
that relates the total moment fractions to the driving tectonic stress. The new stress-strain
characterization (SSC) model relates the short-term seismic response (strain) with the long-term
geological forcing (stress). In Chapter 1, we developed the machinery to test the hypothesis that
mechanism complexity is governed by tectonic stress.
We implement the SSC model in a maximum likelihood estimation technique that uses catalogs of
earthquake mechanisms to estimate the principal-stress directions, the differential stress ratio, and
the strain sensitivity factor that measures the strain dependence on the shear-stress magnitude.
Section 1.2 discusses the data sets we use to characterize the stress fields and analyze biases in the
inversions. In Section 1.3, we develop the maximum likelihood estimation technique to invert for the
xv
stress parameters from observations of earthquake mechanisms. Section 1.4 extensively discusses
how to handle uncertainties in the estimations. In Section 1.5, we analyze possible biases in focal
mechanism orientations that might result from the modeling assumptions or poor data quality used
for estimating them. Observational errors and biases in the orientations of earthquake mechanisms
can be interpreted as physical complexity. Therefore, we investigate the effect of different
parameterizations of orientation uncertainty on the total moment fractions and the estimations of
the stress parameters. We implement the forward and inverse modeling capabilities into a software
package named SSC-Lab.
In Section 1.6, we applied the SSC-Lab to seismic sequences of fault zones in California and developed
a new stress-strain model for Northern and Southern California. We show that our analysis technique
provides new insights into the plate boundary's regional and local tectonics, the internal structure of
faults, and the tectonic stress spatial variation. Furthermore, in Section 1.7, we show the potential of
the SSC-Lab for temporal changes in the stress field.
Sections 1.8 and 1.9 show the stress inversions of 25 fault zones in California and compare our
estimations of stress concentration with estimates of geological offsets. We show that the SSC model
optimally describes the tectonic stress fields of fault zones and that our measures of stress
concentration from mechanism complexity correlate with geological offsets. Furthermore, we find
that distributions of mechanism orientation allow for extracting information about fault plane
orientation concentration in fault zones. As fault zones evolve, fault planes align with the direction of
tectonic motion making these preferred directions weak zones more likely to fail than expected by
the simple stress concentration.
Chapter 2 focuses on understanding the effects of reductions of seismic velocities in the shallow crust
on seismic wavefields. Seismic waves from repeating earthquakes and background seismic noise can
be used to monitor temporal changes in rock properties before, during, and after big earthquakes.
Many studies have used long-period surface seismic waves, and because they have resolution to
xvi
deeper depths, detected changes are interpreted to occur at seismogenic depths. However, it is
unclear whether anomalies detected occur at seismogenic depths, where the confining pressure
promotes rapid healing of the rock, or they are shallow changes, where at low confining pressures,
the spread of rock damage might be more prominent and the recovery slower. The problem is that
small perturbations at depth might have the same effect that strong shallow perturbations. Thus, in
Chapter 2, we elucidate whether changes in mechanical properties of shallow rocks could explain
waveform perturbations over long periods. For that aim, we performed three-dimensional wave
propagation simulations and measured the difference between a reference and a perturbed
wavefield from a velocity model that includes surficial velocity reductions.
Our study concludes that seismic velocity changes at shallow depths have effects that might be
misinterpreted to occur at seismogenic depths. Furthermore, our observations have implications for
waveform tomography, where shallow-low velocities are often ignored.
Chapter 2 has been previously published in the Seismological Research Letters journal (Juarez &
Ben-Zion, 2020).
1
Chapter 1: Stress-Strain Characterization of Complex Seismic Sequences Using
Probabilistic Hierarchical Models
Alan Juárez Zúñiga and Thomas H. Jordan
Department of Earths Sciences, University of Southern California

Abstract
Earthquake ruptures and seismic sequences are the inelastic strain response to tectonic stress fields.
We develop a physics-based probabilistic model to characterize stress-strain fields using moment
measures of mechanism complexity. Distributions of focal mechanisms in seismic sequences are used
to estimate the ratio of the Aki seismic moment to the total moment and the partitioning of the
moment tensor field over an orthonormal basis of five deviatoric mechanisms. These moment
measures of complexity are then used to estimate the principal-stress directions, a differential stress
ratio (𝑅), the complexity factor (𝐹

!
), and a stress-strain sensitivity parameter (𝜅). Large values of 𝐹

!

indicate high mechanism complexity, and high 𝜅 indicates high-stress concentration. Additionally,
we investigate differences between observed and modeled moment measures of complexity to
isolate information on fault zone anisotropy. We carefully quantify the effect of focal mechanism
uncertainty on the moment measures using a hierarchical probabilistic model. We apply the model
to characterize the complexity of stress-strain fields of seismicity in California, and we show that the
observed complexity is related to evolutionary geologic variables such as the cumulative slip in the
fault.

2
1.1 Introduction
Earthquakes and seismic sequences are complex phenomena that involve dynamic stress changes,
fracture, damage generation, and slip. Understanding stress and strain processes that initiate,
propagate and terminate earthquakes and their interseismic evolution is essential for quantifying
seismic hazards. Furthermore, elucidating these processes helps to make inferences about the
dynamics of lithospheric deformation and the mechanical properties of rocks.
Fault zones are complex structures of interconnected surfaces that show roughness at several scales.
Earthquakes occur preferentially on faults optimally oriented with the current, local tectonic stress
field in directions that maximize the shear stress (Fletcher et al., 2016; Harris & Simpson, 1992; King
et al., 1994; Segou & Parsons, 2020; Stein et al., 1994; Stein et al., 1992). However, observations of
complex earthquakes and seismic sequences such as the Mw 7.2 El Mayor earthquake of 2010 and
the Mw 7.8 Kaikoura, New Zealand earthquake of 2016 have shown that failure along non-optimally
oriented surfaces can occur due to slip-plane heterogeneities. During rupture propagation, dynamic
stress changes can critically stress faults not optimally aligned with the background stress and
consequently promote or prevent ruptures in unexpected directions (Ando & Kaneko, 2018; Cochran
et al., 2020; Dieterich & Smith, 2009; Fletcher et al., 2016; Segou & Parsons, 2020; Xu et al., 2018).
Stress heterogeneities can result from geometrical complexities or roughness in the source volume,
rock deformation, and permeability evolution. Heterogeneities significantly affect the nucleation,
propagation, and termination of earthquake ruptures and other characteristics such as frequency-
size distribution and temporal occurrence of earthquakes in faults (de Joussineau & Aydin, 2009;
Wechsler et al., 2010). Moreover, it is uncertain whether the local stress heterogeneities can be more
significant than the background tectonic stress after large earthquakes (Hardebeck, J. L., 2010;
Hauksson, 1994; Li & Cai, 2022; Smith & Dieterich, 2010; Stein et al., 1994; Stein et al., 1992). Thus,
it is crucial to obtain robust measures of seismic source complexity to characterize the evolutionary
stage and geometrical properties of active faults.
3
Detailed mapping of fault zones, fault lengths and stepovers, and other seismological observations,
such as seismicity distribution, have shown that the structural complexity of fault zones decreases
with increasing deformation (Brodsky et al., 2011; Powers & Jordan, 2010; Sagy et al., 2007).
Earthquake ruptures produce fault zone heterogeneities that are smoothed through repeated
rupturing over several earthquake cycles resulting in simpler localized structures. Furthermore,
earthquakes can rupture several faults removing stepovers and constructing larger faults (de
Joussineau & Aydin, 2009). Consequently, stress can concentrate along more localized and larger
fault volumes promoting the occurrence of larger magnitude events (Martínez-Garzón et al., 2015;
Wechsler et al., 2010). Quantifying mechanism complexity as a response to the tectonic stress field
and its relationship with other evolutionary properties of faults would improve our understanding
of the earthquake systems progression and potentially detect stress-strain earthquake precursors
(Li & Cai, 2022).
In two previous papers (Jordan & Juarez, 2019; Jordan & Juarez, 2020), which we refer to as JJ19 and
JJ20 hereafter, we formulated a mathematical representation of seismic sources using orthogonal
moment tensor fields and their polynomial moments. The representation technique partitions the
moment-tensor density or stress glut (Ampuero & Dahlen, 2005; Backus & Mulcahy, 1976; Backus,
1977; Bukchin, 1995) by projections onto an orthonormal basis of moment tensors led by the
mechanism of the centroid moment tensor, CMT (Dziewonski et al., 1981; Ekström et al., 2012). This
representation accounts for the total moment of the source that is not observed in long-period
seismic waves used to calculate the CMT. Furthermore, we demonstrated that seismic wavefields can
be inverted to estimate higher-degree terms in the polynomial moment expansion. Therefore, our
model provides an alternative representation of seismic waveforms that, in general, is simpler than
finite fault models because it contains fewer parameters but goes beyond the traditional point source
representation by a centroid moment tensor (Chen et al., 2005; Clévédé et al., 2004; McGuire et al.,
2001).
4
In a subsequent paper (Jordan and Juarez, 2021), JJ21, we developed a physics-based probabilistic
model to characterize the stress field and its strain response during large earthquakes and seismic
sequences. The moment-tensor density is assumed to be a stochastic field of elementary double
couples distributed in a finite space-time volume with a well-defined CMT. The observables are the
scalar moments obtained by projecting this deviatoric density onto a basis set of five source
mechanisms defined by the CMT and its orthogonal rotations. The parameters of this stress-strain
characterization (SSC) model are the principal-stress directions, the differential stress ratio (𝑅), and
a strain sensitivity factor (𝜅).
The SSC model conforms to the standard assumptions used in stress inversions; (1) the ambient
stress is constant within the volume of interest, (2) the slip vectors of earthquake ruptures are
aligned with the shear stress resolved on the fault surface, and (3) the seismic energy is released on
surfaces with higher shear tractions. The second assumption, the Wallace-Bott condition (Angelier,
1994; Bott, 1959; Célérier et al., 2012; Hardebeck, J. L., 2015b; Wallace, 1951), implies that the local
fault mechanism is uniquely specified by its fault-normal unit vector. Consequently, we can explicitly
incorporate assumption (3) into our model using a strain-sensitivity factor 𝜅 that relates the
probability of slip on a fault element to the magnitude of the shear traction acting across it. Increasing
𝜅 concentrates the energy release into optimal mechanisms and decreases mechanism complexity
(JJ21). Although deviations from the Wallace-Bott condition can be caused by small-scale stress
heterogeneities, rupture interactions, fault-oriented anisotropy, and fault roughness (Lisle, 2013),
various studies have shown that the Wallace-Bott condition accurately describes the average rupture
behavior over seismogenic volumes (Angelier, 1979; Carey-Gailhardis & Louis Mercier, 1987;
Célérier, 1988; Etchecopar et al., 1981; Lisle, 1992; McKenzie, 1969).
Focal mechanisms and moment tensors of earthquakes have been reliably inverted for regional
stress and strain field orientation and magnitude (Abolfathian et al., 2020; Gephart & Forsyth, 1984;
Hardebeck, Jeanne L. & Hauksson, 2001; Hardebeck, Jeanne L. & Michael, 2006; Lisle, 1992; Martínez-
5
Garzón et al., 2016; Matsumoto, S., 2016; Terakawa & Matsu'ura, 2010). The resulting stress models
have helped to understand critical properties of the earthquake phenomena, such as how stress is
distributed in faults and blocks during the earthquake cycle, how stresses initiate, propagate, and
stop seismic ruptures, and how stress is transferred in complex fault systems.
We use the theory developed in our previous papers (JJ19, JJ20, and JJ21) to investigate how the
mechanism complexities observed in large tectonic earthquakes and seismic sequences are related
to tectonic stress and other evolutionary properties of faults, such as cumulative offset. Furthermore,
we expand the SSC model to extract information about tectonic anisotropy observed as preferred
orientations of fault planes. We use ensembles of earthquake focal mechanisms to calculate empirical
distributions of moment measures of mechanism complexity. From them, using a maximum
likelihood estimation (MLE) technique, we estimate probability distributions for the parameters that
characterize the stress-strain fields. We find that errors in the orientations of observed focal
mechanisms, for example, due to poor station coverage and modeling assumptions, contribute to the
observed moment measures of mechanism complexity. Hence, mechanism observational errors bias
the SSC parameter estimations. The apparent complexity is accounted for through a hierarchical
probability model where SSC parameters and observations are random variables. Probability
functions and the dependencies between random variables are computed using Monte Carlo
sampling and integration techniques. Thus, our new SSC+Noise model estimates and propagates the
effects of observational errors into moment measures of complexity and finally into the SSC model
parameters.
The MLE inversion technique uses the distributions from the SSC+Noise model to calculate
probability distributions for the reduced stress tensor and the strain sensitivity parameter.
Estimation uncertainties result from uncertainties in earthquake mechanism data. Comparison
between observed moment tensor fields and SSC+Noise synthetic simulations show that moment
6
fractions also contain information about preferred orientations of failure, which we interpret as
tectonic anisotropy.
In the following sections, we summarize the concept of mechanism complexity of earthquakes and
seismic sequences and their moment measures introduced by JJ20. Then, we reintroduce the stress-
strain characterization theory of JJ21, and from it, we develop the hierarchical probabilistic model to
account for uncertainties in the focal mechanism observation. The computation capabilities,
including the MLE technique and the SSC+Noise numerical simulations of moment tensor fields, are
adapted into MatLab software that we named the SSC-Lab. We present examples of how to use the
SSC Lab and apply it to investigate the stress-strain fields of the Ridgecrest sequence of 2019
(Brandenberg et al., 2020; Cheng, Y. & Ben-Zion, 2020; Lin, G., 2020; Rodriguez Padilla et al., 2022;
Ross et al., 2019; Shelly, 2020). Moreover, we present an application of the SSC-Lab to monitor
temporal changes of stress. Then, we calculate the SSC parameters of 25 fault zones in California and
show that moment measures of mechanism complexity and the strain sensitivity parameter are
correlated with evolutionary parameters of faults such as cumulative displacements. We investigate
differences between observed and modeled moment measures of complexity and find that a simple
anisotropic SSC model with one extra parameter can reproduce the observations. Finally, we present
the new SSC model of Northern and Southern California that contains information about the
orientation of the stress tensor, the strain sensitivity parameter, and the stress differential ratio.
1.1.1 Mechanism Complexity
Earthquakes are a strain response to tectonic stress loading. Therefore, the mechanism complexity
observed in earthquake ruptures and seismic sequences can be used to make inferences about the
complexity of the stress fields that govern the deformation process (Apperson, 1991). The
assumption that slip on faults occurs in the direction of maximum shear stress, known as the Wallace-
Bott condition (Angelier, 1994; Bott, 1959; Wallace, 1951), is the basis for most stress inversion
7
techniques (Célérier et al., 2012; Gephart & Forsyth, 1984; Hardebeck & Michael, 2006; Kassaras &
Kapetanidis, 2018; Lisle, 1992). Here, we formulate a stress inversion scheme that characterizes the
stress-strain complexity from observations of earthquake mechanism complexity.
Previous studies have attempted to quantify the complexity observed in earthquakes and seismic
sequences, e.g., by mapping fault traces and determining relationships between parameters such as
length and number of step-overs (Anderson et al., 2017; de Joussineau & Aydin, 2009; Martínez-
Garzón et al., 2015; Torabi, Anita & Berg, 2011; Wechsler et al., 2010). Other measures of mechanism
complexity of a single earthquake estimate, for example, the invariants of the centroid moment
tensor (CMT) and use them to obtain the ratio of double-couple (DC) and compensated linear vector
dipole (CLVD) components (Cesca & Heimann, 2018; Kagan & Knopoff, 1985; Kagan & Jackson, 2014).
Earthquakes and seismic sequences with any degree of geometrical complexity can be approximated
by discrete distributions of elementary dislocations in a space-time volume. A discrete and finite
moment tensor distribution is
𝐦(𝐱) = 𝑚
"
𝐦
"
𝛿(𝐱−𝐱
"
),
$
"%&
(1)
where 𝐦
"
are unit moment tensors (𝐦
"
:𝐦
"
=1), and 𝑚
"
are the scalar moments of N sources.
𝛿(𝐱) is the Dirac delta function and 𝐱
"
=(𝐫
"
,𝑡
"
) is the spatiotemporal location of the elementary
source indexed by 𝑛. Examples of discrete moment distributions are finite fault models (Hayes, 2017;
Mai & Thingbaijam, 2014) and earthquake mechanism catalogs (Ekström et al., 2012; Hauksson et
al., 2012). In finite-fault models, the set {𝐱
"
} defines the source geometry consisting of 𝑛 discrete
patches over a prescribed fault surface, and the source mechanisms (𝐦
"
) are typically constrained
to double couples. In seismicity catalogs, 𝐱
"
is the earthquake location, and 𝐦
"
the source
mechanism.
8
Equation 1 gives a discrete approximation to the stress-glut density 𝚪
̇(𝐱) of Backus & Mulcahy
(1976). The stress-glut is defined as the difference between the model elastic stress and the true
stress, and its integral over the source volume is the net moment tensor,
𝐌
𝟎
=
1
√2
𝚪
̇(𝐱)𝑑𝐱.
𝐱
(2)
According to our definition, √2𝐌
𝟎
is the centroid moment tensor (CMT) of Dziewonski et al. (1981).
The stress-glut density and its CMT characterize the physical process of elastic wave excitation by
earthquakes and other indigenous seismic sources (Ampuero & Dahlen, 2005; Ben-Zion, Yehuda &
Lyakhovsky, 2019; Dahlen & Tromp, 1998). The stress-glut density is identically zero everywhere
except in the source volume, where the elastic constitutive relations break down during finite strain.
This work focuses on discrete distributions due to the datasets we analyze in the following sections:
earthquake focal-mechanism catalogs and finite fault source models.
The net moment tensor of the discrete distribution is the sum of all elementary sources,
𝐌
𝟎
= 𝑚
"
𝐦
"
,
$
"%&
(3)
and its Euclidean norm, 𝑀
)
=‖𝐌
𝟎
‖
&/+
is the traditional scalar seismic moment, which we call the
Aki moment (Aki, 1966; Jordan & Juarez, 2019).
In JJ19, we begin with the fact that the Aki moment does not measure all the seismic moment for
stress gluts with any degree of mechanism complexity. We defined the scalar total-moment to be the
sum of the scalar moment density, or equivalently, the integral of the norm of the stress-glut field,
𝑀
,
= 𝑚
"
$
"%&
=
1
√2
𝚪
̇(𝐱)𝑑𝐱
𝐱
. (4)
𝑀
,
is computed by integrating the tensor norm of 𝚪
̇(𝐱) whereas 𝑀
)
is the norm of the integral. By the
Cauchy–Schwarz inequality, 𝑀
,
must be greater than or equal to 𝑀
)
. Hence, mechanism complexity
measures the difference between the total scalar moment 𝑀
,
and the Aki moment 𝑀
)
. The
representation of seismic sources using the polynomial moments of orthogonal moment tensor fields
9
of JJ2020 allows quantifying the scalar moment not measured by 𝑀
)
and is contained in higher-
degree moments (Chen et al., 2005; Juarez & Jordan, 2020; McGuire et al., 2001). 𝐌
𝟎
and therefore,
its norm 𝑀
)
are estimated from long-period seismic waves that average over small-scale processes
in the source. Thus, estimations of 𝐌
𝟎
neglect seismic moment that radiates higher-frequency waves
and moment in directions orthogonal to 𝐌
𝟎
. Possibilities to account for the “residual” moment are to
invert for slip distributions on prescribed faults geometries or to expand the moment tensor
representation of seismic sources.
Any moment tensor field can be represented as the sum of up to six orthogonal moment tensor fields.
A discrete field representation is
𝐦= 𝐌

-
.
-%)
𝑚
"
cos𝜃
-,"
$
"%&
, (5)
where 𝐌

0
is a basis set of orthonormal symmetric moment tensors 𝐌

0
:𝐌

1
=𝛿
-2
, 𝑑 is the
reduced dimension, such that 𝑑+1 is the number of moment tensors in the basis set to fully
represent the field. The cosine fields are defined by the projections
cos𝜃
-,"
=𝐌

0
:𝐦
"
. (6)
For every elementary source 𝐦
"
, the direction cosines satisfy
cos
+
𝜃
-,"
.
-%)
=1. (7)
Is there an optimum basis set 𝐌

-
,𝑎 =0,…,𝑑 to represent seismic sources? Any basis set can be
used in the stress-glut representation in equation 5; however, not all of them might have an intuitive
geometrical or physical interpretation. In JJ20, we discussed two useful representations; 1) the
moment-oriented representation (MOR), in which the centroid moment tensor is the leading term,
and subsequent terms are estimated by maximizing the polynomial moments of order equal to the
degree of residual fields, and 2) the principal component analysis (PCA) basis set, whose elements
are the eigentensors of the stress glut density variance over the source volume. In JJ21, we propose
10
the stress-oriented representation (SOR), in which the leading term is the unit CMT mechanism 𝐌

𝟎
,
and subsequent terms are orthogonal rotations about its principal axis.
The coherency in the orientation of 𝐌
𝟎
observed in some regions suggest that source kinematics are
governed by the mean state of stress, and that 𝐌
𝟎
contains information on the average deformation
observed in earthquakes and seismic sequences; 𝐌
𝟎
is proportional to the net strain tensor and the
coefficient of proportionality is the shear modulus (Jackson & McKenzie, 1988; Kostrov, 1976;
Pondrelli et al., 1995). Furthermore, 𝐌
𝟎
is directly related to the stress tensor through an averaging
kernel that can be interpreted as the elastostatic Green’s function (Matsumoto, 2016; Terakawa &
Matsu'ura, 2008).
Following JJ21, here we use the SOR and take 𝐌

𝟎
as a leading term in the representation. The net
moment tensor becomes
𝐌
𝟎
= 𝐌

)
𝑚
"
cos𝜃
),"
.
$
"%&
(8)
The Aki moment is
𝑀
)
= 𝑚
"
cos𝜃
),"
3
"%&
. (9)
To quantify the moment not measured by 𝑀
)
and its difference to 𝑀
,
, in JJ19, we introduced the
moment fractions, which are the average of the scalar moment density over the quadratic direction
cosines,
𝑀

-
= 𝑚
"
cos
+
𝜃
-,"
3
"%&
. (10)
The total moment is
𝑀
,
= 𝑀

-
.
-%)
. (11)
11
A comparison between equations 9 and 10 shows that since |cos𝜃
)
|≤1, then 𝑀

)
≤𝑀
)
, and 𝑀
)
≤
𝑀
,
as we discussed earlier. Hence, a measure of complexity is the ratio of the Aki moment to the total
moment, 𝑀
)
/𝑀
,
. This dimensionless scalar is unity when there is no mechanism complexity and, in
JJ21, we found that it decreases to 𝑀
)
/𝑀
,
~0.6 in the limit of highest complexity for stress-aligned
moment tensor densities. Some studies of mechanism complexity have referred to this ratio as the
seismic consistency index and used it to investigate the complexity of focal mechanisms in volcanic
zones and plate boundaries (Apperson, 1991; Bailey et al., 2010).
For convenience, we defined the unitless moment fractions of the total moment. They are the moment
fractions normalized by the total moment,
𝐹

0
=
𝑀

0
𝑀
,
, 𝐹

-
.
-%)
=1. (12)
The moment fractions of the total moment form a simplex, i.e., they can be modeled statistically using
distributions over proportion variables (Aitchison, J. & Shen, 1980; Aitchison, John, 1986; Barndorff-
Nielsen & Jørgensen, 1991). The moment-fraction simplex allows us to define a measure of
complexity that we call the complexity factor,
𝐹

!
= 𝐹

-
4
-%&
=1−𝐹

)
. (13)
𝐹

!
is a measure of mechanism complexity of the moment tensor field. It quantifies the proportion of
seismic moment not measured by the CMT. A qualitative scale of complexity based on values of 𝐹

!
is
given by three intervals designated as low complexity if 0≤𝐹

!
<0.2, moderate complexity for 0.2≤
𝐹

!
<0.4, and high complexity 0.4≤𝐹

!
<0.6. These values correspond to Aki moment fractions of
90-100% for low-complexity fields, 75-90% for moderate-complexity fields, and 60-75% for high-
complexity fields.

12
1.1.2 Stress-Strain Characterization (SSC) Model
Tectonic stresses govern the processes of deformation, fracturing, and release of seismic energy in
faults. Moreover, fault complexities like roughness, branches, steps, and edges dynamically interact
with tectonic stresses promoting the formation and reactivations of faults and damage generation
(Apperson, 1991; Crider & Pollard, 1998; Kim, Y. et al., 2004; Vavryčuk et al., 2021). Seismic moment
tensors provide information on the complex fracturing and deformation process of earthquakes.
They thus can be used to make inferences about the stress-strain processes that promote, propagate,
and terminate earthquakes.
There are three common assumptions to characterize stress fields from source mechanisms
observed in seismic sequences (Angelier, 1994; Carey-Gailhardis & Louis Mercier, 1987; Gephart &
Forsyth, 1984; Hardebeck & Michael, 2006; Lisle, 1992; Martínez-Garzón et al., 2016; Michael, 1984;
Vavryčuk, 2014): (1) The ambient stress is constant in a volume. If there are variations, they are small
and average to zero. (2) The slip vectors of earthquake ruptures are stress aligned, i.e., their
orientation is parallel to the maximum shear traction across the fault plane, and (3) seismic energy
is preferentially released on surfaces with high shear tractions. The alignment of the fault slip with
the shear traction is known as the Wallace-Bott condition (Bott, 1959; Wallace, 1951).
In JJ21, we use the concept of mechanism complexity to derive a theory that relates stress fields to
the mechanism complexity observed in earthquakes and seismic sequences. Furthermore, we
demonstrated that complexity measures could be used to invert the parameters that characterize the
stress field: the principal stress directions, the differential stress ratio, and the stress concentration
parameter. Consider the deviatoric stress tensor in its principal-axis frame,
𝚺=𝜎
&
𝐫
&
𝐫
&
+𝜎
+
𝐫
+
𝐫
+
+𝜎
5
𝐫
5
𝐫
5
, (14)
in which 𝜎
6
are the principal stresses, such as ∑ 𝜎
6
5
6%&
=0, ∑ 𝜎
6
+
=1
5
6%&
, 𝜎
&
≥𝜎
+
≥𝜎
5
, and the set 𝕽=
(𝐫
&
,𝐫
+
,𝐫
5
) are the stress principal directions. The orientations of 𝕽, and in particular of 𝐫
&
and 𝐫
5

define the expected tectonic regime. If 𝐫
&
and 𝐫
5
are horizontal and 𝐫
+
vertical, transform faulting is
13
expected. If 𝐫
+
and 𝐫
5
are horizontal and 𝐫
&
vertical, normal faulting is expected, and finally, for 𝐫
5

vertical and 𝐫
&
and 𝐫
+
horizontal, a reverse system is expected.
The differential stress ratio is
𝑅 =
𝜎
&
−𝜎
+
𝜎
&
−𝜎
5
, (15)

such as 0≤𝑅 ≤1. Similarly, different values of 𝑅 help interpret the tectonic setting. In general, if
𝑅 <0.5 , a tensional regime is inferred (“transtensional”), and if 𝑅 >0.5 , compressional
(“transpressional”). Stress inversion techniques seek to estimate functionals of the stress tensor, for
example, the set of stress principal direction 𝕽(𝚺

)=(𝐫
&
,𝐫
+
,𝐫
5
) and the differential stress ratio 𝑅(𝚺

),
typically from observations of focal mechanism and other geodetical and geological fault slip data.
Earthquakes are the partial release of stress on a fault, and they occur if the shear stresses overcome
the frictional strength of the fault. The tractions on a fault element with normal unit vector 𝐧 are
𝜏(𝐧 )=𝚺∙𝐧 =𝜎
6

𝑛
6
𝐫
7

5
6%&
. (16)
The tractions in the fault surface can be separated into normal and shear components. The normal
traction is
𝛾(𝐧 )=𝜎
6

𝑛
6
+
.
5
6%&
(17)
The shear traction is
𝑠̂6
=𝜎
6

−𝛾𝑛
6
. (18)
According to the Wallace-Bott hypothesis, the stress-aligned slip vector is oriented in the direction
of 𝐬 . Thus, a stress-aligned double-couple dislocation is
𝐦
8
(𝐧 )=
1
√2
[𝐧 𝐬 (𝐧 )+𝐬 (𝐧 ) 𝐧 ]. (19)
14
If the deviatoric stress tensor 𝚺 is constant over the volume of interest, the stress-aligned mechanism
𝐦
8
is a function of the fault orientation vector, 𝐧 , only. The moment tensor density that satisfies the
Wallace-Bott condition is
𝐦(𝐱)=𝑚(𝐱)𝐦
8
𝐧 (𝐱). (20)
Fault zones are complex systems that hold geometrical heterogeneities over several orders of
magnitude. Some heterogeneities are branches, segments, step-overs, damage zones, and anisotropy.
All these irregularities interact with the stress field and modulate the release of elastic energy and
hence, the occurrence of earthquakes and seismic sequences. Moreover, stress tensor inversion
techniques aim to determine the stress tensor from structural fault data including fault orientation,
slip orientation and the sense of slip. Although huge improvements have been achieved in
geophysical imaging of faults (Anderson et al., 2021; Brengman et al., 2019; Torabi, A. et al., 2020;
Xiong et al., 2018; Yang, H., 2015; Yang, J. et al., 2020; Zhou et al., 2022), detailed deterministic
knowledge of fault structure and earthquake slip distribution is unfeasible currently. Therefore, one
possibility for studying the strain response to stress is to parametrize their relationship with a
probability density function (PDF) that depends on a strain-sensitivity parameter.
In JJ21, we define the strain-sensitivity factor 𝜅, which relates the probability of slip on a fault element
to the magnitude of the shear traction acting on it. On a fault plane, displacement is more probable to
occur in the direction of the highest shear traction. Hence, a PDF of exponential form that depends
on shear traction magnitude 𝑠(𝐧 ) describes this relation. The PDF is
𝑝
9
(𝐧 )=
1
𝒩
e
:9(𝐧 =)
, (21)
where 𝒩 is a normalization constant, such that the probability function integrates to unity,
𝒩(𝑅,𝜅)= e
:9(𝐧 =;@)

𝒮
!
𝑑𝐧 , (22)
𝜅 measures the seismic strain response to stress. Large values of 𝜅 concentrate 𝑝
9
(𝐧 )near the
maxima of 𝑠(𝐧 ), decreasing complexity. In JJ21, we show that, in the limit 𝜅 →∞, the complexity
15
factor is 𝐹

!
=0, and as 𝜅 decreases, the complexity increases. In the limit 𝜅 →0, the PDF is a uniform
distribution in the unit sphere, and 𝐧 can point in any direction with equal probability, which
maximizes the complexity to 𝐹

!
=0.6. The PDF depends on the orientation of the vector normal to
the fault, as such, the integration is performed over the unit sphere 𝒮
&
that supports 𝐧 . Figure A1 in
the Appendix shows 𝑠(𝐧 ) for some selected values of 𝑅.
Earthquakes are samples of the Wallace-Bott moment tensor density (equation 20), consequently,
their expected mechanism under the PDF 𝑝
9
(𝐧 ) is the CMT mechanism,
𝐌

)
=
1
𝒩
e
:9(𝐧 =)
𝐦
8
(𝐧 )

𝒮
!
𝑑𝐧 . (23)
Earthquake sources are the partial release of tectonic stress in a volume, and they are represented
by their CMT, which is proportional to the integral of the moment tensor density. The CMT
mechanism is the expectation of the stress release (stress-drop) in a seismic episode, thus
proportional to the stress-drop tensor. Furthermore, assuming that the stress-drop tensor is
proportional to the stress-tensor, then the CMT mechanism (𝐌

)
) is proportional to the regional
stress tensor (Matsumoto, Satoshi et al., 2012; Terakawa & Matsu'ura, 2008). Under these
assumptions, in JJ21, we showed that
𝐌

𝟎
=𝚺

=
1
¢6(1−𝑅+𝑅
+
)
diag[1+𝑅,1−2𝑅,𝑅−2]. (24)

In terms of the probability distribution, the expected value of an observed moment tensor density is
𝐌

𝟎
, and allows estimating the stress tensor, 𝚺

. The eigenvectors of 𝐌

𝟎
, 𝕰(𝐌

𝟎
)=(𝐞
&
,𝐞
+
,𝐞
5
), are
estimators of the stress tensor principal directions 𝕽(𝚺

)=(𝐫
&
,𝐫
+
,𝐫
5
), and its eigenvalues 𝜆(𝐌

𝟎
)
=(𝜆
&
, 𝜆
+
, 𝜆
5
,) are proportional to the principal stresses 𝜎(𝚺

)=(𝜎
&
, 𝜎
+
, 𝜎
5
,). Physically, the principal
axes of an elementary dislocation do not necessarily correspond to the stress tensor principal axes.
This observation is a limitation of the Wallace-Bott condition, and it has been extensively discussed
in the literature, e.g., (Célérier, 1988; Lisle, 2013). Nevertheless, our model assumes that the
directions of 𝕰 lie in the same quadrants as 𝕽 and follow identical probability distributions. Since
16
E©𝐌

)
ª=𝚺

, the statistical fluctuations in our estimate 𝐌

)
are symmetrically distributed about 𝚺

and
thus average to 𝚺

. We enforce these symmetries by two requirements: (a) the Wallace-Bott condition
and (b) the parameterization 𝑝
9
(𝐧 ) ∝ e
:9(𝐧 =)
, where 𝑠(𝐧 ) is the shear traction magnitude resolved
on a plane perpendicular to 𝐧 . This PDF has the requisite symmetries to ensure E©𝐌

)
ª=𝚺

.
𝑅 by itself does not provide information on the geometrical complexity in the fault system or moment
tensor observation uncertainties, which are fundamental for the correct characterization of the stress
field and its heterogeneity (Kassaras & Kapetanidis, 2018; Lisle, 2013). Mechanism complexity is the
deviation of the orientation of any elementary source relative to the expectation 𝐌

)
, and it can be
measured by the fractional moments, which are the quadratic average of the moment tensor density
projected onto a basis set 𝐌

-
,𝑎 =0,…,4. In JJ21, we proposed the stress-oriented representation
(SOR) basis set 𝕭(𝚺

)={E[𝐌

-
]:𝛼 =0,…4}, where the unit CMT is the leading term, and subsequent
terms are orthogonal rotations about its principal axis.
Consider the principal directions 𝕰(𝐌

)
)=(𝐞
&
,𝐞
+
,𝐞
5
), and the normal and shear directions 𝐧
𝟎
=
&
√+
(𝐞
&
+𝐞
5
), and 𝐬
𝟎
=
&
√+
(𝐞
&
−𝐞
5
), respectively. The SOR moment tensors are
𝐌

𝟏
=
1
√2
(𝐞
&
𝐞
5
+𝐞
5
𝐞
&
), (25𝑎)
𝐌

𝟐
=
1
√2
(𝐧
)
𝐞
+
+𝐞
+
𝐧
)
), (25𝑏)
𝐌

𝟑
=
1
√2
(𝐞
+
𝐬
)
+𝐬
)
𝐞
+
), (25𝑐)
The basis elements 𝐌

𝟏
,𝐌

𝟐
,𝐌

𝟑
are off-diagonal tensors in the principal axis reference frame. The
basis set is then completed by 𝐌

𝟒
, which is the orthogonal complement of 𝐌

𝟎
. From equation 24 we
get
𝐌

𝟒
=
1
¢2(𝑅
+
−𝑅−1)
diag[1−𝑅,−1,𝑅], (26)

In the particular case, when 𝑅 =0.5, 𝐌

)
is DC and 𝐌

4
is a CLVD mechanism,
17
𝐌

)
=
1
√2
[𝐧
)
𝐬
𝟎
+𝐬
𝟎
𝐧
)
], (27𝑎)
𝐌

4
=
®
1
6
(𝐞
&
𝐞
&
−2𝐞
+
𝐞
+
+𝐞
5
𝐞
5
). (27𝑏)

Figure 1.1 shows the SOR basis set elements 𝕭(𝚺

)=𝐌

-
,𝑎 =0,…,4 represented as focal spheres.
𝐌

)
and 𝐌

4
are also shown for selected values of the differential stress ratio, 𝑅. The focal spheres in
Figure 1.1 are calculated for principal horizontal stresses 𝜎
&
and 𝜎
5
. Thus, 𝐌

)
represents strike-slip
faulting. 𝐌

𝟎
and 𝐌

𝟒
depend on 𝑅, and in the limits 𝑅 ⟶0, and 𝑅 ⟶1, 𝐌

)
is a CLVD mechanism and
𝐌

4
is a DC. For 𝑅 ⟶0.5, 𝐌

)
is a DC and 𝐌

4
is a CLVD. The mechanism 𝐌

&
, 𝐌

+
, and 𝐌

5
do not depend
on 𝑅 because they are defined from the principal axis of 𝐌

)
which is fixed in the coordinate system.
Using rupture models with two degrees of freedom of the slip vector direction in strike-slip sources,
JJ19 found that 𝐌

&
describes out-of-plane variations, e.g., strike variations for the strike-slip fault, 𝐌

+

accounts for in-plane variations of the slip vector, i.e., rake variation, 𝐌

5
variations of the slip vector
in the dip direction. Additionally, analyzing realizations of stochastic rupture models (Graves &
Pitarka, 2016; Maechling et al., 2014) for vertical strike-slip faults, we recovered this basis set and
were able to model it in terms of rotations of the strike, dip, and rake.
Mechanism complexity can be measured by the moment fractions. For a Wallace-Bott moment tensor
density, the cosine fields are calculated by the projection cos𝜃
-
=𝐌

0
:𝐦
8
(𝐧 ), and the moment
fractions are their quadratic expectations (10) under the PDF,
𝐹

-
(𝑅,𝜅)=
1
𝒩
e
:9(𝐧 =,@)
cos
+
𝜃
0
(𝐧 )

𝒮
!
𝑑𝐧 . (28)
The Aki moment fraction is
𝐹
)
(𝑅,𝜅)=
1
𝒩
e
:9(𝐧 =,@)
cos𝜃
)
(𝐧 )

𝒮
!
𝑑𝐧 . (29)
Figure 1.2 shows the total moment fractions as functions of the differential stress ratio 𝑅 and the
stress sensitivity factor, 𝜅. JJ21 extensively discuss the properties of these fields and their limiting
18
behaviors. Some relevant characteristics are that the maximum complexity is archived when 𝜅 =0.
In this case, the Aki moment fraction, and the moment fraction of zeroth degree reach the minimum
values and are weak functions of 𝑅, 𝐹
)
→0.6, and 𝐹

)
→0.4. 𝐹
)
and 𝐹

)
increase monotonically with 𝜅;
this means that complexity decreases as 𝜅 increases. In the limiting case, 𝜅 →∞, and for 𝑅 =0.5, 𝐹
)

and 𝐹

)
converge asymptotically to unity, meaning no complexity. It is worth noting that 𝐹

+
and 𝐹

5
are
identical in the SSC model due to the symmetries of 𝐌

+
and 𝐌

5
.
In JJ21, we defined a qualitative scale of complexity based on values of 𝐹

!
=1−𝐹

)
. Three intervals
represent the scale: 1) low complexity if 0≤𝐹

!
<0.2, 2) moderate complexity for 0.2≤𝐹

!
<0.4,
and 3) high complexity 0.4≤𝐹

!
<0.6. These values correspond to Aki moment fractions of 0.9-1 for
low-complexity fields, 0.75-0.9 for moderate-complexity fields, and 0.6-0.75 for high-complexity
fields.

Figure 1.1. Stress-oriented basis sets {𝐌
)
!
:𝛼 =0,…,4} for selected values of the differential stress ratio 𝑅. The
inset plot shows the coordinate system axes. 𝐬 4
𝟎
is the direction of maximum shear traction, 𝐧 6
𝟎
the vector
normal to the fault plane, and 𝐫 4
!
the principal stress directions.

19

Figure 1.2. Aki moment fraction 𝐹
#
and total moment fractions 𝐹
9
!
as functions of the differential stress ratio 𝑅
and the stress sensitivity parameter 𝜅. (a) Aki moment fraction 𝐹
#
. (b) Total moment fraction of zeroth
degree (𝐹
9
#
). Solid lines are the limits between low (𝐹
9
$
=0.2), and high (𝐹
9
$
=0.4) complexity. Dashed lines
are intervals of 0.1 𝐹
9
$
. (c)-(f) Moment fractions of higher-degree (𝐹
9
%
−𝐹
9
&
).

20
1.2 Earthquake Mechanism Datasets
In the previous section, we introduce the stress-strain characterization (SSC) model and solved the
forward problem of calculating the moment measures of mechanism complexity from the SSC model
parameters. Our goals are to (1) test the ability of the SSC model to explain the observed mechanism
complexity, and (2) estimate the SSC parameters from the data for various fault zones, i.e., solve the
inverse problem. Using catalogs of earthquake focal mechanisms, we apply the SSC model to
characterize the stress field in specific fault volumes in Northern and Southern California. Fault-
specific catalogs of focal mechanisms represent the total seismicity in the fault zone. Focal
mechanisms are samples of a stress-align moment tensor density modulated by a physical process
described by the SSC probability function, 𝑝
9
(𝐧 ) ∝ e
:9(𝐧 =)
, and contaminated by noise. Noise is any
non-physical rotation of a mechanism and therefore can be misinterpreted as genuine complexity.
Therefore, advances statistical techniques are required to invert observations of moment measures
of mechanism complexity for the SSC parameters in the presence of noise. In the following sections,
we show the inversion technique and how we deal with errors in observations of focal mechanisms.
In this section, first, we introduce the datasets.
The plate boundary in California is a highly complex fault system with heterogeneities of several
scales and broad distribution of seismicity (Waldhauser, Felix & Schaff, 2008; Yang, W. & Hauksson,
2013). Although the San Andreas fault (SAF) accommodates most inter-plate motion, other faults,
such as the San Jacinto and Elsinore faults in Southern California or the Calaveras Fault in Northern
California (Figure 1.3) are very seismically active. These tectonically active regions experience
deformation from the relative motion between the North American and the Pacific Plates, primarily
right lateral. However, the seismicity shows distinctive deformation styles along the complex plate
boundary.
Stress models of California (Becker et al., 2005; Bird, 2017; Yang & Hauksson, 2013) show that inland
faults systems north of the Gulf of California and in the Coachella Valley show lateral-extensional
21
motion as the continuation of the divergent margin in the Gulf of California. Transform faulting
dominates along the San Andreas fault and subparallel systems in Southern California, including the
San Jacinto and Elsinore faults. Collision on a smaller scale is observed in the Transverse Ranges
(Hauksson, 2011; Shen et al., 2011; Wdowinski et al., 2001; Weldon & Humphreys, 1986).
To the north, in Central California, the SAF is very localized, and estimates from geological and
geomorphological data suggest that it has accommodated over 560 km of slip (Hill & Dibblee, 1953;
Sims, 1993; Stanley, 1987). The plate boundary in the San Francisco Bay Area consists of numerous
large faults, including the San Andreas, Hayward, and Calaveras faults. The Hayward and Calaveras
faults in the eastern and southern Bay area dominate the seismic hazard. According to the third
Uniform California Earthquake Rupture Forecast, UCERF3 (Field, 2015), they can produce
earthquakes of moment magnitude Mw>7. Moreover, UCERF3 estimates indicate a high likelihood of
an Mw6.7 earthquake in the next 25 years. These faults have accumulated up to 175 km offset
(Graymer et al., 2002; Lienkaemper & Borchardt, 1996).
We characterize the stress field complexity around the plate boundary in California. Using the focal
mechanisms of tectonic earthquakes, we investigate how stress orientation and stress regime vary
on various scales. As a master dataset, we use source parameters of 77,337 earthquakes in Northern
California and 144,038 in Southern California. Earthquakes have magnitudes Mw between 1.5-7.3
and occur in the time interval between 1981-2021. Their locations, seismic magnitudes, and focal
mechanisms were downloaded from the Northern California Earthquake Data Center (NCEDC;
available at http://www.ncedc.org/ncedc/catalog-search.html/) for Northern California. From the
Southern California Earthquake Data Center (SCEDC), we obtained the catalog of Hauksson et al.
(2012) extended to 2021 (available at https://scedc.caltech.edu/eq-catalogs/altcatalogs.html/).
Figure 1.3 shows the earthquake epicenters and depth in Northern and Southern California. Major
events between 1983-2019 are also highlighted.
22
The NCEDC uses a double-difference algorithm to locate the events and invert observed P-wave
polarities for the focal mechanisms (Reasenberg & Oppenheimer, 1985; Waldhauser, Felix, 2009).
The NCEDC catalog uses the FPFIT algorithm of Reasenberg and Oppenheimer (1985) to estimate
focal mechanisms. Double-couple fault plane solutions are calculated by fitting sets of observed first-
motion polarities for an earthquake. The inversion minimizes through a multistage grid-search the
first-motion polarity discrepancy. FPFIT formally estimates the uncertainty in the model parameters
(strike, dip, rake) and reports the preferred solution and the 90% confidence interval.
The SCEDC calculates focal mechanisms by grid searching for the best-fitting double-couple focal
mechanism solution to P-wave first motion polarities and the S/P amplitude ratios computed from
three-component seismograms (Hauksson et al., 2012; Yang, W. et al., 2012). The SCEDC catalog
uses the inversion algorithm of Hardebeck and Shearer (HASH; 2002) to estimate focal mechanisms.
HASH is based on FPFIT and improved by inverting P- and S-wave amplitudes in addition to their
polarities. In HASH, the preferred solution is the average of acceptable solutions after removing
outliers iteratively until all mechanisms are within 30° of the average. The averaging operation is
calculated over the normal vectors to the nodal planes in vector coordinates. The catalogs report two-
nodal plane uncertainties (PU) for each mechanism. The PU is one standard deviation of mechanism
uncertainty relative to either the vector normal to the fault 𝐧 or the slip direction, 𝐬 . The root-mean-
squared (RMS) variance angle is 𝜎
+
=
&
+
(PU1
+
+PU2
+
), According to Hardebeck and Shearer (2002),
the variance angle is a measure of the total dispersion of the distribution of the angles between a
proffered mechanism (in tensor space) and the set of possible solutions.

23

Figure 1.3. Seismicity distribution along faults in California. (a) Locations of earthquakes in Northern
California. Dots are colored by earthquake hypocenter depth and sized by earthquake magnitude. Red stars
show the epicenters of large earthquakes since 1981. Black solid lines are major mapped fault traces. The
inset map shows the location of the area of the study. (b) Similar to (a), but for Southern California. SAF is the
San Andreas Fault, and SAF-C is its creeping section in Northern California.

1.2.1 Fault Zone Seismicity Catalogs
In the SSC model, earthquakes are samples of stress-aligned moment tensor density 𝐦
8
(𝐧 )
modulated by a physical process described by the function 𝑝
9
(𝐧 ) ∝ e
:9(𝐧 =)
. 𝑝
9
(𝐧 ) implies that the
probability of failure on planes oriented with normal vector 𝐧 depend exponentially on the shear
traction magnitude 𝑠(𝐧 ) and the strain sensitivity parameter 𝜅. Furthermore, 𝐦
8
(𝐧 ) and 𝑠(𝐧 ) are
functions of the fault orientation direction only, and therefore the term 𝜅𝑠(𝐧 ) alone determine which
orientations are more favorable for failure. We characterize stress-strain fields under the SSC model
assumptions using observations of earthquake mechanisms. For this aim, we constructed 25 catalogs
of focal mechanisms of fault zones in Southern and Northern California. The catalogs are constructed
by selecting earthquakes reported in the SCEDC and the NCEDC that lay inside boxes surrounding the
fault zones.
24
Figure 1.4 shows the boxes enclosing the fault zones, and Table 1.1 contains information about the
box coordinates, dimensions, and the number of events. We selected fault zones based on several
criteria. The primary condition is that fault zones contain segments established in the literature;
therefore, fault traces have been mapped, and geological offsets have been estimated. Quaternary
fault traces are obtained from the United States Geological Survey Quaternary Fault and Fold
Database of the United States (we refer to herein as QFFD) last accessed August 24, 2022, at:
https://www.usgs.gov/natural-hazards/earthquake-hazards/faults.
Boxes have variable lengths, and we determine that extending up to 10km from mapped fault traces,
we capture most of the seismic activity in a fault zone. Therefore, boxes are 20km wide, centered in
the fault trace. Table 1.1 contains the geographical coordinates of the extreme points of each fault
segment. In addition, we avoid joints and branches of other major faults when selecting fault zones
and seismicity.

Fault Zones in Northern California
In Northern California, we chose the seismicity of the Calaveras (CAL) and Hayward (HAY) faults and
two segments of the San Andreas fault: Loma Prieta (SAN) and Parkfield (SAP). CAL intersects HAY
and SAN; therefore, we selected a segment of 53.1 km in the central Calaveras fault zone between the
intersections with HAY and SAN. Similarly, the boxes for HAY and SAN avoid the intersection with
the Northern Calaveras fault.
The SAN catalog is composed mainly of the seismicity following the Mw6.9 Loma Prieta earthquake
of 1989, which ruptured over 30km of this section. The SAP segment of the San Andreas fault is 48km
long and is located between the creeping section and the intersection with the oblique San Juan fault
zone to the east of San Luis Obispo County. There are 5,499 events in the catalog of CAL, 1,548 for
HAY, 5,781 for SAN, and 2,560 for SAP.

25
Fault Zones in Southern California
In Southern California, we considered fault zones along the Elsinore, San Jacinto, San Andreas faults,
Newport-Inglewood, and quaternary faults in the Mojave Block, including faults that ruptured during
the Landers earthquake of 1992 and the Hector Mine earthquake of 1999.
There are two segments of the south San Andreas fault in the database: The San Bernadino Mountains
(SAB) and the Coachella Valley (SAC) fault zones. SAB is a broad region of deformation that goes from
Cajon pass, where the San Jacinto fault intersects the San Andreas fault, to the north of the Coachella
Valley. This section is very complex and consists of several strike-slip, thrust, and oblique-slip faults.
We draw a box that encloses the seismicity around the San Bernardino Mountains trace of the San
Andreas fault as mapped in the QFFD. SAB is delimited to the south by the Banning and the San
Gorgonio fault zones and avoids the intersection with the SJA to the west. The box a is 50.3km long
and has 11,580 events. SAB is the second most active fault zone analyzed after SJB.
The SAC section goes from the southeast San Bernardino Mountains to the Salton Sea and is
characterized by lateral-extensional motion. We draw a box that is 89.4km long and has 2,581 events.
The Brawley seismic zone connects the SAC with the Imperial Valley (IMP) fault south of the
Coachella Valley. IMP is probably a segment of the San Andreas fault; however, it is located between
the southernmost ends of the San Jacinto fault and the San Andreas faults. The last major earthquake
in IMP was the 1979 Mw6.5 Imperial Valley earthquake (Figure 1.1), and seismicity indicates lateral-
extensional motion. The box around IMP is 72.7km long and has 1921 events, but it avoids
earthquakes from the Brawley fault zone, which appear more spread.
We divide the Elsinore fault into three segments: the Granite Mountains (EGM) section on the south,
the Central (ELS) section, and the Whittier (EW) section on the north. The southern segment of the
Elsinore fault is not well localized, and seismicity is spread over a large volume. Although we identify
the segment as the Granite Mountains, we select the box to contain the seismicity of the Coyote
Mountains, San Julian, and the San Felipe sections, which are also considered part of the Elsinore fault
26
zone by the QFFD. The EGM box is 116km long and 20km wide, and the catalog contains 8,529 focal
mechanisms. ELS is 78km long and encloses the Temecula section. The box avoids the bifurcation of
ELS into EW and Elsinore-Chino, located to the east of Los Angeles. The ELS catalog has 773 events
and EW 2,102.
The San Jacinto fault is the most seismically active fault in Southern California, with two major
earthquakes in 1987, the M6.6 Superstition Hills and M6.2 Elmore Ranch earthquakes (Figure 1.3).
We divided the San Jacinto fault into the Borrego (SJB) section, which encloses the broad seismicity
distribution in the Coyote Mountains and the Borrego Badlands, and the Anza (SJC) section that
extends from Anza to San Jacinto Valley. The SJB box is 86.7km long and contains 15,553 events, the
fault zone with the largest number of events. The SJC box is 66.4 km and includes 10,186 earthquakes.
The Newport-Inglewood (NI) fault extends over 209 km from offshore San Diego to Los Angeles
Basin. However, it is less seismically active than the Elsinore and San Jacinto faults. The last major
earthquake recorded was the Mw6.3 Long Beach earthquake of 1933. We selected a segment of
133km that contains 893 events and goes from Oceanside to its end in the Santa Monica fault at the
base of the Santa Monica Mountains. As such, the NI fault encloses most of the seismicity in the Los
Angeles basin.
The Garlock (GAR) fault zone is the east-northeast striking strike-slip fault that marks the north
boundary of the Mojave Desert. It extends from a complex intersection with the San Andreas fault in
the west to near south Dead Valley. We selected a 126km-long segment of the Garlock fault from its
junction with the San Andreas fault up to the Freemont Valley to the South of the Sierra Nevada. This
section is the most seismically active section of the Garlock fault, and the box avoids the seismicity in
the Ridgecrest area to the east. There are 1,888 events in the GAR catalog.
The Eastern Mojave Desert is located between the Salton Trough and the Basin and Range Province.
This region accommodates some of the lateral motion that results from the crustal spreading
observed in the Gulf of California and continues in the South San Andreas fault. Several north-striking
27
lateral-motion faults cross the block. We considered several fault zones in the Mojave Desert region
that are part of the Eastern California Shear Zone, including faults that ruptured during the Landers
and Hector Mine earthquakes. The Black Water (MBW) fault zone is the most extended segment,
which is 93.1km long and contains 1,850 events. To the south, the Calico-Hidalgo (MCB) fault zone is
a northwest striking dextral strike-slip fault located between the ruptures of the Landers and Hector
Mine earthquakes and connected to the MBW.
At around 15km to the west of MCB lays the Camp Rock (MCR) fault zone. MCR is a complex dextral
strike-slip fault zone that includes the Emerson (LE) and North Cross (LNC) faults. LE and LNC
ruptured during the Landers earthquake, but the southern section of MCR that extends to the Pinto
Mountains did not rupture. The LE fault zone is also a northwest striking lateral-motion fault. We
selected a 35.2 km long box that contains 1,883 earthquakes. The Homestead Valley (LHV) fault
connected the Landers' ruptures between the Johnson Valley (LJV) and Emerson faults. The north-
northwest striking fault, LHV, had over 3 m of lateral displacement associated with the Landers
earthquake. We draw a box of 20.1 km long with 3,686 events that avoid the intersections with LE
and LJV. LJV is a north-striking strike-slip fault that ruptured during the Lander earthquake showing
up to 3 m of dextral strike-slip surface rupture. It intersects obliquely LHV to the north and
orthogonally the Pinto Mountains fault zone, and perhaps continuous to the south as the Eureka Peak
or Burnt Mountain fault zone. However, we selected a box 23.5 km long between the Pinto Mountain
fault zone and the junction with the LHV.
We consider three fault segments ruptured during the Mw7.1, 1999 Hector Mine earthquake. From
south to north, they are the Mezquite Lake (HML), Pisgah (HP), and Lavick Lake (HLL) fault zones.
HML is a north-northwest striking dextral strike-slip fault bounded in the south by the Pinto
Mountain fault zone and intersecting the HP fault almost subparallel. The HML segement is 45.7 km,
HP is 20.8 km, and HLL 43 km long. HP lays between the Lavick Lake and Mezquite Lake fault zones
and encloses the section of the Rodman-Pisgah fault that ruptured during the Hector Mine
28
earthquake. MRP is a box that encloses most of the Pisgah-Bullion fault zone, a northwest-trending
dextral strike-slip fault zone of 71.8 km long that goes from the Pinto Mountain fault zone in the south
to the Cady fault in the north.

Table 1.1. Fault zone information. The codes of each fault zone correspond to Figure 1.4. LON1 and LAT1 are
the geographical coordinates (longitude and latitude) of the northern extreme of the fault segment and LON2
and LAT2 the coordinates of the southern extreme. Length in kilometers is the distance between points
(LON1, LAT1) and (LON2, LAT2).
Fault SEGMENT CODE LON1 LAT1 LON2 LAT2
LENGTH
(KM)
N.
EVENTS
Calaveras Calaveras CAL -121.78 37.432 -121.48 37.017 53.1 5499
Hayward Hayward HAY -122.39 38.002 -122 37.595 56.4 1548
Elsinore Granite Mountain EGM -116.92 33.351 -115.96 32.684 116 8529

Central ELS -117.55 33.813 -116.92 33.348 78 773

Whittier EW -118.05 33.995 -117.55 33.813 50.5 2102
Garlock Garlock GAR -118.92 34.824 -117.75 35.423 126 1888
Hector Mine Lavick Lake HLL -116.41 34.863 -116.21 34.511 43 2311

Mesquite Lake HML -116.25 34.471 -115.99 34.121 45.7 918

Pisgah HP -116.38 34.715 -116.25 34.559 20.8 4986
Imperial Valley Imperial Valley IMP -115.53 32.9 -115.07 32.373 72.7 1921
Landers Emerson LE -116.86 34.783 -116.57 34.569 35.2 1883

Homestead Valley LHV -116.54 34.529 -116.45 34.364 20.1 3104

Johnson valley LJV -116.47 34.348 -116.41 34.141 23.5 3686

North Cross LNC -116.67 34.648 -116.44 34.45 30.7 4186
Mojave Block Blackwater MBW -117.41 35.45 -116.69 34.857 93.1 1850

Calico Hidalgo MCB -116.78 34.916 -116.34 34.477 62.9 2544

Camp Rock MCR -116.81 34.752 -116.3 34.308 67.6 1738

Rodman Pisgah MRP -116.45 34.832 -116 34.306 71.8 3523
Newport-Inglewood Newport-Inglewood NI -118.39 34.043 -117.43 33.156 133 893
San Andreas San Bernardino SAB -117.07 34.088 -116.54 33.989 50.3 11580

Coachella Valley SAC -116.29 33.906 -115.69 33.276 89.4 2581

Loma Prieta SAN -121.73 36.989 -121.45 36.744 37.1 5781

Parkfield SAP -120.73 36.174 -120.38 35.843 48.3 2560
San Jacinto Borrego Badlands SJB -116.57 33.466 -115.86 32.967 86.7 15553

Anza SJC -117.09 33.904 -116.55 33.511 66.4 10186

29

Figure 1.4. California faults and fault zones analyzed. The colored dotted boxes delimit fault zones, and the
solid-colored line indicates their traces. Gray dots are epicenters of earthquakes shown in Figure 1.3. The
codes and names of the fault zones are in Table 1.

1.2.2 Cumulative Offsets of California Fault Segments
Previous seismic, geodetic, and geological studies of deformation in California have shown that the
stress and strain fields can vary rapidly in neighboring fault zones. Moreover, fault zones evolve with
increasing cumulative geological offset, structural heterogeneities such as damaged volume and step-
overs tend to decrease, and seismicity concentrates closer to the fault core (Brodsky et al., 2011;
Powers & Jordan, 2010; Sagy et al., 2007; Wesnousky, 1988). Therefore, mechanism complexity
30
might be expected to correlate with the cumulative offset of faults. In this subsection, we describe the
database of cumulative fault offsets that will be used in our later analysis to test this hypothesis.
We compile from published literature estimations of cumulative geological offsets of 25 strike-slip
fault zones in California for comparison with estimates of the strain sensitivity factor 𝜅, and the total-
moment fraction of zeroth degree 𝐹

)
. For this aim, we analyze the seismicity in boxes surrounding
the faults where there are measurements of geological slips. Figure 1.4 shows the fault zones
considered and the main fault trace. Table 1.2 contains the fault sections, the minimum, median, and
maximum reported offsets, and the references where estimations are reported. Some references are
also compilations from other studies, and we cite the compiled database only in such cases.

31
Table 1.2. California faults zones and cumulative offsets. Some faults are divided into sections, as shown in
Figure 1.4. We report the minimum, median, and maximum of all reported values in the literature. Some
references are compilations from other works. Therefore, we cite the compilation only.
FAULT SECTION CODE
OFFSETS: MIN,
MEDIAN, MAX (KM)
REFERENCES
CALAVERAS Calaveras CAL 20 89.5 165
Page (1982), Kelson et al. (1998), Stirling &
Wesnousky (1996), Powers & Jordan (2008)
ELSINORE Granite Mountain EGM 2.5 10 40
WGCEP (1995), Hull and Nicholson (1992),
Magistrale and Rockwell (1996)

Central ELS 9 12.5 15
Bergmann and Rockwell (1989), Powers &
Jordan (2008)

Wittier EW 10 15 40
Hull and Nicholson (1992), Stirling &
Wesnousky (1996), Wesnowsky (1990)
GARLOCK Garlock GAR 12 62 65
Garfunkel (1974); Bryant (2000), Stirling &
Wesnousky (1996), Powers & Jordan (2008)
HAYWARD Hayward HAY 42 70 175
Powers & Jordan (2008), Parson et al. (1999),
Lienkaemper and Borchardt (1996), Graymer et
al. (2002)
HECTOR MINE Lavick Lake HLL 1.2 4.85 20 Jachens et al. (2002), Witkosky et al. (2020)

Mesquite Lake HML 0.01 7.8 14 Jachens et al. (2002), Bryant (2003)

Pisgah HP 3.4 10.5 40 Jachens et al. (2002), Treiman (2003)
IMPERIAL Imperial IMP 0.01 85 85 Treiman (1999), Powers & Jordan (2008)
LANDERS Emerson LE 1.6 4 5 Zachariasen & Sieh (1995), Bryant (2000)

Homestead Valley LHV 0.25 0.39 4.6 Zachariasen & Sieh (1995), Jachens et al. (2002)

Johnson Valley LJV 0.01 3.1 3.1 Zachariasen & Sieh (1995)

Northern Cross LNC 0.13 0.2 0.27 Zachariasen & Sieh (1995)
MOJAVE Blackwater MBW 0.3 1.9 10
Jachens et al. (2002), Glazner et al. (2002),
Oskin and Iriondo (2004)

Calico Hidalgo MCB 8 10 20
Dokka & Travis (1990), Jachens et al. (2002),
Glazner et al. (2002), Stirling & Wesnousky
(1996), Oskin & Iriondo (2004)

Camp Rock MCR 0.95 3.75 10
Dokka & Travis (1990), Jachens et al. (2002),
Stirling & Wesnousky (1996), Glazner et al.
(2002), Bryant (2000)

Rodman-Pisgah MRP 6.3 10.5 40
Dokka & Travis (1990), Jachens et al. (2002),
Stirling & Wesnousky (1996)
NEWPORT-
INGLEWOOD
Newport-Inglewood NI 0.2 7.5 10
Stirling & Wesnousky (1996), Powers & Jordan
(2008)
SAN ANDREAS San Bernardino SAB 3 17.5 48 Weldon & Sieh (1985); McGill et al. (2002)

Coachella Valley SAC 160 197 257
Hill and Dibblee (1953), Crowell (1962), Power
& Jordan (2008)

Loma Prieta SAN 280 307 360 Matthews (1976), Ross (1984)

Parkfield SAP 150 277 325 Powers & Jordan (2008)
SAN JACINTO Borrego Badlands SJB 3 24 28
Treiman & Lundberg (1999), Ostermeijer et al.
(2020)

Anza SJC 1.6 14.4 28
Sharp (1967), Stirling & Wesnousky (1996),
Powers & Jordan (2008)

1.2.3 Ridgecrest Datasets
Observational errors in the focal mechanism orientation can be misinterpreted as rupture
complexity. Therefore, it is important to understand the error statistical properties, including
correlation between errors in the strike, dip, and rake directions or the nodal plane orientations.
32
Additionally, observations of focal mechanism might be biased by the station distribution and the
modeling assumptions.
We investigate systematic biases in the SCEDC catalog by comparing statistical distributions of
reported errors and distributions computed from our noise model. Additionally, we compare an
extract of the SCEDC catalog with to two additional catalogs: (1) the Cheng catalog (Cheng & Ben-
Zion, 2020) and (2) Lin (Lin, 2020). Here after we refer these three catalogs as the Hauksson, Chen,
and Lin catalogs. The three catalogs report focal mechanisms for the Mw 6.4 and 7.2 Ridgecrest
earthquakes of 2019 and their aftershocks from July 4-December 31, 2019 (Figure 1.5). We refer to
these catalog excerpts collectively as the Ridgecrest dataset. From each catalog, we analyzed
earthquake locations, magnitudes, and from fault plane strike, dip, and rake, we calculated the
moment tensors. The Hauksson dataset has 9,081 events, Cheng has 5,866, and Lin has 4,348 for the
Ridgecrest sequence.
The catalogs of Hauksson, Cheng, and Lin used the inversion algorithm of Hardebeck and Shearer
(HASH; 2002) to estimate focal mechanisms. Nevertheless, there are differences in the earthquake
detection algorithms, the data collection, and inversion assumptions: the Hauksson catalog relocated
events using the 3D velocity model of Hauksson (2000) and arrival-time picks found by Southern
California Seismic Network (SCSN) data analyst. The Cheng catalog uses convolutional-neural-
networks (CNN) to detect phases and polarities on continuous waveforms of the SCSN, but the CNN
are trained on earthquakes previously detected by the analysts of the SCSN. Then, detected
earthquakes are located using theoretical arrival times from the 1D velocity model in Hutton et al.
(2010). Lastly, the Lin catalog uses detections and arrival times from the SCSN and relocates
earthquakes using differential times from waveform cross correlation in the 3D model of Zhang and
Lin (2014).
Figure 1.5 shows the earthquake epicenters and mechanism uncertainties reported in the Hauksson,
Cheng, and Lin catalogs. Figure 1.5b shows the empirical distribution of the uncertainties and their
33
fit by two-parameter Gamma distributions, Γ(𝜎;𝛼,𝛽), where the root-mean-squared (RMS) variance
angle is 𝜎
+
=
&
+
(PU1
+
+PU2
+
), and PU1 and PU2 are the two-nodal plane uncertainties reported by
HASH. On average, the estimated quality of focal mechanisms in the Hauksson dataset is better than
in the Lin and Cheng catalogs. Although it is expected that source mechanism estimation is better for
larger events, e.g., over Mw4 at the regional scale, none of the datasets shows this characteristic
clearly. For example, for earthquakes with a magnitude below 4, the mean uncertainties 𝜎 range
between 20
G
to 40
G
and for events above 4, the mean uncertainties are still around 20
G
for the
Hauksson and Cheng catalogs and up to 40
G
for the Lin dataset. We find that PU1 and PU2 follow
identical distributions and are correlated; the Pearson correlation coefficients are 𝑟
H
=0.75, 𝑟
I
=
0.67, and 𝑟
J
=0.81, for Hauksson, Cheng and Lin datasets respectively. Cheng shows more significant
dispersion at larger PU.
Cross-checking of focal mechanism orientations in the three catalogs might reveal biases from the
selection of the velocity model used to locate the events and estimate theoretical polarities, and
biases due to human error against machine-learning algorithms for event detection and the
observation of polarities. In Section 1.3.2 and 1.4, we develop an SSC+Noise model that quantifies the
effect of uncertainties, meaning PU or 𝜎, on the estimations of the SSC parameters. In Section 1.5, we
perform a systematic analysis of earthquake catalog biases using the Ridgecrest dataset. We find that
mechanism uncertainties in the Hauksson catalogs, although large, can be modeled, allowing us to
characterize the stress-strain fields under the SSC+Noise model assumptions.

34

Figure 1.5. Source mechanism uncertainties of the Hauksson, Cheng, and Lin catalogs of the Ridgecrest
sequence. (a) Map showing the mainshock, aftershock, and seismicity epicenters in the Ridgecrest area. Red
stars show the epicenter location of the Mw 6.4 and 7.2 earthquakes of July 2019. Their mechanisms are also
shown as focal spheres. Gray dots are epicenters of earthquakes from 1981 to 2021, and colored dots are the
aftershocks used in this analysis. (b) Earthquake mechanism uncertainty 𝜎. Histograms are empirical
distributions of 𝜎. Solid lines are their fit by two-parameter gamma distributions. (c), (d), and (e) show the
event counts in logarithmic scale as a function of magnitude Mw and uncertainty 𝜎 for the Hauksson, Cheng,
and Lin catalogs, respectively (white areas = no events).

1.3 Inversion of Focal-Mechanism Catalogs for SSC Model Parameters
The key property of the SSC model is that the expected mechanism of earthquakes and seismic
sequences is proportional to the regional stress tensor, E©𝐌

)
ª=𝚺

. The SSC model parameters are
the normalized stress tensor, 𝚺

and the strain sensitivity factor, 𝜅. 𝚺

can be uniquely specified by the
principal stress directions 𝕽(𝚺

)=(𝐫
&
,𝐫
+
,𝐫
5
) and the differential stress ratio 𝑅(𝚺

), and 𝜅 is specified
by the total-moment simplex 𝓕

(𝚺

,𝜅)={𝐹

-
(𝚺

,𝜅):𝛼 =0,…4}.
In this section, we introduce a maximum likelihood estimation (MLE) technique that inverts for the
SSC parameters 𝑅, and 𝜅 from uncertain observations of earthquake focal mechanisms. The idea is
simple: consider a catalog 𝒞 ={𝐦
"
:𝑛 =1,…,𝑁} of 𝑁 observed mechanisms. Mechanism in the
catalog are samples of a stress aligned moment tensor field modulated by a physical process
(a) (b)
(e)
(e)
Hauksson
Cheng
Lin
Mw7.2
Mw6.4
Garlock Fault
Southern Sierra
Nevada Fault Zone
Ash Hill Fault
Panamint Valley
Fault
Airport Lake
Fault Zone
SAF
Pacific
Ocean
Ridgecrest
Nevada
Mexico
Hauksson
Cheng Lin
(c)
(d)
-118 - 117.8 -117.6 -117.4 -117.2
Longitude
36.2
36
35.8
35.6
35.4
La:tude
35
described by the SSC distribution 𝑝
9
(𝐧 )∝e
:9(𝐧 =)
. Therefore, the reduced stress tensor is 𝚺

=𝐌

𝟎
∝
∑ 𝐦
"
$
"%&
. In Section 1.1.2, we show that the total moment fractions, 𝐹

-
are functions of 𝑅 and 𝜅. 𝐌

𝟎

constrains 𝑅 and thus, observations of 𝐹

-
constrain 𝜅.
Mechanism complexity is any physical deviation of an elementary mechanism 𝐦
"
from a mean
mechanism 𝐌

)
, and it is specified by 𝐹

-
. The total moment fractions are estimated by squaring the
direction cosines of 𝐦
"
relative to the SOR basis set 𝕭(𝚺

)={𝐌

-
:𝛼 =0,…4} and summing over the
ensemble: 𝐹

-
=
&
3
∑ 𝐌

-
:𝐦
"

+
$
"%&
. Unfortunately, errors in 𝐦
"
, even if they have zero mean, can bias
estimates of the complexity measures, just as moment-tensor uncertainties can bias the estimation
of the Aki moment (Silver & Jordan, 1983).
Uncertainties in source-mechanism estimation depend on many factors, such as station coverage,
event depth, seismic-velocity model, and estimation techniques, and they can be quite large (Dreger,
2015; Duputel et al., 2012; Julian, Bruce R. et al., 1998; Moghtased-Azar et al., 2022; Rösler & Stein,
2022; Vasyura-Bathke et al., 2021). Uncertainties in mechanism observation are a major
complication for obtaining accurate estimations of 𝜅 . However, statistical properties of the
uncertainties are usually available and therefore help constrain the inversion.
In this section, we develop a maximum likelihood estimation (MLE) technique that inverts
observations of earthquake mechanisms and total moment fractions for the SSC parameters. The MLE
technique considers all the observations and parameters as random variables, and their distributions
are constructed empirically by random sampling of multiple observations with replacement.
Uncertainties in the observations are handled by forward modeling their effect on total moment
fractions of SSC moment tensor densities.

36
1.3.1 Probabilistic Estimation of the SSC Model Parameters
In the SSC model, the differential stress ratio and the strain sensitivity factor govern the distributions
of total-moment simplex 𝓕

(𝑅,𝜅), and mechanism complexity is the deviation of elementary sources
relative to a mean (CMT) mechanism. If observations of elementary mechanisms are uncertain due
to errors or biases, such uncertainties can be misinterpreted as genuine source complexity. We
develop an SSC-model parameter estimation technique that quantifies and corrects for effects of
mechanism uncertainty. The model propagates uncertainties using conditional probability
distributions on observations, model parameters, and their uncertainties.
Let 𝐌

𝟎
be a random variable drawn from a distribution of CMT mechanisms with mean E©𝐌

)
ª=𝚺

.
𝕰
)
𝐌

)
=(𝐞
&
,𝐞
+
,𝐞
5
) and 𝜆
)
𝐌

)
=(𝜆
&
,𝜆
+
,𝜆
5
) are the set of ordered principal axes and principal
values of 𝐌

𝟎
. We calculate 𝐸
)
=(𝜆
&
−𝜆
+
)/(𝜆
&
−𝜆
5
), which is the random variable that estimates the
differential stress ratio. Let 𝓕

=𝐹

-
;𝛼 =0,…,𝑑 be a random variable vector of observed moment
fractions from the distribution of 𝐌

𝟎
. 𝓕

and 𝐸
)
are generated by a stress-strain process governed by
the parameters 𝚺

and 𝜅. We assume 𝓕

and 𝐸
)
are perturbed by random noise processes described
by a random variable 𝜎.
For simplicity, here we assume that 𝜎 represents the total uncertainty on the observation of
elementary mechanisms: if the angular distance between a solution 𝐦 and a preferred solution 𝐌

𝟎

is 𝜑 =arccos(𝐌

𝟎
,𝐦 ) (Hauksson et al., 2012; Reasenberg & Oppenheimer, 1985; Shelly et al., 2016;
Waldhauser, 2009; Yang et al., 2012), then 𝜎 is the standard deviation of the angle 𝜑. We
acknowledge that earthquake mechanisms have five independent parameters, and their
uncertainties might follow a five-dimensional distribution with correlations. The focal mechanism
inversion code of Hardebeck and Shearer (HASH; 2002) reports two nodal-plane uncertainties (PU1
and PU2) and defines 𝜎 as the root mean square uncertainty, 𝜎
KL
= ³
MN&
"
!MN+
"
+
. Furthermore, the
NCEDC catalogs reports the 90% confidence interval for estimated strike, dip, and rake directions.
37
From the confidence interval, we infer 𝜎
OPQ
, 𝜎
R7S
, and 𝜎
TUQV
. In Section 1.3.2, we investigate these
definitions of mechanism uncertainty and find that 𝜎
OPQ
≈𝜎
TUQV
≈𝜎
MN
≈𝜎 and 𝜎
R7S
≈0.4𝜎.
The conditional likelihood probability distribution that specifies the dependencies of all the random
variables in the SSC model and observations, including observational errors of mechanism
orientation is 𝑃
WWX
=𝑃𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅,𝜎. We ignore any correlation between the fluctuations in 𝕰
𝟎

about 𝚺

with those in 𝐸
)
, which we expect to be small because the eigenvalues are first-order
insensitive to errors in the eigenvectors (Rayleigh’s Principle). Hence, the likelihood probability can
then be factorized as
𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅,𝜎=𝑃𝐸
)
,𝓕

;𝕰
𝟎
,𝚺

,𝜅,𝜎 𝑃𝕰
𝟎
; 𝚺

,𝜎. (30)
Following JJ21, we assume 𝓕

and 𝐸
)
are conditionally independent, i.e., for a fixed value of 𝕰
𝟎
, the
fluctuations in 𝓕

and 𝐸
)
are statistically independent. This independence is guaranteed by the
assumption that the distribution of 𝐸
)
is, to first order, independent of 𝕰
𝟎
, as well as 𝜅:
𝑃𝐸
)
| 𝕰
𝟎
; 𝚺

,𝜅,𝜎=𝑃(𝐸
)
;𝑅,𝜎). (31)
Using numerical simulations of moment tensor densities sampled from the SSC model, we find that
there are negligible correlations between 𝐹

)
, 𝐸
)
, and 𝕰
𝟎
(𝑟 <0.1), supporting these assumptions.
Moreover, empirical distributions computed from the Ridgecrest datasets show only weak
correlations between 𝐹

)
and 𝐸
)
, for example, the Pearson correlations are 𝑟 =0.25,0.19, and 0.06
for the Hauksson, Cheng, and Lin datasets, respectively. Assuming independence, the resulting
probability function is
𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅,𝜎=𝑃𝓕

| 𝕰
𝟎
; 𝚺

,𝜅,𝜎 𝑃(𝐸
)
;𝑅,𝜎) 𝑃𝕰
𝟎
; 𝚺

,𝜎. (32)
We further assume that the data averaging to obtain the reduced observable 𝓕

is equivalent to the
marginalization over eigenvector errors, i.e., the joint distribution of the reduced observational
variables is given by the marginal distribution
𝑃
WWX
=𝑃(𝐸
)
| 𝑅,𝜎) 𝑃𝓕

| 𝕰
)
; 𝚺

,𝜅,𝜎 𝑑𝑃𝕰
)
; 𝚺

,𝜎
&
)
=𝑃(𝐸
)
;𝑅,𝜎)𝑃𝓕

; 𝑅,𝜅,𝜎. (33)
38
Furthermore, 𝜎 is an independent random variable with a distribution 𝑃(𝜎). The dependency of the
SSC parameters on 𝜎 can be marginalized, such as
𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅= 𝑃(𝐸
)
;𝑅,𝜎)𝑃𝓕

; 𝑅,𝜅,𝜎𝑑𝑃(𝜎)
&
)
= 𝑃(𝐸
)
;𝑅,𝜎)𝑃𝓕

; 𝑅,𝜅,𝜎𝑑𝑃(𝜎)
&
)
.(34)
In summary, the SSC parameters can be estimated by maximizing the likelihood probability function
𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅, which requires evaluating three probability distributions: (1) the conditional
distribution of the moment fractions, 𝑃𝓕

;𝑅,𝜅,𝜎, (2) the differential stress ratio 𝑃(𝐸
)
;𝑅,𝜎), and (3)
distribution of the uncertainties, 𝑃(𝜎). The independent parameters in these distributions are
obtained by fitting the observations and the probability evaluated on the expectations from the SSC
model (Figure 1.6). The maximum likelihood estimation is thus the set of parameters 𝑅 and 𝜅 that
maximize 𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅.
The moment fractions are the expectations of quadratic directions cosines; they always add to unity
and are distributed in the interval 𝓕

∈[0,1]. Statistically, they form a simplex structure (Aitchison,
1986; Barndorff-Nielsen & Jørgensen, 1991), which inherently implies data correlations. Therefore,
moment fractions cannot be modeled with unconditional distributions and the selection of 𝑃𝓕

is
an important step. A common distribution for simplex data is the multivariate Beta or Dirichlet
distributions. However, the correlation structure of observed and synthetic 𝓕

show strong positive
and negative correlations. Strong negative correlations are not properly modeled by the Dirichlet
distribution and positive correlations are not allowed.
In JJ21, we showed that the logistic normal ( ℒ𝒩) distribution (Frederic & Lad, 2008; Holmes &
Schofield, 2022) accurately models the moment fraction simplex and its correlation structure. The
PDF is
𝑃·𝓕

(𝛍,𝐕)º=ℒ𝒩(𝛙;𝛍,𝐕)=
1
√2𝜋 det(𝐕)
¾¿𝐹
2
Y&
.
2%)
À expÁ−
1
2
(𝛙−𝛍)
Z
𝐕
Y&
(𝛙−𝛍)Â, (35)
where the vector 𝛙 results from the logit normal transformation of the moment fractions,
39
𝜓
-
=lnÅ
𝐹

-
𝐹

)
Æ=lnÅ
𝐹

-
1−∑ 𝐹

2
.
2%&
Æ, 𝛼 =1,…,𝑑. (36)

The inverse transformation is
𝐹

-
=
e
[
#
1+∑ e
[
$
.
2%&
, 𝛼 =1,…,𝑑. (37)

𝛍 and 𝐕 are the mean and covariance of 𝛙. Theoretically, any 𝐹

-
can be chosen for the denominator
in the logit normal transformation. In practice, we choose 𝐹

)
; however, the selection of other 𝐹

-
does
not change the probability estimates.
𝐸
)
is a random variable distributed in the interval (0,1), and a two-parameter Beta distribution
allows us to model it accurately, 𝑃(𝐸
)
|𝑅)=Beta(𝐸
)
;𝑎,𝑏). Finally, we analyze the mechanism
estimation uncertainty. Uncertainty can be parametrized by the total variance 𝜎
+
of the angle 𝜑
between an estimated mechanism and a preferred solution. We assume that 𝜑 is a random variable
that follows a circular normal or von Mises distribution (Gatto & Jammalamadaka, 2007), we equate
𝜏 =𝜎
Y+
and allow 𝜎 →∞,
𝑣𝑀(𝜑|𝜏) =
exp(τcos𝜑)
2𝜋𝐼
)
(τ)
, (38)
where 𝐼
)
(∙) is the modified Bessel function of zeroth order with argument (τ), and 𝜏 is the von Mises
concentration parameter related to the variance angle, which measures the dispersion of 𝜑. For large
values of 𝜏, the von Mises distribution resembles the normal distribution with a standard deviation
𝜎. Where 𝜏 =0, the distribution converges to a uniform distribution in the interval [0,2𝜋).
Catalogs of focal mechanisms report a 𝜎 for each solution. Therefore, 𝜎 is itself a random variable,
and it can be modeled with a two-parameter Gamma distribution, 𝑃(𝜎|𝛼,𝛽)=Gamma(𝜎;𝛼,𝛽).
The two-parameter Beta and Gamma distributions help model the location and spread of the random
variables. Additionally, samples from the Gamma and von Mises distributions properly model angle
uncertainty, and consequently, we can fully quantify its effect on the moment fractions. The logistic
40
normal distribution is flexible enough to model strong positive and negative correlations in the
simplex structure, which a Dirichlet distribution cannot model.
Figure 1.6 shows the workflow for calculating the likelihood function 𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅. We start
with a catalog 𝒞 ={𝐦
"
,𝜎
"
:𝑛 =1,…,𝑁} of 𝑁 observed mechanisms and their uncertainties. Then,
we randomly sample with replacement 𝑞 times a 𝑃-number mechanisms. We typically take 𝑃 =
0.3𝑁, i.e., 30% of mechanisms in the catalog. The 𝑞-th CMT mechanism from the sample is 𝐌

𝟎
(\)
∝
∑ 𝐦
]
K
]%&
, from which we obtain the set of principal directions 𝕰
)
(\)
, the differential stress ratio 𝐸
)
(\)
,
and the SOR basis set, 𝕭𝐌

𝟎
=𝐌

-
,𝛼 =0,…,4. The sample moment fractions are 𝐹

-
(\)
=
&
K
∑ 𝐌

-
:𝐦
]

+
K
]%&
. The ensemble of 𝑞-samples definite the distributions of observables, 𝐸
)
.^_^
and
𝓕

.^_^
.
The next step is constructing the SSC probability distributions given observed uncertainties. This
process is achieved by random sampling synthetic catalogs that obey the SSC+Noise model. The
process for this forward modeling is explained in Section 1.4, but in summary, we want to estimate
𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅 marginalized over 𝜎 . We performed the marginalization implicitly by
aggregating 𝑁 catalogs containing 𝑀
"
mechanism where 𝑀
"
∝𝑝(𝜎
"
|𝛼,𝛽). The mechanism in each
catalog are samples from the SSC distribution 𝑝(𝐦
OO`
|𝜅,𝑅)∝e
:9(@)
, and randomly rotated by an
angle 𝜑 with standard deviation 𝜎
"
. This process is depicted by the green dashed line box in Figure
1.6.
From the aggregated catalog 𝒞 ={𝒞
"
(𝑅,𝜅,𝜎
"
):𝑛 =1,…,𝑁}, we obtain distributions of synthetic 𝐸
)

and 𝓕

given the model parameters 𝑅 and 𝜅. From these synthetic empirical distributions of 𝐸
)
and
𝓕

, we fit Beta(𝐸
)
;𝑎,𝑏) and ℒ𝒩·𝓕

(𝛍,𝐕)º. Finally, the MLE estimates the model parameters by
maximizing the likelihood of the observables Beta(𝐸
)
.^_^
;𝑎,𝑏), and ℒ𝒩(𝓕

.^_^
;𝛍,𝐕).

41

Figure 1.6. SSC model parameter estimation. Input data are a source mechanism catalog 𝓒, from which 𝓕
9
,𝐸
#
,
and 𝕰
𝟎
, are measured, and together with 𝜎, their empirical distributions are fitted: 𝛤, ℒ𝒩, and 𝛣 are the
Gamma, Logistic Normal, and Beta distributions, respectively. The likelihood function is proportional to the
probability of the data given the SSC+Noise model parameters. The calculations in the SSC+Noise box
correspond to the forward modeling of SSC+Noise moment tensor densities, and more details are shown in
Figure 1.11.

1.3.2 Observational Earthquake Mechanism Uncertainties
Uncertainties in moment tensor estimation can arise from observational and modeling errors
(Duputel et al., 2012; Moghtased-Azar et al., 2022; Vasyura-Bathke et al., 2021). Observational errors
are essentially due to imperfect data or measurements, and modeling errors come from the
assumptions made to characterize the strongly anelastic processes in the source and its location and
imperfect knowledge of the structure of the Earth. Therefore, observed source mechanisms are
uncertain, and their uncertainties, biases, and errors contribute to mechanism complexity
observations and inferred stress heterogeneity (Hardebeck, J. L., 2006).
To see this more clearly, let’s consider a simple example where the moment tensor density is 𝐦(𝐱)=
∑ 𝑚
"
𝐦
"
𝛿(𝐱−𝐱
"
)
$
"%&
. Assuming no complexity (𝜅 →∞) and 𝑅 =0.5: 𝐦
"
=𝑚
"
𝐌

)
. Let us assume
random uncertainties in the orientation of the estimated mechanism, such as errors in the
observation data or the moment tensor inversion methodology. Any uncertainty on the directions of
42
𝐌

-
can be expressed in terms of an angle 𝜑
-,"
. For simplicity, consider errors in the orientation 𝐌

&

only. The moment tensor density becomes
𝐦
"
=𝑚
"
cos𝜑
-%&,"
𝐌

)
+sin𝜑
-%&,"
𝐌

&
, (39)
In JJ19, we modeled the angle 𝜑 as a random variable that follows the von Mises distribution of
equation 38. The circular moments of the von Mises distribution are functions of the modified Bessel
functions of first kind. The Aki moment fraction is the linear expectation of the zeroth-degree
direction cosine under the von Mises distribution,
𝐹
)
=E[cos(𝜑)]=
𝐼
&
𝐼
)
, (40)
where E represents the expectation integral. The first-degree term arises from the uncertainties on
𝜑, which we assume to have zero mean and angular variance 𝜎
+
. Note that E[sin(𝜑)]=0. As such,
mechanism uncertainty does not contribute to the Aki moment fraction, but due to the constraint of
direction cosines simplex, cos
+
(𝜑)+sin
+
(𝜑)=1, it reduces it. The total moment fractions are the
quadratic expectations of the direction cosines. We get,
𝐹

)
=E[cos
+
(𝜑)]=
1
2
Ì1+
𝐼
+
𝐼
)
Í,and (41𝑎)
𝐹

&
=𝐹

!
=E[sin
+
(𝜑)]=
1
2
Ì1−
𝐼
+
𝐼
)
Í. (41𝑏)
Figure 1.7a shows the behavior of the Aki moment fraction and the total moment fractions as a
function of the variance 𝜎
+
of the uncertainty angle 𝜑. In this simple model, there are uncertainties
only in the direction of 𝐌

&
and as such, 𝐹

&
is equal to the complexity factor 𝐹

!
. As shown in the
following section, high-quality seismic catalogs in Southern California have average uncertainties
𝜎 =25
G
, thus the Aki moment fraction 𝐹
)
=0.9, the zeroth-degree moment fraction is 𝐹

)
=0.84, and
the apparent complexity factor 𝐹

!
=0.16. For 𝜎 =27
G
, the complexity factor reaches the limit
between low and moderate complexity, and for 𝜎 =48
G
, high complexity is achieved by noise
43
perturbations only. We emphasize that these values have been observed in other catalogs of Southern
California (e.g., Hardebeck, 2006).
This example demonstrates that uncertainties in the determination of a mechanism 𝐦
"
increase the
complexity measures, and they propagate onto estimates of the SSC model parameters. Equations 40
and 41 were obtained by JJ19 (equations 70-72 there). In JJ19, the moment fraction in the direction
𝐌

&
arises from in-plane variations of the slip vector on a fault plane instead of uncertainties.
Therefore, mechanism uncertainties are indistinguishable from source complexity in this framework,
and special care must be taken when modeling stress-strain fields from observed uncertain moment
tensors.

Figure 1.7. Aki moment fraction 𝐹
#
, zeroth-degree moment fraction 𝐹
9
#
, and complexity factor 𝐹
9
$
expectations
from a moment tensor density with uncertainties. (a) No complexity (𝜅 →∞). Uncertainties are one-
dimensional only. (b) 𝜅 =20, 𝑅 =0.5, and 𝜎 is the total standard deviation of four-dimensional noise. (c) 𝜅 =
0, 𝑅 =0.5, and 𝜎 is the total standard deviation of five-dimensional noise. The dashed grey lines divide the
low, moderate, and high complexity regions in all panels.

1.3.3 Probability Distributions of Focal-Mechanism Orientation
In the previous section we show that mechanism uncertainty is basically undistinguishable from
physical mechanism complexity reflected in the total moment fractions, 𝓕

. Therefore, it is important
to understand different parameterizations of mechanism uncertainty and ultimately, account for it
on observations of 𝓕

. In this section we show that it is possible to correct for bias in the estimation
44
of the SSC parameters due to noise by modeling 𝓕

from SSC+Noise mechanism to match
observations and then, compare it against noise-free simulations. Hence, we investigate several
parametrizations of mechanism orientation uncertainty, but focus on nodal plane uncertainties
because it is the uncertainty measure reported by HASH.
The seismicity catalogs that we analyze report mechanism for double couple dislocations. Therefore,
we investigate three parametrizations of uncertainties in double-couple mechanism orientation:
(1) the angular distance between a preferred average mechanism and any acceptable solution 𝜑
(Kagan & Jackson, 2014), parametrized by the total variance 𝜎
,
+
,
(2) the angles between the mechanism principal vectors and/or the vectors orthogonal to the nodal
planes (Hardebeck, Jeanne L. & Shearer, 2002; Hauksson et al., 2012; Yang et al., 2012), parametrized
by three variances 𝜎
𝐧 =
+
, 𝜎
𝐬 b
+
, and 𝜎
𝐛
d
+
, and
(3) the distribution of resolved strike, dip, and rake angles (Reasenberg & Oppenheimer, 1985),
parametrized by three variances 𝜎
9_e
+
, 𝜎
.6]
+
, and 𝜎
f^eg
+
.
Additionally, we investigate how these parametrizations are reflected on distributions of the angles
between uncertain mechanisms and the SOR basis set, 𝜑
-
=arccos𝐌

0
,𝐦 , and therefore, onto the
total moment fractions.
Angular distances between mechanisms can be parametrized by probability distributions on the unit
circle, distributions over 3D rotation matrices, or 5D unit sphere. Although complex probability
distributions can be used to model uncertainties and correlations in n-dimensions, here we focus on
finding the equivalences between 𝜎
,
+
and 𝜎
𝐧 =
+
, 𝜎
𝐬 b
+
, and 𝜎
𝐛
d
+
and 𝜎
9_e
+
, 𝜎
.6]
+
, and 𝜎
f^eg
+
. The description of
uncertainties with one parameter 𝜎
,
is possible if the source mechanism parameters, either full
tensor, strike-dip-rake, or nodal plane vectors, are independent and uncorrelated.
Consider, for example, 3D rotations of source mechanisms given by the Rodrigues rotation matrix
(Dai, 2015; Goldstein et al., 2002),
45
𝐑(𝜑,𝐮)=
Ð
𝑢
&
+
+(1−𝑢
&
+
)cos𝜑 𝑢
&
𝑢
+
(1−cos𝜑)−𝑢
5
sin𝜑 𝑢
&
𝑢
5
(1−cos𝜑)+𝑢
+
sin𝜑
𝑢
&
𝑢
+
(1−cos𝜑)+𝑢
5
sin𝜑 𝑢
+
+
+(1−𝑢
+
+
)cos𝜑 𝑢
+
𝑢
5
(1−cos𝜑)−𝑢
&
sin𝜑
𝑢
&
𝑢
5
(1−cos𝜑)−𝑢
+
sin𝜑 𝑢
+
𝑢
5
(1−cos𝜑)+𝑢
&
sin𝜑 𝑢
5
+
+(1−𝑢
5
+
)cos𝜑
Ò. (42)

𝜑 is the rotation angle and 𝐮 =(𝑢
&
,𝑢
+
,𝑢
5
) is the 3D unit vector in the direction of the rotation axis.
If 𝐦 is an uncertain estimation of a stress-aligned mechanism, 𝐦
hhI
, that contains random errors,
then 𝐦 =𝐑𝐦
hhI
𝐑
,
, and the angle between mechanisms is 𝜑 =arccos(𝐦
hhI
:𝐦 ) . Modeling
mechanism orientation uncertainty requires parametrizing the angular distances between a
preferred solution and an ensemble of possible solutions. Therefore, it is likely that the marginal
distribution of 𝜑 is symmetric on (−𝜋,𝜋) with zero mean and conditionally independent of the
rotation axis. Let’s assume, for example, that 𝜑 follows a normal circular von Mises probability
distribution,
𝑝(𝜑;𝜏)=𝑣𝑀(𝜑|𝜏)=
1
𝑐
)
(𝜏)
exp(𝜏cos𝜑), (43)
where 𝑐
)
(𝜏) is a normalizing constant, and 𝜏 =1/𝜎
,
+
is the von Mises distribution concentration
parameter. Furthermore, assume that the rotation axis vector is uniformly distributed on the 3D
sphere. We can construct this distribution by expressing the rotation-axis vector in polar-spherical
coordinates such as 𝐮 =(sin𝜙cos𝜓,sin𝜙sin𝜓,cos𝜙), 𝜙 ∈[0,𝜋] is the colatitude, and 𝜓∈[0,2𝜋)
is the azimuth. The uniform probability density function of a vector orientation in the unit sphere is
𝑝𝐮(𝜙,𝜓)=
1
4𝜋
sin𝜙. (44)
Therefore, the probability of random Rodrigues’ rotation matrices is the product of the probability
of the rotation axis and the probability of the rotation angle (Miles, 1965; Roberts & Winch, 1984),
𝑝(𝐑|𝜑,𝐮)= 𝑝(𝜑;𝜏)𝑝𝐮(𝜙,𝜓). (45)
From 𝑝(𝐑|𝜑,𝐮) we can sample random rotation matrices and use them to rotate mechanism sampled
from the SSC model distributions, 𝐦 =𝐑𝐦
hhI
𝐑
,
. This process allows computing stress-aligned
moment tensor densities that contain noise, and from them, one can compute total moment fraction
46
distributions. Additionally, we can measure the effect of random rotations of focal mechanism in the
direction of SOR basis elements, 𝔅(𝐦
hhI
)=𝐌

-
;𝛼 =0,…,4. The uncertainties of 𝐦 in the
direction of 𝐌

-
are measured by the rotation angles 𝜑
-
=arccos 𝐌

-
:𝐦 . The angles 𝜑
-
satisfy the
direction cosine constraint ∑ cos
+
(𝜑
-
)
4
-%&
=1, and their expectation integral is
E[cos
"
(𝜑
-
)]=
1
4𝜋𝑐
)
(𝜏)
exp(𝜏cos𝜑)sin𝜙 𝐌

-
:[𝐑𝐦 𝐑
,
]
"
𝐑∈h)
%
𝑑𝜑𝑑𝜙𝑑𝜓. (46)
The integral is performed over the space of orthonormal rotation matrices of three dimensions S0
5

(special orthonormal group). E[cos
"
(𝜑
-
)] is then a function of the rotation angle 𝜑, its concentration
parameter 𝜏, and the rotation axis parameters (𝜙,𝜓) and the parameters of their distributions. The
concentration parameter 𝜏 can be estimated from reported uncertainties in focal mechanism
catalogs.
Figure 1.8 shows the total moment fractions and the Aki moment fraction expectations as functions
of the angular standard deviation of the rotation angle of random Rodrigues matrices. Random
isotropic rotations of focal mechanisms increase the complexity factor 𝐹

!
=∑ 𝐹

-
4
-%&
from 0 up to
𝐹

!
=0.5 for 𝜎
,
=45
G
and 𝐹

!
=0.64, at 𝜎
,
=90
G
. The Aki moment fraction decays to 𝐹
)
=0.28
when 𝜎 =90
G
. For higher degree moment fractions, 𝐹

&
grows faster than 𝐹

+Y4
, from 𝐹

&
=0 at 𝜎
,
=
0
G
up to a maximum 𝐹

&
=0.2048 at 𝜎
,
=66
G
. 𝐹
)
=0.2875, 𝐹

)
→0.3644, 𝐹

&
→0.2044, 𝐹

+
=𝐹

5
→
0.1497, and 𝐹

4
→0.1315, as 𝜎
,
=90
G
. 𝐹

&
increases rapidly when 𝜎
,
<15
G
then slows, its first
derivative reaches a maximum at 𝜎
,
=15
G
, this value roughly corresponds to rotations up to
𝜑 ≤45
G
in the 95% interval of a von Mises distribution 𝑣𝑀(𝜑|𝜏). If a rotation angle is 𝜑 >45
G
in
the direction of 𝐌

&
, the elements 𝐌

)
, and 𝐌

&
of the SOR basis set interchange. A similar nonlinearity
occurs when 𝜑 >90
G
which has over 5% of probability for 𝜎
,
~28
G
and either 𝐌

+
or 𝐌

5
switch
places with 𝐌

)
. Rotations in 3D do not turn 𝐌

)
into 𝐌

4
, thus, 𝐹

4
increase more rapidly up to 𝜎
,
~38
G

which roughly corresponds to rotations 𝜑 ≤180
G
in the 95% confidence interval. This nonlinear
behavior is likely to be considered in focal-mechanism inversion algorithms to produce stable
47
solutions. For example, HASH truncates the distribution of acceptable solutions at 𝜑 =30
G
. The
effect of such truncation is depicted by the dashed lines in Figure 1.8.

Figure 1.8. Total moment fraction and Aki moment fraction expectations for 3D isotropic uncertainties on the
orientation of double-couple mechanism as functions of the variance angle. (a) Aki moment fraction 𝐹
#
,
zeroth-degree moment fraction 𝐹
9
#
, and complexity factor 𝐹
9
$
=1− 𝐹
9
#
for different values of angular
standard deviation 𝜎, such as the von Mises concentration parameter is 𝜏 =1/𝜎
'
. (b) Higher-degree total
moment fraction 𝐹
9
!(%
. In both panels, dashed lines are computed from truncated distributions with |𝜑
!
| ≤
30
)
.

Consider the two vectors 𝐧 and 𝐬 that are mutually orthogonal (𝐧 ∙𝐬 =0) and orthogonal to the
focal-mechanism nodal planes. Also, consider the perpendicular vector 𝐛
Ö
=−𝐧 ×𝐬 . The vectors 𝐧 , 𝐬 ,
and 𝐛
Ö
are the normal, slip, and null directions of a fault dislocation. A typical parameterization of
rotations of a rigid body is the Euler angles. Thus, we now investigate how this parametrization
would affect the expectation of uncertain moment fractions. Let 𝐑 be the rotation matrix of the
classical Euler sequence 𝐑=[𝐑𝟏(𝛾
&
)][𝐑𝟐(𝛾
+
)][𝐑𝟑(𝛾
5
)] such as,
𝐑=©𝐧 ,𝐛
Ö
,𝐬 ª=
Û
cos𝛾
+
−cos𝛾
5
sin𝛾
+
sin𝛾
+
sin𝛾
5
cos𝛾
&
sin𝛾
+
cos𝛾
&
cos𝛾
+
cos𝛾
5
−sin𝛾
&
sin𝛾
5
−cos𝛾
5
sin𝛾
&
−cos𝛾
&
cos𝛾
+
sin𝛾
5
sin𝛾
&
sin𝛾
+
cos𝛾
&
sin𝛾
5
+cos𝛾
+
cos𝛾
5
sin𝛾
&
cos𝛾
&
cos𝛾
5
−cos𝛾
+
sin𝛾
&
sin𝛾
5
Ü. (47)
The columns of 𝐑 specify the locations ©𝐧 ,𝐛
Ö
,𝐬 ª of rotated vectors relative to mean vectors ©𝐧
𝟎
,𝐛
Ö
𝟎
,𝐬
𝟎
ª
of a focal mechanism 𝐦
𝟎
, and 𝛾
&
=𝛾
5
=[0,2𝜋), and 𝛾
+
=[0,𝜋] are the Euler angles. Multiple rotated
48
directions ©𝐧 ,𝐛
Ö
,𝐬 ª can be seen as an ensemble of acceptable estimations of a mechanism 𝐦
𝟎
. For
notation simplicity, let the elements be 𝐑=[𝑛
6
,𝑠
6
,𝑏
6
],𝑖 =1,2,3, ordered column-wise and let 𝐦 be
an uncertain estimation of the 𝐦
99k
that contains random errors. The uncertain mechanism is then
𝐦 =𝐑𝐦
99k
𝐑
,
and, of course, 𝐑 is a random Euler orthogonal rotation matrix, and 𝐑
,
is its transpose.
If 𝐑 is a simple rotation in three dimensions, from Euler’s theorem (Goldstein et al., 2002;
Stuelpnagel, 1964) the total rotation angle is
cos𝜑 =
1
2
𝑡𝑟(𝐑)−1. (48)
Therefore, the total rotation angle, 𝜑 is related to the Euler angles through 𝑡𝑟(𝐑). With some algebra
and basic trigonometric identities, we find that
cos
𝜑
2
=cos
𝛾
&
+𝛾
5
2
cos
𝛾
+
2
. (49)
Substituting equation 49 into 43 gives the probability distribution of the Euler angles (𝛾
&
,𝛾
+
,𝛾
5
) that
is equivalent to the distribution of Rodrigues rotation matrices:
𝑝(𝛾
&
,𝛾
+
,𝛾
5
|𝜏)=
1
𝑐
&
(𝜏)
expÌ𝜏Ácos
+
Ì
𝛾
&
+𝛾
5
2
Ícos
+
·
𝛾
+
2
ºÂÍ. (50)
The distribution 𝑝(𝛾
&
,𝛾
+
,𝛾
5
|𝜏) is not a von Mises distribution, strictly speaking, because of the
quadratic cosine arguments. Furthermore, the Euler angles might not be conditionally independent
if biases exist in the mechanism inversion.
Inversion algorithms that estimate focal mechanisms and moment tensors might not provide enough
information to justify the choice of the Euler angle parametrization over the rotation angle-axis of
the Rodrigues formula. For example, the inversion algorithm HASH of Hardebeck and Shearer reports
estimations of the nodal plane uncertainties, and the FPFIT algorithm of Reasenberg and
Oppenheimer (1985) reports uncertainties of the strike, dip, and rake directions. Thus, we are
interested in the relation between the total rotation angle and the nodal plane uncertainties.
49
Without loss of generality, we select the coordinate system oriented in directions parallel to the
normal, null, and shear directions such as 𝐈
5l
=©𝐧
𝟎
,𝐛
Ö
𝟎
,𝐬
𝟎
ª is the identity matrix. The angles between
the vectors 𝐧
𝟎
,𝐬
𝟎
, and 𝐛
Ö
𝟎
and their rotated versions are
cos𝛿
&
=𝐧
𝟎
∙𝐧 =𝑛
&
=cos𝛾
+
, (51𝑎)
cos𝛿
+
=𝐛
Ö
𝟎
∙𝐛
Ö
=𝑏
+
=(cos𝛾
&
cos𝛾
+
cos𝛾
5
−sin𝛾
&
sin𝛾
5
),and (51𝑏)
cos𝛿
5
=𝐬
𝟎
∙𝐬 =𝑠
5
=cos𝛾
&
cos𝛾
5
−cos𝛾
+
sin𝛾
&
sin𝛾
5
. (51𝑐)
The focal-mechanism inversion software HASH reports two plane uncertainties only, PU1, and PU2,
which are then equivalent to two angles, 𝛿
&
and 𝛿
+
. Furthermore, 𝑡𝑟(𝐑)=(cos𝛿
&
+cos𝛿
+
+cos𝛿
5
).
If the rotations about the axis ©𝐧
𝟎
,𝐬
𝟎
,𝐛
Ö
𝟎
ª are isotropic, then rotation matrices, 𝐑 belong to the special
orthonormal group of three dimensions (SO3). Therefore, 𝐑 follows a von Mises-Fisher distribution
in the 3D unit sphere (Bingham et al., 2009; James, 1954; Kent, 1978; Khatri & Mardia, 1977; Wood,
1994). The distribution is
𝑣𝑀𝐹
5l
(𝐑|𝜏)=
1
𝑐
5
(𝜏)
expÅ
𝜏
2
(cos𝛿
&
+cos𝛿
+
+cos𝛿
5
−1)Æ, (52)
where 𝑐
5
(𝜏) is the normalizing constant. This distribution is obtained by substituting 𝑡𝑟(𝐑)=
2cos𝜑+2 in the von Mises distribution of 𝜑, in equation 43. 𝜑 is again the total rotation angle from
Euler’s theorem. Consequently, the concentration parameter of the von Mises-Fisher distribution is
𝜏 =1/𝜎
+
=2/(PU1
+
+PU2
+
), and PU3 (uncertainty of the plane normal to 𝐛
Ö
𝟎
) follows the same
distribution as PU1 and PU2.
Figure 1.9 shows examples of the marginal distributions of the angles 𝛿
6
between mean vectors
©𝐧
𝟎
,𝐬
𝟎
,𝐛
Ö
𝟎
ª and uncertain observations ©𝐧 ,𝐬 ,𝐛
Ö
ª for different values of the von Mises concentration
parameter 𝜏. For illustrative purposes, we plot the sum of the marginals 𝑝(𝐧 |𝜏)+𝑝(𝐬 |𝜏)+𝑝𝐛
Ö
ß𝜏. As
𝜏 increases, the distribution concentrates on the mean directions.

50

Figure 1.9. Visualization of the marginal probability distributions of the normal, shear, and null vectors
(𝐧 6,𝐬 4,𝐛
R
) of a double couple mechanism for different values of the von Mises concentration parameter 𝜏 =
1/𝜎
'
, (a) 𝜎 =5
)
, (b) 𝜎 =10
)
, and (c) 𝜎 =15
)
.

Binham et al. (2009) showed that if the rotation matrix 𝐑~𝑣𝑀𝐹(𝐈|𝜏) follows a von Mises-Fisher
distribution, a rotated matrix (or a focal mechanism in our case) follows the same distribution, i.e.,
𝐦 ~𝑣𝑀𝐹(𝐑𝐦
99k
𝐑
,
|𝜏). We are interested in modeling the effect of random rotations of focal
mechanism on the total moment fractions and therefore is convenient to estimate the directions
cosines relative to the SOR basis set 𝕭={𝐌

-
;𝛼 =0,…4}, where the unit CMT is the leading term.
The direction cosines are cos𝜑
-
=𝐌

-
:𝐑𝐦 𝐑
,
, and taking 𝐦 =𝐌

𝟎
=
&
√+
diag[1,0,−1], and 𝐑=
[𝑛
6
,𝑠
6
,𝑏
6
],𝑖 =1,2,3, we get
cos𝜑
)
=
1
2
(𝑛
&
+
−𝑠
&
+
−𝑛
5
+
+𝑠
5
+
), (53𝑎)
𝑐𝑜𝑠𝜑
&
=𝑛
&
𝑛
5
−𝑠
&
𝑠
5
, (53𝑏)
cos𝜑
+
=
1
√2
(𝑛
&
𝑛
+
−𝑠
&
𝑠
+
+𝑛
+
𝑛
5
−𝑠
+
𝑠
5
), (53𝑐)
cos𝜑
5
=
1
√2
(𝑛
&
𝑛
+
−𝑠
&
𝑠
+
−𝑛
+
𝑛
5
+𝑠
+
𝑠
5
), (53𝑑)
cos𝜑
4
=
1
2√3
(𝑛
&
+
−𝑠
&
+
−2𝑛
+
+
−2𝑠
+
+
+𝑛
5
+
−𝑠
5
+
). (53𝑒)
The uncertain mechanism might also be written as,
51
𝐦 =
𝟏
√𝟐
[𝐧 𝐬 +𝐬 𝐧 ]=
1
√2
Û
2𝑛
&
𝑠
&
𝑛
&
𝑠
+
+𝑠
&
𝑛
+
𝑛
&
𝑠
5
+𝑠
&
𝑛
5
𝑛
&
𝑠
+
+𝑠
&
𝑛
+
2𝑛
+
𝑠
+
𝑛
+
𝑠
5
+𝑠
+
𝑛
5
𝑛
&
𝑠
5
+𝑠
&
𝑛
5
𝑛
+
𝑠
5
+𝑠
+
𝑛
5
2𝑛
5
𝑠
5
Ü. (54)
In which case the SOR direction cosines become
cos𝜑
)
=𝑛
&
𝑠
&
−𝑛
5
𝑠
5
, (55𝑎)
cos𝜑
&
=𝑛
&
𝑠
5
+𝑠
&
𝑛
5
, (55𝑏)
cos𝜑
+
=
1
√2
(𝑛
&
𝑛
+
−𝑠
&
𝑠
+
+𝑛
+
𝑠
5
+𝑠
+
𝑛
5
), (55𝑐)
cos𝜑
5
=
1
√2
(𝑛
&
𝑠
+
+𝑠
&
𝑛
+
−𝑛
+
𝑠
5
−𝑠
+
𝑛
5
), (55𝑑)
cos𝜑
4
=
1
√3
(𝑛
&
𝑠
&
−2𝑛
+
𝑠
+
+𝑛
5
𝑠
5
). (55𝑒)
The two descriptions in equations 54 and 56 are equivalent if we deal with focal mechanisms only.
Figure 1.10 shows the equivalency between 𝜎
,
’s from different parametrizations of mechanism
orientation uncertainty. For each value of𝜎
,
, we generated 10,000 samples of the total rotation angle
𝜑 drawn from a von Mises distribution 𝑣𝑀(𝜑|𝜏) and rotation axes, 𝐮 from a uniform distribution in
the sphere, 𝑈𝐮 (𝜙,𝜓). The random Rodrigues rotation matrix is 𝐑(𝜑,𝐮 ), and the rotated
mechanism is 𝐦 =𝐑𝐦
𝟎
𝐑
,
, where 𝐦
𝟎
is a double-couple mechanism 𝐦
𝟎
=
&
√+
diag[1,0,−1].
We calculate the angles 𝜑
-
between 𝐦 and the SOR basis set, whose leading term is 𝐦
𝟎
. Then, we
aggregated the angles into empirical distributions and calculated their standard deviations. Figure
1.10-a shows the standard deviation 𝜎
-
, 𝛼 =1,…4. Similarly, we compute the angles between the
normal, null, and slip directions [𝐧 ,𝐛
Ö
,𝐬 ] of 𝐦 and those of 𝐦
𝟎
, [𝐧
𝟎
,𝐛
Ö
𝟎
,𝐬
𝟎
]. The dispersion of these
angles are given by 𝜎
𝐧 =
, 𝜎
𝐬 b
, and 𝜎
𝐛
d
in Figure 1.10-b. Similarly, we computed the strike, dip, and rake
directions of 𝐦 and aggregate them into empirical distributions and estimate the standard
deviations. They are shown in Figure 1.10-c.
From our numerical simulations of mechanism uncertainty, we find that the dispersion of the vectors
[𝐧 ,𝐛
Ö
,𝐬 ] are 𝜎
𝐧 =
≈𝜎
𝐬 b
≈𝜎
𝐛
d
≈𝜎
,
for small rotations. The dispersions of 𝐧 and 𝐬 can be interpreted as
52
the nodal plane uncertainty reported by HASH. For 𝜎
,
>15
G
, 𝜎
𝐧 =
and 𝜎
𝐬 b
decrease faster than 𝜎
𝐛
d
. This
behavior is explained by the nodal plane ambiguity and large rotations: if the rotation angle is too
large, the resulting 𝐬 vector might be at a smaller angle from 𝐧
𝟎
than to 𝐬
𝟎
and the opposite for 𝐧 ,
thus, reducing the total dispersion of the distribution. Furthermore, the angular distance between
the principal directions of a rotated mechanism and a mean mechanism cannot exceed 90
G
, while the
total rotation angles are distributed in the interval [−180
G
,180
G
). Consequently, 𝜎
𝐧 =
, 𝜎
𝐬 b
, and 𝜎
𝐛
d
are
only a fraction 𝜎 for large rotations. For small rotations, 𝜎(𝜑
)
)→𝜎
,
, 𝜎(𝜑
&
)→2𝜎
,
, 𝜎(𝜑
+
)=
𝜎(𝜑
5
)→𝜎
,
, and 𝜎(𝜑
4
)→0. This behavior is also seen in the total moment fractions computed from
the expectation integral in equation 46 and shown in Figure 1.8. Thus, for random isotropic rotations
and 𝜎
,
→∞, 𝜎(𝜑
-
)→
&
m
𝜎
,
.
The standard deviations of strike and rake directions, 𝜎
9_e
and 𝜎
f^eg
, respectively, are similar to the
standard deviation of the total rotation angle and
n
&'(
n
~1. However, 𝜎
.6]
is only about 0.4𝜎
,
. Thus,
random rotations of focal mechanisms produce a smaller dispersion in the dip direction, mainly
because the dip angle is only between 0
G
and 90
G
.

Figure 1.10. Comparison of parametrizations of mechanism orientation uncertainty from numerical
simulations: A mechanism 𝐦 6
𝟎
is rotated by a random Rodrigues rotation matrix 𝐑(𝜑,𝐮 6), whose rotation axis
𝐮 6 is uniformly distribted in the unit sphere, and rotation angle 𝜑 follows a von Mises distribution with zero
mean and concentration parameter 𝜏 =
1
𝜎
*
' V . (a) Standard deviation of the angles between a rotated
mechanism 𝐦 6 and the SOR basis set whose leading term is the mean mechanism 𝐦 6
𝟎
. (b) Standard deviations
of the angles between the normal, null, and slip directions of the rotated mechanism and the mean
mechanism. (c) Standard deviations of the rotated mechanism’s strike, dip, and rake directions.
53
1.4 SSC+Noise Model
In the previous section, we introduced three possible parameterizations of source mechanism
uncertainties and their probability functions: (1) total angle-rotation axis, (2) nodal plane
uncertainties, and (3) uncertainties on the resolved strike, dip, and rake angles. We also investigated
the equivalencies between their probability distributions using Euler’s theorem for simple 3D
rotations. Furthermore, using numerical simulations of random rotations of focal mechanisms, we
showed that the dispersion of nodal planes is equal to the distribution of the total rotation angle for
small rotations. Similarly, the randomly rotated mechanism's dispersion of strike and rake directions
is about the same as the total rotation angle. However, we found that the dip angle dispersion is about
a factor of 0.4. We also found that the standard deviations of the angular distances between rotated
mechanisms and each of the members of the SOR basis sets are different: for small angles, 𝜎(𝜑
)
)~𝜎,
but 𝜎(𝜑
&
)→2𝜎. 𝜎(𝜑
+
)=𝜎(𝜑
5
)→𝜎 and 𝜎(𝜑
4
)→0.
We note, however, that the distributions of 𝜑
-
as well as those of the strike, dip and rake are not
Gaussian or von Mises distributions as the estimation of standard deviations might suggest. Kagan
(1990) found that circular Cauchy distributions better approximate the angular distributions of
random mechanisms. Nevertheless, we find that the circular von Mises distributions are suitable for
modeling the uncertainties of the [𝐧 ,𝐛
Ö
,𝐬 ] directions as predicted by equations 43, 48, and 52.
Therefore, we model random rotations using Rodrigues’ matrices and their probability distributions
in equations 43-45, and for simplicity, we use 𝜎 =𝜎
,
.
In this section, we merge the probability distributions of mechanism uncertainty and the probability
distributions of the SSC model. The goal is to estimate synthetic empirical distributions of the total
moment fractions that can be compared against observations. A simple forward modeling technique
draws random samples from the SSC and noise distributions and constructs synthetic moment tensor
densities. From these densities, then one can estimate empirical distributions of 𝓕

and 𝐸
)
. This
process is depicted in Figure 1.11 and explained in Section 1.4.1. Another possibility is to combine
54
the probability functions analytically and solve for the expectation integrals of the direction cosines
to obtain the total moment fractions (e.g., equation 46), as we show in Section 1.4.2.

Figure 1.11. Modeling of SSC+Noise mechanisms and estimation of synthetic moment fractions. Input data are
values of 𝑅 and 𝜅, and the distribution of mechanism errors. The SSC and rotation-matrix distributions are
sampled to generate a synthetic moment tensor catalog. From the catalog, empirical distributions of 𝓕
9
and 𝐸
#

are estimated.

1.4.1 Monte Carlo Simulation of SSC+Noise Mechanism Catalogs
The probability density function of the SSC moment tensor field is
𝑝𝐦
OO`
|𝜅,𝑠(𝐧 )=
1
𝒩
e
:9(𝐧 =)
, (56)
from which we draw a random sample 𝐦
99k
. The moment magnitude of this elementary source, 𝑀
o

is also a random sample of a Gutenberg-Richter distribution,
𝑁 =10
(^Ypq
)
)
, (57)
with 𝑏 =1, and 𝑎 adjusted to the desired number of events in the magnitude range. From the 𝑀
o
we
estimate the event scalar seismic moment as 𝑀
)
=10
%
"
(q
)
! &).s5)
(10
Ys
Nm).
55
Assume that 𝐦 is an uncertain estimation of the 𝐦
99k
that contains random errors, biases, and noise
due to assumptions made in the mechanism estimation technique and imperfect data. These noise
processes change the orientation and amplitude of the observed moment tensor relative to a true
mechanism. However, we assume that the amplitude (tensor norm, scalar seismic moment, or
seismic magnitude) uncertainty is not too relevant and consider normalized tensors. In the following
sections, we investigate the effect of mechanism amplitude or inversion weighting schemes during
the maximum likelihood estimation technique.

Isotropic Noise on the 3D Rotation Sphere
Uncertainties in the orientation of focal mechanisms can be represented by distributions over the
total rotation angle, Euler angles, or nodal plane uncertainties (e.g., 𝑣𝑀𝐹
5l
(𝐑|𝜏) of equation 52). We
assume that 𝜎, the standard deviation of the angular distances between a mechanism solution and a
proffered solution, can parametrize uncertainties. The probability distribution 𝑝(𝐑|𝜎) is sampled to
obtain random rotation matrices 𝐑
"
. The mechanism contaminated by noise is 𝐦
"
=𝐑
"
𝐦
99k
𝐑
"
,
,
from which we calculate directions cosines relative to the SOR basis set 𝕭={𝐌

-
;𝛼 =0,…4}:
cos𝜑
-,"
=𝐌

-
:𝐦
"
.
The resulting SSC+Noise catalog of focal mechanisms is
𝒞(𝑅,𝜅,𝜎)=𝑀
o,"
,𝐦
"
:𝑛 =1,…,𝑁. (58)
Note that this catalog depends on 𝑅 and 𝜅 through equation 56. We construct synthetic catalogs for
values of 𝑅 ∈(0,1) sampled at 𝛿𝑅 =0.01, 𝜅 ∈(0,20) sampled at 𝛿𝜅 =0.2, and 𝜎 ∈(0,60) sampled at
𝛿𝜎 =1. The workflow is shown in Figure 1.11. In total, we simulate 622,261 catalogs with 𝑁 =
10,000 mechanisms. Figure 1.12 shows the total-moment fraction distributions as functions of 𝜎,𝜅,
and 𝑅 =0.1,0.5. Since the catalogs are independent for each combination of parameters, we
implemented the sampling subroutines in a parallel algorithm using the MatLab Parallel Toolbox
(https://www.mathworks.com/products/parallel-computing.html; last accessed on Jun 28, 2022),
56
which allows for reducing the computation time almost proportional to the number of available
computing cores.
Distributions of total moment fractions as functions of 𝑅, 𝜅, and 𝜎 can be computed from the
synthetic catalogs as follows:
• For each catalog 𝒞(𝑅,𝜅,𝜎), average moment tensors to obtain the catalog CMT mechanism,
𝐌

)
=∑ 𝐦
"
3
"%&
.
• Calculate the remaining SOR basis set elements 𝐌

-
;𝛼 =0,…,4, and the direction cosines
cos𝜃
-,"
=𝐌

0
:𝐦
"
.
• Calculate the Aki moment fraction 𝐹
)
(𝑅,𝜅,𝜎)=
&
3
∑ cos𝜃
),"
3
"%&
and the total moment
fractions 𝐹

-
(𝑅,𝜅,𝜎)=
&
3
∑ cos
+
𝜃
-,"
3
"%&
.

Figure 1.12. Total moment fractions (𝐹
9
!
(𝑅,𝜅,𝜎)) calculated from Monte Carlo simulations of the SSC and
noise distributions. The top panels (a-d) show 𝐹
9
!
for 𝑅 =0.1, and the bottom panels (e-h) for 𝑅 =0.5. In each
panel, colored lines are for selected values of 𝜅. In (c) and (g) solid lines are 𝐹
9
'
and dashed lines 𝐹
9
+
.

57
Isotropic Noise on the 5D Unit Sphere
The measure of orientation uncertainty is the variance of the angles between an acceptable solution
𝐦 and the actual mechanism 𝐦
99k
, 𝜑 =arccos (𝐦 :𝐦
99k
). We now want to quantify the uncertainties
in the direction of SOR basis elements 𝕭(𝐦
99k
)=𝐌

-
;𝛼 =0,…,4, where 𝐌

)
=𝐦
99k
. The rotation
angles measure such deviations: 𝜑
-
=arccos 𝐌

-
:𝐦 . The orthogonal angles satisfy the direction
cosine constraint ∑ cos
+
(𝜑
-
)
4
-%&
=1. Although {𝜑
-
,𝛼 =0,…,4} can follow a joint probability
distribution with correlations, we assume isotropic noise and model them as if they are uncorrelated
and identically distributed. Furthermore, we assume that each 𝜑
-
follows a von Mises distribution
with zero mean and standard deviations 𝜎
-
(equation 38), where cos(𝜎
-
)=cos
&/4
(𝜎).
Forward modeling of SSC mechanism uncertainty thus requires modeling two random variables
hierarchically: (1) sample variance angles from a Gamma(𝜎;𝛼,𝛽) distribution to obtain 𝜎
-
, and (2)
sample four independent error angles {𝜑
-
,𝛼 =1,..,4} from the von Mises distribution, 𝑣𝑀(𝜑
-
|𝜎
-
).
The uncertain stress-aligned mechanism is
𝐦 =á1− sin
+
(𝜑
-
)
4
-%&
â
&
+
𝐌

)
+ sin(𝜑
-
)
4
-%&
𝐌

-
. (59)

This equation is valid only if ∑ sin
+
(𝜑
-
)
4
-%&
<1, which imposes an external constraint for the joint
distribution of the angles 𝜑
-
, and suggests that a joint distribution over the independent moment
tensor components, cos(𝜑
-
) rather than 𝜑
-
, would be more appropriate. Nevertheless, in this
empirical exercise, the distributions 𝑣𝑀(𝜑
-
|𝜎
-
) can be sampled multiple times if required until the
angles satisfy the constraint. The resulting SSC+Noise catalog is
𝒞
ml
=𝑀
o,"
,𝐦
"
:𝑛 =1,…,𝑁. (60)
Figure 1.13 shows the expected value of total moment fractions estimated from the 𝒞
ml
catalog as
functions of 𝑅, 𝜅, and a few selected values of 𝜎. These synthetic three-dimensional moment fraction
maps can be used in the probabilistic inversion technique. In the likelihood function, 𝐹

-
and 𝐹
)
are
conditional on the SSC parameters 𝑅 and 𝜅. Figure 1.7 shows that for the moment fraction of degree
58
zero, 𝐹

)
decreases with increasing 𝜎. In the low complexity limit, 𝑅 =0.5 and 𝜅 =20, 𝐹

)
goes from
0.9 to 0.6 at 𝜎 =40
t
. In response, the higher degree fractions increase.

Figure 1.13. Synthetic Aki moment fraction 𝐹
#
and total moment fraction 𝐹
9
!
as functions of 𝑅, 𝜅 , and 𝜎. The
gray contour line in (a) and (b) corresponds to 𝐹
#
=0.6,0.8 and 𝐹
9
#
=0.6,0.8, respectively. In (c)-(f) is 𝐹
9
!
=
0.1.

1.4.2 Numerical Integration of SSC+Noise PDFs on the Sphere Surface
We investigate the problem of estimating total moment fractions when a mechanism contaminated
by errors is available. Probability distributions on the three-dimensional (3D) unit sphere are
common to model orientation data in geosciences, such as paleomagnetic vectors or lattices of
minerals. Any rotation of the oriented 3D Euclidean space is defined by an angle 𝜃 ∈[0,𝜋] and a 3D
unit vector on the 2-space, 𝐮 ∈𝑆
+
(The unit vector has two degrees of freedom; for example,
59
colatitude and azimuth). A random rotation matrix 𝐑 is a random variable distributed in the special
orthonormal group of three dimensions (3D), 𝐑∈S03.
Euclidian operations of probability functions on the d-sphere surface require further considerations
depending on the parametrization. For example, the area element of the distribution space, known
as the normalized Haar measure of the SO3, is 𝑑𝐴=
&
+u
"
sin
+
v
+
𝑑𝜃𝑑𝐮 . If the rotation axis is 𝐮 =
(sin𝜙cos𝜓,sin𝜙sin𝜓,cos𝜙), 𝜙 ∈[0,𝜋] is the colatitude, and 𝜓∈[0,2𝜋) is the azimuth, then
𝑑𝐮 =𝑑𝜙𝑑𝜓 is the area element of the unit sphere 𝑆
+
.
The expected value integrals of many probability functions on the sphere do not have a closed
analytical form. As such, they must be evaluated numerically using either Monte Carlo integration or
integration of analytical representations (Ley & Verdebout, 2018; Mardia, 1975). In the previous
section, we implemented a simple Monte Carlo integration technique to obtain the total moment
fractions by sampling the involved distributions.
Several studies use Monte Carlo sampling of spherical distributions to investigate biases due to
parametrization (Bingham et al., 2009; Sato & Yamaji, 2006; Stuelpnagel, 1964; Wood, 1994). For
example, it is well known that uniform sampling of the spherical polar angles 𝜙 and 𝜓 does not give
a uniform distribution on the sphere. Instead, samples must be 𝜓~𝑈[0,2𝜋) and 𝜙~
&
4u
sin𝜙, 𝜙 ∈
[0,𝜋], because the Haar measure of the uniform distribution on 𝑆
+
is 𝑑𝐴=
&
4u
sin𝜙𝑑𝜙𝑑𝜓. Although
typically easy to implement, a disadvantage of the Monte Carlo integration technique is that it
becomes computationally demanding. In particular, when the number of independent parameters
and their dependencies in the PDFs increases: the number of operations multiplies by expanding the
function domain, its discretization resolution, and the number of samples required to accurately
model probabilities empirically. Moreover, the SSC+Noise model also contains hierarchical
dependencies.
60
The PDF of an SSC mechanism is 𝑝𝐦
99k
|𝜅,𝑠(𝐧 )∝e
:9(𝐧 =)
, and the probability of a random rotation
matrix is 𝑝(𝐑|𝜃,𝐮)=𝑝(𝜃|𝜏)𝑝𝐮 (𝜙,𝜓), where 𝑝(𝜑|𝜏)∝exp(𝜏cos𝜑) and 𝑝𝐮 (𝜙,𝜓)∝sin𝜙. Thus,
the total moment fraction expectation integrals are
𝐹

-
=
1
𝒩
exp𝜅𝑠(𝐧 )exp(𝜏cos𝜑)
𝐑∈h)
%
𝐧 =∈ h
"
𝐌

-
:𝐑𝐦
99k
𝐑
,

+
sin𝜙𝑑𝜑𝑑𝐮 𝑑𝐧 . (61)
where 𝒩 is the normalization constant, such as
𝒩 = exp𝜅𝑠(𝐧 )exp(𝜏cos𝜑)
𝐑∈h)
%
𝐧 =∈ h
"
sin𝜙𝑑𝜑𝑑𝐮 𝑑𝐧 , (62)

The probability of observing an uncertain mechanism 𝐦 =𝐑𝐦
99k
𝐑
,
is a function of the orientation
vector 𝐧 on 𝑆
+
and the SSC parameter 𝜅 and 𝑅 through 𝑠, and additionally, a function of the error
rotation angle 𝜑 ∈(−𝜋,𝜋) and the rotation axis 𝐮 ∈𝑆
+
. Thus, numerical integration would require
constructing a 5D grid whose parameters are the rotation angle 𝜑 and the orientation vectors 𝐧 and
𝐮 independent components (𝜙′,𝜓′), and (𝜙,𝜓). This integration is repeated at each node of the 3D
grid of SSC parameters and noise 𝜅, 𝑅, and 𝜏. There are eight independent variables in total, and
solving the integration at each point becomes computationally expensive.
We develop a numerical algorithm that reduces the computational requirements by taking advantage
of the conditional independency of SSC and noise probability distributions. A stress aligned
mechanism is
𝐦
99k
(𝐧 )= 𝐌

-
cos𝜑
-
(𝐧 )
4
-%)
, (63)
where 𝕭𝚺

=𝐌

-
;𝛼 =0,…,4 is a SOR basis set with 𝐌

)
=𝚺

, the reduced stress tensor. An
uncertain estimate of 𝐦
99k
is
𝐦 (𝜑,𝐮 ,𝐧 )= 𝐑(𝜑,𝐮 )𝐌

-
𝐑(𝜑,𝐮 )
,
cos𝜑
-
(𝐧 )
4
-%)
. (64)
The projections of the rotated basis set elements onto the original set 𝕭𝚺

are
cos𝜃
-,2
=𝐌

-
:𝐑𝐌

2
𝐑
,
; 𝛼,𝛽 =0,…4. (65)
61
Thus, the components of the uncertain mechanism 𝐦 (𝜑,𝐮 ,𝐧 ) are
𝐦 = cos𝜑
-
cos𝜃
-,2
4
-%)
𝐌

2
4
2%)
. (66)
Using a norm-preserving tensor-vector isomorphism such that, if 𝐘=𝑥
6w
;𝑖,𝑗 =1,2,3 is a
symmetric second-order tensor, then 𝐲=vec(𝐘)=©𝑥
&&
,𝑥
++
,𝑥
55
,√2𝑥
&+
,√2𝑥
&5
,√2𝑥
+5
ª is a vector
with the independent components of 𝐘. We have
𝐱 =vec(𝐦
99k
)=[cos𝜑
)
,cos𝜑
&
,…,cos𝜑
4
], (68𝑎)
𝐀=é
cos𝜃
),)
cos𝜃
&,)
⋯ cos𝜃
4,)
cos𝜃
),&
cos𝜃
&,&
⋯ cos𝜃
4,&
⋮ ⋮ ⋱ ⋮
cos𝜃
),4
cos𝜃
&,4
⋯ cos𝜃
4,4
í, (67𝑏)
𝐲 =vec(𝐦 )=𝐀𝐱 . (68𝑐)
The matrix 𝐀 is then a rotation matrix in the 5D space of symmetric moment tensors equivalent to a
3D rotation of a matrix in the 3D space. The direction cosines cos𝜃
-,2
(𝜑,𝐮 ) are the projections of
basis set 𝕭𝚺

onto the 𝕭(𝐦
99k
) coordinates, and these projections do not depend on the direction
cosines cos𝜑
-
(𝐧 ) of 𝐦
99k
. In consequence, their expectations are conditionally independent because
isotropic rotations of 𝐦
99k
do not change the expected 𝕭(𝐦
99k
) and 𝕭𝚺

; If 𝜑 follows a von Mises
distribution on the circle with mean 𝜑 =0 and concentration parameter 𝜏, and 𝐮 is uniformly
distributed in 𝑆
+
, then 𝐑 follows a matrix von Mises-Fisher distribution (Bingham et al., 2009),
𝑣𝑀𝐹(𝐑|𝜏)=
1
𝑐(𝜏)
exp·
𝜏
2
𝑡𝑟(𝐑)−1º. (68)
where 𝑐(𝜏) is a normalization constant. The first and second moments of the 𝑣𝑀𝐹 distribution are,
E(𝐑)=Å
1
3
+
2
3
𝐼
&
(𝜏)
𝐼
)
(𝜏)
Æ𝐈
5l
, (69𝑎)
E(𝐑
+
)=Å
1
3
+
2
3
𝐼
+
(𝜏)
𝐼
)
(𝜏)
Æ𝐈
5l
, (70b)
62
where 𝐼
6
(∙) is the modified Bessel function of order 𝑖. Hence, random isotropic rotations of a tensor
do not change the orientation expectation, which is proportional to 𝐈
5l
, only its norm changes.
Therefore, the expected orientations of 𝕭(𝐦
99k
), and under the same argument 𝕭𝚺

, are not
sensitive to uncertain estimations of 𝐦
99k
. Consequently, the total moment fraction expectations can
be expressed as the inner tensor product of a tensor whose components are the noise expectations
and a second tensor whose components are the SSC expectations.
Consider the vectors 𝐱 =[cos𝜑
)
,cos𝜑
&
,…,cos𝜑
4
] , 𝐳
-
=[cos𝜃
-,)
,cos𝜃
-,&
,…,cos𝜃
-,4
] , with
cos𝜃
-,2
=𝐌

-
:𝐑𝐌

2
𝐑
,
; 𝛼,𝛽 =0,…4. The quadratic tensors are
𝐂
x
=𝐱
+
=é
cos
+
𝜑
)
cos𝜑
)
cos𝜑
&
⋯ cos𝜑
)
cos𝜑
4
cos𝜑
)
cos𝜑
&
cos
+
𝜑
&
⋯ cos𝜑
&
cos𝜑
4
⋮ ⋮ ⋱ ⋮
cos𝜑
)
cos𝜑
4
cos𝜑
&
cos𝜑
4
⋯ cos
+
𝜑
4
í, (70𝑎)
𝐂
v
-
=𝐳
-
+
=
⎝
⎜
⎛
cos
+
𝜃
-,)
cos𝜃
-,)
cos𝜃
-,&
⋯ cos𝜃
-,)
cos𝜃
-,4
cos𝜃
-,)
cos𝜃
-,&
cos
+
𝜃
-,)
⋯ cos𝜃
-,&
cos𝜃
-,4
⋮ ⋮ ⋱ ⋮
cos𝜃
-,)
cos𝜃
-,4
cos𝜃
-,&
cos𝜃
-,4
⋯ cos
+
𝜃
-,)
⎠
⎟
⎞
. (71𝑏)
The expectation of 𝐂
v
-
over the distribution of rotation matrices is
E[𝐂
v
-
]= exp(𝜏cos𝜃)
𝐑∈h)
%
𝐂
v
-
sin𝜙𝑑𝜃𝑑𝐮 . (71)

With some algebra, we can show that the expectation of the SSC tensor 𝐂
x
is
𝐹

-
= 𝑝𝜅,𝑠(𝐐)÷𝐂
x
:E[𝐂
v
-
]ø𝑑𝐧
𝐧 =∈ h
"
. (72)

𝑝𝜅,𝑠(𝐐) is the SSC probability function that results from adding the uncertainties in the angles 𝜃
-,2
.
Assume that 𝐑~ 𝑣𝑀𝐹(𝐈
5l
|𝜏), then (𝐑𝐐)~ 𝑣𝑀𝐹(𝐐|𝜏), if we can take 𝐐=[𝐧 ,𝐛
Ö
(𝐧 ),𝐬 (𝐧 )] the matrix
whose columns are the SSC normal, null, and shear traction directions. Then, the expectations of
cos𝜃
-,2
over the 3D sphere that contains 𝐑 can be evaluated on the sphere supporting 𝐧 ∈𝑆
+
(rather
than on 𝐑∈S03). The density function, up to a normalization constant, simply is the spherical
convolution of the SSC density and the probability function of 𝐐,
63
𝑝(𝜅,𝑠(𝐐))∝ exp𝜅𝑠(𝐧 −𝐧
y
)
𝐧 =y∈ h
"
exp·
𝜏
2
𝑡𝑟(𝐐
y
)−1º𝑑𝐧
y
. (73)

This representation of the expectation also allows computing the integrals for each element of the
outer matrices 𝐂
x
and 𝐂
v
-
independently, which is ideal for the computational parallelization using
nested loops that require no inter-core communication, and as a result, reducing idle times.
We implemented subroutines in MatLab that use the parallel toolbox utility
(https://www.mathworks.com/help/parallel-computing/). The spherical integrals of 𝑑𝐮 and 𝑑𝐧 are
solved even more efficiently using Fibonacci spherical grids (Marques et al., 2021) rather than
regular 2D grids. The Fibonacci grid gives an extremely uniform, but irregular point set distribution
on the sphere and has the advantage that the Haar measure of the probability space becomes 𝑑𝐴=
&
3
, where 𝑁 is the number of points on the grid. We set 𝑁 =10
4
, and in an identical grid, we solve the
convolution integrals using the spherical harmonic transform (Di Marzio et al., 2022; Driscoll &
Healy, 1994; Wieczorek & Meschede, 2018) with complex harmonics up to degree 𝑙 =50, which
approximates the density functions reasonably well for small angular standard deviations 𝜎 <10.
For 𝜎 >15, we reduce 𝑙 =30, and when 𝜎 >30, 𝑙 =20 to speed calculations. We adapt and
optimize the algorithms of Wieczorek & Meschede (2018) to speed the spherical harmonic transform
and the spherical convolution in the MatLab parallel loops.
Figure 1.14 and Figure 1.15 show synthetic total moment fractions 𝐹

-
calculated using the procedure
described above. The expectations agree notably well with the values obtained from Monte Carlo
sampling for small values of angular standard deviation (𝜎 <25) as shown in Figure 1.12. In the same
way, for large angle variance (𝜎 >45
G
), the moment fractions from the numerical solutions and
random sampling approach similar values. For intermediate values of 𝜎, we find that random
sampling is affected by random nonlinearities from the interchanging of SOR tensors: a number of
random rotations give cos𝜑
-z&
>cos𝜑
)
, changing the leading term. This effect can be corrected by
64
imposing non-linear constraints in the sampling algorithm. The expectation integrals consider this
effect inherently and thus provide a smoother continuous solution.
The following sections show that mechanism uncertainties in California, given by the distribution of
angles between the proffered mechanism and possible solutions, have average standard deviations
over 𝜎 =25
t
. Our calculations show that these uncertainties can reduce 𝐹

)
over 15%, and it has an
important issue for stress inversion procedures. For example, Lisle (2013) used kinematic
simulations of stress fields in faults with different degrees of geometrical complexity and anisotropy
and showed that the magnitude of the shear stress resolved on a fault plane is a function poorly
defined maximum direction. Consequently, the shear stress direction can vary over 26
t
on the 90%
confidence interval. Uncertainty values on the mechanism estimation propagate onto the stress
tensor, and they are often not considered in traditional stress inversions. Hardebeck (2006) showed
that the average uncertainty of focal mechanism in California could be as large as 𝜎 =32
t
. The
uncertainties are mainly due to the velocity model and source-station coverage. Other studies of
stress rotations report variations more negligible than the uncertainties in the mechanisms
(Hardebeck, 2010; Hauksson, 1994). We show that uncertainties could heavily impact
interpretations of stress states and the seismic response to stress.

Figure 1.14. Synthetic total moment fraction 𝐹
9
!
as functions of 𝑅, 𝜅 , and 𝜎.

65

Figure 1.15. Synthetic total moment fraction 𝐹
9
!
as functions of 𝜅 , and 𝜎, for two selected values of 𝑅. Top
panels are 𝑅 =0.1, and bottom panes for 𝑅 =0.5.

66
1.5 Earthquake Catalog Biases
Determining how errors and biases in focal mechanisms impact stress inversions is essential because
errors can be interpreted as apparent mechanism complexity. Systematic errors might result from
the inversion technique and dataset, e.g., P- and S-wave polarities and amplitudes, source and station
configuration, and velocity model. One possibility for detecting biases is to model synthetic catalogs
by adding random perturbations to the data and model parameters and investigating the
asymmetries in the output distributions (Hardebeck, 2006). Alternatively, catalogs with overlapping
events estimated from different datasets and inversion techniques can be directly compared using
statistical methods to identify biases. We follow this second approach and compare the moment
tensor orientations from three different catalogs of the Ridgecrest earthquakes and their aftershocks:
Hauksson (H), Chen (C), and Lin (L) datasets.
We are particularly interested in identifying biases in the Hauksson catalog because it is an extract
of the SCEDC catalog, which we extensively use to investigate the stress fields in Southern California.
Therefore, we assume the Ridgecrest dataset is representative of the entire SCEDC dataset. In this
section, we show that there are no systematic biases in the Hauksson catalog, and simple random
rotations can model errors. Additionally, we find that the Hauksson catalog has mechanisms with
better qualities on average than the other two datasets. Furthermore, we investigate the effects of
different data weighting schemes and sensitivity to noise levels. We demonstrate that the Hauksson
dataset allows to safely invert for the strain sensitivity factor 𝜅, which is the parameter most sensitive
to uncertainties.

1.5.1 Orientation Biases
For every cross-pair of catalogs (H-C, H-L, and C-L), we find all the coinciding events and estimate the
angle between the focal mechanism. Each dataset reports earthquakes detected with different
67
techniques, data is processed differently, and locations, magnitudes, and mechanisms are calculated
using different models. As a result, reported parameters might show differences between catalogs, in
particular earthquake spatial location. Therefore, to classify an event as coincident, we enforce these
conditions: (1) the origin-time difference is smaller than 10 seconds, (2) hypocenters are within 500
m, and (3) the magnitude differences are smaller than 0.1 magnitude unit. For the pair Hauksson-
Chen, there are 1156 coinciding earthquakes, Hauksson-Lin 2490 events, and Cheng-Lin 586. These
numbers reflect similarities and differences between datasets. Lin used the earthquake detections
and P- and S-waves polarities from Hauksson. Then, earthquakes are relocated, and focal
mechanisms are inverted using a different three-dimensional velocity model. Therefore, systematic
biases between these catalogs might reflect the epistemic uncertainties of the geologic structure and
additional assumptions for the modeling technique.
Earthquakes in the Cheng catalog are detected by applying machine learning algorithms on
continuous waveforms from the SCSN. These algorithms use previously detected earthquakes as
training data. Source parameters, including locations and mechanisms, are inverted using the same
inversion technique as Hauksson and Lin. Hence, the differences between Cheng and the other
datasets would reflect biases in the earthquake detection algorithm and the P- and S-wave polarities
and amplitudes used to estimate the focal mechanisms.
A reference frame is needed to investigate systematic biases in the mechanism orientation. We
concatenate the three datasets, estimate a SOR basis set, and use it as a reference coordinate system.
Furthermore, we assume that mechanisms in the catalogs are randomly oriented following a von
Mises distribution with zero-mean relative to the SOR 𝐌

)
. We are interested in investigating the
errors relative to the SOR coordinate system, and therefore mechanisms in each catalog are projected
onto the elements of the basis set, 𝑥
-,"
(H)
=𝐌

-
:𝐦
"
(H)
. The superscript (𝐻) stands for the Hauksson
catalog, and similarly, we obtain 𝑥
-,"
(I)
and 𝑥
-,"
(J)
for the Chen and Lin catalogs, respectively.
68
Additionally, we calculate the angles between the coinciding mechanism in pairs of catalogs, e.g.,
𝜂
I,H
=𝑎𝑟𝑐𝑐𝑜𝑠(𝐦
"
(I)
:𝐦
"
(H)
).
Figure 1.16 shows the distributions of the projections 𝑥
)
(∙)
and the angles 𝜂 between the coinciding
mechanism. Ideally, if two mechanisms were identical in two catalogs, pairs ·𝑥
-
(H)
,𝑥
-
(I)
º would lay in
the 1:1 line, meaning that individual mechanisms have similar orientation relative to 𝐌

)
. If there are
no biases or errors, the Pearson correlation coefficient should be 𝑟 =1. However, projection pairs
show that there are random errors, and if they were independently distributed, pairs of 𝑥
)
(∙)
should
be symmetrically distributed over the 1:1 line. Any systematic biases cause asymmetry and reduce
the correlation coefficient. The pair Hauksson-Cheng has the highest Pearson correlation coefficient,
𝑟 =0.81. Additionally, the distribution of angles between the Hauksson-Cheng mechanisms has a low
mean, 𝜇 =23
G
, indicating that the mechanisms in these two catalogs are closely oriented. Thus, we
conclude that there is no indication of biases in the detection method and data processing for the
mechanism observation.
The angles between the Hauksson-Lin and Cheng-Lin mechanisms have means 𝜇 =32
G
and 𝜇 =33
G
,
respectively. The Pearson correlation coefficients are 𝑟 =0.75 and 𝑟 =0.74 . Due to larger
dispersion, the correlations are significantly high but smaller than in the H-C case. This dispersion
effect is also reflected by the mean and standard deviations of the angles. It is worth noticing that
the Lin catalog has more significant uncertainties on average (Figure 1.5b & e). Furthermore, the
marginal distributions of 𝑥
)
(H)
and 𝑥
)
(I)
are consistently skewed towards one, while 𝑥
)
(J)
has a mode
at about 0.5. The model in 𝑥
)
(J)
suggests an orientation bias in the Lin catalog, presumably from the
three-dimensional velocity model used for the inversions compared to the 1D models of Cheng and
Hauksson. They use a one-dimensional for locating and inverting source mechanisms.
Our stress-strain characterization technique assumes that the stress field is and that the source
mechanism reflects the stress field state. Large earthquakes can locally perturb stress fields, and to
69
avoid such perturbations that do not reflect background stress, some sort of data filtering and
declustering might be required to ensure the independency of the data (Abolfathian et al., 2018). To
investigate any dependencies within datasets, we calculated the joint distributions
𝑝·𝑥
-
(H)
,𝑥
2
(H)
º,𝛼,𝛽 =0,…,4, and their correlations. Figure A5 of the appendix shows that, for the
Hauksson dataset, there is no strong evidence of a correlation between focal mechanisms (𝑟 <0.35).
Additionally, we show the scatter plots of higher degree 𝑥
-|)
(∙)
. In general, H-C has the highest
correlation coefficient 𝑟, between 0.55-0.87, while H-L range between 0.53 and 0.85, and C-L between
0.51-0.84.

70

Figure 1.16. Projections and angles between the focal mechanism of Hauksson and those of Cheng and Lin. (a)
Empirical distribution of the angle 𝜂
, ,.
between the focal mechanisms of Hauksson and Cheng. 𝜇 and 𝜎 are
the mean and standard deviations, respectively. (b) Distribution of 𝑥
#
(, )
=𝐌
)
#
:𝐦 6
1
(, )
vs. 𝑥
#
(.)
=𝐌
)
#
:𝐦 6
1
(.)
. The
solid stair-like lines are the scaled kernel density estimate (histograms) (c) and (e) are similar to (a), but for
the pairs Hauksson-Lin and Cheng-Lin. (d) and (f) are similar to (b).

1.5.2 Biases From Mechanism Weighting Scheme
In the SSC model, the reduced stress tensor is proportional to the CMT. The CMT is an average over
the moment tensor density and, as such, depends on the scalar moment of elemental sources.
Furthermore, in the stress inversion algorithm, mechanisms can be weighted, for example, to include
a factor that expresses confidence in the moment tensor. Therefore, the selection of the weighting
71
scheme might change the distributions of total moment fractions and influence the estimation of the
SSC parameters.
We investigate the effect of three different weighting schemes on measurements of the moment
fractions and, thus, on the SSC parameters and their uncertainties. We show that the mechanism
weighting by magnitude (Mw) gives stable solutions and accommodates the assumption that the
mechanisms of larger events are better determined.
Consider the Ridgecrest dataset of Hauksson and three catalogs constructed from different weighting
schemes,
𝒞
}"6_
={𝐦
"
:𝑛 =1,…,𝑁}, (74𝑎)
𝒞
q
)
=𝑀
o,"
𝐦
"
:𝑛 =1,…,𝑁,and (74b)
𝒞
q
*
={ 𝑚
"
𝐦
"
:𝑛 =1,…,𝑁}. (74𝑐)

where 𝑀
o,"
and 𝑚
"
are the earthquake magnitude and the seismic scalar moment of the n-th event,
respectively.
We tested the effects of using unit mechanisms (𝐦
"
), weighting them by magnitude 𝑀
o,"
𝐦
"
, or
scalar moment ( 𝑚
"
𝐦
"
). For each weighting scheme, we estimate a SOR basis set. Then, empirical
distributions of the total moment fractions are calculated by randomly sampling with replacement
30% of the mechanisms in a catalog 𝒞
(∙)
and projecting them onto the SOR basis sets (e.g., equation
6). Additionally, mechanisms in the sample are averaged to obtain the sample 𝐌

𝟎
. The eigenvalues
of 𝐌

𝟎
allow estimating the sample eigenvalue ratio 𝐸
)
=(𝜆
&
−𝜆
+
) (𝜆
&
−𝜆
5
) ⁄ . Its eigenvectors 𝕰
𝟎
=
(𝐞
&
,𝐞
+
,𝐞
5
), are random variables that estimate the principal stress directions.
The random sampling and subsequent operations are repeated 10,000 times, and estimated variables
are aggregated on empirical distributions for moment fractions 𝑝𝓕

, differential stress ratio 𝑝(𝐸
)
),
and the stress tensor directions 𝑝(𝕰). Figure 1.17 shows the effect of different choices of the
weighting of the source mechanism on the moment fraction empirical distributions and the
differential stress ratio. Additionally, we tested the sensitivity to different earthquake magnitude
72
intervals. We bin earthquakes in magnitude intervals between 1.5 and 𝑀
o
, ranging between 2-8 with
one magnitude unit increment. In Figure 1.17, markers indicate the distribution median and bars one
standard deviation.
The mean 𝐸
)
changes significantly from moment weighting to the other weighting schemes.
However, it stays within one standard deviation. The moment weighted averaging increases the
variance dramatically, e.g., for earthquakes with 𝑀
o
∈ [1.5,4.5], 𝐸
)
(q
)
)
=0.36±0.03,𝐸
)
(}"6_)
=
0.36±0.03,and 𝐸
)
(q
*
)
=0.37±0.11. The selection of the mechanism weighting heavily biases the
moment fractions because large events dominate the averaging, and thus, the CMT is dominated by
the larger event mechanisms. As a result, the moment fraction empirical distributions have larger
variances, and the higher degree fractions are skewed towards zero. The magnitude weighting gives
similar variances as the normalized mechanism distributions, e.g., for seismicity with 𝑀
o
∈ [1.5,4.5],
𝐹

)
(q
)
)
=0.54±0.14,𝐹

)
(}"6_)
=0.49±0.02,and 𝐹

)
(q
*
)
=0.5±0.02.
Based on these results, we choose earthquake magnitude 𝑀𝑤 as the weighting scheme. The
magnitude weighted distributions of total moment fractions are within one standard deviation of the
other weighting schemes and have smaller variances. Moreover, the average values are stable with
respect to the maximum magnitude of the interval. The stability of the total moment fractions with
magnitude indicates that they are a robust measure of complexity. Furthermore, it is consistent with
our assumption that small earthquakes, like bigger events, are samples of the inelastic strain due to
a homogeneous background stress field (Hardebeck, J. L., 2015a; Smith & Heaton, 2011).
𝐸
)
is a weak function of magnitude range independently of the weighting scheme. Thus, it is a robust
parameter for inferring the tectonic setting from earthquake focal mechanisms. Several studies have
successfully inverted focal mechanisms for parameters of the stress tensor. The CMT from the
moment weighting scheme (the CMT, rigorously) is proportional to the average strain tensor
(Jackson & McKenzie, 1988; Kostrov, 1976) where the coefficient of proportionality is the shear
modulus. Furthermore, Matsumoto (2016) determined that stress-free strain produced by
73
earthquakes in a volume is proportional to the background deviatoric stress. Similarly, Terakawa &
Matsu'ura (2008) showed that the information obtained from a CMT solution is independent of the
scale of events, consistent with the assumption of self-similarity of the stress tensor at different scales
(Amelung & King, 1997; Seredkina & Melnikova, 2018; Sipkin & Silver, 2003). Therefore, magnitudes
weighting on our maximum likelihood estimation agrees with our prior assumption that mechanisms
of larger magnitude events have better quality, and observational data demonstrates that mechanism
quality is a weak function of magnitude and far from exponential as the moment weighting would
imply.

Figure 1.17. Sensitivity of 𝐸
#
and 𝓕
9
to mechanism weighting scheme using the Ridgecrest dataset of
Hauksson. We bin earthquakes in magnitude intervals between 1.5 and the upper cutoff 𝑀
2
, which ranges
between 2-8 with one unit increment. (a) Empirical distributions of the stress differential ratio as a function
of the upper cutoff magnitude. (b)- (g) Distributions of total moment fractions measured from different
weighting schemes of the mechanisms 𝐦
1
: purple stars and error bars are the average and one standard
deviation confidence interval from the distributions for magnitude weighted mechanism, 𝐦
1
=𝑀
2,1
𝐦 6
1
. In
blue, the average and confidence interval from moment weighted mechanisms, 𝐦
1
=𝑀
2,1
𝐦 6
1
, and in green
from unit tensors, 𝐦 6
1
.

74
1.5.3 Sensitivity to Noise Levels
If the mechanism orientation uncertainty is too large, the complexity factor and the strain sensitivity
parameter might not be resolvable by the available data. Figure 1.12 shows that the total moment
fractions 𝐹

-
are more sensitive to noise, described by the parameter 𝜎, when 𝜅 is large. To compute
the total moment fractions of Figure 1.12, we assume that 𝜎 is the standard deviation of the angular
distance between a mean mechanism and an uncertain estimation. If mechanism uncertainty
increases, larger values of 𝜅 become unresolvable. For example, a typical value of 𝜎 from seismicity
catalogs is 𝜎~30, and assuming 𝑅 =0.5, 𝜅 =20, and then 𝐹

)
=0.636. However, at 𝜎~37, observing
such 𝐹

)
would imply 𝜅 →∞.
We analyzed the sensitivity of our maximum likelihood estimation technique shown to different
noise levels 𝜎 and for different combinations of the SSC parameters {𝑅,𝜅}. For this purpose, we
simulated SSC+Noise synthetic catalogs of focal mechanisms following the procedure in Section 1.4.1.
We simulated catalogs for all combinations of 𝑅 =0.5, 𝜅 =5,10,20, and 𝜎 =10, 15, 20,…, 50, and
each catalog contains 10,000 mechanisms. Figure 1.18 shows the inverted parameters {𝑅,𝜅} and
their 67% confidence interval as functions of 𝜎. Additionally, we show the residuals computed as the
difference between measured total moment fractions (data) and the inverted values (model): 𝛿𝐹

-
=
𝐹

-
(.^_^ )
−𝐹

-
(~G.g )
.
We find that for small values of 𝜅, for example, 𝜅 =5, there are no considerable biases or errors on
the estimated parameters independently of the noise levels. For example, for 𝜎 =50, 𝜅 =5.5±3.5.
For 𝜅 =10, estimations are accurate for uncertainties up to 𝜎 =35, where we get 𝜅 =12±8.
Therefore, observing large values of 𝜅 requires datasets with small uncertainties. For example, we
find that for 𝜅 =20, ideally 𝜎 <20; otherwise, the model has no resolution. We have confidence that
typical catalogs in California allow observing values of the strain sensitivity factor 𝜅 <15 and, in
special cases, up to 𝜅 =20.
75
The estimation of 𝑅 is robust even with high noise levels. As expected, the uncertainty on the
estimation increases but can be solved for 𝜎 <40 at any 𝜅. Similarly, the residuals between observed
and inverted total-moment fractions do not correlate with 𝜎.

Figure 1.18. Sensitivity of the likelihood inversion scheme to catalog uncertainty calculated from synthetic
catalogs with 𝜅 =5,10,20, 𝑅 =0.5, and 𝜎 =5,20,25,…,50. (a) Estimated strain sensitivity factor, 𝜅 for
different levels of simulated noise uncertainty 𝜎. (b) Estimated differential stress ratio 𝑅 for different levels of
simulated noise uncertainty 𝜎. (c)-(f) Difference between expected moment fractions from synthetic
simulations and the inverted from the likelihood function: 𝛿𝐹
9
!
=𝐹
9
!
(3454)
−𝐹
9
!
(6)378)
.

Table 1.3. Baseline values of the total-moment fractions 𝐹
9
!
for selected values of 𝜎, 𝜅, and for 𝑅 =0.5. Figure
1.18 shows the residuals between these values and inverted.
𝝈
𝜿=𝟓 𝜿=𝟏𝟎 𝜿=𝟐𝟎
𝑭
+
𝟎
𝑭
+
𝟏
𝑭
+
𝟐,𝟑
𝑭
+
𝟒
𝑭
+
𝟎
𝑭
+
𝟏
𝑭
+
𝟐,𝟑
𝑭
+
𝟒
𝑭
+
𝟎
𝑭
+
𝟏
𝑭
+
𝟐,𝟑
𝑭
+
𝟒

0 0.66 0.11 0.085 0.061 0.8 0.078 0.05 0.015 0.87 0.057 0.034 0.0046
5 0.65 0.12 0.087 0.061 0.79 0.085 0.052 0.016 0.86 0.065 0.036 0.0048
10 0.63 0.13 0.09 0.061 0.76 0.1 0.057 0.018 0.82 0.087 0.042 0.0065
15 0.59 0.15 0.095 0.065 0.72 0.13 0.064 0.021 0.78 0.12 0.049 0.0098
20 0.56 0.17 0.1 0.067 0.67 0.16 0.073 0.027 0.72 0.14 0.059 0.015
25 0.52 0.19 0.11 0.073 0.62 0.18 0.083 0.034 0.66 0.17 0.07 0.023
30 0.49 0.2 0.12 0.081 0.57 0.2 0.093 0.043 0.61 0.19 0.082 0.032
35 0.46 0.21 0.12 0.087 0.54 0.21 0.1 0.055 0.57 0.21 0.091 0.043
40 0.43 0.21 0.13 0.096 0.5 0.21 0.11 0.064 0.54 0.21 0.1 0.055
45 0.41 0.21 0.14 0.11 0.48 0.21 0.12 0.075 0.5 0.21 0.11 0.066
50 0.39 0.22 0.14 0.11 0.45 0.22 0.13 0.086 0.48 0.21 0.12 0.075

76
1.6 The SSC-Lab: Application to Seismic Sequences in California
We implement our maximum likelihood estimation technique into a software package named SSC-
Lab. The SSC-Lab can use catalogs of earthquake focal mechanisms and models of earthquake
ruptures to estimate the moment measures of mechanism complexity and to calculate probability
distributions of the differential stress ratio 𝑅, the stress tensor principal directions 𝐫
-
, and the strain
sensitivity factor, 𝜅. The inversion technique is developed by a rigorous analysis of data and model
uncertainties and their effect on estimated parameters. Furthermore, the SSC-Lab allows computing
synthetic catalogs of focal mechanisms that obey the probability functions of the SSC model, 𝑝
9
(𝐧 )∝
e
:9(𝐧 =)
and different parametrizations of mechanism orientation uncertainty, such as nodal plane
uncertainty or the distribution of the angular distance between two mechanisms. Given the
equivalencies between distributions of mechanism orientation uncertainty in Section 1.3.3, we use
the parametrization over the Rodrigues rotation matrices to model focal mechanism errors.
In this section, we show some examples of the application of the SSC-Lab to three seismic sequences:
(1) seismicity in the San Jacinto fault, (3) the seismicity in the Parkfield segment of the San Andreas
fault, and (2) the 2019 Ridgecrest sequence. We show that our analysis technique provides new
insights into the plate boundary's regional and local tectonics, the internal structure of faults, and the
stress field's spatial variation.

1.6.1 Seismicity of the San Jacinto Fault Zone
In JJ21, we analyze the SSC of the seismicity in the San Jacinto fault without considering uncertainties
in mechanism determination. Therefore, it implied that measured complexity was attributable to
stress-strain processes only and, consequently, overdetermined. We performed the analysis again
using the same dataset shown in JJ21-Figure 11. The data set comprises the focal mechanisms of 1330
earthquakes in the magnitude range 2.5–4.5 that occurred between 1981–2020 with epicenters in a
77
147 km × 22 km box centered on the San Jacinto Fault. The new SSC+Noise model and the resulting
distributions are shown in Figure 1.19. Modeled distributions in Figure 1.19 are calculated from
synthetic catalogs constructed from the MLE of 𝑅 and 𝜅 (Figure 1.19-i) and the observed mechanism
uncertainty distribution is shown in Figure 1.19-g. The total moment fractions are biased for noise,
and such bias is observed by the difference between the mean of the empirical distributions and the
values corrected for noise shown as a red dot in panels (b)-(f).
The SSC+Noise model estimates 𝑅 =0.47±0.09, and 𝜅 =3.2±1.2, while in JJ21, we obtained 𝑅 =
0.45±0.04, and 𝜅 =5.7±1.5. The difference between the estimated 𝜅 clearly comes from modeling
uncertainties in the mechanism orientation and, additionally, in JJ21, mechanisms were weighed by
scalar seismic moment rather than magnitude. Weighting mechanisms by their scalar moment rather
than magnitude gives 𝜅 =11±7.5. As shown in Figure 1.17, using mechanisms weighted by their
scalar moment typically gives bigger values of 𝜅, and thus smaller complexity estimates; larger
events, usually in smaller numbers, dominate the average. Additionally, moment weighting of
mechanisms also results in larger uncertainties on the estimations.
Distributions in Figure 1.19 (a-f) show that the SSC+Noise model accurately simulates observations
of the differential stress ratio and the total moment fractions. For example, the observed zeroth-
degree moment fraction is 𝐹

)
=0.45±0.074 and the model gives 𝐹

)
=0.48±0.054, which is
slightly overdetermined. In contrast, the first-degree total moment fraction from the data is 𝐹

&
=
0.193±0.05 and the model gives 𝐹

&
=0.175±0.039, which is slightly underdetermined. The MLE
technique considers the distributions of total moment fractions, including the entire correlation
structure. Such correlations might bias the estimation of total moment fractions: the model tries to
fit all of them with the same weight, but it might not be able to recover the correlation structure. This
effect can be stronger if there are preferred orientations on observed mechanisms that the SSC model
does not predict. For example, in the next section, we discuss the stress characterization of the
78
Parkfield segment of the San Andreas fault, where the SSC+Noise model cannot accurately recover
the distributions of observed total moment fractions.

Figure 1.19. SSC modeling of the seismicity in the San Jacinto fault from the dataset in JJ21. (a) Observed (red)
and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and
modeled (blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions
corrected for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma
distribution. For this dataset, Γ(𝜎|𝛼 =8.75,𝛽 =2.48), which gives mean(𝜎)=21.75
)
. (h) Distribution of
measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅) estimated from the
observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the Pearson correlation
matrices of observed and modeled moment fractions. (l) SOR basis set for the San Jacinto fault seismicity.

79
1.6.2 Seismicity in the Parkfield Segment of the San Andreas Fault
The Parkfield segment of the San Andreas fault is geologically and tectonically simple (Eberhart-
Phillips & Michael, 1993; Waldhauser, F. et al., 2004): the fault is very linear, well localized, and
earthquake hypocenters do not show any major structures such as branches or bending.
Furthermore, focal mechanisms are dominated by pure strike-slip, and the nodal planes are aligned
with the fault trace (Thurber et al., 2006). Geological observations estimate that the San Andreas fault
in the Parkfield segment has experienced over 570 km of lateral displacement. The last major
earthquake was the Mw6.0 event of September 2004.
We test the SSC model against the seismicity in the Parkfield segment (SAP in Figure 1.4). Figure 1.20
shows the inversion results and the modeling from the SSC-Lab. We obtain 𝑅 =0.55±0.06 for the
differential stress ratio and 𝜅 =11.25±5.75. The zeroth-degree moment fraction corrected for
noise is 𝐹

)
=0.86±0.068, which implies low complexity. The principal stress 𝐩 =𝐞
&
is oriented
north-south horizontally. Its azimuth is 𝐴(𝐩 )=1.2
G
±1.7
G
, and its plunge, measured downwards
from the horizontal plane, is 𝐷(𝐩 )=5.1
G
±3.8
G
(Figure 1.20-h). We typically find a good agreement
between modeled and observed distributions of the stress differential ratio. However, total moment
fractions, which determine 𝜅, have some biases due to the correlation structure of the total moment
simplex that our modeling cannot fully replicate. The seismicity in Parkfield is an example.
A comparison of the correlation matrices of total moment fractions from the data and the model in
Figure 1.20 (j) and (k) show the differences. For the observed 𝐹

-
, there are negative correlations
between 𝐹

)
and higher degree total moment fractions, which is expected from the simplex structure.
However, there is a strong negative correlation between 𝐹

)
and 𝐹

4
(𝑟 =−0.64) that the SSC+Noise
model cannot reproduce. The SSC+Noise model shows the strongest negative correlation between 𝐹

)

and 𝐹

&
. The effect can also be seen in the empirical distributions in panels (b)-(f) in Figure 1.20. We
find that the SSC+Noise model largely overpredicts 𝐹

&
and slightly underpredicts the rest of the
moment fractions 𝐹

-z+
. We get 𝐹

&
=0.054±0.013 from the data and 𝐹

&
=0.19±0.038 from the
80
model. These results indicate that mechanisms in the Parkfield area are more “organized” than
predicted by the SSC model. In particular, there is less dispersion in the strike directions than
predicted by the model.
In the SSC model, the probability of failure in a plane with a normal vector 𝐧 , depends exponentially
on the shear traction amplitude and the strain sensitivity factor, 𝑝
9
(𝐧 )∝e
:9(𝐧 =)
. Therefore, there is
no constraint on the orientation of 𝐧 other than through the stress field. In the Parkfield fault zone,
however, the fault planes are well localized and organized in preferred directions, which could be
described as “tectonic grain” or anisotropy resulting from the long-term evolution of the fault.
Therefore, earthquakes occur in such preferred orientations. In Section 1.8.1, we investigate a
modification to the SSC model that allows us to model and make inferences about the anisotropy of
fault zones and its relation to long-term tectonic forcing.

81

Figure 1.20. SSC modeling of the seismicity in the Parkfield segment of the San Andreas Fault as shown in
Figure 1.4. (a) Observed (red) and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f)
show the observed (red) and modeled (blue) distributions of the total moment fractions. The red dot is the
MLE of total moment fractions corrected for noise. (g) Distribution of observed mechanism uncertainty 𝜎, and
its fit by a gamma distribution. For this dataset, Γ(𝜎|𝛼 =34.3,𝛽 =0.82), which gives mean(𝜎) =28.4
)
. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the San Jacinto
fault seismicity.

82
1.6.3 The 2019 Ridgecrest Sequence
Locations of mainshocks and aftershocks of the Mw6.4 and Mw7.2 Ridgecrest earthquakes of 2019
and geological and geodetical data show that the Ridgecrest earthquakes ruptured a complex system
of crosscutting faults oriented towards the northwest mostly and minor structures towards the
northeast (Hough et al., 2020). The Ridgecrest area wasn’t as seismically active as other regions of
Southern California, such as the San Jacinto Fault. Most of the seismic activity before the mainshocks
was recorded to the north and northeast of Ridgecrest, on the Coso geothermal field and the southern
Sierra Nevada fault zone. The Mw 6.4 event was preceded by a few small events hours before,
including an Mw 4 a day earlier. These foreshocks appear to align on a northeast-striking fault
perpendicular to the prevalent northwest trend of faults in the Little Lake fault zone south of Walker
Lane. Unexpectedly, the Mw 7.2 and its aftershocks ruptured a fault orthogonal to the rupture of the
Mw 6.4 and intersected multiple smaller sub-orthogonal faults, generating wide damage zones
(Cheng & Ben-Zion, 2020; Lin, 2020; Shelly, 2020).
The southern end of the Ridgecrest mainshock ruptured sets of sub-parallel faults, and the rupture
seemed to stop before the Garlock fault. To the north, aftershocks reached the Coso Range, and focal
mechanisms and moment tensor inversions show different seismicity styles, including shear strike-
slip failure and events with large CLVD components, and even volumetric sources (Cheng, Y. et al.,
2021; Rodriguez Padilla et al., 2022; Ross et al., 2019). The complexity observed in this sequence
shows a highly heterogenous stress-strain field and strong dynamic evolution of the material
properties. However, spatiotemporal monitoring of stress-strain parameters, such as strain energy
and strength, months before the Ridgecrest earthquakes of 2019 showed no clear indication of
preparation phase in the hypocenters area (Bondur et al., 2020; Kim, J. et al., 2021).
We apply our inversion technique to the Ridgecrest dataset of Hauksson and compare estimations
against inversions from the Cheng and Lin datasets. Figure 1.21 shows the SOR basis set 𝕭𝐌

𝟎
for
the Hauksson dataset. 𝐌

𝟎
has a similar orientation to the mechanism of the two largest earthquakes
83
of the sequence shown in Figure 1.5a. The angles between them are 𝜂
qos.+
=
𝑎𝑟𝑐𝑐𝑜𝑠·𝐌

)
(h@)
:𝐌

)
(qos.+)
º=8.3
G
, and 𝜂
qo.4
=𝑎𝑟𝑐𝑐𝑜𝑠·𝐌

)
(h@)
:𝐌

)
(qo.4)
º=12
G
, where 𝐌

)
(h@)
is
the seismicity CMT mechanism, and 𝐌

)
(qo.4)
the foreshock and 𝐌

)
(qos.+)
the mainshock CMT
mechanisms, respectively. The seismicity CMT mechanism is consistent with previous stress models
that estimate the principal compressive stress 𝜎
&
oriented almost north-south and subhorizontal
(Hauksson & Jones, 2020).
Figure 1.22-a shows the empirical distribution of the stress differential ratio from data and from
synthetic simulations of SSC+Noise catalogs using the maximum likelihood estimates. Figure 1.22b-f
show the empirical distributions of the moment fractions measured from data, sampled from the
maximum likelihood function, and calculated from a synthetic catalog with the MLE. Figure 1.22g
shows the Pearson correlation matrix of observed and modeled 𝓕

. Panels a-f of Figure 1.22 show the
effect of noise on the distributions of total moment fractions. When distributions are corrected for
noise, the distribution spread is reduced, but more importantly, the mean value is shifted. For
example, the data 𝐹

)
=0.4698±0.069, and the corrected is 𝐹

)
=0.664. Figure 1.22i shows the
distributions of the principal stress directions estimated from the principal directions 𝕰 of the CMT
mechanism. The principal stress 𝐩 =𝐞
&
is oriented north-south sub-horizontally. Its azimuth
is 𝐴(𝐩 )=7.0
G
±2.2
G
, and its plunge, measured downwards from the horizontal plane, is 𝐷(𝐩 )=
7.6
G
±4.7
G
(Figure 1.22i). The complexity factor corrected for noise is 𝐹

!
=0.335, which implies a
moderate complexity.
Figure 1.23 shows the likelihood function 𝐿𝐹 ∝𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅 computed from the three
Ridgecrest datasets: Cheng, Hauksson, and Lin. Although the uncertainty distributions on the three
catalogs are considerably different, they are correctly modeled, and the inverted parameters agree
well. However, Figure 1.22 shows that the distributions of the total moment fractions do not match
perfectly. For example, the observed zeroth degree moment fraction is 𝐹

)
=0.4698±0.069 and the
modeled 𝐹

)
=0.554±0.05. Therefore, the misfit of the mean value is 𝛿𝐹

)
=−0.0840. The higher
84
degree misfits are 𝛿𝐹

&
=0.0982, 𝛿𝐹

+
=-0.0239, 𝛿𝐹

5
=-0.0231, and 𝛿𝐹

4
=0.0289. In comparison with
the residuals from the Parkfield dataset, the model underpredicts 𝐹

&
and overpredicts 𝐹

)
, 𝐹

+
, and 𝐹

5
.
Indicating that there is more variability of slip vectors component along the strike directions.
The SSC parameter estimations, with one standard deviation confidence interval, are 𝑅
I
=0.42±
0.07, 𝑅
H
=0.41±0.12, and 𝑅
J
=0.25±0.11, for Cheng, Hauksson, and Lin, respectively, and 𝜅
k
=
5.8±1.8, 𝜅
H
=5.2±1.2, and 𝜅
J
=4.2±1.8. Notice that the smallest estimated value of strain
sensitivity comes from the Lin dataset, which has the highest uncertainties. Moreover, the Lin dataset
also gives the bigger difference in 𝑅.

Figure 1.21. Stress-oriented representation (SOR) basis set 𝕭f𝐌
)
𝟎
g=h𝐌
)
!
,𝛼 =0,…,4i from the Hauksson
dataset (ℋ). Moment tensors are represented as focal spheres with compressional quadrants in color and
tensional in white.

85

Figure 1.22. Empirical distributions of the total stress differential ratio (𝐸
#
), the total moment fractions (𝐹
9
!
),
and stress principal directions calculated from the Ridgecrest catalog of Hauksson. (a) Distributions of 𝐸
#
. The
red histogram is calculated from measured data. The blue histograms is the empirical distribution of 𝐸
#

estimated from a synthetic catalog whose input parameters are the maximum likelihood estimations: R=
0.42, 𝜅 =5.2, and Γ(𝜎|𝛼 =12.6,𝛽 =1.9) where 𝛼 and 𝛽 are estimated from the data uncertainties. The red
dot is the estimation corrected for noise. Panels (b)-(f) show the histograms are empirical distributions of
𝐹
9
!
,𝛼 =0,…,4. The blue histogram is the distribution of 𝐹
9
!
computed from the maximum-likelihood estimate
catalog. The red dot is the estimation corrected for noise (g) Pearson correlation matrix of the observed 𝐹
9
!
.
(h) is the Pearson correlation matrix from modeled 𝐹
9
!
. (i) Distributions of the orientation of the stress
principal axis 𝕰
𝟎
.

86

Figure 1.23. Likelihood function 𝐿𝐹f𝐸
#
,𝓕
9
,𝕰
𝟎
; 𝚺
)
,𝜅g of the data given the SSC model parameters computed
from three Ridgecrest datasets: (a) Cheng, (b) Hauksson, and (c) Lin.

1.6.4 SSC Model of Southern California
We construct an SSC model of the upper crust of Southern California using the focal mechanism of
over 144,000 earthquakes of magnitudes Mw 1.5-7.3 that occurred between 1980-2021 (Figure 1.3).
We divided the region into boxes of 0.1°x0.1° size with 0.05° overlapping in the longitude and latitude
directions and performed independent inversions with the focal mechanism lying in each box. For
the inversions, we required boxes to contain at least 50 mechanisms with maximum plane
uncertainty of 40°. The grid size and mechanism distribution parameters are similar to the ones
chosen by Hardebeck and Michael (2006), who observed that the average mechanism difference
decreases from ∼60° for events >10 km apart to ∼40° for events ∼1 km apart. They concluded that
the resolved stress variability is constrained robustly with this parameter set.
Figure 1.24 shows the stress-strain characterization of seismicity in Southern California. Figure 1.24-
a shows the spatial distribution of the total-moment fraction of zeroth degree, 𝐹

)
corrected for noise,
which is related to the complexity factor, 𝐹

!
=1−𝐹

)
. Thus, high values of 𝐹

)
imply low complexity
and low 𝐹

)
implies high complexity. Zones with higher complexity and low 𝐹

)
(typically 𝐹

)
<0.65) in
Southern California are typically associated with large earthquakes, such as Northridge, Ridgecrest,
Hector Mine and Landers, and El Mayor. Large earthquakes relax the regional stress field, and
aftershock sequences usually show more complexity. Regions with large earthquakes and long
87
aftershock sequences also show low values of strain sensitivity parameter, 𝜅 (Figure 1.24b). High
values of 𝜅 might indicate high stress concentration or strain weakening and could be an indication
of a preparatory phase for a large earthquake or background seismicity in a well-developed fault.
Figure 1.24-c shows the expected stress tensor plotted as the source mechanism and colored by
differential stress ratio, 𝑅.
Stress patterns along Southern California's San Andreas fault (SAF) are diverse, showing a high strain
sensitivity in the Mojave section and the Coachella Valley. These two regions are tectonically
different. The Mojave section of the SAF is mainly compressional (𝑅 >0.7), exhibiting well-localized
seismicity with mostly reverse mechanisms. On the other hand, the Coachella Valley is governed by
extensional tectonics (𝑅~0.4) and more diffuse seismicity distributed over a larger volume.
High mechanism complexity along the SAF is observed near its intersection with the Garlock and San
Jacinto faults (also known as Tejon Pass and Cajon Pass, respectively). In these regions, most of the
seismicity occurs at deeper depths (>15Km) compared with the overall seismicity in Southern
California, which is shallower (<15km).
The Eastern California Shear Zone (ECSZ), just south of the Eastern section of the Garlock fault,
known as the Black Water fault zone, shows low complexity and high 𝜅. Estimated values in this area
are interesting because the Harper and Helendale faults are aligned with ruptures of the Ridgecrest
on the north and Hector Mine and Landers earthquakes to the south. Moreover, there is no record of
major earthquakes on these faults recently. Hence, higher values of strain sensitivity might indicate
high-stress concentration expected before the occurrence of a large earthquake.
The southern section of the Elsinore fault is not well localized on the surface, and seismicity is spread
over a wide region. Nevertheless, the Elsinore fault, near the border between Mexico and the USA,
shows low complexity and high-stress concentration. The El Mayor-Cucapah earthquake of 2010
occurred just to the southern continuation of this fault in the Laguna Salada section.

88

Figure 1.24. Stress and strain characterization of seismicity in Southern California. (a) Stress differential ratio
𝑅. (b) Strain sensitivity parameter 𝜅. (c) Stress tensor plotted as source mechanism and colored by
differential stress ratio. Moment tensors of large earthquakes (Mw>6) from 1974-2019 are indicated by the
black and white source mechanisms.

89
1.6.5 SSC Model of Northern California
Figure 1.25 shows the stress-strain characterization from seismicity in Northern California and the
Greater Bay Area of San Francisco. Overall, there are stress variations along all the major faults: San
Andreas, Hayward, and Calaveras. For example, the Parkfield section of the San Andreas fault shows
very low complexity and high strain sensitivity (𝐹

)
>0.8 and 𝜅~20), which agrees with the
observations of localized seismicity in the fault structure inferred from hypocenter locations (Schaff
et al., 2002; Thurber et al., 2006; Waldhauser et al., 2004; Waldhauser, Felix et al., 1999). In contrast,
the creeping section of the SAF to the north of Parkfield shows high complexity and low strain
sensitivity (𝜅 <4 and 𝐹

)
<0.5), suggesting that failure is controlled by material weaknesses rather
than stress concentration in the fault. Furthermore, mechanism complexity presumably results from
the heterogeneous branching of the SAF and the Calaveras fault. The seismicity style is mostly strike-
slip and 𝑅~0.5 in Parkfield and the central San Andreas fault. To The north, in the Santa Cruz
Mountains section, where the Mw6.9 1989 Loma Prieta earthquake occurred, the stress field is more
transpressional. It evolves to more transtensional in the San Francisco Peninsula.
The Hayward fault, located on the eastern margin of the San Francisco Bay, show high-stress
concentration (𝜅 >10), low complexity (𝐹

)
>0.7), and tectonically, mostly shear lateral motion.
However, to the north, the tectonic style changes to more transpressional where the rupture of the
Mw 6.0 Napa earthquake of 2014. Low values of 𝜅 and 𝐹

)
indicate high complexity and are observed
in the regions around major earthquakes, including the Loma Prieta, San Simeon, and Napa
earthquakes of 1989, 2003, and 2014 respectively. Fault zones with big earthquakes and their
aftershock sequences typically exhibit higher complexity. The high complexity can be explained by
strong stress perturbations of a mainshock, which changes the orientations of the local stress field
and might activate faults that were not optimally oriented for failure under the background stress.
Therefore, aftershock sequences dominate the complexity.
90
Figure 1.25-c shows the expected stress tensor in Northern California plotted as the source
mechanism and colored by the differential stress ratio. We find that the CMT mechanisms of large
earthquakes are consistent with the stress field. For example, the Mw6.7 Coalinga earthquake of 1983
shows a reverse mechanism and agrees with the background stress. However, this indicates a rapid
spatial change of the stress field in the proximity of the San Andreas fault, from lateral shear motion
to a transpressional regime less than 15 km to the east. The stress field is transpressional to the west
of the San Andreas fault in the Coastal Ranges, and the seismicity indicates moderate complexity.

91

Figure 1.25. Stress and strain characterization of seismicity in Northern California. (a) Stress differential ratio
𝑅. (b) Strain sensitivity parameter 𝜅. (c) Stress tensor plotted as source mechanism and colored by
differential stress ratio. Moment tensors of large earthquakes (Mw>6) from 1983-2014 are indicated by the
black mechanisms.

92
1.7 Temporal Monitoring of Stress Fields
Temporal variations of stress fields might provide insights into the estate of faults at different stages
of the seismic cycle. Earthquakes are the inelastic response to stress; therefore, they relax the
regional stress field. Additionally, the heterogeneous co-seismic deformation produces changes in
the local stress field.
Stress field changes can be measured in boreholes (Zoback, Mark D. & Healy, 1992). Unfortunately,
borehole measurements have resolution only at shallow depths and poor coverage. Alternatively,
stress field perturbations after earthquakes can be estimated using stress transfer models (Harris &
Simpson, 1992; King et al., 1994; Segou & Parsons, 2020; Stein et al., 1992) and inferred from
inversions of focal mechanisms. Stress inversions using focal mechanisms have been used to detect
rotations of the principal stress directions from before and after large earthquakes.
Stress rotations after large earthquakes indicate that pre-earthquake differential stress (𝜎
&
−𝜎
5
) is
on the order of the earthquake stress drop. On the other hand, if the pre-earthquake differential
stress is much larger than the stress drop, no rotations are expected (Faulkner et al., 2006). Several
studies have shown large earthquakes can produce large rotations of the stress field, for example, on
the order of 10
G
after the Landers earthquake of 1992 or the Northridge earthquake of 1994
(Hauksson, 1994; Zhao, D. et al., 1997), and up to 20
G
after large subduction zone earthquakes, such
as the Mw9.4 Sumatra earthquake of 2004 (Hardebeck, Jeanne L., 2012; Hardebeck, Jeanne L. &
Okada, 2018).
We investigate the temporal variations of the stress fields in the Ridgecrest fault zone using the
dataset of Hauksson. For this aim, we order the focal mechanism by the earthquake origin time and
select them sequentially in groups of 200 with ten mechanisms overlapping. The estimated
parameters are assigned to the time of the last event in the group.
Figure 1.26 shows the temporal evolution of the stress differential ratio 𝑅, the strain sensitivity factor
𝜅, and the azimuth of maximum horizontal compressive stress 𝐴𝑧(𝐩 ). Before the Mw6.4 and 7.2
93
earthquakes of 2019, the strain sensitivity factor varies between 1.5-4. In fact, inversion from all the
events before July 2019 gives 𝜅 =2.5±0.64. Between 2010 and June 2019, 𝜅 increases from 𝜅 =
2.1±0.8 to 𝜅 =3.75±0.75 indicating a gradual stress concentration or strain weakening. In
general, the tectonics of Ridgecrest show an extensional strike-slip regime typical of the Walker Lane.
Between 2010-2019, the stress differential ratio reflects this tectonic setting, but it increases from
𝑅 =0.15±0.05 to 𝑅 =0.4±0.12. Cheng and Ben-Zion (2020) analyzed focal mechanisms as well,
and they found that during this period, the seismicity changed from more normal-faulting to more
strike-slip, also consistent with our observations.
An inversion of focal mechanisms of earthquakes that occurred between the Mw6.4 and 7.2 events
also shows an increase of 𝜅, from 𝜅 =3.75±0.75 pre Mw6.4 to 𝜅 =5.4±0.8 before the Mw7.2
event. These results suggest that the Mw6.4 earthquake increased the stress concentration,
triggering the Mw7.2 event. After the Mw7.2 earthquake, 𝜅 decreases rapidly to 𝜅 =1.25±0.5 and
is followed by a slow increase until the Mw5.5 aftershock of August 19, 2019. On average, the
aftershocks of the Mw7.2 earthquake after July 4, 2019, give 𝜅 =2.1±0.6, 𝑅 =0.3±0.11.
The azimuth of maximum horizontal compressive stress Az(𝜎
&
) estimated from events before the
Mw6.4 earthquake varies between 4
G
and 20
G
, giving in average Az(𝜎
&
)=11.5
G
±4.6
G
. Between the
Mw6.4 and 7.2 events, there is a slight counterclockwise rotation, Az(𝜎
&
)=3.8
G
±4.1
G
, and after the
Mw7.2, there is a clockwise rotation that results in Az(𝜎
&
)=9.8
G
±5.8
G
, which is similar to pre-
earthquake azimuths. Furthermore, as the aftershock sequence progresses, there is a slight
counterclockwise rotation.

94

Figure 1.26. Temporal monitoring of the stress field in the Ridgecrest fault zone using the mechanism in the
catalog of Hauksson. (a) Evolution of the strain sensitivity parameter 𝜅 between the years 2002-2022. (b)
Change of the strain sensitivity parameter 𝜅 after the Mw6.4 foreshock on July 2, 2019. (c) and (d) show the
evolution of the differential stress ratio. (e) and (f) is the azimuth of the principal compressive stress Az(𝜎
%
).
(g) and (h) show the depth distributions of earthquakes. Dots are the epicenters in time and depth, colored
and sized by earthquake magnitude.

95
1.8 Mechanism Complexity and Cumulative Offsets
Several studies have demonstrated that fault zone heterogeneity is correlated to tectonic variables
such as cumulative slip, slip rate, and fault zone misalignment from the plate motion direction. Fault
heterogeneity refers to offsets of the fault surface (step-overs), fault orientation changes, fault
surface roughness, and overlapping fault segments. Therefore, fault heterogeneity can be inferred
from the geological mapping of fault zones, which contain robust information on shallow
heterogeneities, and the fault properties at depth can be extrapolated. Additionally, the spatial
distribution of hypocenters and earthquake mechanism variability contains information on fault
zone heterogeneities at depth. Fault zones evolve with increasing geological displacement, and
structural heterogeneities such as damaged volume and step-overs tend to decrease, and seismicity
concentrates closer to the fault surface (Brodsky et al., 2011; Powers & Jordan, 2010; Sagy et al., 2007;
Wesnousky, 1988). As a result, faults with larger cumulative displacements typically exhibit smaller
geometrical complexity. Hence, it is expected that the moment measures of mechanism complexity
correlate with the cumulative offset of faults.
For each fault zone in Table 1, we estimate distributions of total moment fractions and stress
differential ratio and invert them to estimate the zeroth-degree total moment fraction 𝐹

)
corrected
for noise and the strain sensitivity factor 𝜅. 𝐹

)
is related to the complexity factor as 𝐹

!
=1−𝐹

)
.
Figure 1.27 shows the estimations of 𝑅 and 𝜅, and Figure 1.28 shows the comparison between 𝜅 and
𝐹

)
and estimations of cumulative offsets reposted in the literature.
Fault zones in California show a wide range of mechanism complexity, measured by 𝐹

)
, and tectonic
regimes represented by values of 𝑅. We find a strong correlation (Pearson’s correlation coefficient
𝑟 =0.69), between cumulative displacements and maximum likelihood estimations of 𝜅, and also
between slip and 𝐹

)
corrected for noise (𝑟 =0.67).
High complexity (𝐹

)
<0.6) is observed in faults that rupture during the Landers and Hector Mine
earthquakes. For example, the Mezquite Lake fault (HML) estimate 𝜅 =3.0±1.5, and 𝐹

)
=0.59±
96
0.093, and Lavick Lake-Bullion (HLL) has 𝜅 =2.0±1.25, and 𝐹

)
=0.51±0.072. These faults
ruptured during the Mw7.2 Hector Mine earthquake of 1999, and both fall in the high complexity
level (𝐹

)
<0.6). The Homestead Valley (LHV) and North Cross (LNC) faults, which ruptured during
the Mw7.4 Landers earthquake of 1992, estimate 𝜅 =2.0±1.25, and 𝐹

)
=0.55±0.078 and 𝜅 =
0.5±0.25, and 𝐹

)
=0.487±0.050, respectively. Furthermore, the stress differential ratio estimated
for faults that rupture during the Landers earthquake indicates a more extensional regime (0.1<
𝑅 <0.4) compared to the Hector Mine earthquake faults with 𝑅~0.5. These faults have estimates of
less than 10 km of geological offset, and the high complexity measures can also be attributed to the
aftershock sequences. Large earthquakes produce strong perturbations on the stress field, resulting
in changes in seismicity patterns in space and time. Variations on the local stress field are likely to
activate faults with orientations more favorable for failure under the perturbed stress field than in
the background stress pre-mainshock.
Faults in the Mojave Block (code names begin with M) have estimated between 1 to 40km of total
displacement. Faults such as Calico Hidalgo (MCB), Camp Rock (MCR), and Rodman-Pisgah (MRP)
display moderate complexity (0.6≤𝐹

)
<0.8), and estimate strain sensitivity factors between 3 to 8.
The stress differential ratio indicates, as expected, a transtensional regime typical of the Mojave block
and the Eastern California Shear Zone, with stress differential rations of 0.4<𝑅 <0.5.
Most sections of the San Andreas fault have accumulated over 150km of slip, with estimates up to
570 km. The Parkfield segment (SAP) shows a large strain sensitivity factor, 𝜅 =11.75±5.75, and
𝐹

)
=0.85±0.067, both parameters indicate low mechanism complexity. Other fault zones with low
mechanism complexity (𝐹

)
>0.8) are the Loma Prieta (SAN) section of the SAF and the Hayward
fault (HAY) in Northern California. In Southern California, SAF-Coachella Valley (SAC), the San
Jacinto-Borrego fault (SJB), and the Granite Mountains section of the Elsinore fault (EGM) also show
low mechanism complexity and high values of strain sensitivity, 𝜅 >7. Figure 1.27 demonstrates the
change in the tectonic regime along the San Andreas faults. The most extensional section occurs in
97
the Imperial Valley (IMP) and the Coachella Valley in Southern California, estimating 𝑅 =0.09±
0.04 and 𝑅 =0.29±0.09. To the northwest of SAC, the bending of the SAF produces oblique strike-
slip and reverse motion in the San Bernardino Mountain section (SAB). The differential stress ratio
of this fault zone indicates a transpressional regime (𝑅 =0.64±0.1). Similarly, the Parkfield and
Loma Prieta sections are transpressional, 𝑅 =0.5±0.05 and 𝑅 =0.63±0.09 respectively.
High strain sensitivity factor values do not necessarily imply low mechanism complexity. For
example, the Imperial Valley fault (IMP) in Southern California has 𝜅 =14.75±9, the highest
estimate out of the dataset. However, the noise-free zeroth degree moment fraction is 𝐹

)
=0.72±
0.051, which implies a moderate, rather than low, mechanism complexity. This behavior is explained
by the dependency of the total moment fractions on 𝑅, and 𝜅, as shown in Figure 1.2. Where 𝑅 →0,1,
the 𝐹

-
are not very sensitive to 𝜅. The measured stress differential ratio of the IMP is 𝑅 =0.1±0.04,
being the most extensional fault zone of the analyzed dataset. In contrast, the Newport-Inglewood
fault is the most compressional with 𝑅 =0.85±0.06.
Figure 1.29 shows the distributions of residuals between measured and modeled total moment
fractions, 𝛿𝐹

-
=𝐹

-
(.^_^ )
−𝐹

-
(~G.g )
,𝛼 =0,…,4.
Systematic variations on the residuals provide information contained in the data that is not modeled
by the SSC model. For example, higher degree residuals and their correlations reveal information
about preferred failure directions.
The SSC model assumes that fault planes are isotropically oriented (the fault-normal vectors follow
a uniform distribution in the sphere), and the stress tensor through the probability of shear traction
(𝑝
9
(𝐧 ) in equation 21) give which orientations are more favorable for failure. This assumption is
valid for intact rocks of isotropic material. However, fault zones sustaining repeated deformation and
failure tend to develop preferred orientations of tectonic grain, and such particular orientations
become more favorable for failure. As a result, the observed total moment fractions of degrees 1 to 3,
98
which depend on the in-plane or out-of-plane variations, can systematically deviate from model
expectations.
Total moment fraction residuals in Figure 1.29 show a strong negative correlation between 𝛿𝐹

)
and
𝛿𝐹

&
(𝑟 =−0.9) and moderate negative correlations between 𝛿𝐹

&
and 𝛿𝐹

+
(𝑟 =−0.56),𝛿𝐹

5
(𝑟 =
−0.48), and 𝛿𝐹

4
(𝑟 =−0.29). Most fault zones analyzed are strike-slip and, as such, 𝐹

&
measures the
variability of focal mechanism along the strike direction represented by 𝐌

&
. 𝛿𝐹

&
>0 implies that
observed focal mechanisms show more strike variability than predicted by the SSC model, and if
𝛿𝐹

&
<0, the model is predicting more variability than observed. We find that fault zones that have
sustained large cumulative displacements, such as SAP, SAN, SAC, HAY, and CAL, typically have 𝛿𝐹

&
>
0, implying that the SSC model predicts higher variability along strike than the observed. On the
contrary, younger faults, such as those in the Mojave Block, including Landers and Hector Mine
earthquake faults, typically give 𝛿𝐹

&
<0.
Another interesting result is that Pearson’s correlation coefficients between 𝛿𝐹

)
with 𝛿𝐹

+
, and 𝛿𝐹

5

is 𝑟 =−0.48 in both cases and between 𝛿𝐹

&
with 𝛿𝐹

+
, and 𝛿𝐹

5
is fairly close: 𝑟 =−0.56 and 𝑟 =
−0.48. In the SSC model, variations along 𝐌

+
and 𝐌

5
are related to variations along rake and dip, and
if they have the same variance, they produce the same effect in the total moment fractions of degrees
2 and 3. Our results confirm this prediction.

99
Table 1.4. Fault zone inversion results. The columns for 𝑅, 𝜅, and 𝐹
9
#
show the MLE value and the 67%
confidence interval. 𝐹
9
#
is the estimation corrected for noise, and 𝛿𝐹
9
!
=𝐹
9
!
(3454)
−𝐹
9
!
(6)378)
,𝛼 =0,…,4, are the
residuals of data minus model. 𝐹
9
!
(6)378)
are calculated from the SSC+Noise model.

FAULT 𝑹 𝜿 𝑭
+
𝟎
𝜹𝑭
+
𝟎
𝜹𝑭
+
𝟏
𝜹𝑭
+
𝟐
𝜹𝑭
+
𝟑
𝜹𝑭
+
𝟒

CAL 0.6±0.06 7.25±2.75 0.76±0.086 0.02 -0.13 0.076 0.035 -0.0087
HAY 0.51±0.1 6±2.25 0.81±0.089 0.067 -0.13 0.048 -0.015 0.015
EGM 0.42±0.09 7.75±2.75 0.84±0.068 -0.015 0.0071 -0.023 -0.027 0.042
ELS 0.44±0.15 4.75±2.25 0.69±0.1 -0.013 -0.064 -0.02 0.0076 0.079
EW 0.7±0.08 7.75±2.5 0.76±0.053 0.014 -0.044 -0.035 0.0055 0.054
GAR 0.46±0.13 6.25±2.5 0.78±0.097 -0.02 -0.0081 -0.025 -0.018 0.064
HLL 0.55±0.14 2±1.25 0.51±0.072 -0.078 0.11 0.0079 -0.044 -0.0085
HML 0.46±0.13 3±1.5 0.59±0.093 -0.027 0.042 -0.03 0.029 -0.02
HP 0.48±0.12 4.5±2 0.63±0.099 -0.1 0.12 -0.037 0.0006 0.0049
IMP 0.09±0.04 14.8±5.25 0.72±0.037 -0.0059 -0.074 0.034 0.037 -0.002
LE 0.4±0.19 4.25±2 0.67±0.1 -0.037 0.052 0.013 -0.051 0.013
LHV 0.21±0.09 2±1 0.55±0.078 -0.00065 -0.019 0.0047 0.013 -0.0043
LJV 0.16±0.06 4±1.75 0.64±0.071 -0.036 0.065 -0.05 -0.019 0.032
LNC 0.31±0.1 0.5±0.25 0.49±0.06 -0.014 0.012 0.0017 0.0099 -0.022
MBW 0.39±0.12 7.75±3.25 0.81±0.084 -0.034 0.012 -0.028 0.0079 0.027
MCB 0.39±0.16 3.75±2 0.73±0.13 -0.031 0.025 -0.032 -0.018 0.023
MCR 0.4±0.19 4.25±1.5 0.7±0.092 -0.023 0.023 0.019 -0.047 0.016
MRP 0.47±0.09 5.75±2 0.73±0.1 -0.078 0.09 -0.02 -0.017 0.0011
NI 0.85±0.06 7.75±4.25 0.7±0.065 -0.0031 -0.044 0.012 0.0082 0.0081
SAB 0.63±0.11 5.25±2.25 0.69±0.084 -0.021 -0.016 0.0037 -0.039 0.055
SAC 0.29±0.09 13±6 0.85±0.056 0.032 -0.061 -0.0067 -0.014 0.034
SAN 0.62±0.09 9.75±3.5 0.86±0.051 0.045 -0.11 0.018 -0.0088 0.03
SAP 0.5±0.05 11.8±5.75 0.85±0.063 0.031 -0.14 0.027 0.038 0.039
SJB 0.39±0.08 10.8±3.75 0.86±0.052 -0.042 -0.004 -0.0054 -0.024 0.051
SJC 0.49±0.09 3.25±1.25 0.59±0.069 -0.022 0.017 -0.0075 0.0093 -0.005

Figure 1.27. SSC inversion results for the fault zones in Table 1. Colored markers are the maximum likelihood
estimation of the SSC parameters 𝑅 and 𝜅, and gray error bars are their 67% confidence interval. Gray
segmented lines are the limits between low, moderate, and high complexity intervals of the complexity scale.

100

Figure 1.28. Correlation between fault cumulative offsets and measures of mechanism complexity for
different fault zones in California. (a) Cumulative slip against strain sensitivity factor. (b) Cumulative slip
against zeroth-degree moment fraction. Horizontal gray bars indicate the minimum and maximum reported
offsets. Vertical bars are the 67% confidence interval on the estimated 𝜅 and 𝐹
9
#
corrected for noise. 𝑟 is the
Spearman correlation coefficient.

101

Figure 1.29. Scatter plot showing the correlations between residuals of total moment fractions. Residuals are
𝛿𝐹
9
!
=𝐹
9
!
(3454)
−𝐹
9
!
(6)378)
,𝛼 =0,…,4. 𝑟 is the Pearson’s correlation coefficient.

1.8.1 Tectonic Anisotropy
The residuals of the total moment fractions in Figure 1.29 show that the SSC model, on average,
successfully models the mechanism complexity of the dataset (the average 𝛿𝐹

)
~0). Furthermore, it
demonstrates that failure orientation can be modeled reasonably well with an exponential function
that depends on the shear traction magnitude multiplied by the strain sensitivity factor. However,
higher-order correlations on the residuals exist, indicating that the probability of failure direction
102
has a higher-order dependency on other variables. In particular, the correlation between 𝛿𝐹

)
and
𝛿𝐹

&
, and between 𝛿𝐹

&
with 𝛿𝐹

+
, 𝛿𝐹

5
, and 𝛿𝐹

5
suggest that, as fault zones evolve, fault planes localize
and organize in preferred directions aligned with the tectonic stress. Preferred orientations of failure
might occur due to material rheological anisotropy and other crustal geological and morphological
properties of faults that have sustained multiple episodes of deformation.
The organization of faults in preferred directions reduces the dispersion of out-of-plane variability,
which are variations of the strike direction for strike-slip systems. A good example of a fault zone
with less-than-expected strike variability is the seismicity of the Parkfield segment of the San
Andreas fault. Figure 1.20 shows that the SSC model predicts higher-than-observed strike variability
reflected by the distributions of 𝐹

&
. Moreover, the compilation of residuals from individual fault zones
in California (Figure 1.29) also show that lower-than-predicted values of 𝐹

&
occur on fault zones that
have large cumulative displacements, such as HAY, CAL, and SAN. These fault segments are typically
well localized, show less heterogeneity, and fault planes follow similar orientations.
Several studies of spatiotemporal patterns of seismicity in Nothern California fault zones such as SAP,
HAY, and CAL have found patters of seismicity and repeating earthquakes that often occur in well-
confined horizontally aligned streaks and display similar source mechanisms (Schaff et al., 2002;
Waldhauser et al., 2004; Waldhauser et al., 1999). Moreover, such lineations are parallel to the slip
direction of faults, suggesting that they result from the long-term anelastic evolution and might
enhance the stress concentration in the fault. The simple e
:9(𝐧 =)
function cannot model such preferred
orientations.
Figure 1.30 shows an example of the spherical distributions of nodal plane vectors, 𝐧 computed from
the focal mechanisms of earthquakes in EGM, SAP, and IMP. Due to the nodal plane ambiguity, we
count both nodal planes for the empirical distribution. The column 𝑝(𝐧 ) shows the empirical
distribution of observed vectors and 𝑝
9
(𝐧 ;𝜅,𝑠) is the SSC probability distribution from the best fit of
parameters 𝜅, and 𝑅. The third column shows the residuals between distributions. We selected those
103
fault zones because they are typical examples; the residuals for EGM show that 𝑝(𝐧 ) is more spread
in the plane containing (𝐫
&
,𝐫
5
) with respect to 𝑝
9
(𝐧 ;𝜅,𝑠) and less concentrated in the planes
containing the vectors (𝐫
&
+𝐫
5
,𝐫
5
) and (𝐫
&
−𝐫
5
,𝐫
5
). SAP, on the other hand, is more concentrated on
the planes containing the vectors (𝐫
&
+𝐫
5
,𝐫
5
) and (𝐫
&
−𝐫
5
,𝐫
5
). The residuals of IMP are more
peculiar because 𝐸
)
qK
=0.09±0.04, but it shows that the model has more dispersion than the
observations.
The residuals in Figure 1.30 show patterns that hint at the modifications to the SSC model to include
preferred predetermined failure directions. Additional data such as mapped fault plane orientations
and orientations of slip vectors inferred from geological markers can be included in the inversion
scheme by modifying the SSC probability function 𝑝
9
(𝐧 ) as well. However, here we investigate
theoretical forms of SSC probability distributions that allow inferring information of fault zone
localization and alignment. Mathematically, the correlations between 𝛿𝐹

&
and the other total moment
fractions can be modeled by a probability density function that amplifies or reduces the contribution
of the first-degree direction cosine cos𝜃
&
, and thus, the expectation of 𝐹

&
.
Statistical analysis of 3D rotations of focal mechanism due to random stresses arising in fault
heterogeneities shows that focal mechanism orientations can be modeled with symmetric spherical
distributions such as the Cauchy distribution (Kagan, 1990). We investigate biases in the inversions
due to tectonic grain anisotropy not considered by the original SSC model. The SSC model is isotropic
in the sense that it assumes that fault plane orientations (described by 𝐧 ) depend only on the shear
traction magnitude. The SSC probability function determines the orientations that are more favorable
for failure, 𝑝
9
(𝐧 )=
&
𝒩
e
:9(𝐧 =)
, where the probability depends exponentially on the shear traction
amplitude 𝑠(𝐧 ) and the strain sensitivity factor. However, studies of the geometrical complexity of
fault zones demonstrate that 𝐧 has preferred directions typically well aligned with mapped fault
traces, and their distribution deviates from the exponential form.
104
In the JJ21, we showed that, 𝑠(𝐧 )=cos𝜃
)
(𝐧 ), where cos𝜃
)
(𝐧 )=𝐌

)
:𝐦
8
(𝐧 ). The SSC distribution
can be written as
𝑝(𝐧 )=
1
𝒩
e
:`tOv
*
(𝐧 =)
. (75)
We look for a modification of 𝑝(𝐧 ) that amplifies or reduces the contribution of cos𝜃
&
(𝐧 ) to the first-
degree total moment. One option is to include an additional term in the argument of the exponential
function. Considering that cos𝜃
)
(𝐧 )∈[0,1], and cos𝜃
&
(𝐧 )∈[−1,1], a suitable distribution is
𝑝(𝐧 ,𝜅′,𝜖) =
1
𝒩
exp[𝜅′cos
+
𝜃
)
(𝐧 )+𝜖cos
+
𝜃
&
(𝐧 )], (76)
where 𝜖 is a fault-orientation concentration parameter, and 𝜅′ is the strain sensitivity factor
calibrated for cos
+
𝜃
)
rather than cos𝜃
)
. The normalization constant is
𝒩 = exp[𝜅′cos
+
𝜃
)
(𝐧 )+𝜖cos
+
𝜃
&
(𝐧 )]

𝒮
!
𝑑𝐧 . (77)
Equation 76 is an anisotropic SSC model. Figure 1.31 shows the anisotropic SSC probability function
𝑝(𝐧 ,𝜅′,𝜖) for 𝑅 =0.5, and some selected values of 𝜅 and 𝜖. As in the isotropic SSC model, larger
values of 𝜅′ in 𝑝(𝐧 ,𝜅′,𝜖) concentrate the probability near the maximum of cos
+
𝜃
)
=𝑠
+
. 𝜖 →−∞
concentrates the probability on the planes orthogonal to 𝐧
)
and 𝐬
)
, and 𝜖 >0 increases the
probability towards the plane spanned by 𝐫
&
and 𝐫
5
. We impose 𝜖 ≤1 to guarantee that the expected
values of the zeroth- and first-degree direction cosines under 𝑝(𝐧 ,𝜅′,𝜖) satisfy E[cos
+
𝜃
)
]≥
E[cos
+
𝜃
&
] for any 𝜅′, i.e., 𝐹

)
≥𝐹

&
. Figure 1.32 shows the total moment fractions estimated for
𝑝(𝐧 ,𝜅′,𝜖) and for 𝑅 =0.5. 𝐹

&
is sensitive to both, 𝜅′ and 𝜖. As 𝜅′ increases, 𝐹

&
decreases similarly to
the isotropic SSC. Moreover, 𝐹

&
decreases with decreasing 𝜖 <0. 𝐹

)
, 𝐹

+
, 𝐹

5
, and 𝐹

4
are not very
sensitive to large negative values of 𝜖. Therefore, 𝐹

&
would drive the correlation structure of the total
moment simplex.

105

Figure 1.30. Examples of spherical distributions of observed and modeled fault orientation vectors. We
present data only for the EGM, SAP, and IMP fault segments. 𝑝(𝐧 6) is the empirical distribution of observed
vectors and 𝑝
9
(𝐧 6;𝜅,𝑠) is the SSC probability distribution from the best fit of parameters 𝜅, and 𝑅. Spherical
distributions are plotted in the orthogonal projection perpendicular to 𝐫 4
%
.

106

Figure 1.31. Probability density function 𝑝(𝐧 6,𝜅,𝜖) ∝exp[𝜅cos
'
𝜃
#
(𝐧 6)+𝜖cos
'
𝜃
%
(𝐧 6)], for selected values of 𝜖
and 𝜅, and R = ½. The projection is orthogonal to 𝐫 4
%
⁠. The PDF is scaled to a maximum value of unity.

Figure 1.32. Total moment fractions 𝐹
9
!
as functions of the stress sensitivity parameter 𝜅 and fault
concentration parameter 𝜖. We set R = ½.

107
1.8.2 𝜿
y
𝒗𝒔.𝜿
In the anisotropic SSC model, the total moment fractions, 𝓕

and the stress differential ratio 𝐸
)
are
generated on a stress-strain process described by the parameters 𝑅, 𝜅, and 𝜖, and perturbed by
random noise processes described by a parameter 𝜎. In Section 3, we describe the maximum
likelihood estimation technique to estimate the SSC parameters from observations of 𝓕

and 𝐸
)
. The
likelihood function 𝑃
WWX
is easily modified to include the dependency on the new variable 𝜖. The joint
distribution of the observational variables is given by,
𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅,𝜎=𝑃(𝐸
)
;𝑅,𝜎)𝑃𝓕

; 𝑅,𝜅,𝜖,𝜎. (78)
As in equation 34, the dependency on 𝜎 can be marginalized by integrating 𝑃
WWX
𝐸
)
,𝓕

,𝕰
𝟎
; 𝚺

,𝜅,𝜎
over 𝑃(𝜎). The densities 𝑃(𝐸
)
;𝑅) and 𝑃𝓕

; 𝑅,𝜅,𝜖 can be computed numerically using the Monte
Carlo integration technique, as shown in the workflow in Figure 1.6.
One difference between the isotropic and anisotropic SSC models is that the probability of failure on
faults with normal vectors 𝐧 , is now proportional to the squared shear traction magnitude, 𝑠
+
(𝐧 ).
Therefore, we first investigate the differences between estimations using these two SSC models,
neglecting, for now, the contribution 𝜖. Thus, we set 𝜖 =0, and invert the moment fractions for 𝜅
y
.
Figure 1.33 shows the comparison between estimates of 𝜅 and 𝜅
y
, and the zeroth-degree moment
fractions corrected for noise. These results show that there is approximately a linear relation
between 𝜅 and 𝜅
y
with the constant of proportionality 𝑐 =0.4. The total moment fractions of zeroth-
degree, follow a similar distribution and the fit by a line gives 𝑐 =0.98, implying that estimations of
complexity are insensitive to changing the density function.

108

Figure 1.33. Effect of 𝑝(𝐧 6) ∝e
:9(𝐧 <)
and ∝e
:=>?@
!
A
"
(𝐧 <)
in the estimation of the strain sensitivity factor and the
zeroth-degree moment fraction corrected for noise. In the inversions for 𝜅
=
we set 𝜖 =0. (a) 𝜅 against 𝜅′. (b)
𝐹
9
#
′ against 𝐹
9
#
. The gray dashed line is the fit by a line to the estimations.

1.8.3 Estimation of the Fault Orientation Concentration Parameter
We implemented the new probability density function in the maximum likelihood estimation
technique of the SSC parameters that characterize the stress field and the fault zone anisotropy.
Figure 1.34 summarizes the results for the fault zones in Northern and Southern California and
compares the SSC parameters against total displacements of faults.
Figure 1.34-a shows estimations of the new parameter 𝜖 which measures the fault orientation
concentration. Faults that have experienced small offsets, such as faults in the Mojave block, the
Landers, and the Hector Mine earthquake, typically show values of 𝜖>0, implying strong variability of
mechanism along the strike direction. Faults such as LNC, MCR, and HML are in the limit 𝜖 =1, which
we set to satisfy that modeled total moment fractions are 𝐹

)
≥𝐹

&
. These fault zones also estimate
very low values of strain sensitivity factor, generally 𝜅 <1, and zeroth-degree total moment fraction
𝐹

)
<0.5, falling in the high complexity zone. Although estimations for the Mojave Block fault zones
are reasonable and agree between them, we interpret them cautiously because these faults'
109
mechanism uncertainties are quite significant. For example, for LNC, Γ(𝜎
J3I
|𝛼 =11.22,𝛽 =2.81),
which gives E[𝜎
J3I
]=31.52
G
. We estimate E[𝜎
qI@
]=40.57
G
and E[𝜎
HqJ
]=40.49
G
for MCR and
HML as well. As we show in Section 1.5.3, ideally E[𝜎]<30; otherwise, the model has little to no
resolution.
Our results show that faults with geological offsets over approximately 10 km show preferential
directions of failure (𝜖 <0) that can be explained by tectonic anisotropy, and it is not modeled by the
isotropic SSC model that depends on the shear traction magnitude and the strain sensitivity factor
only. Furthermore, the fault orientation parameter rapidly decreases with increasing slip. The
strongest concentrations are observed in the CAL, HAY, and SAP. We truncate our estimations to 𝜖 >
−10 because smaller values cannot be resolved for the noise levels.
We find that the inversions with the new parameter 𝜖 give smaller estimations of 𝐹

)
corrected for
noise.
The total moment fraction residuals between the data and model predictions are shown in Figure
1.35. We find that the anisotropic SSC model successfully accounts for the dependencies observed in
the residuals from the isotropic inversion (Figure 1.29). For example, from the isotropic model, there
is a strong negative correlation between 𝛿𝐹

)
and 𝛿𝐹

&
(𝑟 =−0.9), we now find 𝑟 =−0.24. The more
moderate negative correlations between 𝛿𝐹

&
and 𝛿𝐹

+
,𝛿𝐹

5
, and 𝛿𝐹

4
are also smaller, falling from 𝑟 =
−0.56 to 𝑟 =0.32, 𝑟 =−0.48 to 𝑟 =−0.16, and 𝑟 =−0.29 to 𝑟 =−0.28, respectively.

110

Figure 1.34. Anisotropic SSC inversion results for the fault zones in Table 1. Colored markers are the
maximum likelihood estimation of the SSC parameters 𝑅 and 𝜅, and 𝜖. Gray error bars are their 67%
confidence interval. (a) Cumulative slip against fault orientation concentration parameter. (b) Estimations of
stress-differential ratio and strain sensitivity factor. The low-moderate complexity boundary corresponds to
𝐹
9
#
=0.8 and the moderate-high to 𝐹
9
#
=0.6 computed for 𝜖 =0. (c) Cumulative slip against strain sensitivity
factor. (b) Cumulative slip against zeroth-degree moment fraction corrected for noise. 𝑟 is the Pearson’s
correlation coefficient.

111

Figure 1.35. Correlations between residuals of total moment fractions. Residuals are 𝛿𝐹
9
!
=𝐹
9
!
(3454)
−
𝐹
9
!
(6)378)
,𝛼 =0,…,4. 𝑟 is the Pearson’s correlation coefficient. 𝐹
9
!
(6)378)
are estimated from numerical
simulations using the MLE estimations of the SSC parameters 𝑅, 𝜅, and 𝜖.

112
1.9 Discussion and Conclusions
Knowledge of tectonic stress provides essential information about regional tectonic background,
crustal deformation, fault evolution, and earthquake occurrence. Earthquakes and seismic sequences
are complex physical phenomena involving dynamic stress changes, damage generation, and slip.
Furthermore, fault zones show rheological and structural heterogeneities on various scales. Such
heterogeneities strongly influence the initiation, propagation, and resting of seismic ruptures and the
intrinsic properties of seismicity. We investigated how the mechanism complexity observed in
earthquakes and seismic sequences is governed by tectonic stress. Furthermore, we characterized
mechanism complexity and used complexity measures to make inferences about the stress
concentration in fault zones.
In a series of previous papers (Jordan and Juarez, 2019, 2020, 2021), we investigated the concept of
mechanism complexity and moment measures of mechanism complexity. In particular, we
introduced the total moment fractions that measure the moment portioning of a moment tensor
density over a basis set composed of up to six orthogonal moment tensors. In Jordan and Juarez
(2021), we developed the Stress-Strain Characterization (SSC) model that relates the total moment
fractions computed from earthquakes and seismic sequences to the stress concentration parameter
𝜅. Therefore, the SSC model links the short-term seismic response to the long-term tectonic forcing.
The SSC model is consistent with the standard assumptions used in stress inversions: (1) the ambient
stress is constant within the volume of interest, (2) the slip vectors of earthquake ruptures are
aligned with the shear stress resolved on the fault surface (the Wallace-Bott condition), and (3) the
seismic energy is released on surfaces with higher shear tractions (Angelier, 1994; Bott, 1959;
Célérier et al., 2012; Hardebeck, 2015; Wallace, 1951). The Wallace-Bott hypothesis is the basis for
inverting seismic observations for stress. Many studies of stress characterization from seismic
observations have proved the effectiveness of the Wallace-Bott hypothesis (Angelier, 1979; Angelier,
1994; Célérier, 1988; Hardebeck & Michael, 2006; Hauksson & Jones, 2020; Lisle, 2013; McKenzie,
113
1969; Michael, 1984; Pollard et al., 1993; Zoback, Mary Lou, 1992). However, some authors argue
against its validity because it ignores fault heterogeneities, such as roughness and anisotropy that
might affect the characterization of regional stress fields (Kassaras & Kapetanidis, 2018; Lisle, 2013;
Pollard et al., 1993).
We acknowledge that local heterogeneities such as fault roughness, branching, and step-overs might
affect estimations of the regional stress tensor. However, we assume that the background stress is
homogeneous, and heterogeneities and geometrical complexities of fault zones result from a physical
process governed by stress. Earthquakes release seismic energy on planes with high shear traction
and the direction of failure reflects the geometrical complexity of fault zones. The SSC model handles
the complexity of mechanism orientation as follows: elementary source mechanisms are samples of
a stress-aligned moment tensor field (𝐦
8
) modulated by a stress-strain process in which the
direction of failure is described by a probability density function of exponential form
𝑝𝐦
8
|𝜅,𝑠(𝐧 )∝e
:9(𝐧 =)
. 𝑠(𝐧 ) is the shear traction magnitude resolved in a fault plane with normal
vector 𝐧 . One result is that the average mechanism 𝐌

𝟎
is proportional to the stress tensor, 𝚺

(Jordan
& Juarez, 2021; Matsumoto, 2016; Terakawa & Matsu'ura, 2008).
We applied the SSC model to characterize stress fields using focal mechanisms of earthquakes in
Northern and Southern California (Figure 1.19, Figure 1.20, and Figure 1.24). We implemented a
maximum likelihood estimation (MLE) technique that uses observations of earthquake mechanisms
to estimate the stress parameters: the principal stress orientations 𝕽𝚺

=(𝐫
&
,𝐫
+
,𝐫
5
) , the
differential stress ratio 𝑅𝚺

=
n
!
Yn
"
n
!
Yn
%
, and the strain sensitivity factor, 𝜅 that measures the strain
response to stress. The SSC-Lab is the software package that contains the inversion algorithm and
the forward simulation subroutines to compute synthetic moment tensor densities that obey the SSC
probability functions and different parametrizations of errors in mechanisms orientation, such as in
observed datasets. The SSC-Lab MLE technique accounts for errors in focal mechanism estimations
that can be interpreted as complexity. Noise is any non-physical deviation of a mechanism from its
114
expectation. The SSC-Lab models the effect of noise on the total moment fractions, then the MLE
algorithm evaluates the probability that observations are samples from the SSC+Noise distributions.
The solution is the set of SSC parameters that maximize that probability.
The analysis of 25 fault zones in Northern and Southern California (Figures A7-A29 of the Appendix)
shows that the SSC model, on average, successfully characterizes stress fields (Figure 1.27, Figure
1.28, and Figure 1.29). Fault zones such as the Parkfield (SAP) and Coachella Valley (SAC) segments
of the San Andreas fault give large values of strain sensitivity parameter (𝜅 >10), and are
categorized in the low complexity level of our qualitative scale. In comparison, fault zones in the
Mojave Desert, including the ruptures of the Landers and Hector Mine earthquakes of 1992 and 1999,
respectively, typically show 𝜅 <5 and are classified as moderate to high complexity. These results
suggest that, as fault zones evolve with increasing deformation, surfaces of high strain align with the
driving stress. Laboratory experiments and numerical simulations of rock fracturing show that
surfaces of shear deformation form from the interaction and coalescence of shear fractures. As
deformation continues, strain localizes and narrows the fault-zone volume (Ben-Zion, Y. & Sammis,
2003; Faulkner et al., 2006; Healy et al., 2006; Wibberley et al., 2008). Therefore, we expect our
estimations of mechanism complexity to correlate with evolutionary properties of faults such as
cumulative displacement. To test this hypothesis, we compiled geological offsets from 25 fault zones
in California (Table 1.2) and compared them with our estimations of the strain sensitivity factor and
the zeroth-degree total moment fraction.
We find a strong correlation between geological displacements with the strain sensitivity factor (𝑟 =
0.69) and the zeroth-degree moment fraction (𝑟 =0.67; Figure 1.25). The zeroth-degree moment
fraction is related to the complexity factor (𝐹

!
) of JJ2021 as 𝐹

)
=1−𝐹

!
. Estimations suggest that the
San Andreas fault has sustained over 500 km of cumulative offset, and we find that sections such as
Parkfield (SAP), Loma Prieta (SAN), and the Coachella Valley (SAC) are in the low-complexity
category. The seismicity in the Granite Mountain section of the Elsinore fault (EGM) and the Borrego
115
Badlands segment of the San Jacinto fault (SJB) show lower complexity and relatively higher stress
alignment than expected from their lateral displacements. Moreover, the San Jacinto fault is the most
active in Southern California and there have not been any significant earthquakes. Only the 1987
Mw6.6 Superstition Hills and Mw6.2 Elmore Ranch earthquakes to the south of SJB. Similarly, the
Mw7.2 El Mayor-Cucapah earthquake of 2010 occurred south of EGM. The occurrence of those
earthquakes might have increased the stress in the SJB and EGM fault zones. We speculate that high
values of the strain-sensitivity parameter and the low complexity might indicate a preparation phase
for a large earthquake in these fault segments.
Using the seismicity of the Ridgecrest fault zone, which includes the aftershock sequences of the
Mw6.4 and Mw7.2 earthquakes of 2019, we investigate temporal changes in the SSC parameters
(Figure 1.26). We find an increase in the strain sensitivity factor from 2010 to July 2019. Similarly,
there was a change in the differential stress ratio from a transtensional regime to a more lateral shear.
Furthermore, our results suggest that the Mw6.4 earthquake increased the stress concentration
triggering the Mw7.2 event. Immediately after the Mw7.2 earthquake, 𝜅 decreases drastically,
followed by a recovery until the Mw5.5 aftershock of August 19, 2019. We also detect small stress
rotations. After the Mw6.4 shock, there was a counterclockwise rotation of about 8
G
. After the Mw7.2
mainshock, a clockwise rotation of ~5
G
, followed by a slow counterclockwise change over the two
following years.
Our SSC time monitoring technology has the potential to detect changes in stress-strain patterns that
might indicate preparatory phases of large earthquakes. Furthermore, complementing observations
from studies investigating seismicity localization and rock damage coalescence in fault zones before
large earthquakes (Ben-Zion, Yehuda & Zaliapin, 2020; Cheng & Ben-Zion, 2020; Lyakhovsky & Ben-
Zion, 2009).
Analysis of the residuals between observed and modeled total moment fractions shows that there
are properties of the seismicity that the SSC+Noise model does not properly describe. For example,
116
Figure 1.20 shows that the model overpredicts the first-degree total moment fraction in the Parkfield
segment. On the other hand, it underpredicts 𝐹

&
for the Ridgecrest dataset (Figure 1.22). Close
inspection of the results in Figure 1.29 reveals that fault zones with large cumulative displacements,
such as San Andreas, Hayward, and Calaveras faults, typically have a smaller than predicted 𝐹

&
.
Furthermore, faults with smaller offsets and recent large earthquakes, such as Landers, Hector Mine,
and Ridgecrest, show larger than expected 𝐹

&
. These observations are consistent among faults that
show predominantly strike-slip motion and where 𝐹

&
measures variability along the strike direction.
Therefore, the differences between model and data 𝐹

&
shows that there are preferred orientations of
faults that result from tectonic anisotropy or rheological properties of the rock that reflects the long-
term evolution of fault zones, such as biomaterial interfaces.
In the SSC model, failure in a given direction only depends on the shear traction magnitude. This
assumption seems valid for fault zones with small cumulative offsets, seismicity distributed over
large volumes, and without recent big earthquakes or aftershock sequences in the catalog. Examples
are the San Jacinto, Elsinore, and Garlock faults and the San Bernardino Mountains section of the San
Andreas fault, which show a good fit between observed and modeled distributions.
Shear deformation promotes the creation of fractures that interact and grow, creating localized zones
of deformation. As deformation increases, fault structural heterogeneities are smoothed, and in
general, the rheology of the rock changes, creating structures aligned with the direction of tectonic
motion. Consequently, elementary earthquake fault planes are aligned with preferred directions,
reducing the observed variability out-of-plane. Therefore, fault zones such as Parkfield, Hayward,
Calaveras, and the Coachella Valley, which have large offsets and are well localized, show little out-
of-plane variability. On the other extreme, we find that faults with smaller offsets and big earthquakes
followed by aftershock sequences in the recorded catalogs might perturb the background stress field
and consequently show more out-of-plane variability than predicted. Examples are the faults that
117
ruptured during the Landers and Hector Mine earthquakes. However, an exception is the Loma Prieta
segment of the San Andreas fault that ruptured during the Mw6.9 Loma Prieta earthquake of 1996.
We conclude that the SSC model is a good baseline for studying the stress field from earthquake focal
mechanisms, but a modification is needed to model the interfault variability. Therefore, we proposed
a new anisotropic SSC probability density function 𝑝(𝐧 ,𝜅′,𝜖)∝exp[𝜅′cos
+
𝜃
)
(𝐧 )+𝜖cos
+
𝜃
&
(𝐧 )],
that stills depends exponentially on the strain sensitivity factor and the shear traction magnitude,
now squared, and has extra parameter 𝜖, the fault orientation concentration parameter. 𝜖 <0
reduces the contribution of cos
+
𝜃
&
, and consequently 𝐹

&
, by concentrating the probability along the
planes spanned by the vectors (𝐧
)
,𝐫
+
) and (𝐬
)
,𝐫
+
). 𝐧
)
and 𝐬
)
are the normal and slip directions of the
CMT mechanism (Figure 1.31). 𝜖 >0, increases the probability in the plane (𝐧
)
,𝐬
)
).
Results from the new model show that strain sensitivity still correlates with cumulative offsets
(Figure 1.34), suggesting that the long-term tectonic forcing is the primary controller of the short-
term seismic response. Furthermore, the fault orientation concentration also increases with
cumulative offsets. Hence, supporting the hypothesis that as fault zones evolve with anelastic
deformation, heterogeneities tend to concentrate close to the fault core and fault planes align with
the tectonic stress (Brodsky et al., 2011; Powers & Jordan, 2010; Wesnousky, 1988). Moreover, other
factors, such as the misalignment of fault planes with plate motion and the rheology of fault zones,
such as biomaterial interfaces, fluid content, and heat regime, also contribute to mechanism
complexity (Lyakhovsky & Ben-Zion, 2009; Wechsler et al., 2010).
Spatiotemporal analysis of seismicity patterns in well-developed faults have found that earthquakes
typically occur in horizontal streaks aligned with the fault slip direction. However, from hypocenter
distributions, it is difficult to infer weather earthquakes occur in streaks of weak brittle material or
the stress is concentrated in these streaks (Waldhauser et al., 1999). Our results suggest that in fault
zones such as Parkfield, Hayward and Calaveras, where large negative values of 𝜖 are observed,
seismicity is controlled by failure in weak material. In contrast, fault zones such as the Coachella and
118
Imperial Valleys and Loma Prieta moderate values of 𝜖, but large values of 𝜅 are observed, suggesting
that failure is controlled by stress concentration.

Acknowledgements
Earthquake catalog of Northern California for this study were accessed through the Northern
California Earthquake Data Center (NCEDC), doi:10.7932/NCEDC. The SCEDC and SCSN are funded
through U.S. Geological Survey Grant G20AP00037, and the Southern California Earthquake Center,
which is funded by NSF Cooperative Agreement EAR-0529922 and USGS Cooperative Agreement
07HQAG0008.

119
Chapter 2: Effects of Shallow Velocity Perturbations on Three-Dimensional Propagation
of Seismic Waves
Alan Juárez Zúñiga
1
and Yehuda Ben-Zion
1,2
1
Department of Earth Sciences, University of Southern California, Los Angeles, CA
2
Southern California Earthquake Center, University of Southern California, Los Angeles, CA,

Abstract
We perform 3D simulations of seismic wavefields to clarify effects of strong reductions of shallow
velocities on long-period seismic waves. The simulations use a reference community velocity model
of Southern California and a modified version with strong velocity reductions in the top 500 m of the
Los Angeles basin. Differences between wavefields generated by ten earthquakes in the reference
and perturbed models are analyzed. Velocity changes are estimated by measuring relative time shifts
between reference and perturbed seismograms using wavelet cross-correlation spectra. The results
indicate that strong localized temporal velocity drops near the surface, such as those observed during
strong ground motions, may generate regional perturbations of wavefields at periods up to 20 s.
These perturbations may be misinterpreted as generated by temporal changes at seismogenic
depths. The results also have important implications for waveform tomography studies.

2.1 Introduction
Temporal variations of seismic velocities have been observed in association with diverse settings
including earthquake ruptures (Hobiger et al., 2016; Peng & Ben-Zion, 2006; Wegler & Sens-
Schönfelder, 2007), volcanic activity (Brenguier, Florent et al., 2016; Sens-Schönfelder & Wegler,
2006), seasonal changes of precipitation and temperature (Hillers et al., 2015; Meier et al., 2010;
Wang et al., 2017), and fluid injection or extraction (Taira et al., 2018). Such processes produce
120
changes of strain and stress that can modify the elastic properties of soils and rocks at different
locations and depth sections. In particular, rock properties are expected to change significantly
within earthquake rupture zones (Aben et al., 2019; Ben-Zion, Yehuda & Ampuero, 2009; Lockner et
al., 1991; Lyakhovsky et al., 2016), as well as in broad shallow regions during the passage of strong
ground motions generated by earthquakes (Bonilla et al., 2019; Brenguier, F. et al., 2014; Nakata &
Snieder, 2011; Viens et al., 2018). The simultaneous occurrence of velocity changes during large
earthquakes within the source volumes and regional-scale surficial layers, coupled with the faster
recovery of properties with increasing normal stress and hence depth (Brantut, 2015; Dieterich &
Smith, 2009; Pei et al., 2019), present significant challenges for resolving spatiotemporal changes of
seismic properties at depth.
Seismic studies of temporal velocity changes usually assume that the locations of the velocity changes
and sources of seismic waves are fixed. If the velocity perturbation is homogeneously distributed, the
relative velocity variation (𝑑𝑣 𝑣 ⁄ ) can be estimated by measuring relative travel time differences
(𝑑𝑡 𝑡 ⁄ ) between a perturbed wave and a reference wave (Poupinet et al., 1984). Measurements can
be made in seismograms generated by repeating earthquakes (Peng & Ben-Zion, 2006; Schaff &
Beroza, 2004), and waveforms from ambient-noise Green functions (Planès et al., 2020; Rivet et al.,
2011). Other techniques use spectral ratios of earthquake waveforms at reference and target stations
(Nakata & Snieder, 2011; Sawazaki et al., 2006; Wu et al., 2009), and time delays from
autocorrelations of moving windows within earthquake waveforms (Bonilla et al., 2019; Qin et al.,
2020). Methods such as time-dependent tomography complement the measurements and help to
infer the spatiotemporal locations of the velocity changes (Cheng, X. et al., 2010; Julian, B. R. &
Foulger, 2010; Koulakov et al., 2013; Pei et al., 2019).
The different monitoring approaches involve collecting and processing seismic data at different time
scales, from seconds to weeks after the temporal changes of seismic properties, and various locations
from near-fault to regional stations. Measurements of time delays are, therefore, sensitive to different
121
time and space scales and might result from a combination of different processes (Wang et al., 2017;
Yang, C. et al., 2018). The depths of temporal velocity changes are usually inferred from the depth
sensitivity of the waves used in the analysis, since long-period waves are sensitive to deeper
structural sections than short period waves. Consequently, analyses of time delays based on long-
period surface or coda waves are usually interpreted in terms of velocity changes at seismogenic
depths (Froment et al., 2013; Obermann et al., 2019).
While long-period waves are sensitive to deep structures, it is unclear whether significant velocity
changes at the subsurface, like those observed during strong ground motions, could explain the small
changes estimated using long period waves. Yang et al. (2019) performed numerical simulations
using several one-dimensional (1D) velocity models to estimate the effects of velocity changes,
attenuation, and velocity-stress relations on surface-wave dispersion curves. They found that
significant changes of shallow seismic velocities can modify the dispersion curves of surface waves
at periods up to 20 s. Obermann et al. (2016; 2019) analyzed the sensitivity to localized velocity
changes using scattering theory and three-dimensional (3D) wavefield simulations in a medium with
uniform scattering heterogeneities. They concluded that the combined sensitivity of body, surface,
and coda waves is robust and decays with the lapse time. Taborda and Bielak (2014) analyzed the
sensitivity of ground motions at different frequencies to velocity models of Southern California with
different resolutions including a geotechnical layer. They performed simulations of wavefields at
frequencies up to 4 Hz, and using goodness-of-fit criteria they found considerable differences in
wavefields generated with and without the geotechnical layer even at frequencies as low as 0.1-0.25
Hz.
The studies of Taborda and Bielak (2014), Obermann et al. (2016), and Yang et al. (2019) clarified
various aspects of the seismic observations associated with changes in shallow seismic properties.
However, Taborda and Bielak (2014) focused on evaluating velocity models with different
resolutions, rather than analyzing the impact of shallow velocity perturbations on inferred temporal
122
changes of velocities, while Obermann et al. (2016) and Yang et al. (2019) used simplified models
that do not account for effects associated with 3D wave propagation in realistic structures. In the
present paper, we seek to complement these studies by clarifying effects of strong localized
reductions of shallow seismic velocities on measurements of time delays using realistic regional
wave propagation. Toward this end, we design a synthetic experiment to investigate effects of
shallow velocity changes, such as expected during strong ground motions generated by major
earthquakes, on regional wavefields. We perform simulations of ground motion generated by
earthquakes using a 3D community velocity model (CVM) of Southern California, and estimate the
effects of prescribed velocity changes on time delays and amplitudes measured on long-period
seismic waves.
We follow the approach of Mao et al. (2020) to measure time shifts between original and perturbed
wavefields in the cross-spectra of pairs of synthetic seismograms at different locations, and to
estimate from the measurements relative velocity perturbations. Spectral amplitude anomalies in the
simulation domain are also measured. The results indicate that shallow velocity drop localized in the
Los Angeles (LA) basin can produce variations in the amplitude and phase spectra of seismic waves
in the surrounding region at periods up to 20 s. The simulations and analyses have important
implications for interpretations of the structures responsible for time delays and amplitude
variations in seismic wavefields. The results are also relevant for waveform tomography inversions
and evaluation of velocity models. The synthetic calculations highlight the usefulness of data
recorded by dense arrays for estimating velocity changes generated by earthquakes.

123
2.2 Material and Methods
2.2.1 Numerical Modeling
The geological complexity of Southern California provides a good natural laboratory for testing
methods to detect temporal changes of seismic velocities produced by ruptures and strong ground
motions during large earthquakes. This region has a highly heterogeneous crust with significant
lateral and vertical variations of seismic velocities. The LA basin is of particular interest because it is
formed by low-velocity sediments, which can be subject to strong ground motions during moderate
and large earthquake. We perform numerical simulations of wave propagation from ten randomly
located earthquakes around the LA basin, using the Community Velocity Model (CVM) S4.26 of
Southern California (Lee et al., 2014). The CVM-S4.26 was constructed from full-waveform
tomography using data of several earthquakes and noise cross-correlations in the area.
Figure 2.1 illustrates the CVM-S4.26 at the surface of the simulation region, along with locations and
source mechanisms of the earthquakes simulated. The model was retrieved from the UCVM Software
(Small et al., 2017) on a grid with a resolution of 1 km. The earthquakes are modeled as point sources
with moment magnitude Mw 5, and their locations and source mechanism are assigned randomly
(Table 2.1). The source time function is a triangular pulse with 1 s duration. For each earthquake, we
compute a reference wavefield using the CVM-S4.26 model, and a perturbed wavefield in a modified
velocity model. The modified model has reduced shear wave velocity in the LA basin, with a 30%
reduction at the surface that is tapered linearly to zero reduction at 500 m depth (
Figure 2.1b). Since we focus on effects of superficial velocity changes, we do not model velocity
perturbations at seismogenic depths.
The simulations use the Hercules Toolchain (Bielak et al., 2005; Tu et al., Nov 2006) at frequencies
up to 0.25 Hz with 12 points per wavelength. Wavefields are calculated for 100 s in a volume of 200
km long, 120 km wide, and 60 km depth without topography. Hercules has a spatial database
manager that linearly interpolates the model grid to the simulation mesh. Wavefields for each
124
simulation are saved on a grid of 2 km spacing at the surface and with 2 Hz sampling. The grid
resolution in our simulation is much larger than the station distributions in Southern California, but
the interest is to analyze an idealized case that may be achieved with dense array data (Ben-Zion,
Yehuda et al., 2015; Bielak et al., 2005; Lin, F. et al., 2013; Tu et al., Nov 2006). The analysis aims to
provide a basis for interpretation of field observations, accounting for the dynamic response of
shallow soft sediments.

Table 2.1. Earthquake locations and source parameters.
NO. LONGITUDE LATITUDE DEPTH (KM) STRIKE DIP RAKE
1 -118.48 34.09 19.66 111 13 -132
2 -117.52 34.26 6.03 -38 59 -118
3 -117.80 34.36 14.02 187 47 -39
4 -117.50 33.31 13.33 -44 88 119
5 -118.94 34.72 10.78 -33 58 109
6 -117.66 33.61 13.96 129 72 -158
7 -117.46 33.68 13.33 -24 41 -36
8 -117.89 34.02 3.56 236 39 10
9 -117.35 34.16 2.56 235 74 -30
10 -118.18 34.02 19.98 201 8 56

Figure 2.1. Community Velocity Model (CVM-S4.26) of Southern California and earthquake locations in the
simulation region. (a) Shear wave velocity (Vs) at the surface of the simulation region. The source locations
and mechanisms are indicated by the focal mechanisms (black-compression, white-extension). The shear-
wave velocity is reduced by 30% on the surface within the Los Angeles basin. The star in the LA basin
indicates the location of the profiles in (b). Gray-thin lines are faults in Southern California. (b) Velocity and
density profiles on the top 5 km of the crust on the LA basin region. The shear wave velocity is reduced by
30% on the top 0.5 km, The P-wave speed and density are not modified.
125
2.2.2 Spectral Lag-Time Measurements
In-situ observations of velocity variations generated by earthquakes indicate that coseismic velocity
changes involve rapid velocity drop followed by a recovery proportional to the logarithm of time
(Bonilla et al., 2019; Viens et al., 2018; Wu et al., 2009). For simplicity, we adopt the common
assumption of velocity variations (𝑑𝑣 𝑣 ⁄ ) that are constant over the recording time. In this case, the
relative travel time shift between the perturbed and reference waveform (𝛿𝑡 𝑡 ⁄ ) is also constant, and
𝑑𝑣
𝑣
=−
𝛿𝑡
𝑡
. (1)
Seismograms have many phases with different amplitudes that interfere with each other, and
dispersion relations that are primarily affected by velocity variations in the propagation medium.
These factors complicate simple time delay measurements with conventional techniques such as
windowed cross-correlation or the stretching method. However, the time-frequency representation
of seismograms is useful to identify seismic phases even in the presence of noise (Castellanos et al.,
2020; Parolai, 2009), to determine their arrival times and to analyze their dispersion relations. Here
we follow the approach of Mao et al. (2020) to estimate the relative velocity perturbations by
measuring relative time delays in the wavelet time-frequency spectra. The wavelet time-frequency
representation of a time-dependent function ℎ(𝑡) is a function 𝐻(𝜏,𝜔)=𝑊(ℎ(𝑡) ) that allow us to
determine the dominant modes and how they vary with time. The wavelet cross-spectrum of the
functions ℎ(𝑡) and 𝑔(𝑡) is
𝐶
,
(𝜏,𝜔)=𝐻(𝜏,𝜔)𝐺
̅(𝜏,𝜔), (2)
where 𝐺
̅ is the complex conjugate of 𝐺. The phase cross-spectrum, 𝜑
,
=arg(𝐶
,
), measures the
phase difference between two signals, and hence the time shift from the phase cross-spectrum is
𝛿𝑡(𝜔,𝑡)=
𝜑
,
(𝜔,𝑡)
𝜔
. (3)
Equations 1 and 3 allow estimating frequency-dependent velocity variations by analyzing the time-
frequency cross-correlation spectrum between two signals. For the measurements, we utilize the
126
wavelet spectrum calculated from the S-transform, which uses a Gaussian modulated by a phase
factor as a mother wavelet (Mansinha et al., 1997; Stockwell et al., 1996; Ventosa et al., 2008). Figure
2.2 illustrates the steps for estimating relative velocity variations from phase cross-spectrum. We
utilize the vertical component of velocity waveforms filtered at periods between 4-40 s with a 4
th
-
order causal Butterworth filter. Figure 2.2a shows two synthetic seismograms computed with the
reference and modified velocity models at receiver location shown in Figure 4 The waveform
differences in this pair of seismograms are noticeable at times after about 75 s. Time-shifts are
evident in the coda and there are also differences in the amplitudes, e.g., for phases arriving at about
90 s and 115 s. Figure 2.2b displays the S-transform phase cross-spectrum of the pair of seismograms,
and Figure 2.2c shows the time-delays calculated from the cross-spectrum in Figure 2.2b using
equation 3. We corrected for cycle skipping in the phase cross-spectrum by unwrapping the time
delays at each frequency band as shown in Figure 2.3 (Mao et al., 2020). The dashed lines in Figure
2.2b and c indicate the cone of influence (Torrence & Compo, 1998). The region outside the cone of
influence can have large amplitudes due to the nonperiodic nature of the signals, but since it has no
influence on the estimates it is tapered. Figure 2.2d shows an example of the corresponding time shift
at the 10 s period obtained from the time-frequency time shifts in Figure 2.2c (solid black line).
An important observation discernible even in the pair of seismograms in Figure 2.2a is that there are
time shifts at periods up to 30 s. The time delays should increase linearly with time, but due to the
complexity of wave propagation and the cycle skipping effect, the phase cross-spectrum is
complicated (Figure 2.2c). At a given period, the cycle skipping produces a sudden change of sign in
the relative time delay. Hence, to obtain a reliable measurement, we assume that the time delays
increase approximately linearly and measure it before the cycle skipping effect (Figure 2.2d). We
discard values of relative time lag after the first cycle skipping, which usually includes measurements
in the coda. The obtained average amplitude of the relative time delays is stable except near the
jumps due to the cycle skipping.
127
To validate the technique for estimating velocity perturbations, we perform a one-dimensional
checkerboard-type test. The test consists of modeling signals with known time-shifts from a synthetic
velocity perturbation (
𝛿𝑣
𝑣
'
) model and performing the measurements using the algorithm. We use
a red noise signal (Hristo Zhivomirov, 2018) as the reference seismogram and construct perturbed
seismograms by assuming
𝛿𝑣
𝑣
'
=0.1sin(10𝜔), i.e., the velocity perturbation is constant in time and
oscillates in frequency with a maximum absolute amplitude of 10%. The perturbed seismograms are
computed by stretching and compressing the reference seismogram at different frequencies
according to the expected time-shifts. We apply the measurement algorithm to the reference and
perturbed seismograms and reconstruct the theoretical time delays (Figure 2.3). The results
demonstrate that the algorithm provides accurate and robust measurements of time-frequency
dependent time-shifts, and thus can be used to estimate relative velocity perturbations.

Figure 2.2. Time-lag measurements using the phase cross-correlation spectrum. (a) Synthetic seismograms
computed with the reference velocity model and the modified model. (b) Wavelet phase cross-spectrum
𝜑
B,C
(𝜔,𝑡). The dashed line is the cone of influence. (c) The relative time-shift calculated from the cross-
spectrum in (b). Red and blue colors indicate positive and negative time lags, respectively. We taper the time-
frequency dependent time shifts outside the cone of influence. The solid line indicates the profile at 10 s
period used in (d). (d) Timeshift at 10 s period. The orange dots show the sections that the algorithm
considers for the averaged measurement.

128

Figure 2.3. Validation test of the measuring technique. (a) Time-frequency dependent velocity change model.
The
𝛿𝑣
𝑣
V model is constant in time and in frequency varies as 0.1sin(10𝜔). (b) Theoretical phase cross-
spectrum. (c) Expected time-frequency dependent time-shifts the synthetic model of
𝛿𝑣
𝑣
V . (d) Reference and
perturbed signals. Both signals have the power spectra of red noise. (e) Calculated phase cross-spectrum. (f)
Reconstructed time-shifts.

2.3 Effects of Shallow Velocity Perturbations on Seismic Waves
In the previous section we show measurements of relative time lags only for a pair of seismograms.
Although the results demonstrate that long-period seismic waves are sensitive to temporal velocity
changes in the shallow crust, they are not sufficient to derive conclusions about 3D effects of localized
velocity perturbations on regional scale measurements. To obtain more evidence, we calculate
relative time delays and amplitude reduction times (Chen et al., 2010) of the wavefields on a grid of
2 km at the model surface. The amplitude reduction times are
𝛿𝐴=−
𝑙𝑜𝑔(𝐴
&
𝐴
+
⁄ )
𝜔
, (4)
where 𝐴
&
and 𝐴
+
are the amplitudes of the reference and perturbed waveforms at a given angular
frequency 𝜔.
Figure 2.4-a, b, and c show the spatial distribution of relative velocity perturbations estimated at
different periods for one pair of simulations, and Figure 2.4-d, e, and f show the spatial distribution
129
of the frequency-dependent amplitude reductions times. These results indicate that large relative
velocity perturbations can be detected from measuring relative time delays of seismic waves at
periods of 5, 10, and 20 s. The most considerable velocity variations are estimated at 5 s period, which
corresponds to seismic waves of short wavelength (~3.5 km) that are sensitive to small scale
variations. Although the velocity changes of seismic properties are limited to the top 500 m of the LA
basin, we observe relative time lags across the entire simulation domain, leading to small estimated
velocity changes everywhere in the simulation region. These large measurements in the entire
simulation domain are explained by waveform dissimilarities found mainly in the coda. While the
waves between the source and basin are unchanged, once they interact with the basin they travel to
the whole domain with signatures of the velocity perturbation. The signatures are more evident in
the coda, since it contains reverberations of the basin and the recording time is long enough to be
observed everywhere in the simulation domain. Similar observations are discussed in the waveform
tomography of the LA area by Tape et al. (2010).
The spatial distribution of the amplitude reduction times at the three examined frequencies is
complex. The shallow velocity perturbation imposed in the model has a strong signature in seismic
waves of periods up to 20 s. The largest amplitude perturbations are measured in the LA basin area,
and 𝛿𝐴~1 𝑠 observed in the basin indicate amplifications with factors of two; however, there are
some anomalies everywhere in the simulation region. Similar results in seismic monitoring studies,
aiming to detect locations of velocity changes, can be misinterpreted to imply changes over larger
areas. Furthermore, waveform tomography studies use frequency-dependent phase delay times and
amplitude reduction times at low frequencies to model seismic structures. Our results show that
strong shallow localized perturbations of seismic velocities can have comparable effects to reduced
velocities everywhere in the model. Thus, inversions of low-frequency data could be biased if they do
not account for shallow localized low velocities.
130
The calculated velocity perturbations in the LA basin are around 10%, which is smaller than the
changes imposed in the shallowest crust. These results are consistent with the estimations of Yang
et al. (2018) using 1D models and measuring travel time shifts of Rayleigh waves and phase velocity
perturbations in dispersion curves. The maximum measurable time shift depends on the period of
the seismic waves; the measurements of time shift are smaller than half the used period due to the
cycle skipping effect. Bootstrapping, smoothing filters, and weighting factors can account for cycle
skipping effects (Mao et al., 2020). However, developing more sophisticated techniques is outside
the scope of this study. Instead, we simply account for cycle skipping effects by performing the
measurements early in the seismograms before the phase jumps.
As pointed out by Obermann et al. (2016, 2018), the amplitude of the measured velocity variations
decays with distance from the region sustaining the velocity perturbations. Therefore, the most
significant velocity drops should be measured at points located within or close to the area sustaining
the changes, such as fault damaged zones or the surficial layers subject to significant strains. Our
numerical experiment demonstrates that velocity changes are estimated with similar amplitudes
outside the LA basin and do not necessarily decay with distance. This is because waves propagating
through the zone sustaining the velocity drop gain a time shift, and then travel outside that region
with a relative phase shift that it is constant at far distances (assuming there are no additional
velocity changes along the path).
The complexity of wave propagation in heterogeneous media produce tangled waveforms and simple
phase shifts and amplification factors are unlikely to apply. Figure 2.4 shows that considerable
velocity changes may be estimated in the westernmost part of the volume. Apparent velocity changes
are observed even at a period of 20 s. After the direct waves pass through the LA basin, they
propagate to the south-east region with a large time shift. Waves propagating in different directions
are likely to have different resolutions due to the interaction of wave propagation effects and the
131
velocity structure. These effects produce interference of multiple arrivals that are observed at longer
periods but with smaller amplitude.
The resolution of the analysis techniques and uncertainties could lead to erroneous inferences that
velocity perturbations occur at seismogenic depths even if the actual variations of seismic properties
are limited to a small volume near the surface. Having this in mind, and to better quantify the regional
effects of localized-shallow velocity drop to seismic waves, we account for directional propagation
effects by averaging the estimated velocity perturbations in ten pairs of simulations associated with
ten different earthquakes.
Figure 2.5 shows the average relative velocity perturbations and amplitude changes in the simulation
region at 5, 10, and 20 s periods. The most significant relative velocity changes are measured as
expected in the LA basin and around it. The estimated velocity changes gradually decay with distance,
but perturbations are observed again in the entire region. Significant relative velocity drops of about
1% are detected in the 20 s period. The maximum inferred relative velocity perturbation in the
analyzed cases is smaller than the 30% prescribed value at the subsurface. Notice that the averaging
over all the simulations has a smoothing effect on the highly heterogeneous spatial distribution of
measurements. The amplitude reductions times show a more complex structure, and the averaging
does not smooth the spatial variations of the measurements, indicating that the amplitude of the
wavefields vary strongly over short distances. The amplitude reduction times are measured on the
spectrum of the seismograms in the vertical component, and they show the overall changes in the
recording time. The estimation technique could be improved by measuring amplitude anomalies for
particular arrivals only.
132

Figure 2.4. Period-dependent distributions of velocity perturbations and spectral amplitude reduction times.
The top row shows the estimated relative velocity perturbation (
𝛿𝑣
𝑣
V ) at (a) 5 s, (b) 10 s, and (c) 20 s. The
bottom row shows the measured amplitude reduction times (−
8)B(D
#
D
!
⁄ )
F
) at (a) 5 s, (b) 10 s, and (c) 20 s. In
all the subplots, the red triangle indicates the location of the record in Figure 2.2. The thick red line is the
border of the LA basin, where the velocity was reduced by 30%. The focal mechanism indicates the
earthquake location and source mechanisms.

133

Figure 2.5. Average estimations of relative velocity perturbations and amplitude reduction times from the ten
pairs of simulations. The top row is the average velocity perturbations measured at (a) 5s, (b) 10 s, and (c) 20
s. The bottom row is the average amplitude anomalies at (d) 5 s, (c) 10 s, and (f) 20 s. The thick red line is the
border of the LA basin, where the velocity was reduced by 30%. The focal mechanism indicates the
earthquake location and source mechanisms.

2.4 Discussion and Conclusions
We perform 3D numerical simulations of wave propagation to investigate effects of a shallow
localized velocity drop on the propagation of seismic waves in a complex velocity model with realistic
lateral and depth variations of properties. We model waves generated by moderate earthquakes with
random source mechanisms and locations within the CVM S4.26 of Southern California (
Figure 2.1 and Figure 2.4). The main goal is to clarify the impact of shallow localized velocity
reductions in time on inferences about the depth extent, location, and amplitude of the region
sustaining the changes from observations associated with long-period seismic waves (Figure 2.3).
Given the focus of the study, we limit the simulations to a single case of assumed variations of material
properties: a localized strong velocity perturbation in the top 500 m of the sedimentary structure in
the LA basin. Similar conditions are expected during strong ground motions of earthquakes in
134
Southern California (Qin et al., 2020). We estimate the velocity changes by measuring time shifts on
the cross-spectrum of synthetic seismograms at different locations (Figure 2.2). As described in detail
by Mao et al. (2020), this technique has higher time-frequency resolution than other traditional
methods. It allows measuring time-frequency dependent time shifts efficiently, and it accounts for
the interference of phases with similar arrival times and different dispersions relations, along with
cycle skipping.
Although the assumed velocity drop is localized and shallow, changes in the seismic wavefield occur
at most locations in the simulation region at periods up to 20 s. The amplitudes of the inferred
velocity changes are considerably smaller than the maximum 30% prescribed change at the
subsurface. However, the estimated values are similar to and larger than the very small values (e.g.
0.1% and less) found in some observational studies of coseismic velocity changes following moderate
and large earthquakes (Brenguier, F. et al., 2008; Brenguier et al., 2014; Cheng et al., 2010; Froment
et al., 2013). Our numerical simulations are not fully realistic in some respects, e.g., lack of
attenuation, small-scale heterogeneities and source-related complexities. Nevertheless, they
demonstrate that localized velocity reduction in the very shallow crust can account for observed
coseismic variations of wavefields at periods up to 20 s. The results imply that it is essential to
consider the complexity of wave propagation in heterogeneous structures and uncertainties of the
estimated values for proper interpretation of temporal variations of travel time shifts in seismic data.
Coseismic velocity changes are expected to be significant in earthquake source volumes (Gupta,
1973; Hamiel et al., 2009; Lockner et al., 1991), but are followed by rapid recoveries at depth because
of the high confining pressure (Brantut, 2015; Dieterich & Kilgore, 1996; Johnson & Jia, 2005).
Significant changes are also expected to occur at shallow depths; in soft sediments and soils
sustaining large dynamic strains during strong ground motion (Bonilla et al., 2019; Nakata & Snieder,
2011; Qin et al., 2020; Sawazaki et al., 2006), and in the top portions of damaged fault zone rocks
(Peng & Ben-Zion, 2006; Rubinstein & Beroza, 2005; Wu et al., 2009). Since the recovery process at
135
shallow depth is considerably slower than in the seismogenic zone, measurements done at the
surface after the occurrence of earthquakes are likely to be dominated by changes of seismic
velocities at shallow depth.
Simulations using 3D velocity models of the type done in this study can also help to understand many
of the difficulties faced when developing tomographic models from regional seismic observations at
the surface. Our results demonstrate that the time delay measurements and the amplitude anomalies
depend on the size and shape of the region having velocity changes, the amplitude of the
perturbation, other structural heterogeneities in the model, the direction of wave propagation, and
the distance from the region having the velocity changes. In addition to analyses of coseismic velocity
perturbations, these compounded factors also have important implications for derivations of seismic
velocity models from waveform inversions and simulations of ground motion for earthquake hazard
analysis.
Waveform tomography studies use 3D simulations of ground motion that require large amounts of
computational resources for high-frequency simulations (Lee et al., 2014; Tape et al., 2010). For this
reason, waveform inversions limit measurements to a low-frequency band, and the constructed
velocity models do not consider the strong velocity reductions in the subsurface. A common
assumption is that long period waves are not sensitive to the very shallow structure, and models
usually have fixed minimum velocities at values considerably larger than the actual velocities in the
shallow layers. Consequently, measurements of differences between simulated and recorded
wavefields made at long periods can be mapped into apparent structures and velocities somewhere
else in the model.
It might be possible to resolve localized changes of velocities at different depth sections by
constructing 3D kernels for different phases in the coda of seismograms or seismic noise (Liu, 2020;
Zhao, L. et al., 2005), and estimating the sensitivity to local changes using scattering theory. These
techniques, combined with statistical models that estimate the sensitivity of body waves, surface
136
waves and coda (Kanu & Snieder, 2015; Obermann et al., 2019), could improve the accuracy of
locating regions sustaining velocity changes. Such methods have the advantage of intrinsically
accounting for the complexity of wave propagation in heterogeneous velocity structures and could
be useful for estimating uncertainties of the observations. Double beam-forming techniques that
track seismic waves following specific paths (Brenguier, F. et al., 2019; Brenguier et al., 2016) also
have a high potential of resolving temporal changes of properties at different locations and depth
sections.
Additional simulations with different cases of velocity changes can be used to develop strategies for
improved estimates of temporal changes of velocities different depths, and for resolving detailed
structural units at different depth sections. This is relatively easy to do with simple representative
velocity models. However, 3D simulations with realistic velocity models such as the CVM-S4.26 for
Southern California are computationally very challenging and require systematic analysis of seismic
data at different scales. A combination of additional simulations of the type done here and further
analysis developments might allow resolving the space-time distributions of velocity changes that
occur at different portions of the crust.

Data and Resources
No observational data is used in this study. The synthetic data used for this study are available
through Open Science Framework at
https://osf.io/n24qk/?view_only=4481a63733964d4ab6cbb38f1821fb3d. The CVM-S4.26 was
retrieved with the UCVM v19.4.0 software (Small et al., 2017) available at
https://scec.usc.edu/scecpedia/UCVM#Current_UCVM_Software_Releases.

137
Acknowledgments
We thank Leonardo Ramirez-Guzman for providing the simulation code and for his support with the
installation and troubleshooting. The paper benefitted significantly from useful comments by Robert
Graves and Kyle Withers. The study was supported by the Southern California Earthquake Center
(based on NSF Cooperative Agreement EAR-1600087 and USGS Cooperative Agreement
G17AC00047) and the U.S. Department of Energy (Award #DE-SC0016520).

138
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156
Appendix A. Supplemental Material to Chapter 1

Figure A1. Shear traction 𝑠(𝐧 6) for selected values of 𝑅. 𝑠(𝐧 6) is plotted on the sphere that supports the
orientation vector 𝐧 6. The projection is orthogonal to 𝐫 4
'
⁠.

Figure A2. Stress-Oriented Representation basis set h𝐌
)
!
i from the concatenated catalogs of Hauksson, Cheng
and Lin.

Figure A3. Projections and angles between the focal mechanism of Hauksson and Cheng. (a) Empirical
distribution of the angular distance 𝜂
, ,.
=arccos (𝐦 6
,
,𝐦 6
.
) between the focal mechanisms of Hauksson and
Cheng. 𝜇 and 𝜎 are the mean and standard deviations, respectively. (b)-(f) show the projections of Hauksson
157
and Cheng mechanisms onto the SOR basis set: 𝑥
!
(, )
=𝐌
)
!
:𝐦 6
,
vs. 𝑥
!
(.)
=𝐌
)
!
:𝐦 6
.
. The solid stair-like lines are
the scaled histograms and 𝑟 is the Pearson correlation coefficient.

Figure A4. Projections and angles between the focal mechanism of Hauksson and Lin. (a) Empirical
distribution of the angular distance 𝜂
, ,G
=arccos (𝐦 6
,
,𝐦 6
G
) between the focal mechanisms of Hauksson and
Lin. 𝜇 and 𝜎 are the mean and standard deviations, respectively. (b)-(f) show the projections of Hauksson and
Lin mechanisms onto the SOR basis set: 𝑥
!
(, )
=𝐌
)
!
:𝐦 6
,
vs. 𝑥
!
(G)
=𝐌
)
!
:𝐦 6
G
. The solid stair-like lines are the
scaled histograms and 𝑟 is the Pearson correlation coefficient.

158

Figure A5. Projections and angles between the focal mechanism of Lin and Cheng. (a) Empirical distribution
of the angular distance 𝜂
G,.
=arccos (𝐦 6
G
,𝐦 6
.
) between the focal mechanisms of Lin and Cheng. 𝜇 and 𝜎 are
the mean and standard deviations, respectively. (b)-(f) show the projections of Lin and Cheng mechanisms
onto the SOR basis set: 𝑥
!
(G)
=𝐌
)
!
:𝐦 6
G
vs. 𝑥
!
(.)
=𝐌
)
!
:𝐦 6
.
. The solid stair-like lines are the scaled histograms
and 𝑟 is the Pearson correlation coefficient.

159

Figure A6. Distributions of the projections 𝑥
!
(, )
=𝐌
)
!
:𝐦 6
1
(, )
vs. 𝑥
H
(.)
=𝐌
)
H
:𝐦 6
1
(, )
for the Hauksson dataset. The
solid stair-like lines are the scaled histograms and 𝑟 is the Pearson correlation coefficient.

160

Figure A7. SSC modeling of the seismicity in the Calaveras (CAL) fault zone. (a) Observed (red) and modeled
(blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled (blue)
distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected for
noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

161

Figure A8. SSC modeling of the seismicity in Elsinore Granite Mountain (EGM) fault zone. (a) Observed (red)
and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and
modeled (blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions
corrected for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma
distribution. (h) Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC
parameters (𝑅,𝜅) estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j)
and (k) show the Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set
for the fault zone seismicity.

162

Figure A9. SSC modeling of the seismicity in Elsinore Tamecula (ELS) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

163

Figure A10. SSC modeling of the seismicity in Elsinore-Whittier (EW) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

164

Figure A11. SSC modeling of the seismicity in Garlock (GAR) fault zone. (a) Observed (red) and modeled
(blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled (blue)
distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected for
noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

165

Figure A12. SSC modeling of the seismicity in Hayward (HAY) fault zone. (a) Observed (red) and modeled
(blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled (blue)
distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected for
noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

166

Figure A13. SSC modeling of the seismicity in the Lavick Lake (HLL) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

167

Figure A14. SSC modeling of the seismicity in the Mezquite Lake (HML) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

168

Figure A15. SSC modeling of the seismicity in the Pisgah (HP) fault zone. (a) Observed (red) and modeled
(blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled (blue)
distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected for
noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

169

Figure A16. SSC modeling of the seismicity in the Imperia Valley (IMP) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

170

Figure A17. SSC modeling of the seismicity in the Emerson (LE) fault zone. (a) Observed (red) and modeled
(blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled (blue)
distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected for
noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

171

Figure A18. SSC modeling of the seismicity in the Homestead Valley (LHV) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

172

Figure A19. SSC modeling of the seismicity in the Johson Valley (LHV) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

173

Figure A20. SSC modeling of the seismicity in the North Cross (LHV) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

174

Figure A21. SSC modeling of the seismicity in the Blackwater (MBW) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

175

Figure A22. SSC modeling of the seismicity in the Calico-Hidalgo (MCB) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

176

Figure A23. SSC modeling of the seismicity in the Camp Rock (MCR) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

177

Figure A24. SSC modeling of the seismicity in the Rodman-Pisgah (MRP) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

178

Figure A25. SSC modeling of the seismicity in the Newport-Inglewood (NI) fault zone. (a) Observed (red) and
modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the observed (red) and modeled
(blue) distributions of the total moment fractions. The red dot is the MLE of total moment fractions corrected
for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a gamma distribution. (h)
Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC parameters (𝑅,𝜅)
estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j) and (k) show the
Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set for the fault zone
seismicity.

179

Figure A26. SSC modeling of the seismicity in the San Bernardino Mountains (SAB) section of the San Andreas
Fault. (a) Observed (red) and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show
the observed (red) and modeled (blue) distributions of the total moment fractions. The red dot is the MLE of
total moment fractions corrected for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit
by a gamma distribution. (h) Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the
SSC parameters (𝑅,𝜅) estimated from the observed shape factor and total moment fractions in panes (a)-(f).
(j) and (k) show the Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis
set for the fault zone seismicity.

180

Figure A27. SSC modeling of the seismicity in the Coachella Valley (SAC) section of the San Andreas Fault. (a)
Observed (red) and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the
observed (red) and modeled (blue) distributions of the total moment fractions. The red dot is the MLE of total
moment fractions corrected for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a
gamma distribution. (h) Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC
parameters (𝑅,𝜅) estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j)
and (k) show the Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set
for the fault zone seismicity.

181

Figure A28. SSC modeling of the seismicity in the Loma Prieta section (SAN) of the San Andreas Fault. (a)
Observed (red) and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the
observed (red) and modeled (blue) distributions of the total moment fractions. The red dot is the MLE of total
moment fractions corrected for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a
gamma distribution. (h) Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC
parameters (𝑅,𝜅) estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j)
and (k) show the Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set
for the fault zone seismicity.

182

Figure A29. SSC modeling of the seismicity in the Borrego Badlands section (SJB) of the San Jacinto Fault. (a)
Observed (red) and modeled (blue) distributions of the stress differential ratio (𝐸
#
). (b)-(f) show the
observed (red) and modeled (blue) distributions of the total moment fractions. The red dot is the MLE of total
moment fractions corrected for noise. (g) Distribution of observed mechanism uncertainty (𝜎) and its fit by a
gamma distribution. (h) Distribution of measured stress principal axis 𝕰
𝟎
. (i) Likelihood function of the SSC
parameters (𝑅,𝜅) estimated from the observed shape factor and total moment fractions in panes (a)-(f). (j)
and (k) show the Pearson correlation matrices of observed and modeled moment fractions. (l) SOR basis set
for the fault zone seismicity.

Asset Metadata

Creator Juarez Zuñiga, Alan (author)

Core Title Stress-strain characterization of complex seismic sources

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Geological Sciences

Degree Conferral Date 2022-12

Publication Date 10/17/2022

Defense Date 09/30/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag earthquake mechanism,oai:digitallibrary.usc.edu:usctheses,OAI-PMH Harvest,seismic source complexity,seismic velocities,stress inversion,tectonic stress,wave propagation simulations

Format theses (aat)

Language English

Advisor Jordan, Thomas H. (committee chair), Ben-Zion, Yehuda (committee member), Jha, Birendra (committee member)

Creator Email alanjuar@usc.edu,zu.alan.zu@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC112114341

Unique identifier UC112114341

Identifier etd-JuarezZuig-11267.pdf (filename)

Legacy Identifier etd-JuarezZuig-11267

Document Type Dissertation

Format theses (aat)

Rights Juarez Zuñiga, Alan

Internet Media Type application/pdf

Type texts

Source 20221017-usctheses-batch-986 (), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Abstract (if available)

Abstract Earthquakes are complex physical systems that involve dynamic stress changes, fracture and damage generation, and slip. Furthermore, they release accumulated strain energy in the Earth's crust as heat and seismic waves. Understanding the physical processes that initiate, propagate, and terminate earthquake ruptures, as well as the interaction of seismic waves with complex geological structures, is essential for modeling seismic hazards and improving earthquake forecasting systems.
In this work, I investigate two aspects of the earthquake problem: (1) the characterization of complex earthquakes and seismic sequences and the tectonic stress fields that drive them, and (2) understanding the effects of changes in rock properties on the propagation of seismic waves.
Chapter 1 is devoted to characterizing the mechanism complexity observed in earthquakes and seismic sequences. Section 1.1 summarizes our mathematical theory for representing complex seismic sources using orthogonal moment tensor fields (Jordan & Juarez, 2019; Jordan & Juarez, 2020). Moreover, we introduce the concept of earthquake mechanism complexity and the total moment fractions that measure the seismic moment partitioning into a basis set of up to six-moment tensors. The total moment fractions are a measure of mechanism complexity.
In a subsequent paper (Jordan & Juarez, 2021), we developed a physics-based probabilistic model that relates the total moment fractions to the driving tectonic stress. The new stress-strain characterization (SSC) model relates the short-term seismic response (strain) with the long-term geological forcing (stress). In Chapter 1, we developed the machinery to test the hypothesis that mechanism complexity is governed by tectonic stress.
We implement the SSC model in a maximum likelihood estimation technique that uses catalogs of earthquake mechanisms to estimate the principal-stress directions, the differential stress ratio, and the strain sensitivity factor that measures the strain dependence on the shear-stress magnitude. Section 1.2 discusses the data sets we use to characterize the stress fields and analyze biases in the inversions. In Section 1.3, we develop the maximum likelihood estimation technique to invert for the stress parameters from observations of earthquake mechanisms. Section 1.4 extensively discusses how to handle uncertainties in the estimations. In Section 1.5, we analyze possible biases in focal mechanism orientations that might result from the modeling assumptions or poor data quality used for estimating them. Observational errors and biases in the orientations of earthquake mechanisms can be interpreted as physical complexity. Therefore, we investigate the effect of different parameterizations of orientation uncertainty on the total moment fractions and the estimations of the stress parameters. We implement the forward and inverse modeling capabilities into a software package named SSC-Lab.
In Section 1.6, we applied the SSC-Lab to seismic sequences of fault zones in California and developed a new stress-strain model for Northern and Southern California. We show that our analysis technique provides new insights into the plate boundary's regional and local tectonics, the internal structure of faults, and the tectonic stress spatial variation. Furthermore, in Section 1.7, we show the potential of the SSC-Lab for temporal changes in the stress field.
Sections 1.8 and 1.9 show the stress inversions of 25 fault zones in California and compare our estimations of stress concentration with estimates of geological offsets. We show that the SSC model optimally describes the tectonic stress fields of fault zones and that our measures of stress concentration from mechanism complexity correlate with geological offsets. Furthermore, we find that distributions of mechanism orientation allow for extracting information about fault plane orientation concentration in fault zones. As fault zones evolve, fault planes align with the direction of tectonic motion making these preferred directions weak zones more likely to fail than expected by the simple stress concentration.
Chapter 2 focuses on understanding the effects of reductions in seismic velocities in the shallow crust on seismic wavefields. Seismic waves from repeating earthquakes and background seismic noise can be used to monitor temporal changes in rock properties before, during, and after big earthquakes. Many studies have used long-period surface seismic waves, and because they have resolution to deeper depths, detected changes are interpreted to occur at seismogenic depths. However, it is unclear whether anomalies detected occur at seismogenic depths, where the confining pressure promotes rapid healing of the rock, or they are shallow changes, where at low confining pressures, the spread of rock damage might be more prominent and the recovery slower. The problem is those small perturbations at depth might have the same effect that strong shallow perturbations. Thus, in Chapter 2, we elucidate whether changes in the mechanical properties of shallow rocks could explain waveform perturbations over long periods. For that aim, we performed three-dimensional wave propagation simulations and measured the difference between a reference and a perturbed wavefield from a velocity model that includes surficial velocity reductions.
Our study concludes that seismic velocity changes at shallow depths have effects that might be misinterpreted to occur at seismogenic depths. Furthermore, our observations have implications for waveform tomography, where shallow-low velocities are often ignored.
Chapter 2 has been previously published in the Seismological Research Letters journal (Juarez & Ben-Zion, 2020).