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Finite-Fault Bayesian Inversion of Teleseismic Body Waves

by Brandon S. Clayton, Stephen H. Hartzell, Morgan P. Moschetti, and Sarah E. Minson

Abstract Inverting geophysical data has provided fundamental information aboutthe behavior of earthquake rupture. However, inferring kinematic source model param-eters for finite-fault ruptures is an intrinsically underdetermined problem (the problem ofnonuniqueness), because we are restricted to finite noisy observations. Although manystudies use least-squares techniques to make the finite-fault problem tractable, thesemethods generally lack the ability to apply non-Gaussian error analysis and the impo-sition of nonlinear constraints. However, the Bayesian approach can be employed to finda Gaussian or non-Gaussian distribution of all probable model parameters, while uti-lizing nonlinear constraints. We present case studies to quantify the resolving power andassociated uncertainties using only teleseismic body waves in a Bayesian framework toinfer the slip history for a synthetic case and two earthquakes: the 2011Mw 7.1 Van, eastTurkey, earthquake and the 2010 Mw 7.2 El Mayor–Cucapah, Baja California, earth-quake. In implementing the Bayesian method, we further present two distinct solutionsto investigate the uncertainties by performing the inversion with and without velocitystructure perturbations. We find that the posterior ensemble becomes broader when in-cluding velocity structure variability and introduces a spatial smearing of slip. Using theBayesian framework solely on teleseismic body waves, we find rake is poorly con-strained by the observations and rise time is poorly resolved when slip amplitude is low.

Electronic Supplement: Figures of histograms of slip and rise time, waveformcomparisons between data and synthetics, and slip velocity along the fault plane for asynthetic case, as well as for the 2011 Mw 7.1 Van, east Turkey, earthquake, and the2010 Mw 7.2 El Mayor–Cucapah, Baja California, earthquake.

Introduction

Understanding earthquake rupture behavior is offundamental importance for earthquake-hazard analysis.Inferences on the evolution of slip on a fault are vital forscenario event modeling and the prediction of ground-motion parameters through strong ground motion simula-tions. Current understanding of the behavior of earthquakesis primarily informed by inversions of geophysical data.These deductions are made possible through the utilizationof observations of seismic ground motions, which have beencommonly used in inverse problems for fault rupture history(Trifunac, 1974; Hartzell and Heaton, 1983; Olson and An-derson, 1988; Das and Kostrov, 1990, 1994; Hartzell et al.,1991, 1996, 2007; Zeng and Anderson, 1996; Sekiguchiet al., 2000; Graves and Wald, 2001; Olson and Apsel, 2001;Wald and Graves, 2001; Ji et al., 2002; Liu and Archuleta,2004; Custódio et al., 2005; Liu, 2006; Piatanesi et al., 2007;Liu et al., 2008). However, underdetermined model param-eters lead to a nonunique ill-posed problem.

To make this problem tractable, regularization techniqueshave been employed to transform an ill-conditioned inverse

problem into a well-conditioned one. These techniques includesmoothing, moment minimization, and positivity constraintson the distribution of slip. Mendoza and Hartzell (2013) ap-plied the L-curve methodology from Tikhonov regularizationto obtain an optimal smoothing weight for the finite-fault tele-seismic inversion problem. In this method, a trial-and-errorsearch defines the trade-off curve between fitting the data andsatisfying the constraint and thereby defines the optimal con-straint weight. Alternatively, Asano et al. (2005) employedAkaike’s Bayesian information criterion (Akaike, 1980) ap-plied to the inversion of strong ground motion waveforms toselect optimum constraint weights.

Although these types of regularization techniques seek awell-posed inverse problem, application of the regularizationconstraints alters the solution space. How this solution spaceis altered can be fundamentally different given different regu-larization techniques, yielding contrasting results. However,Bayesian inference can be used to avoid these problems. In-version schemes that employ regularization techniques mostoften conceive a single solution based on a single set of

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Bulletin of the Seismological Society of America, Vol. 107, No. 3, pp. 1526–1544, June 2017, doi: 10.1785/0120160268

http://www.bssaonline.org/lookup/suppl/doi:10.1785/0120160268/-/DC1

model parameters. Bayesian inference yields a set of poten-tial models, using physical a priori constraints throughoutthe process, to form posterior probability density functions(PDFs) of the model parameters. The ability to infer a setof potential models through forward model evaluation elim-inates the need for regularization techniques, while illuminat-ing the uncertainties in the solution.

Though Bayesian theory has a long history (Bayes,1763; Laplace, 1812; Jeffreys, 1931, 1939; Tarantola, 2005),its application to finite-fault problems has historically beenlimited by computational power. However, increasing com-putational power has made tractable Bayesian analysis oflarger scale geophysical problems. This has led to a widerange of Bayesian geophysical inverse problems, for exam-ple, from utilizing lunar seismic and gravity data (Khan et al.,2007), to reservoir monitoring measurements (Xuan andSava, 2010), to frequency-domain electromagnetic data(Minsley, 2011). In earthquake seismology, specifically withreference to finite-fault inversions, a multiplicity of differentdata sets has been incorporated into a Bayesian framework.With the inclusion of strong ground motion data (Monelliand Mai, 2008; Fan et al., 2014); Global Positioning System(GPS) offsets, offshore seafloor geodesy, and tsunami re-cords (Minson et al., 2014); W-phase waveforms (Dettmeret al., 2014); and geodetic, tsunami, and strong ground mo-tion records (Duputel et al., 2015), Bayesian inference hasbeen widely used to access the uncertainty question and yieldsolutions unaffected by regularization. To quantify such un-certainties in the Bayesian framework, Dettmer et al. (2014)presented 95% credibility intervals for slip magnitude andrupture velocity, as well as the marginal densities for the hy-pocenter location, rupture velocity, and moment magnitude.Duputel et al. (2015) quantified the uncertainties by present-ing the posterior mean slip model with 95% confidence errorellipses, the PDFs for stress drop, rupture velocity, rise time,and seismic potency, and additionally highlighted the obser-vational errors and prediction uncertainty using sensitivityplots for each of the data sets used. Despite the growing bodyof literature employing Bayesian methods for finite-sourceinversions, we know of no work that has modeled only tele-seismic body-wave data within a Bayesian framework; thus,we seek to better understand their resolution and uncertainty.

In this study, we focus on uncertainty analysis fromBayesian inference using teleseismic body waves to infer theslip history for a synthetic case, as well as for the 2011Mw 7.1 Van, east Turkey, earthquake and the 2010 Mw 7.2El Mayor–Cucapah, Baja California, earthquake. Our objec-tive in this study is to quantify the resolving power ofteleseismic body waves in determining finite-fault slip.

Methodology

Data

The teleseismic body-wave data sets were obtained fromthe Incorporated Research Institutions for Seismology Data

Management Center. The P waves used in the study are fromthe vertical component, whereas the S waves are from thetransverse component. The records are first processed by re-moving the instrument response, over the frequency range of0.0166 Hz (60 s) to 2 Hz (0.5 s). The instrument-correcteddata are then resampled to a time step of 0.1 s, and a band-pass Butterworth filter from 0.0166 Hz (60 s) to 1 Hz (1 s) isapplied. This filter is applied to aid in the evaluation of signaland noise present in the records and to help define an appro-priate frequency band to use for the inversion. The data re-cords are then transformed into the frequency domain, inwhich the Bayesian inversion is conducted. The frequencyband used for the Bayesian inversion depends on the mag-nitude of the earthquake, with longer periods required forlarger magnitude events.

The record length used in the inversion is chosen basedon a consideration of the characteristics of the waveforms,the fault dimensions, and the assumed rupture velocity, toencompass the entire source process. Generally, the recordlength should not exceed the length of the synthetic seismo-gram for a subfault furthest from the hypocenter, accountingfor the time it takes the wave front to propagate to that dis-tance (Hartzell, 1989).

Model Parameterization

Fault planes for our finite-fault inversions are parame-terized using fixed orientation (strike and dip), dimensions(along strike and down dip), and depth, embedded in a seis-mic velocity model. For a more complex rupture, like the ElMayor–Cucapah earthquake, we can consider multiple faultplanes. The determination of the fault-plane parameters isbased on moment tensor solutions (e.g., U.S. GeologicalSurvey, Global Centroid Moment Tensor [CMT], SouthernCalifornia Seismic Network, and other published studies).The fault plane is discretized into a specified number ofequal-sized subfaults or patches. The size and total numberof subfaults are chosen based on several considerations.With a linear formulation (Hartzell and Heaton, 1983), thefault plane is generally overparameterized using a relativelylarge number of subfaults and then applying smoothingconstraints to adjacent slip. In this study, we wish to avoidsmoothing constraints, and our choice of subfault size isdriven by the time resolution between nearby subfaultsand the frequency content of the data. There is also a prac-tical limitation, based on computational effort, to the num-ber of model parameters in a Bayesian style inversion. Wewill discuss this topic further in the Results and Discussionssection.

Our finite-fault spatiotemporal slip history is defined bythe following set of parameters. There is a finite ruptureduration to quantify the time it takes for a subfault to rupture,known as rise time. The rise time and its functional formdefine the source time function (STF) for each subfault. Thedirection of slip on a subfault defines the rake angle. The

Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1527

magnitude of slip for each subfault completes the parameter-ization of the rupture.

To find the spatiotemporal parameters, we first computedisplacement waveform synthetics for two orthogonal unitsteps in slip on a subfault known as the Green’s functions(Fig. 1). During the Bayesian process, two weights for theorthogonal Green’s functions and one for the STF durationare chosen to evaluate the goodness of fit between theobserved and modeled data. In this study, we use a box-carSTF, in which the width of the box-car is a parameter in theinversion. The two weights that scale the orthogonal Green’sfunctions define the slip amplitude and rake angle. In thisstudy, we fix the rupture velocity to be a constant. This sim-plification is made because variations in rupture velocity canhave a significant impact on the slip model (Lay et al., 2010)and can lead to a model parameterization that greatly in-creases the computational effort. We choose to fix rupturevelocity to highlight the uncertainties associated with slipand rise time.

The two orthogonal Green’s functions are computedsuch that they are �45° from the inferred average rake. Thisrake estimate is generally from moment tensor analysis. TheGreen’s functions are calculated using the ak135 velocitystructure (Kennett et al., 1995).

Bayesian Inference

Bayes Theorem. The Bayesian approach is used to estimatethe posterior model distribution that defines the solutionspace. However, to reach the posterior model distributionwe must define the prior PDF and data likelihood function,which are defined over both the model and data spaces. The

prior and theoretical PDFs are then used to develop the pos-terior model distribution to infer the solution.

In a Bayesian inference framework, there are two mainmanifolds (or dimensions) that represent the model spaceM and the data space D. Within each respective spacelie individual models m � fm1; m2;…; mng and datad � fd1; d2;…; dng. Given these manifolds, a new state ofinformation about a model m can be achieved given datad using Bayes theorem:

EQ-TARGET;temp:intralink-;df1;313;625P�mjd� � P�djm�PM�m�PD�d�

�1�

(Tarantola, 2005), in which P�djm� is the conditional PDF ofthe data d given a model m, PM�m� is the model marginalPDF over the model space M, PD�d� is the data marginalover the space D, and P�mjd� is the conditional PDF ofthe model m given the data d. Bayes theorem is the genericsolution to probabilistic inverse problems and can be brokendown into three fundamental parts: prior, theoretical, andposterior joint PDFs.

Prior. The prior joint PDF encapsulates both the a prioriinformation on the models and the observed data, defined as

EQ-TARGET;temp:intralink-;df2;313;448ρ�d;m� � ρD�d�ρM�m�; �2�

in which ρD�d� is the marginal prior probability of the data,ρM�m� is the marginal prior probability of the model, andρ�d;m� is the resultant prior joint PDF defined over themanifold D ×M.

The a priori information on the models, or the marginalprior probability of the model ρM�m�, captures the priorknowledge of the model independent of all observed data.In this study, the a priori information on the models is rep-resented by a truncated uniform distribution, meaning that allvalues are equally probable, which only reflects the range ofpossible values in each model m:

EQ-TARGET;temp:intralink-;df3;313;282ρM�m��1

�mmaxi −mmini�Nformmini ≤m≤mmaxi ; �3�

in which mmaxi and mmini are the bounds chosen for themodel parameterm, and N is the total number of parameters.In our parameterization, N equals three times the total num-ber of subfaults in the finite-fault problem, accounting for thetwo slip weights and rise-time values for each subfault. Thelimits for both the slip and rise time are chosen based onpreviously published studies and the magnitude of the event.Table 1 highlights the bounds chosen for each of the casespresented in the Results and Discussions section.

With each observed value in the data there are associateduncertainties. These uncertainties in the data can be repre-sented by a PDF, the prior marginal of the data space,ρD�d�. In this study, the state of information on the observeddata is considered a Gaussian PDF given observed data, dobs:

–10 –5 0 5 10

0

5

10

15

Along Strike (km)

Alo

ng D

ip (

km)

1

2

3

4

5

6

7

8

9

10

11

12

Rake

Figure 1. Simplified example of the discretization for the finite-fault problem with two Green’s functions; U� represents the �45°Green’s function component from the average rake and U− repre-sents the −45° Green’s function component from the average rake.These two components sum to achieve the rake of the subfault.Subfaults are numbered starting with the subfault in the upper leftand increase along dip.

1528 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

EQ-TARGET;temp:intralink-;df4;55;604ρD�d� �1���������������������

�2π�N jCdjp exp

�−1

2�d − dobs�tC−1d �d − dobs�

�;

�4�

in which j · j represents the determinate, and Cd is the datacovariance matrix.

Theoretical. The theoretical PDF captures the physicaltheory that relates the model with the data, defined as

EQ-TARGET;temp:intralink-;df5;55;491Θ�d;m� � θ�djm�μM�m�; �5�

in which θ�djm� is the conditional PDF of the data given themodel, μM�m� is the marginal homogeneous PDF of themodel, and Θ�d;m� is the resultant joint theoretical PDF.

In this study, the theoretical conditional PDF θ�djm�links the slip and rise time (model parameters) to the teleseis-mic body waves (data). It is assumed to be a Gaussian PDF,

EQ-TARGET;temp:intralink-;df6;55;385θ�djm� � 1����������������������2π�N jCT j

p

× exp�−1

2�d −G�m��tC−1T �d − G�m��

�; �6�

in which G�·� represents the forward operator, and CT is thetheory covariance matrix. The forward operator is applied togiven models m to construct plausible data, or the forwardproblem.

The Posterior and Likelihood Function. The prior andtheoretical states of information combine, through a conjunc-tion, to form the a posteriori state of information,

EQ-TARGET;temp:intralink-;df7;55;212σ�d;m�� 1ν

ρ�d;m�Θ�d;m�μ�d;m� �

1

ν

ρD�d�ρM�m�θ�djm�μD�d�

; �7�

in which ρ�d;m� is the prior joint PDF, Θ�d;m� is the theo-retical PDF, ν is a normalizing constant

EQ-TARGET;temp:intralink-;df8;55;142ν �ZD×M

ρD�d�ρM�m�θ�djm�μD�d�

dd dm; �8�

defined over the joint manifold D ×M, μ�d;m� is thehomogeneous joint PDF

EQ-TARGET;temp:intralink-;df9;313;733μ�d;m� � μD�d�μM�m�; �9�and σ�d;m� is the posterior joint PDF. To arrive at the a pos-teriori state of information about the model, or the solution,a posterior marginal PDF of the model can be constructed,

EQ-TARGET;temp:intralink-;df10;313;675σM�m� �1

νρM�m�

ZD

ρD�d�θ�djm�μD�d�

dd; �10�

in which σM�m� is the resultant a posteriori state of infor-mation on the model represented by a PDF, given the priorand theoretical states of information. This refined state ofinformation on the model can be written as

EQ-TARGET;temp:intralink-;df11;313;581σM�m� �1

νρM�m�L�m�; �11�

in which L�m� is the likelihood function.The likelihood function describes how well a model m

explains data d. Using the prior marginal of the data(equation 4) and the theoretical (equation 6), the likelihoodfunction used is constructed asEQ-TARGET;temp:intralink-;df12;313;478

L�m� � 1�2π�N

������������������jCdjjCT j

p

×ZDexp

�−1

2��d − G�m��tC−1T �d − G�m��

� �d − dobs�tC−1d �d − dobs���dd: �12�

Equation (12) highlights the convolution of two Gaussianprobability densities and can be simplified (e.g., Tarantola,2005) toEQ-TARGET;temp:intralink-;df13;313;340

L�m�� 1�����������������������2π�N jCDj

p

×exp�−1

2�dobs−G�m��tC−1D �dobs−G�m��

�; �13�

in which CD is the covariance that encapsulates the uncer-tainties in the data Cd and the uncertainties in the theory CT,given as CD � CT � Cd.

Covariance Matrix. The covariance of a multidimensionalGaussian, synonymous to variance in 1D, describes both theuncertainties (diagonal elements) and correlations (off-diagonal elements),

EQ-TARGET;temp:intralink-;df14;313;176C��x� �Z�x − �x��x − �x�tf�x�dx; �14�

in which f�x� is the PDF on which to define the covariance, �xis the mean of that PDF, and C��x� is the resulting covarianceat �x. The covariance needed for Bayesian inference, shown inthe likelihood function (equation 13), includes the covari-ance matrix CD, which combines the associated uncertaintiesand correlations in the theory and data. In this study, we

Table 1Physical a priori Constraints for the Prior Marginal

Probability Density Function (PDF)

Event Amplitude (cm)Orientation

(Green’s Functions)Rise

Time (s)

Synthetic 0–1000 105°/15° 1–10Turkey 0–500 105°/15° 1–10Baja subevent 0–200 −135°/−45° 1–10Baja main event 0–1000 135°/−135° 1–10

Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1529

adopt a diagonal form for the covariance matrix. Thissimplification invokes the uncertainties, saving the computa-tional expense of updating a covariance matrix.

The data covariance Cd describes the uncertainties asso-ciated with the observations or the teleseismic body waves.To capture the uncertainties in the seismograms, the pre-event noise was used to determine the fractional differencebetween the peak amplitude of the pre-event noise and thepeak amplitude of the P waves,

EQ-TARGET;temp:intralink-;df15;55;442Cid � 1 −ME −MP

ME; �15�

in which Cid represents the uncertainty of the data for a spe-cific station,ME represents the maximum amplitude of the Pwave present in the waveform, and MP represents the maxi-mum amplitude of the pre-event noise.

The covarianceCT describes the uncertainty in the theory,which can include incorrect source geometry or hypocenterlocation and poorly chosen Earth structure. In this study,we chose to highlight the effects of the Earth structure,specifically the velocity model, on the a posteriori state ofinformation on the model. To assess this uncertainty, 10 ran-dom crustal model perturbations were drawn from a Gaussiandistribution with a 5% standard deviation in wave velocity andlayer thickness from the reference P- and S-wave velocitymodel (Fig. 2; Razafindrakoto and Mai, 2014). For eachperturbation we maintain the ratio 1:60 ≤ VP=VS ≤ 1:90.Table 2 gives the reference velocity model values.

CATMIP: Algorithm for Bayesian Inference

Our Bayesian inference employs an implementation of aparallel Markov chain Monte Carlo (MCMC) algorithm:Cascading Adaptive Transitional Metropolis in Parallel(CATMIP; Minson et al., 2013). The Metropolis algorithm(Metropolis et al., 1953; Hastings, 1970) uses a Markovchain (random walk) to draw samples of a target PDF. AMarkov chain is a stochastic sequence that transitions be-tween different states, in which the probability of each stateis dependent on the previous state. The utilization of a singleMarkov chain consequently is a nonparallelized algorithm.

However, CATMIP can employ thousands of Markov chainssimultaneously, allowing it to be developed in a parallelframework and permitting solutions of higher dimensionalmodeling problems that once were computationally intrac-table. To ensure higher efficiency for higher dimensionalspaces, CATMIP also uses transitioning as well as resam-pling (Ching and Chen, 2007).

CATMIP is a generic Bayesian MCMC sampler thatemployees transitioning, which shares several traits withsimulated annealing optimization. Algorithms that employannealing start off from an initial hot state and cool to a finalcold state. In this particular implementation, the initial stateis the prior PDF that cools to the final posterior PDF. Toensure that a proper cold state has been reached, CATMIPsamples from intermediate PDFs during cooling steps, ortransitional stages, which are controlled by an annealingparameter. During transitioning, resampling is conducted inwhich less probable models from the previous transitioningstep are replaced with more probable models. This allowsCATMIP to relocate models that are trapped in areas of lowprobability to become seeds for new Markov chains in re-gions of higher probability. For a complete discussion ofCATMIP, see Minson et al. (2013)

To ensure that the posterior PDF is stable, the total num-ber of samples is regulated by the total number of modelparameters, in which an increase in dimensionality requiresan increase in samples. In this study, the values chosen forthe number of Markov chains and the length of each chainare chosen purely on the basis of repetition of syntheticcases. The values are chosen such that the solution stabilizedand did not change significantly with further sampling of themodel space.

Uncertainty Analysis

To investigate the effect of variability in seismic velocitymodels, the Bayesian inversion is run 11 times, for a refer-ence model and 10 velocity model perturbations. For each ofthe velocity models, the inversion is run using 1000 Markovchains with 200,000 steps per Markov chain, in which 1000Markov chains with 200,000 steps was found to stabilize thesolution after repetition of different combinations of Markovchains and steps. Once all 11 velocity models have been run,the resultant posterior PDFs are combined, giving a total of11,000 estimates for each of the model parameters: weightsfor the two orthogonal slip components and rise time for eachsubfault. Then, the most probable solution is given by themean of this stacked posterior PDF.

Wavespeed (km/s)3 4 5 6 7 8 9

Dep

th(k

m)

0

10

20

30

40

50

60sevaW PsevaW S

Figure 2. Velocity model perturbations for uncertainties invelocity and layer thickness with S- and P-wave velocities; thickblack solid line represents the reference S- and P-wave velocities.

Table 2ak135 Reference Velocity Model (Kennett et al., 1995)

P Wave (km=s) S Wave (km=s) Density (g=cc) Layer Thickness (km)

5.80 3.46 2.45 20.006.50 3.85 2.71 15.008.04 4.48 3.30 50.00

1530 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

To highlight the single velocity model, or the referencemodel, the inversion is also run using 11,000 Markov chainswith 200,000 steps per Markov chain, equating to an equiv-alent number of Markov chains as for the multiple velocitymodels.

After performing multiple inversions for all three faultconfigurations (synthetic case, 2011 Van earthquake, and the2010 El Mayor–Cucapah earthquake), it was found that therake for each subfault was an unstable parameter, varyingunrealistically between adjacent subfaults. To overcome thisdifficulty, we constructed the rake such that each subfaultcould vary �10° from the average inferred rake (Table 3).The variance of �10° allows for a significant variation inrake that is typically not exceeded in moment tensor esti-mates. As an alternative, the inversion could be conductedusing a single Green’s function; however, there is a clear biasin this parameterization, because the inferred rake is assumedto be correct with no uncertainty. There is no constraintplaced on the moment or any other constraint on the two slipcomponents or the rise time.

Results and Discussions

Here, we present the Bayesian inversion results for threedifferent fault configurations, including a synthetic test, the2011 Mw 7.1 Van, east Turkey, earthquake, and the 2010Mw 7.2 El Mayor–Cucapah, Baja California, earthquake. Foreach of these cases, we present two distinct solutions for com-parison, one utilizing a single given velocity model and an-other to show the effect of uncertainty in velocity model. Thesolutions presented here are obtained by calculating the meanvalue from the distribution of slip or rise time for each subfault.Rise-time values in the plots corresponding to values of slip20 cm and less are shown in gray (Figs. 4, 8, and 12).

Synthetic Tests

We first consider a synthetic test case, using teleseismicP waves, for a thrust fault with 36 idealized stations, all at adistance of 60°, uniformly spaced in azimuth every 10°. Thesynthetic model consists of a single 120 km (along strike) ×16 km (along dip) fault plane with 4 km × 4 km subfaults(120 total subfaults). We construct the synthetic model tohave four major asperities, containing a total of 16 subfaultswith slip of 500 cm per subfault (Fig. 3a), a rake of 60°, a rise

time of 3 s (Fig. 4a), and a constant rupture velocity of2:5 km=s. The slip history for each subfault is constructedfrom two orthogonal components of slip, at 105° and 15°(�45° from the true rake). The inversion is conducted usinga frequency range from 0.0167 Hz (60 s) to 1 Hz (1 s).

Although our synthetic data are noise free, we constructa data covariance Cd (Minson et al., 2013) using equa-tion (15) that is representative of the quietest stations for boththe 2011 Van, east Turkey, and 2010 El Mayor–Cucapah,Baja California, data sets. The data covariance consists ofonly diagonal elements with each element consisting of anuncertainty of 0.02 for all observations.

Synthetic: Single Velocity Model with No Angle Con-straint. Figures 3b and 4b present the posterior mean sol-ution given a single velocity model for slip and rise time,respectively. The solution is obtained using no angle con-straint, meaning that the weights found in the inversion forthe two orthogonal Green’s function have open range, andthat the resulting rake can vary from 15° to 105°. In thisparticular scenario, the change in rake between adjacent sub-faults is reasonable, with no sudden jumps in rake. However,given real data, the behavior of the rake is unreasonable, aswe show below.

Synthetic: Single Velocity Model with No Angle Constraintand Addition of SH Waves. The solution obtained in Fig-ures 3b and 4b only considers P waves. Figures 3c and 4cpresent the posterior mean solution when an additional 36idealized teleseismic SHwaves are added. The additional sta-tions reside at the same locations as the 36 stations for Pwaves. The solution is obtained using a single velocity modelwith no angle constraint. In this particular implementation,the addition of the SH waves does not help resolve the rake;because the behavior of the rake is more erratic, evidenced bythe approximately 90° jump in rake near the hypocenter. Tobe consistent throughout the remaining synthetic scenarios,we only implement teleseismic P waves with a �10° vari-ance allowed from the 60° rake.

Synthetic: Single Velocity Model. With the implementationof a single velocity model with a�10° angle constraint on theinferred rake, the slip distribution (Fig. 3d) and rise-time dis-tribution (Fig. 4d) highlight characteristics similar to the true

Table 3Source Parameters Used for Synthetic Case, the 2011 Turkey Earthquake, and the 2010 Baja California

Earthquake (BC)

EventHypocenter

(Latitude, Longitude; Depth)Fault Geometry(Strike/Dip/Rake)

Fault Dimension(Along Strike × Along Dip)

Subfaults(Length × Width)

RuptureVelocity (km=s)

Synthetic NA, NA; 10 km 313°/36°/60° 120 km × 16 km 30 × 4 2.5Turkey 38.7°, 43.5°; 15 km 246°/38°/60° 100 km × 40 km 20 × 8 2.5BC subevent 32.3°, −115.3°; 10 km 335°/45°/−90° 32 km × 16 km 8 × 4 2.0BC main event 32.3°, −115.3°; 10 km 313°/88°/−174° 120 km × 16 km 30 × 4 2.0

NA, Not applicable.

Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1531

model (Fig. 3a). The distribution of slip is well resolved exceptnear the hypocenter, where there is a smoothing effect. Thepoor resolution near the hypocenter is due to the smallarrival-time difference at teleseismic distances between sub-faults at similar but opposite directions from the hypocenter.However, as the rupture propagates further, the difference inarrival times increases, allowing the slips to be differentiatedfrom one another. The rise times, as shown in Figure 4d, arealso not as well resolved near the hypocenter and tend to bemore consistent with the true model further from the hypocen-ter. In addition, subfaults with longer rise times and smallerslips make negligible contribution to the total synthetic. Theoverall rake obtained is consistent with the rake of the syn-thetic model, given the angle constraint. The mean of the pos-terior model PDF also does a good job recovering the momentof 3:98 × 1019 N·m (Mw 7.0) compared with the correct mo-ment of 3:75 × 1019 N·m (Mw 7.0).

Synthetic: Multiple Velocity Models. The implementation ofmultiple velocity models is presented in Figures 3e and 4e forslip and rise time, respectively. The results highlight similar

characteristics with (Figs. 3d and 4d) and without (Figs. 3eand 4e) velocity model variability. Here, we see a similar lackof resolution near the hypocenter as a consequence of thesmall arrival-time difference between certain subfaults. How-ever, in this scenario with velocity model variability, there issomewhat greater smoothing of slip on different regions ofthe fault. With velocity model variations, changes in timingof phases cause subfaults to acquire slip that are not near theactive subfaults of the true model (Fig. 3a). With velocitymodel variability, the moment increases slightly to4:11 × 1019 N·m (Mw 7.0).

Synthetic: Uncertainties. To investigate the uncertainties,Figure 5a presents scatter plots of rise time versus slip forthe four numbered subfaults (subfaults 6, 51, 66, and 91), inFigure 3d and 3e, for a single velocity model (shown in blue)and multiple velocity models (shown in red), respectively.Each scatter plot represented in Figure 5a shows the posteriorensemble of 11,000 possible solutions for each subfault. Theposterior ensemble is no longer confined to a single regionbut contains multiple areas. These represent the different

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Figure 3. Slip distributions for the synthetic scenario: (a) synthetic model input for the Bayesian inversion, (b) distribution with a singlevelocity model and no angle constraint, (c) distribution with a single velocity model and no angle constraint with SH waves in addition to Pwaves, (d) distribution with a single velocity model and an angle constraint of �10° from the correct rake, and (e) distribution when usingmultiple velocity models throughout the inversion process with the angle constraint applied. Numbered subfaults are further explored (Fig. 5a).

1532 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

velocity models used in the inversion, because each velocitymodel creates its own distribution of solutions. In general,the scatter plots for the single and multiple velocity modelsshow an independence between slip and rise time; however,there is a slight linear trend on some subfaults, such as insubfault number 66, which suggest that slip and rise time arecodependent. The rise times are generally better constrainedthan slip, except for low-slip areas of the fault plane. Thisconclusion is based on the composite of the multiple velocitymodel solutions, in which changes in the velocity model havea much larger effect on slip than on rise time.

To highlight the uncertainties between the single andmultiple velocity models, a comparison of the model slip his-tograms for the four numbered subfaults in Figure 3d and 3eis conducted (see Ⓔ the electronic supplement to this ar-ticle). The distribution of slip for the single velocity modelshows highly peaked and well-defined distributions relativeto the PDFs with velocity model variability. With multiplevelocity models, the distribution of slip for each subfault

is greater with accompanying larger standard deviations,on the order of 3.3 times greater on average.

Table 4 highlights the values for the mean slip and stan-dard deviations for the subfaults compared. Taking subfaultnumber 6 as an example, the mean slip value correspondingto a single velocity model is 468 cm, with an accompanyingstandard deviation of 49 cm, compared with the multiplevelocity model solution of 403 cm, with an accompanyingstandard deviation of 250 cm. In general, the uncertaintiesincrease with multiple velocity models, altering the solutionspace by having a wider range of slip values in the posterior(Razafindrakoto and Mai, 2014).

Given velocity model variability, certain distributionsare best represented by a multimodal distribution, not aunimodal one, as highlighted by the scatter plots. This phe-nomenon can be explained by two effects: the timing differ-ence caused by using different velocities and the use of afinite number of velocity models. Each of the 11 velocitymodels contains different P- and S-wave velocities, which

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Figure 4. Rise-time distributions for the synthetic scenario: (a) synthetic model input for the Bayesian inversion, (b) distribution with asingle velocity model and no angle constraint, (c) distribution with a single velocity model and no angle constraint with SH waves in additionto P waves, (d) distribution with a single velocity model and an angle constraint of�10° from the average inferred rake, and (e) distributionwhen using multiple velocity models throughout the inversion process with the angle constraint applied. Numbered subfaults are furtherexplored (Fig. 5a).

Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1533

causes timing differences between solutions. This timing dif-ference causes slip to reside on different subfaults.

The phenomenon of the multimodal distribution is fur-ther explained by considering an additional 10 velocity mod-els (see Ⓔ the electronic supplement to this article). Whenmore velocity models are used, the histograms start to fill in.As an example, consider the histograms for subfault number6 for 11 and 21 velocity models. There is also a resultingsmall change in the mean and standard deviation. From thispoint forward, we opt to consider velocity model variabilityusing a total of 11 models, due to computational expenses.

A comparison of the model rise-time histograms for thesolution containing a single velocity model (Fig. 4d) andmultiple velocity models (Fig. 4e) can be seen inⒺ the elec-tronic supplement to this article. The standard deviations cor-responding to the rise times show similar characteristics tothe slip standard deviations, with an increase in standarddeviation when considering multiple velocity models, in-creasing on average by a factor of 1.3. Table 4 shows therise-time mean values and associated standard deviationsfor the four subfaults. With the inclusion of multiple velocity

models, the rise-time distributions are also best representedby a multimodal distribution.

Synthetic: Waveform Fit. The calculated waveforms fromthe true model (Figs. 3a and 4a), the inversion results for themean posterior model from the reference velocity model in-version (Figs. 3d and 4d), and the inversion including velocitymodel variability (Figs. 3e and 4e) are shown in Ⓔ the elec-tronic supplement to this article. The waveforms with andwithout velocity model variability show no visible deviationsfrom the true data (see Table 5 for L2 norms). This resultpoints out the limitations in resolution of teleseismic bodywaves given the differences in these fault models, as well asillustrating how the inverse problem is underdetermined.

2011 Van Earthquake

Now, we turn to the inversion of real earthquake dataand the 2011 Mw 7.1 Van, east Turkey, earthquake with55 teleseismic P-wave records azimuthally dispersed withdistances from 31° to 94°, as shown in Figure 6a. The rupture

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1534 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

is modeled on a single plane that is 100 km (along strike) ×40 km (along dip), having 5 km × 5 km subfaults (160 totalsubfaults), with a strike of 246°, a dip of 38°, and a rupturevelocity of 2:5 km=s (Mendoza and Hartzell, 2013). Thestrike and dip values used in the inversion refer to the GlobalCMT solution (see Data and Resources). Konca (2015) usedteleseismic P and SH waves, GPS displacements, strong-mo-tion data, and high-rate GPS station waveforms for the 2011Mw 7.1 Van earthquake to resolve the uncertainty in faultgeometry by performing a grid search using different combi-nations of strike and dip. Their analysis showed that the tele-seismic and GPS data were unable to definitively constrain thefault geometry, demonstrating that different combinations ofstrike and dip had little impact on the solutions. We opt touse the strike and dip given by the Global CMT. Given themagnitude of the 2011 Van earthquake, a range of frequenciesfrom 0.0167 Hz (60 s) to 1 Hz (1 s) is used in the inversion.The a priori values (Table 1) chosen for the 2011 Van earth-quake come from the magnitude of the event as well as pub-lished studies (Hayes, 2011; Fielding et al., 2013; Mendozaand Hartzell, 2013; Utkucu, 2013; Konca, 2015).

2011 Van Earthquake: Single Velocity Model with No AngleConstraint. The posterior mean solution, for both the slipdistribution (Fig. 7a) and rise-time distribution (Fig. 8a), isobtained using a single velocity model with no angle con-straint. With no constraint, the rake is allowed to vary from15° to 105°. In this scenario, as seen in Figure 7a, the rake hasa sudden change of 90° from adjacent subfaults. This behav-ior is unrealistic and highlights the lack of rake resolutionand the requirement for an angle constraint.

2011 Van Earthquake: Single Velocity Model with No AngleConstraint and Addition of SHWaves. In an attempt to help

constrain the rake, 18 teleseismic SHwaveforms (Fig. 6a) areadded together with the P waves. The posterior solution forthe slip distribution (Fig. 7b) and rise time (Fig. 8b) is ob-tained with the use of a single velocity model and no angleconstraint. The rake shows similar results to the solution withonly P waves. Given that the rake still has poor resolution,we conclude that the addition of SH waves does not helpconstrain the finite-fault problem in this application of theBayesian inference. From this point forward, we only con-sider scenarios using teleseismic P waves with an angle con-straint applied, allowing the rake to vary �10° from theinferred rake of the earthquake.

2011 Van Earthquake: Single Velocity Model. Utilizing asingle velocity model and the angle constraint of �10°,Figures 7c and 8c show the slip and rise-time distribution,respectively. Using a single crustal model, the peak slipachieved is 411 cm near the hypocenter and a maximum slipof 320 cm at the surface. Given the dip of the fault of 38° andrake of 60°, the surface slip is equivalent to a vertical dis-placement of 160 cm and a horizontal displacement of171 cm. These values are greater than those from field re-ports (Doğan and Karakaş, 2013), in which the maximumvertical and horizontal offsets were found to be 150 and90 cm, respectively. However, the surface measurementsmay not be an accurate representation of slip over the depthrange of the subfault, and our estimate of slip is dependent onthe velocity model. One or two subfaults at the boundaries ofthe model show larger slip. These subfaults only contributeto the late portion of the teleseismic records, in which theyare tapered for the inversion. Arrivals around this time areprobably not source related. The posterior mean solutiongives a moment of 5:24 × 1019 N·m (Mw 7.1) for a shearmodulus of 29.3 GPa, based on the velocity model. The aver-

Table 4Mean Slip, Rise Time, and Associated Standard Deviations for the Synthetic Cases, 2011 Van Earthquake, and the 2010 El Mayor–

Cucapah Earthquake

Single Velocity Model Multiple Velocity Models

EventSubfaultNumber

MeanSlip (cm)

StandardDeviations (cm)

Mean RiseTimes (s)

StandardDeviations (s)

MeanSlip (cm)

StandardDeviations (cm)

Mean RiseTimes (s)

StandardDeviations (s)

Synthetic case 6 468 49 2.76 0.10 403 250 2.72 1.6051 363 74 2.47 0.18 312 155 1.84 0.5066 323 60 2.05 0.23 413 154 1.92 0.8491 303 30 1.34 0.14 468 221 2.85 1.70

Van earthquake 35 214 19 1.89 0.13 161 100 2.64 1.2457 109 26 1.31 0.13 102 82 2.50 2.2586 298 7 1.01 0.00 239 87 1.68 1.66131 64 27 2.91 0.08 131 91 3.11 2.12

2010 El Mayor–Cucapah subevent

5 86 10 6.18 0.14 75 25 2.89 2.1714 41 6 1.51 0.11 39 19 1.70 0.7319 127 4 1.01 0.01 140 16 1.01 0.0126 13 6 1.48 1.01 18 11 1.95 2.05

2010 El Mayor–Cucapah mainevent

27 473 52 2.54 0.11 556 212 3.11 0.6064 861 12 2.94 0.06 882 21 2.68 0.2967 916 8 2.19 0.05 904 74 2.27 0.1273 577 49 4.86 0.08 391 199 4.88 0.24

Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1535

age slip velocities, found by dividing the slip by the rise timefor each subfault (Fig. 9a), show a general trend of higherslip velocities near the hypocenter that decrease radiallyaway from the hypocenter. This suggests an energetic initia-tion followed by decreasing radiation of high frequencies asthe rupture propagates away from the hypocenter.

2011 Van Earthquake: Multiple Velocity Models. The slipdistribution (Fig. 7d) and rise-time distribution (Fig. 8d) ob-tained using multiple velocity models show similar charac-teristics to that of the single velocity model (Figs. 7c and 8c).With uncertainty in velocity model, the peak slip decreasesfrom 411 to 361 cm, and the peak surface slip has decreasedfrom 320 to 180 cm, equivalent to 90 and 96 cm of verticaland horizontal offsets, respectively. The patch of higher slipamplitude near the hypocenter obtained in the single velocitymodel solution (Fig. 7c) decreases in amplitude. The mostprominent consequence of including velocity model uncer-tainty is the activation of slip on more subfaults, as we saw

in the synthetic case. As observed in Figures 7d and 8d,compared with the results of a single velocity model (Figs. 7cand 8c), a smoothing effect causes the peak slip values tolower by distributing the slip among more subfaults. The mo-ment rises slightly from 5:24 × 1019 N·m (Mw 7.1) to 5:70 ×1019 N·m (Mw 7.1), using the same shear modulus of29.3 GPa. The rise times, as shown in Figure 8d, show sim-ilar patterns to that of using a single velocity model (Fig. 8c),starting at lower values near the hypocenter and increasingaway from the hypocenter. The slip velocities (Fig. 9b) showa similar distribution to a single velocity model (Fig. 9a).

2011 Van Earthquake: Solution Comparison. Figure 10presents a comparison of four solutions for slip distributions:Figure 10a shows the single velocity model solution,Figure 10b shows the multiple velocity model solution,Figure 10c presents the solution obtained by Mendoza andHartzell (2013), and Figure 10d presents the solution obtainedby Hayes (2011). Mendoza and Hartzell (2013) used teleseis-

Table 5L2 Norms for the Synthetic Case, the 2011 Turkey Earthquake, and the 2010 Baja California

Earthquake (BC)

P Waves P and SH Waves

Angle Constraint No Angle Constraint No Angle Constraint

Event Single Velocity Model Multiple Velocity Models Single Velocity Model Single Velocity Model

Synthetic 0.051 0.152 0.048 0.093Turkey 0.475 0.521 0.453 0.455BC subevent 0.442 0.480BC main event 0.752 0.783

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Figure 6. Station map with the red star indicating the epicenter of the event with lines depicting every 30° on the map for (a) the 55teleseismic P-wave (orange triangles) and 18 SH-wave (yellow squares) records used for the 2011 Van, east Turkey, earthquake and (b) forsub- and main event of the 2010 El Mayor–Cucapah, Baja California, earthquake. The yellow squares represent the station locations for the47 teleseismic P-wave records for the initial subevent, and the orange triangles represent the station locations for the 40 teleseismic P-waverecords for the main event.

1536 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

mic P waves to obtain a slip distribution using the linear least-squares inversion scheme of Hartzell and Heaton (1983).Hayes (2011) incorporated teleseismic P waves, S waves,and long-period surface waves, using the finite-fault algorithmof Ji et al. (2002), which is a simulated annealing regularizedinversion in the wavelet domain, to show a main area of sliparound the hypocenter with a peak slip of 400 cm. All foursolutions show similar characteristics, highlighting a majorpatch of slip near the hypocenter as well as surface slip.Mendoza and Hartzell (2013) obtained a moment of5:3 × 1019 N·m, similar to our estimate. The Global CMTestimate of 6:3 × 1019 N·m is somewhat larger, probablydue to the incorporation of longer period data.

2011 Van Earthquake: Uncertainties. Figure 5b presentsthe posterior ensemble of rise time versus slip as scatter plotsfor the four numbered subfaults (subfaults 35, 57, 86, and131), in Figure 7c,d, for a single velocity model and multiplevelocity models. Figure 5b highlights the same attributes asshown in the synthetic case (Fig. 5a), because each velocitymodel tends to create its own solution space. The linearrelationship between rise time and slip is also depicted incertain subfaults. Subfault number 57, when using multiplevelocity models, shows that rise times corresponding tolower slip values are poorly resolved.

The uncertainties associated with the four numberedsubfaults for a single velocity model and multiple velocitymodels are shown as histograms in Ⓔ the electronic supple-ment to this article. The distribution of slip for both a singlevelocity model and multiple velocity models shares the samecharacteristics as discussed in the synthetic case. With theimplementation of velocity model uncertainty, the slip distri-bution becomes broader, with an increase in standarddeviation by a factor of 4.7. Each subfault compared has dif-ferent mean values and accompanying standard deviations,as shown in Table 4.

The associated rise-time uncertainties of the four num-bered subfaults are additionally shown in Ⓔ the electronicsupplement to this article for a single velocity model andmultiple velocity models. Rise-time standard deviations in-crease by an average factor of 2.8 when considering uncer-tainty in the velocity model. The rise-time mean values andassociated standard deviations for the subfaults compared aregiven in Table 4.

2011 Van Earthquake: Waveform Fit. Comparison of the55 teleseismic P-wave records with the waveforms calcu-lated from the slip distribution of the single velocity model(Figs. 7c and 8c) and the waveforms calculated from the slipdistribution using velocity model variability (Figs. 7d and8d) can be seen in Ⓔ the electronic supplement to this ar-ticle. Both solutions fit the data well, showing a small misfitbetween the data and synthetics for the two solutions (seeTable 5 for L2 norms). In general, the synthetics tend tomatch each other for the first 10 s and then slightly deviatefrom one another. The ability of both solutions to representthe data well highlights the limitation in resolution of tele-seismic body waves. Although the solution is altered givenuncertainty in velocity model, the waveform fits for bothsolutions are comparable.

2010 El Mayor–Cucapah Earthquake

We now turn to the last, and more complicated, scenarioof the 2010 Mw 7.2 El Mayor–Cucapah, Baja California,earthquake. Wei et al. (2011) showed that the earthquakeinitially ruptured as a normal fault lasting ∼15 s with anMw 6.3, then evolving into an Mw 7.2 earthquake on themain strike-slip fault. Wei et al. (2011) used GPS, Interfero-metric Synthetic Aperture Radar, subpixel correlation,

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Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1537

synthetic aperture radar, and teleseismic body waves in thesimulated annealing regularized inversion of Ji et al. (2002)for both the sub- and main event of the 2010 El Mayor–Cucapah earthquake. Initial modeling was unable to achievegood waveform fits when using a single source.

2010 El Mayor–Cucapah Earthquake: Subevent. Becausethe earthquake rupture initiated as a normal fault of Mw 6.3,our modeling approach is to initially invert only the first 15 sof 47 teleseismic P-wave records azimuthally dispersed indistance from 31° to 93°, as shown in Figure 6b. Because ofthe relatively small magnitude of the initial rupture, spectralvalues of 0.067 Hz (15 s) to 1 Hz (1 s) are chosen for theinversion. The initial rupture is considered to occur on a sin-gle plane of 32 km (along strike) × 16 km (along dip), having

4 km × 4 km subfaults (32 total subfaults), with a strike of335°, a dip of 45°, and rupture velocity of 2:0 km=s (Weiet al., 2011). The a priori values (Table 1) are chosen basedon the magnitude of the event and from Wei et al. (2011).

Subevent: Single Velocity Model. The posterior meansolution for slip and rise-time distribution are shown inFigures 11a and 12a, respectively, for the subevent of the2010 El Mayor–Cucapah, Baja California, earthquake usinga �10° constraint on the inferred rake. Using a single veloc-ity model, the peak slip near the hypocenter is 135 cm, whichis close to the peak value found by Wei et al. (2011) of160 cm, also near the hypocenter. Wei et al. (2011) foundhigher amplitude slip near the hypocenter and on the south-east side of the fault. We also find high slip near the hypo-center with a more bilateral distribution. The posterior meansolution obtained a seismic moment of 3:86 × 1018 N·m(Mw 6.3), consistent with that of the 6.3 moment magnitudefound by Wei et al. (2011). The rise times (Fig. 12a) show asimilar pattern to the Van earthquake results with lower risetimes near the hypocenter and generally increasing rise timesradially away from the hypocenter. The slip velocity distri-bution (see Ⓔ the electronic supplement to this article)shows similar characteristics to that of the 2011 Van earth-quake, suggesting an energetic initiation.

Subevent: Multiple Velocity Models. With the utilization ofmultiple velocity models, Figures 11b and 12b show the pos-terior mean solution for slip and rise time, respectively. Theresults are similar to that of a single velocity model (Figs. 11aand 12a). The seismic moment slightly rises from3:68 × 1018 N·m (Mw 6.3) to 4:16 × 1018 N·m (Mw 6.4).The rise times obtained (Fig. 12b) using velocity variabilityshow a very similar pattern as with a single earth structure.The slip velocity distribution (see Ⓔ the electronic supple-ment to this article) obtained from multiple velocity modelshighlights similar patterns to that of a single velocity model.

Subevent: Uncertainties. The posterior ensemble of thesolution for rise time and slip are plotted against one another(Fig. 5c) for the single velocity model solution (shown inblue) and multiple velocity model solution (shown in red),for the four numbered subfaults (5, 14, 19, and 26) in Fig-ure 11a and 11b. Figure 5c shows similar attributes to that ofthe synthetic case (Fig. 5a) and the 2011 Van earthquake(Fig. 5b), highlighting multiple solution spaces when usingmultiple velocity models. Figure 5c additionally shows thatthere are broader distributions when using multiple velocitymodels. The subevent of the 2010 El Mayor–Cucapah earth-quake further shows the lack of resolution of rise times forsmall slip, as seen in subfaults 14 and 26.

The model slip histograms (seeⒺ the electronic supple-ment to this article) compare the four numbered subfaults forthe solution utilizing a single velocity model (Fig. 11a) andvariability in velocity model (Fig. 11b). With the implemen-tation of velocity model uncertainty, the slip distributions

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Figure 8. Rise-time distributions for the 2011 Van, east Turkey,earthquake: (a) distribution with a single velocity model and no an-gle constraint, (b) distribution with a single velocity model and noangle constraint with SH waves in addition to P waves, (c) distribu-tion with a single velocity model and an angle constraint of �10°from the average inferred rake, and (d) distribution when usingmultiple velocity models with an angle constraint. Numbered sub-faults are further explored (Fig. 5b).

1538 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

become broader, with an increase in standard deviations by afactor of 2.2. Table 4 shows the mean values and standarddeviations for the subfaults compared.

The model rise-time histograms (see Ⓔ the electronicsupplement to this article) compare the four numbered sub-faults for a single velocity model (Fig. 12a) and multiplevelocity models (Fig. 12b). Table 4 shows the mean valuesand standard deviations of rise time associated with the sub-faults. Standard deviations increase by a factor of 1.3 whenconsidering velocity model variability.

Subevent: Waveform Fit. The comparison of the 47 tele-seismic P-wave records with the forward-modeled wave-forms calculated from the slip distribution of the singlevelocity model (Figs. 11a and 12a) and the forward-modeledslip distribution using velocity model variability (Figs. 11band 12b) are shown in Ⓔ the electronic supplement to thisarticle. In general, waveforms for both solutions are verysimilar (see Table 5 for L2 norms). Both slip distributionsyield comparable waveform fits to that of Wei et al. (2011).

2010 El Mayor–Cucapah Earthquake: Main Event. Theinitial rupture of Mw 6.3 was followed by the main strike-slip Mw 7.2 event. The inversion of the main event of theEl Mayor–Cucapah earthquake uses 40 teleseismic P-waverecords azimuthally dispersed at distances from 31° to 94°, asshown in Figure 6b, with spectral values chosen for the in-version from 0.0167 Hz (60 s) to 1 Hz (1 s). P-wave model-ing of the main event waveforms starts where the initialsubevent waveforms end. Following the work of Wei et al.(2011), the main event hypocenter is the same as the sube-vent hypocenter. The geometry of the main event’s faultplane is simplified to be a single plane of 120 km (alongstrike) × 16 km (along dip), having 4 km × 4 km subfaults

(120 total subfaults), with a strike of 313°, a dip of 88°, and arupture velocity of 2:0 km=s. The Green’s functions for eachsubfault are �45° from the inferred rake of −180°. The apriori values (Table 1) are chosen based on the magnitudeof the event as well as on previous published studies (Weiet al., 2011; Mendoza and Hartzell, 2013).

Main Event: Single Velocity Model. The posterior solutionobtained by implementing a single velocity model is shownin Figures 11c and 12c for slip and rise-time distribution,respectively, for the main event of the 2010 El Mayor–Cucapah earthquake. Using a single velocity model, the peakslip inferred is 983 cm near the hypocenter, which is higherthan the peak value found by Wei et al. (2011) of 700 cm butconsistent with the value of Mendoza and Hartzell (2013).Wei et al. (2011) shows higher amplitude slip near the hypo-

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Figure 9. Slip velocity distributions for the 2011 Van, eastTurkey, earthquake: (a) distribution with a single velocity modeland an angle constraint of �10° from the average inferred rake,and (b) distribution when using multiple velocity models with anangle constraint. Numbered subfaults are further explored (Fig. 5b).

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Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1539

center with another strong source 30 km to the northwest anda third source 50 km to the southeast. The Bayesian solutionshows a major source at the hypocenter with another source35 km to the southeast. There tend to be more nonzero slipsubfaults in the solution of Wei et al. (2011), which could bea consequence of their Laplacian smoothing. The differenceswith Wei et al. (2011) could also be due to their more com-plex fault parameterization. They subdivided the main eventof the El Mayor–Cucapah earthquake into three distinct faultplanes with different strikes and dips. Mendoza and Hartzell(2013) obtained a slip distribution using an even simpler faultgeometry, consisting of a single fault plane for both the initialsubevent and the main event. The P waves for the El Mayor–Cucapah earthquake are long and complicated, requiring that

they be tapered at a point at which direct source arrivals end.Our solution shows a few subfaults at the far end of the faultwith larger slip. These subfaults only contribute to the ex-treme ends of the P waveforms, near where they are taperedfor the inversion, and may not be source related. Theposterior mean solution (Fig. 11c) obtained using a singlevelocity model gives a seismic moment of 8:79 × 1019 N·m(Mw 7.3), which is slightly lower than that of the9:90 × 1019 N·m (Mw 7.3) found by Wei et al. (2011).The rise-time distribution (Fig. 12c) shows a similar patternto the subevent. The slip velocity distribution (see Ⓔ theelectronic supplement to this article) is characterized by themost energetic source at the hypocenter, with laterally de-creasing slip velocities away from the hypocenter.

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Figure 11. Slip distributions for the 2010 El Mayor–Cucapah, Baja California, earthquake: (a) initial subevent slip distribution with asingle velocity model and an angle constraint of�10° from the average inferred rake, (b) subevent distribution when using multiple velocitymodels, (c) main event distribution with a single velocity model with an angle constraint, and (d) main event distribution when using multiplevelocity models. Numbered subfaults are further explored (Fig. 5c,d).

1540 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

Main Event: Multiple Velocity Model. Figures 11d and 12dpresent the posterior mean solution, given velocity modelvariability in velocity and layer thickness for slip and risetime, respectively. The slip distribution shares similar char-acteristics with a single velocity model (Fig. 11c). The peakslip of 984 cm is nearly identical. The most prominent con-sequence of including velocity-model uncertainty is thesmoothing effect, causing slip to be distributed among agreater number of subfaults. The seismic moment increasesfrom 8:79 × 1019 N·m (Mw 7.3) to 9:35 × 1019 N·m(Mw 7.3), which is more consistent with the finding ofWei et al. (2011). The rise-time (Fig. 12d) and slip velocitydistributions (see Ⓔ the electronic supplement to this ar-

ticle), given velocity model uncertainty, are similar to thoseof the single velocity solution.

Main Event: Uncertainties. Figure 5d presents scatter plotsof rise times versus slip for the four numbered subfaults (27,64, 67, and 73), in Figure 11c and 11d, for a single velocitymodel and multiple velocity models, respectively. Figure 5dshows further similarities to that of the previous cases, high-lighting the slight linear trend between slip and rise time withmultiple regions representing the multiple velocity models.

The associated uncertainties for the subfaults for the sin-gle velocity solution (Fig. 11c) and multiple velocity solution(Fig. 11d) are also plotted as histograms (see Ⓔ the elec-

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Figure 12. Rise-time distributions for the 2010 El Mayor–Cucapah, Baja California, earthquake: (a) initial subevent rise-time distri-bution with a single velocity model and an angle constraint of �10° from the average inferred rake, (b) subevent distribution when usingmultiple velocity models, (c) main event distribution with a single velocity model with an angle constraint, and (d) main event distributionwhen using multiple velocity models. Numbered subfaults are further explored (Fig. 5c,d).

Finite-Fault Bayesian Inversion of Teleseismic Body Waves 1541

tronic supplement to this article). The histograms show sim-ilar characteristics to the synthetic case, the 2011 Van earth-quake, and the subevent of the 2010 El Mayor–Cucapahearthquake, showing broader distributions when consideringmultiple velocity models. With the implementation of veloc-ity model variability, the uncertainty in slip increases by afactor of 3.9. Table 4 shows the mean values and standarddeviations for the subfaults compared.

The model rise-time histograms for a single velocitymodel and multiple velocity models are shown inⒺ the elec-tronic supplement to this article, highlighting the same attrib-utes presented in the slip distributions. Given multiplevelocity models, the distribution of rise time broadens.Table 4 shows the mean values and corresponding standarddeviations for the rise time of the subfaults compared.Introducing velocity model variability causes uncertaintyto increase by a factor of 1.3.

Main Event: Waveform Fit. The comparison of the 40 tele-seismic P-wave records with the forward-modeled wave-forms calculated from the slip distribution of the singlevelocity model (Figs. 11c and 12c) and the forward-modeledslip distribution using velocity model variability (Figs. 11dand 12d) are shown in Ⓔ the electronic supplement to thisarticle. Given the complexity of the event, waveforms forboth solutions do not fit the data as well as Wei et al. (2011;see Table 5 for L2 norms). To improve the model misfit, amore complex fault parameterization may be required.

Conclusions

We conducted a finite-fault analysis of the resolvingpowers and associated uncertainties of teleseismic bodywaves in a Bayesian framework using a synthetic case, the2011 Van, east Turkey, earthquake, and the 2010 El Mayor–Cucapah, Baja California, earthquake. The Bayesianframework (CATMIP) allows us to refrain from using regu-larization techniques, dismissing smoothing, moment mini-mization, and positivity constraints on the solution, all ofwhich are fairly standard in other common source inversiontechniques. For each scenario, we present two distinct solu-tions to explore some of the epistemic uncertainty. The firstsolution utilizes a single velocity model, whereas the secondsolution utilizes 11 velocity models, varying the P- and S-wave velocity and layer thickness.

The synthetic case shows degraded resolution near thehypocenter, with and without velocity model variability, as aconsequence of small arrival-time differences at teleseismicstations between subfaults, resulting in a spatial smearing.The rake is reasonably well resolved when no angle con-straint is applied for the synthetic case, as is the moment.The addition of SH waves did not help constrain the rake.With the inclusion of multiple velocity models, the standarddeviations for slip values increase by a factor of 3.3, increas-ing the uncertainty of the solution while sampling a widerrange of values for each subfault. For all solutions (single

velocity model with and without angle constraint and multi-ple velocity models), the waveform fits to the data are nearlyidentical.

The 2011 Van earthquake shows a spatial smoothing inslip when including velocity model variability, which in-creases the total number of nonzero slip subfaults. The rakeis unresolvable when no angle constraint is applied when us-ing P or P and SH waves. Standard deviations for the slipincrease by a factor of 4.7 when considering velocity modelvariability compared with using a single velocity model. Ingeneral, the waveforms representing both solutions fit thedata well, highlighting qualitatively small misfits betweenthe data and solution waveforms. Both solutions show sim-ilar slip distributions to those presented by Hayes (2011) andMendoza and Hartzell (2013), each using a different inver-sion scheme.

The 2010 El Mayor–Cucapah earthquake is the mostcomplicated scenario, because the event is divided into aninitial subevent and a main event. The subevent accountsfor the first 15 s of record and contains relatively simplisticwaveforms. Solutions with and without velocity model un-certainty show similar attributes to those of Wei et al. (2011).With the implementation of multiple velocity models, thestandard deviations increase on average by a factor of 2.2,when compared with using a single velocity model. Giventhe simplistic nature of the subevent, both solutions with andwithout velocity model variability explain the data equiva-lently. The overall waveform fits are similar to that found byWei et al. (2011).

The main source of the El Mayor–Cucapah earthquakehighlighted similar characteristics to the previous scenarios,obtaining a slip distribution containing a spatial smearingeffect when considering multiple velocity models. The slipdistribution, with and without velocity model variability andan angle constraint, shows significant differences from theslip distributions found by Wei et al. (2011) and Mendozaand Hartzell (2013). However, Wei et al. (2011) utilized amore complicated fault geometry, employing three faultplanes with different strikes and dips for the main event.Mendoza and Hartzell (2013) used a simpler fault modelwith a single fault plane for the entire event. When imple-menting multiple velocity models, the standard deviation in-creases by a factor of 3.9, in comparison with using a singlevelocity model. The synthetic waveforms, with and withoutmultiple velocity models, do not fit the data as well as Weiet al. (2011), who used multiple data sets to help constrainthe complexity of the earthquake.

Although each scenario shows unique attributes of theresolution and uncertainties, there are similar characteristicbehaviors throughout all scenarios. Given a fixed rupturevelocity with a specific fault geometry and parameterization,the small timing difference between subfaults around thehypocenter can induce a spatial smearing and degraded res-olution. This is an issue with teleseismic data that can beremoved if near-field records are available. Velocity modelvariability introduces a greater number of nonzero subfaults,

1542 B. S. Clayton, S. H. Hartzell, M. P. Moschetti, and S. E. Minson

also creating a spatial smoothing effect. A multimodal dis-tribution is best representative of the PDFs of slip and risetimes, for certain subfaults, when considering velocity modelvariability. However, this is an artifact of the timingdifferences caused by the velocity models and the model pa-rameterization, as well as by only considering a few velocitymodels. When a change in velocity structure is introduced,the timing difference can shift in which slip resides to adja-cent subfaults. Rake is generally not well resolved without anangle constraint, except given a simple source; as such, weusually allow the rake to diverge�10° from the inferred rakeof the earthquake. The total seismic moment for each eventis well resolved. The rise times corresponding to lower slipvalues are poorly resolved when considering either single ormultiple velocity models. For the two earthquakes consid-ered, there is a tendency for shorter rise times and large slipvelocities near the hypocenter. The slip distributions withand without velocity model variability contain no dramaticdifferences, resulting, generally, in equivalent waveforms.The standard deviations of slip and rise time increase whenconsidering uncertainty in the velocity model. The slip dis-tributions obtained from teleseismic body waves in theBayesian inference, excluding the main event of the 2010El Mayor–Cucapah earthquake, were comparable to otherstudies using different and multiple data sets with differentinversion schemes. The utilization of both teleseismic P andSH waves provided no additional constraints to help resolvethe rake, possibly reflecting the difficulty of obtaining cleanSH waveforms.

To foster the knowledge of the resolving power of tele-seismic body waves in a Bayesian framework with associ-ated uncertainties, further exploration should be conductedto see the effect on the posterior, including the effect of usingdifferent functional forms for STFs, choosing differentmodel parameterizations, and utilizing more complicatedfault geometries. With a Bayesian framework, the numberof model parameters can be restricted compared with moreconventional approaches, due to the computational effort in-volved. Even with a few hundred parameters, a supercom-puter is desirable. However, significant insight is gainedfrom an examination of the model parameter posterior PDFsthat is not obtained from other methods.

Data and Resources

All data used in this study were obtained through Incor-porated Research Institutions for Seismology (IRIS) and canbe downloaded from the IRIS Data Management Center atwww.iris.edu (last accessed July 2016). Source mechanismsare from the Global Centroid Moment Tensor (CMT) Projectdatabase (www.globalcmt.org, last accessed July 2016). Thealgorithm for the Bayesian inversion is published in Geo-physical Journal International (Minson et al., 2013). Theteleseismic body-wave synthetics code is available athttps://github.com/usgs/finite-fault (last accessed July 2016).This work used the Extreme Science and Engineering

Discovery Environment (XSEDE), which is supported byNational Science Foundation Grant Number ACI-1053575.The authors acknowledge the Texas Advanced ComputingCenter (TACC) at The University of Texas at Austin for pro-viding high performance computing resources (www.tacc.utexas.edu, last accessed July 2016).

Acknowledgments

We would like to acknowledge Carlos Mendoza for helpful discus-sions; and thoughtful reviews from Martin Mai and an anonymous reviewer.

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