Ⓔ

Multicomponent C3 Green’s Functions for Improved

Long-Period Ground-Motion Prediction

by Yixiao Sheng, Marine A. Denolle, and Gregory C. Beroza

Abstract The virtual earthquake approach to ground-motion prediction usesGreen’s functions (GFs) determined from the ambient seismic field to predictlong-period shaking from scenario earthquakes. The method requires accurate relativeGF amplitudes between stations and among components; however, the amplitudes ofambient-field GFs are known to be subject to biases from uneven source distribution.We show that multicomponent, higher order cross correlations are significantly lessbiased than the conventional first-order cross correlation, and we demonstrate thatthey provide a more reliable prediction of observed ground-motion amplitudes fora recent moderate earthquake on the San Jacinto fault in southern California.

Electronic Supplement: Examples of observed and virtual earthquake-derivedthree-component velocity amplitude spectrum.

Introduction

Accurately predicting the strength of ground shaking inearthquakes, an essential part of seismic-hazard analysis, istypically carried out using empirical ground-motion predic-tion equations (GMPEs) (e.g., Boore and Atkinson, 2008).GMPEs are constructed through fitting observations ofground-shaking intensity with parametric equations repre-senting source, path, and site contributions. Lack of data inclose to large earthquakes leads to significant uncertainty inGMPEs for events that pose the greatest threat. This situationmotivated seismologists to develop physics-based ap-proaches for scenario ground-shaking simulations (Olsenet al., 2006; Day et al., 2012) and to use such simulationsfor probabilistic seismic-hazard analysis (Graves et al.,2011). Accurate ground-motion quantification requires bothexceptional computational effort (Cui et al., 2013) and com-prehensive knowledge of 3D crustal structure (Shawet al., 2015).

Denolle et al. (2013) proposed an alternative ground-motion prediction method, the virtual earthquake approach(VEA), which does not require comprehensive knowledge ofcrustal structure, nor exceptional computational effort. VEAtakes the surface impulse response (Green’s function [GF])proportional to the time derivative of cross correlation of theambient seismic field and corrects it for source depth, sourcemechanism, and finite-fault effects of earthquakes of interest.This method captures long-period sedimentary basin ampli-fication effects (Denolle, Dunham, et al., 2014; Denolle,Miyake, et al., 2014) and has shown promise for character-izing the strength of long-period earthquake shaking.Moreover, Viens et al. (2014) coupled the deterministic

low-frequency ground motion with a nonstationary stochas-tic model and successfully extended ground-shaking simula-tions to 50 Hz.

Accurate ambient-field GFs are foundational to reliableground-motion prediction, and studies of the ambient fieldhave shown steady progress and innovation resulting insteadily improving GF estimates. Snieder and Safak (2006)pointed out that deconvolution can also be used to retrievethe impulse response of a system. Prieto and Beroza (2008)used deconvolution to extract not only phase but also relativeamplitude and showed that relative amplitudes of GFs re-flected ground-motion amplification observed from earth-quake records. Amplitude measurements can be difficultto interpret (Prieto et al., 2011), and the uneven distributionof noise sources can bias such amplitude measurements(Tsai, 2011; Stehly and Boué, 2017). Typically, in southernCalifornia, noise signals generated by nonlinear interactionof the ocean swell dominate the 5–10 s frequency band,forming a strongly directional noise wavefield (Longuet-Higgins, 1950; Stehly et al., 2006). Symmetry in the cross-correlation function, a metric for stability of the GF ampli-tude, is affected by this directionality (Paul et al., 2005; Yaoet al., 2006). Wapenaar et al. (2008) showed that interferom-etry by multidimensional deconvolution can mitigate theeffects of source irregularity.

Campillo and Paul (2003) demonstrated that it is alsopossible to use the diffuse waves of the earthquake coda toconstruct the station-to-station response. This is possible be-cause the scattered waves of the coda form a diffuse wave-field with scatterers acting effectively as secondary sources.

BSSA Early Edition / 1

Bulletin of the Seismological Society of America, Vol. XX, No. XX, pp. –, – 2017, doi: 10.1785/0120170053

2017053_esupp.zip

Such use of earthquake coda can be extended to the coda ofcorrelation as well. Stehly et al. (2008) used the correlationof the coda of the vertical–vertical ambient-field correlation,which they referred to as C3, to demonstrate that the scat-tered waves of the correlation coda also form a diffuse wave-field even if the original-wave excitation (raw seismic noise)is highly directional. Froment et al. (2011) further tested theC3 method and suggested that the relatively even distributionof secondary source stations would lead to more symmetricGF estimates. Ma and Beroza (2012) and Spica et al. (2016)showed that C3 can bridge seismic networks operating atdifferent times by estimating the GFs between stations thatdid not record simultaneous seismic noise. Zhang and Yang(2013) suggested that C3 greatly reduces amplitude bias.

In this article, we construct C3 GFs using the east–vertical and north–vertical components, in addition to thevertical–vertical component in correlation (C1). We test themulticomponent C3 results with stations near the San Jacintofault zone, California. Though noisier than C1 GFs, theestimated GFs from multicomponent C3 are comparable inshape and substantially more symmetric than those from C1.We extend the vertical–vertical GFs to nine-componentGreen’s tensors and incorporate them into VEA to synthesizeseismograms, which we compare with recorded strong-motion records of the 2016 Mw 5.2 Borrego Springs earth-quake. We find that the virtual earthquake seismogramsderived by the multicomponent C3 method lead to a morereliable estimate of the observed earthquake records.

Data and Methodology

The San Jacinto fault has a high long-term slip rate, in-dicating that the hazard from the fault is high (Rockwell et al.,1990). Lozos (2016) showed that the San Jacinto fault mightrupture together with the San Andreas fault, leading to verylarge events. The San Jacinto fault zone (SJFZ) is one of themost seismically active regions in southern California withabundant microseismicity that includes five Mw >5 earth-quakes since 1981. The most recent of these was the 10 June2016 Mw 5.2 Borrego Springs earthquake, which provides agood opportunity to use VEA to assess earthquake ground-motion hazard from SJFZ events.

To represent the shaking from a real earthquake, VEArequires a reference broadband station close to the epicenter,such that GFs estimated from the reference station are rep-resentative of ground motion excited by the earthquake. Wechoose station YN.TR02 (from YN network; see Data andResources), which is about 2.5 km away from the epicenter,as the earthquake virtual source. We use all 230 availablebroadband stations in the Southern California Seismic Net-work (CI network; see Data and Resources). We focus ourground-motion comparison on those stations closest to theBorrego Spring earthquake that have high-quality C1 results.This excludes five stations in the Salton trough that werecharacterized by strong acausal arrivals in the C1 results(Fig. 1), possibly due to either local, or directionally biased,

noise sources. We use the seismic ambient-field data of allthree components (vertical Z, north N, east E) recorded con-tinuously for the year 2015 to estimate GFs. To mitigate thebias in GFs estimated by C1, we use the multicomponent C3method to construct the Green’s tensors as described below.Although we explain the methodology to obtain the vertical–vertical (ZZ) C3, the approach is identical for the other eightcomponents of a Green’s tensor (ZE, ZN, EZ, EN, EE, NZ,NN, and NE). The instrumental responses are removed. Wedivide the data into 30-min-long time series and discardthose with spikes larger than 8 times the standard deviationof the window.

We first extract the ZZ GF between the virtual sourceYN.TR02, marked as R1 below, and a receiver station R2(any inverted triangle in Fig. 1) by taking the following steps(shown schematically in Fig. 2).

1. We select an intermediate station, denoted as S, differentfrom R1 and R2, and from the network (stars as well asthe rest of the inverted triangles in Fig. 1) to act as a sec-ondary source. We Fourier transform the selected dataand compute the S-R1 and S-R2 noise correlations basedon deconvolution interferometry to preserve their relativeamplitudes (Prieto and Beroza, 2008; Denolle et al.,2013). The equation we use is

EQ-TARGET;temp:intralink-;df1;313;109C1S−Ri�ω� ��u�xRi;ω�u��xS;ω�

fju�xS;ω�jg2�; �1�

−118°

−118°

−117°

−117°

−116°

−116°

−115°

−115°

33° 33°

34° 34°

35° 35°

10 June 2016 Mw 5.2

YN.TR02

Background station

Receiver station

Virtual earthquake source YN.TR02

10 June, 2016 Mw 5.2 Borrego Springs earthquake focal mechanism

Figure 1. Reference map showing location and focal mecha-nism of the 2016 Mw 5.2 Borrego Springs earthquake and the sta-tions used in this study. The epicenter is indicated by the focalmechanism of the event. The earthquake virtual source is markedwith a triangle overlapped by the focal mechanism plot. Invertedtriangles represent the receiver stations in this study. The other sta-tions in the CI network are denoted as background stations, markedby stars in the figure. These background stations, along with mostreceiver stations, other than the chosen one, serve as secondarysources in constructing C1 Green’s functions (GFs). The colorversion of this figure is available only in the electronic edition.

2 Y. Sheng, M. A. Denolle, and G. C. Beroza

BSSA Early Edition

in which h·i denotes stacking over all time windows, ωrefers to the angular frequency, Ri is either R1 or R2, xRirefers to the location of the station, asterisk denotes thecomplex conjugate, and f·g represents 10 points smooth-

ing over the spectrum, with df � 0:005 Hz. We use thevertical component recorded at R1 and R2 to get the ZZGF but include the north, east, and vertical componentsfor S. The notation C1XZS−Ri�ω� stands for using the Xcomponent of S and the Z component of either R1 or R2.We take the inverse Fourier transform of C1XZS−Ri�ω� andobtain the time series C1XZS−Ri�t�.

2. Following Stehly et al. (2008) and Froment et al. (2011),we select a 1200-s-long time window in the late coda ofeach C1 correlation, starting at twice the Rayleigh-wavetravel time. The Rayleigh-wave velocity we choose isfixed at 3 km=s. By doing so, we ensure that the codawindow starts well after the surface-wave arrival at allstations.

3. We cross correlate the positive part of the coda in theS-R1 correlation, with the positive part of the coda in theS-R2 correlation, as well as the negative part of the codain the S-R1 correlation, with the negative part of the codain the S-R2 correlation. We denote their average timeseries as C3ZXXZS �t�. We whiten the coda spectra between4 and 10 s and use Welch’s method to speed the conver-gence of the estimated GFs (Seats et al., 2012). We dividethe 1200-s-long time series into 500-s-long 50% overlap-ping windows. Cross correlation is then constructed ineach subwindow and the average is kept as the result.

4. We repeat this process for all N (229 in our study) sec-ondary sources from the network and stack all the C3functions to obtain C3ZXXZ�t� between R1 and R2:

EQ-TARGET;temp:intralink-;df2;313;397C3ZXXZ�t� � 1N

XNi�1

C3ZXXZSi �t�; X∈ fZ;N;Eg:

�2�

C3ZXXZ�t� corresponds to the ZZ GF between R1 and R2but with the excitation at the secondary source stationtaken in X direction.

5. To improve the signal-to-noise ratio (SNR), we furthertake the average of C3ZZZZ, C3ZEEZ, and C3ZNNZ toobtain an improved estimate that we denote as C3ZΣZ:

EQ-TARGET;temp:intralink-;df3;313;259C3ZΣZ�t� � 13�C3ZZZZ�t� � C3ZNNZ�t� � C3ZEEZ�t��:

�3�

Through this processing, we preserved the amplitude forC3 by preserving the amplitudes for the individual C1contributions. The relative amplitudes of C3ZΣZ GFs areshown in Figure 3c.

The difference between the C3 method from Stehly et al.(2008) and Froment et al. (2011) and our method is that in-stead of using only the vertical components at the secondarysource stations, we incorporate all components, that is, weconsider all the possible orientations of the virtual impulse-force sources that could contribute to the wavefield of the C1coda. Figure 3 compares the ZZ GFs extracted by different

C1S-R1

C1S-R2

Time (s)–400 –300 –200 –100 0 100 200 300 400

coda coda

x5

S

C1S-R1C1S-R2

R1

R2

C3 R1-R2

S

C1S-R1C1S-R2

R1

R2

Z N

E

(a)

(b)

(c)

Figure 2. Schematic diagram for multicomponent C3, showingvertical response at R2 from a vertical impulse at R1. (a) We firstcalculate the cross correlations (C1) between a virtual source (S)and two receivers (R1 and R2). In addition to using vertical com-ponent, we also use north or east component of the virtual source.Note that we only use vertical components of R1 and R2. (b) We useboth the causal and anticausal parts of the C1 coda. The coda partshave been amplified by a factor of 5 in the figure for clarity. (c) Wecompute the cross correlations between coda waves from tworeceivers, separately on causal and anticausal parts, and take aver-age of them to estimate the GF from R1 to R2. The color version ofthis figure is available only in the electronic edition.

Multicomponent C3 Green’s Functions for Improved Long-Period Ground-Motion Prediction 3

BSSA Early Edition

methods. Figure 3a shows that, though noisier, the GFs forC3ZZZZ�t� are more symmetric than those for C1ZZ�t�. Toquantify the symmetry of the GFs, we calculate the correlationcoefficients between the causal and flipped anticausal parts ofeach C1ZZ�t� and C3ZZZZ�t�. More symmetric GFs wouldhave higher correlation coefficient values. The distributionsof the coefficients are shown in Figure 4. Because C3ZNNZ�t�and C3ZEEZ�t� also contain highly coherent signals, they

contribute to a more stable overall C3 result. Following Liuet al. (2016), we define the SNR as the ratio of peak amplitudeof the wave-packet envelope, which is calculated with the Hil-bert transform, over the root mean square of the coda noise.The 300-s-long noise window starts at 100 s after the arrival ofthe peak amplitude. For each station pair, we compute theSNR both on the causal (SNR�) and anticausal parts (SNR−)and take the ratio of SNRs for C3ZΣZ�t� to those for

–100 –50 0 50 100Time (s)

0

50

100

150

200

Dis

tanc

e (k

m)

C1ZZ

–100 –50 0 50 100Time (s)

0

50

100

150

200C3ZZZZ

–100 –50 0 50 100Time (s)

0

50

100

150

200 C3ZEEZ

–100 -50 0 50 1000

50

100

150

200C3ZNNZ

(a)

–100 –50 0 50 100Time (s)

0

50

100

150

200C3ZΣZ(b) Amplitudes of causal parts

Amplitudes of anticausal parts

Average amplitudes

(c)

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

C1ZZC3ZΣZ

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

Rel

ativ

e am

plitu

de

C1ZZC3ZΣZ

0 5 10 15 20 25 30 35 40 45 50

Number of stations

0

0.02

0.04

0.06

0.08

C1ZZC3ZΣZ

Dis

tanc

e (k

m)

Figure 3. Comparison of GFs retrieved using different methods. All waveforms are band-pass filtered between 4 and 10 s and normalizedto their peak amplitude. (a) Comparison between C1ZZ and multicomponent C3s (C3ZZZZ, C3ZEEZ, and C3ZNNZ). Note that multicomponentC3s are more symmetric than C1s. (b) Seismograms of C3ZΣZ, the summation of C3ZNNZ, C3ZEEZ, and C3ZNNZ. The SNR of C3ZΣZ isimproved for both the acausal artifacts and coda. (c) Relative amplitudes of C1ZZ and C3ZΣZ. The stations are arranged with increasingdistance from the earthquake virtual source. The general decreasing amplitude from left to right is associated with geometric spreading.Local amplifications are captured by both C1 and C3 GFs, showing that relative amplitudes are preserved through C1 and multicomponentC3 method. The color version of this figure is available only in the electronic edition.

4 Y. Sheng, M. A. Denolle, and G. C. Beroza

BSSA Early Edition

C3ZZZZ�t�, that is, SNR��C3ZΣZ�t��=SNR��C3ZZZZ�t�� andSNR−�C3ZΣZ�t��=SNR−�C3ZZZZ�t��. The average ratios overall station pairs are 1.29 for the causal part and 1.31 for theanticausal part (Fig. 5c), demonstrating the improvement inSNR for C3ZΣZ�t�. Improvements in SNR can also be ob-served visually by comparing C3ZΣZ�t� in Figure 3b withC1ZZZZ�t� in Figure 3a.

We repeat the same procedure to obtain all the compo-nents of the Green’s tensors and rotate the coordinate systemfrom north–east–vertical (vertical positive downward) to ra-dial–transverse–vertical. Figure 6 shows examples of com-parisons between horizontal–horizontal (e.g., radial–radial[R–R] and transverse–transverse [T–T]) GFs extracted fromC3 with vertical impulse source and those with multi-component sources. Horizontal components particularly ben-efit from the multicomponent C3 estimation (Fig. 5a,b). Theaverage of C3s from different source components strength-ens coherent signals and suppresses incoherent ones,improving the reliability of the estimated GFs. It is interest-ing to note that C3TZZT�t�, constructed from C1ZT�t�,depends on P=SV to SH coupling. In layered media, the cou-pling between Rayleigh and Love waves does not exist, andtherefore the ZT component of the Green’s tensor is in theory

zero (Aki and Richards, 2002). That is, weassume the P–SV and SH fields are excitedseparately. The clear signals given byC3TZZT�t� result from the complex 3Dcrustal structure of the study area that cou-ples P–SV into SH motion and vice versa.We observe perceptible improvements onT–T GFs by incorporating horizontalsource components in C3. To maximizeSNRs and to retain relative amplituderelationships among components of aGreen’s tensor, it is necessary to use allthree source components.

Virtual Earthquake Seismograms

We exploit the symmetry of the GFsand average the causal and flipped anti-causal time series. We make the approxi-mation that a Green’s tensor’s cross termsZT, TZ, RT, and TR are zero and only useZZ, RR, RZ, ZR, and TT components inthe excitation of the virtual earthquakeseismograms. The time-domain GFs,taken as the time derivatives of the C3s,are Fourier transformed into the frequencydomain, noted as GRR�ω�;GRZ�ω�;GZR�ω�, GZZ�ω�, and GTT�ω�. We applyequations (4)–(6) to correct surface-im-pulse responses to displacements radiatedfrom a buried double-couple point source(Denolle et al., 2013). We choose a Fou-

rier transform convention such that an outward propagatingwave (in the x direction) can be written proportional toexp�iωt − ikx�. To simplify the notation, we suppressexplicit ω dependence while retaining the source-depthdependence h. For the Love-wave component:

EQ-TARGET;temp:intralink-;df4;313;293uT ≈1

l1�0��−ikLMTRl1�h� �MTZl′1�h��GTT ; �4�

and for Rayleigh-wave components:

EQ-TARGET;temp:intralink-;df5;313;236

uZ ≈1

r1�0��−ikRMRRr1�h� �MRZr′1�h��GZR

� 1r2�0�

�−ikRMZRr2�h� �MZZr′2�h��GZZ; �5�

EQ-TARGET;temp:intralink-;df6;313;163

uR ≈1

r2�0��−ikRMZRr2�h� �MZZr′2�h��GRZ

� 1r1�0�

�−ikRMRRr1�h� �MRZr′1�h��GRR: �6�

l1�h� is the fundamental Love-wave displacement eigenfunc-tion at depth h, whereas r1�h� and r2�h� are the horizontal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8(a)

(b)

Cou

nts

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

Cou

nts

CC of C1ZZ+ and C1ZZ-

CC of C3ZZZZ+ and C3ZZZZ-

Correlation coefficient (CC)

Figure 4. (a) Histogram of correlation coefficients of the causal and flipped anti-causal part of C1ZZ GFs. Horizontal axis indicates the coefficient value, varying from 0to 1. We divide the coefficients into 20 bins and each bin has equal width of 0.05. Thevertical axis gives the number of the GFs in each coefficient bin. (b) Same as (a) but forC3ZZZZ GFs. C3ZZZZ GFs have consistently and systematically much higher correlationcoefficients between the causal and anticausal waveforms, indicating they are more sym-metric than C1ZZ GFs. The color version of this figure is available only in the electronicedition.

Multicomponent C3 Green’s Functions for Improved Long-Period Ground-Motion Prediction 5

BSSA Early Edition

and vertical displacement eigenfunctions for fundamental-mode Rayleigh waves. h is the earthquake source depth,which is taken to be 12.3 km for the Borrego Springs event(from U.S. Geological Survey catalog; see Data and Resour-ces). M is the earthquake moment spectrum tensor, whereaskL and kR are the Love and Rayleigh wavenumbers. The cor-rection terms at different frequencies for each component areshown in Figure 7a. The corner frequency fc used in ourstudy is 0.52 Hz, estimated based on the model proposedin Hanks and Thatcher (1972), with an assumed stress dropΔσ � 3 MPa, the observed seismic moment M0, and anaveraged shear velocity β � 3 km=s. We use an omega-squared source model as the moment rate function, shiftedby the source duration:

EQ-TARGET;temp:intralink-;df7;55;190S�ω� � e− iω2πfc

1� � ω2πfc

�2 : �7�

Even though relative amplitudes among different compo-nents at different stations are preserved, the retrieved GFsare dimensionless. Because of the different and unknowninput source energy for Rayleigh and Love waves in theambient wavefield, we use separate normalization factors

for Rayleigh and Love waves to calibratethe virtual earthquake seismograms toreal earthquake displacement amplitudes.The factors are determined by comparingthe observed and synthetic seismograms ofanother earthquake that occurred near thevirtual source. We take the transverseamplitude ratio as the Love-wave normali-zation factor, whereas the average of thevertical and radial amplitude ratios is theRayleigh-wave normalization factor.

We compare the waveforms inFigure 7b. The seismograms we constructmatch the observations reasonably well.They capture the amplitude differenceamong different components (e.g., thesmall amplitudes of Rayleigh and largeamplitudes of Love waves on stationsCI.MTG and CI.EML), which highlightsthe importance of making source depthand mechanism corrections in the VEA.To highlight the accuracy of the VEA-pre-dicted amplitudes with multicomponentC3, we show in Figure 8 peak amplitudesfor the virtual and real earthquake wave-forms for the Borrego Spring earthquake.To make a better comparison, we alsoshow the results of using the C1 Green’stensor for constructing virtual earthquakeseismograms. There is a good matchbetween the observed and predicted ampli-

tudes, both for the C1 and multicomponent C3 results. Weestimate the best-fitting linear trend, with L1-norm minimi-zation between the predicted and observed peak amplitudeson a log scale. The corresponding slopes, shown in Figure 8,are close to one for the two cases. Multicomponent C3, how-ever, yields a smaller sum squared residual, indicating itshigher reliability in predicting strong ground motions. Wealso plot several velocity amplitude spectra in Ⓔ Figure S1(available in the electronic supplement to this article). Bothmulticomponent C3 and C1 virtual seismograms match wellwith the observed strong motions in the 4–10 s period bandwith similar spectral amplitudes.

Conclusions

Because coda waves are more diffuse than the raw am-bient seismic field, we use the coda of the first-order corre-lations (C1) to improve our estimate of GFs, therebyimproving the symmetry and amplitude stability. As sug-gested by Froment et al. (2011), we use widely distributedbackground stations as secondary sources and enhance thesource distribution, therefore reducing the bias caused by un-even noise source excitation in the first-order correlation. Weshow that it is possible to reconstruct an improved Green’stensor with this approach if we use all three components of

0.8 1 1.2 1.4 1.6 1.80

5

10

15

0.8 1 1.2 1.4 1.6 1.80

5

10

15

0.8 1 1.2 1.4 1.6 1.80

5

10

0.8 1 1.2 1.4 1.6 1.80

5

10

0.8 1 1.2 1.4 1.6 1.80

5

10

15

0.8 1 1.2 1.4 1.6 1.80

5

10

15

Mean : 1.30 Mean : 1.28

Mean : 1.31 Mean : 1.38

Mean : 1.31 Mean : 1.29

SNR+(C3RΣR)/SNR+(C3RZZR)SNR-(C3RΣR)/SNR-(C3RZZR)

SNR+(C3TΣT)/SNR+(C3TZZT)SNR-(C3TΣT)/SNR-(C3TZZT)

SNR+(C3ZΣZ)/SNR+(C3ZZZZ)SNR-(C3ZΣZ)/SNR-(C3ZZZZ)

(a)

(b)

(c)

SNR ratio SNR ratio

Figure 5. Histograms of the ratio of the multicomponent C3 signal-to-noise ratios(SNRs) and single-component C3 SNRs, for all three components (from a to c: RR, TT,ZZ) on both (right) causal and (left) anticausal parts. Multicomponent C3 GFs generallyhave higher SNR than original C3 GFs, particularly for TT components. The colorversion of this figure is available only in the electronic edition.

6 Y. Sheng, M. A. Denolle, and G. C. Beroza

BSSA Early Edition

the wavefield. Multicomponent C3 GFs have more reliableamplitudes than C1 GFs and higher SNR than the original C3GFs. When combined with VEA, we find an improved ac-curacy in ground-motion predictions. We compare syntheticseismograms with recorded waveforms from the 2016Mw 5.2 Borrego Springs earthquake. The virtual earthquakeseismograms match well with the real displacement recordsboth in phase and amplitude, which verifies that VEAprovides reliable prediction of long-period ground motions(4–10 s) for the SJFZ earthquake.

In this study, we neglect possible coupling betweenP–SV and SH waves, which are expressed in the cross termsZT, TZ, RT, and TR of the Green’s tensors. We observeenergy coupling across these components in the 4–10 speriod range. Such wave conversions have been predictedat the edge of sedimentary basins in numerical simulation ofwave propagation (Day et al., 2012). Essentially, consider-ing this basin edge effect would yield more reliable ground-

motion predictions; in constructing C3,we use all available stations from theCI network, other than the virtual sourceR1 and the receiver station R2, as thesecondary sources. For different pairsof (R1, R2), the distributions of the sec-ondary sources are different. We ignorethe possible influence on C3s fromsuch changes in the secondary sourcedistributions. Incorporating these effectsinto VEA is a possible future researchdirection.

Constructing multicomponent C3 iscomputationally somewhat more expensivethan the first-order correlation, especiallywith a large number of background sta-tions. With C1 already computed, however,the increase of the computation time ismodest. Assume that T is the unit time tocompute a single cross correlation. With Ntime windows of seismic noise, a numberof stations (Nsta), the total computationalcost to calculate the nine-component C1between each nonredundant station pairsis Nsta × �Nsta − 1�=2 × 9 × N × T. Con-sidering that we use both causal and anti-causal parts of C1 GFs and all threecomponents of the secondary source, andthat there are Nsta-2 secondary sourcesper station pair, the additional cost toconstruct multicomponent C3 Green’s ten-sor between all station pairs is 2× 3×�Nsta− 2�×Nsta × �Nsta− 1�=2× 9× T.The ratio of the additional cost to calculatethe C3s from the C1s is 6 × �Nsta − 2�=N,and taking the practical example thatNsta � 200 and N � 48 × 365 (48 win-

dows of 30 min per day for 1 year) with Nsta ≪ N, the in-crease is only about 7% the computation time of the C1s.Withgrowing seismic networks and duration of continuous records,high-performance computing is becoming necessary to proc-ess ambient noise cross correlation.

Prieto et al. (2009) and Lawrence and Prieto (2011)showed that spatial coherency of the ambient seismic fieldcan be used for attenuation tomography. Stehly and Boué(2017) suggested that there could be a strong trade-off be-tween the noise source distribution and the attenuation ex-tracted from the amplitude decay of the noise correlations,even for a linear array. On the other hand, Zhang and Yang(2013) demonstrated that C3 yields more reliable attenua-tion estimates for linear arrays, which could result from theenhanced source distribution with the C3 method. Ourstudy shows considerable improvement in the estimatesof the GFs at all components of the Green’s tensor andin particular for the Love waves. With our multicomponentC3 technique, it would be straightforward to generalize the

–200 –100 0 100 200–1

0

1

–200 –100 0 100 200–1

0

1

–200 –100 0 100 200–1

0

1

–200 –100 0 100 200–1

0

1

Time (s) Time (s)

C3RZZR

C3TZZT

C3RΣR

C3TΣT

–200 –100 0 100 200–1

0

1

–200 –100 0 100 200–1

0

1

–200 -100 0 100 200–1

0

1

–200 –100 0 100 200–1

0

1C3TZZT

C3RZZR

C3TΣT

C3RΣR(a)

(b)

SNR+ = 18.11SNR- = 6.88 SNR+ = 19.93SNR- = 11.11

SNR- = 7.74 SNR- = 12.14SNR+ = 15.27 SNR+ = 16.45

SNR- = 11.55 SNR+ = 11.57 SNR+ = 12.43SNR- = 17.17

SNR- = 6.93 SNR+ = 10.81 SNR+ = 13.13SNR- = 12.07

Time (s) Time (s)

Figure 6. Horizontal–horizontal C3s for a vertical impulse source and multi-component sources for station pair (a) YN.TR02-CI.RAG and (b) YN.TR02-CI.GMR.The waveforms are band-pass filtered at 4–10 s and normalized to their peak amplitudefor comparison. Multicomponent C3s have better SNR, especially for TT components.SNRs of the causal part (SNR+) and anticausal part (SNR−) are shown in the plots.

Multicomponent C3 Green’s Functions for Improved Long-Period Ground-Motion Prediction 7

BSSA Early Edition

above attenuation studies from only using ZZ GFs to allnine components. Finally, we note that although our appli-cation has been for ground-motion prediction, improved es-timates of GFs resulting from multicomponent C3 shouldprovide improved results for any application of ambient-field GFs.

Data and Resources

Data used in this study were obtained from the SouthernCalifornia Seismic Network CI (doi: 10.7914/SN/CI) andfrom the San Jacinto Fault Zone Experiment YN (doi:10.7914/SN/YN_2010). U.S. Geological Survey (USGS)

0 0

Time (s)

0

Radial Trans Vertical

MTG

JEM

PLM

MGE

IDO

BLA2

MUR

BBS

HAY GOR

Virtual seismogram Real seismogram

Radial correction

30

210

60

240

90

270

120

300

150

330

180 0

Transverse correction

30

210

60

240

90

270

120

300

150

330

180 0

Vertical correction

30

210

60

240

90

270

120

300

150

330

180 0

MTG

JEM

PLM

POB2EML

GMR

MTG

MTG

JEM

JEM

PLM

PLM

POB2

POB2

EML

EML

GMR

GMR

10 s8 s5 s

(a)

POB2

WWC

KYV

DGR

EML

Radial Trans Vertical

50 50 50 0 0

Time (s)

0

PER

MCT

SVD

RAGGMR

50 50 50

(b)

Figure 7. (a) Correction terms for all components varying with azimuth and the relative locations of some stations. For station CI.EML,for example, the transverse correction is large, whereas the vertical and radial corrections are close to zero (nodes of the focal mechanism),corresponding to large amplitude for the synthesized Love wave but small amplitudes for synthesized Rayleigh waves, matching well withobserved records shown in (b). (b) Comparison between virtual earthquake seismograms and observed earthquake waveforms for the 2016Mw 5.2 Borrego Springs earthquake. The waveforms are band-pass filtered between 4 and 10 s. The color version of this figure is availableonly in the electronic edition.

8 Y. Sheng, M. A. Denolle, and G. C. Beroza

BSSA Early Edition

http://dx.doi.org/10.7914/SN/CIhttp://dx.doi.org/10.7914/SN/CIhttp://dx.doi.org/10.7914/SN/YN_2010

catalog was searched using http://earthquake.usgs.gov (lastaccessed January 2017).

Acknowledgments

The authors thank Fabian Bonilla and Luis A. Dalguer for their valu-able comments, which helped improve and clarify this article. This work wassupported by the Southern California Earthquake Center (SCEC; Contribu-tion Number 7365). SCEC is funded by National Science Foundation (NSF)Cooperative Agreement EAR-1033462 and U.S. Geological Survey (USGS)Cooperative Agreement G12AC20038. This work was also supportedthrough NSF Awards EAR-1520867, titled “Ground Motion Prediction Us-ing Virtual Earthquakes.”

References

Aki, K., and P. G. Richards (2002). Quantitative Seismology, Second Ed.,University Science Books, Sausalito, California.

Boore, D. M., and G. M. Atkinson (2008). Ground-motion prediction equa-tions for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s, Earthq.Spectra 24, no. 1, 99–138.

Campillo, M., and A. Paul (2003). Long-range correlations in the diffuseseismic coda, Science 299, no. 5606, 547, doi: 10.1126/sci-ence.1078551.

Cui, Y., E. Poyraz, K. B. Olsen, J. Zhou, K. Withers, S. Callaghan, J. Larkin,C. Guest, D. Choi, A. Chourasia, et al. (2013). Physics-based seismichazard analysis on petascale heterogeneous supercomputers, Proc. SC'13: Proc. of the International Conf. on High Performance Computing,Networking, Storage and Analysis, Denver, Colorado, 18–21 Novem-ber 2013.

Day, S. M., D. Roten, and K. B. Olsen (2012). Adjoint analysis of the sourceand path sensitivities of basin-guided waves, Geophys. J. Int. 189,1103–1124, doi: 10.1111/j.1365-246X.2012.05416.x.

Denolle, M., E. M. Dunham, G. A. Prieto, and G. C. Beroza (2013). Groundmotion prediction of realistic earthquake sources using the ambientseismic field, J. Geophys. Res. 118, 2102–2118, doi: 10.1029/2012JB009603.

Denolle, M., E. M. Dunham, G. A. Prieto, and G. C. Beroza (2014). Strongground motion prediction using virtual earthquakes, Science 343, 399–403, doi: 10.1126/science.1245678.

Denolle, M., H. Miyake, S. Nakagawa, N. Hirata, and G. C. Beroza (2014).Long-period seismic amplification in the Kanto basin using the ambi-ent seismic field, Geophys. Res. Lett. 41, 2319–2325, doi: 10.1002/2014GL059425.

Froment, B., M. Campillo, and P. Roux (2011). Reconstructing the Green’sfunction through iteration of correlations, C. R. Geosci. 343, no. 8,623–632.

Graves, R., T. H. Jordan, S. Callaghan, E. Deelman, E. Field, G. Juve, C.Kesselman, P. Maechling, G. Mehta, K. Milner, et al. (2011).CyberShake: A physics-based probabilistic hazard model for southernCalifornia, Pure Appl. Geophys. 168, 367–381, doi: 10.1007/s00024-010-0161-6.

Hanks, T., and W. Thatcher (1972). A graphical representation of seismicsource parameters, J. Geophys. Res. 77, no. 23, 4393–4405, doi:10.1029/JB077i023p04393.

Lawrence, J. F., and G. A. Prieto (2011). Attenuation tomography in thewestern United States from ambient seismic noise, J. Geophys. Res.116, no. B06302, doi: 10.1029/2010JB007836.

Liu, X., Y. Ben-Zion, and D. Zigone (2016). Frequency domain analysis oferrors in cross-correlations of ambient seismic noise, Geophys. J. Int.207, 1630–1652, doi: 10.1093/gji/ggw361.

Longuet-Higgins, M. S. (1950). A theory of the origin of microseisms, Phil.Trans. Roy. Soc. Lond. A 243, 1–35.

Lozos, J. (2016). A case for historic joint rupture of the San Andreas and SanJacinto faults, Sci. Adv. 2, no. 3, E1500621.

Ma, S., and G. C. Beroza (2012). Ambient-field Green’s functions fromasynchronous seismic observations, Geophys. Res. Lett. 39, L06301,doi: 10.1029/2011GL050755.

Olsen, K. B., S. M. Day, J. B. Minster, Y. Cui, A. Chourasia, M. Faerman, R.Moore, P. Maechling, and T. Jordan (2006). Strong shaking in LosAngeles expected from southern San Andreas earthquake, Geophys.Res. Lett. 33, L07305, doi: 10.1029/2005GL025472.

Paul, A., M. Campillo, L. Margerin, E. Larose, and A. Derode (2005).Empirical synthesis of time-asymmetrical Green function from thecorrelation of coda waves, J. Geophys. Res. 110, no. B08302, doi:10.1029/2004JB003521.

10–5 10–4 10–3

Predicted PGD (m)

10–5

10–4

10–3

Obs

erve

d P

GD

(m

)

10–5 10–4 10–3

Predicted PGD (m)

10–5

10–4

10–3

Obs

erve

d P

GD

(m

)

C3C1Slo

pe =

1.023

Total

resid

ual =

0.01

12

Slope

= 0.9

94

Total

resid

ual =

0.00

92

RTZ

RTZ

Figure 8. Observed and predicted peak displacement amplitudes, band-pass filtered at 4–10 s. We compare the radial (circle), transverse(square), and vertical (triangle) observed peak ground displacements (PGDs) with the virtual earthquake approach waveforms constructedfrom (a) C1 Green’s tensor and (b) multicomponent C3 Green’s tensor. The black line in each panel represents the L1-linear regression, witheach slope and total residual to ideal fit. The total residual from (b) is smaller than that from (a), indicating that the multicomponent C3Green’s tensor yields more reliable amplitude prediction in constructing virtual earthquake seismograms.

Multicomponent C3 Green’s Functions for Improved Long-Period Ground-Motion Prediction 9

BSSA Early Edition

http://earthquake.usgs.govhttp://dx.doi.org/10.1126/science.1078551http://dx.doi.org/10.1126/science.1078551http://dx.doi.org/10.1111/j.1365-246X.2012.05416.xhttp://dx.doi.org/10.1029/2012JB009603http://dx.doi.org/10.1029/2012JB009603http://dx.doi.org/10.1126/science.1245678http://dx.doi.org/10.1002/2014GL059425http://dx.doi.org/10.1002/2014GL059425http://dx.doi.org/10.1007/s00024-010-0161-6http://dx.doi.org/10.1007/s00024-010-0161-6http://dx.doi.org/10.1029/JB077i023p04393http://dx.doi.org/10.1029/2010JB007836http://dx.doi.org/10.1093/gji/ggw361http://dx.doi.org/10.1029/2011GL050755http://dx.doi.org/10.1029/2005GL025472http://dx.doi.org/10.1029/2004JB003521

Prieto, G. A., and G. C. Beroza (2008). Earthquake ground motion predic-tion using the ambient seismic field, Geophys. Res. Lett. 35, L14304,doi: 10.1029/2008GL034428.

Prieto, G. A., M. Denolle, J. F. Lawrence, and G. C. Beroza (2011).On amplitude information carried by the ambient seismic field,C. R. Geosci. 343, 600–614.

Prieto, G. A., J. F. Lawrence, and G. C. Beroza. (2009). Anelastic earthstructure from the coherency of the ambient seismic field, J. Geophys.Res. 114, no. B07303, doi: 10.1029/2008JB006067.

Rockwell, T., C. Loughman, and P. Merifield (1990). Late Quaternaryrate of slip along the San Jacinto fault zone near Anza, southernCalifornia, J. Geophys. Res. 95, 8593–8605, doi: 10.1029/JB095iB06p08593.

Seats, K. J., J. F. Lawrence, and G. A. Prieto (2012). Improved ambient noisecorrelation functions using Welch’s methods, Geophys. J. Int. 188,513–523, doi: 10.1111/j.1365-246X.2011.05263.X.

Shaw, J. H., A. Plesch, C. Tape, M. P. Suess, T. H. Jordan, G. Ely, E.Hauksson, J. Tromp, T. Tanimoto, R. Graves, et al. (2015). UnifiedStructural Representation of the southern California crust and uppermantle, Earth Planet. Sci. Lett. 415, 1–15, doi: 10.1016/j.epsl.2015.01.016.

Snieder, R., and E. Safak (2006). Extracting the building response usingseismic interferometry: Theory and application to the Millikan Libraryin Pasadena, California, Bull. Seismol. Soc. Am. 96, 586–598.

Spica, Z., M. Perton, M. Calò, D. Legrand, F. Córdoba-Montiel, and A.Iglesias (2016). 3-D shear wave velocity model of Mexico and southUS: bridging seismic networks with ambient noise cross-correlations(C1) and correlation of coda of correlations (C3), Geophys. J. Int. 206,no. 3, 1795–1813.

Stehly, L., and P. Boué (2017). On the interpretation of the amplitude decayof noise correlations computed along a line of receivers, Geophys. J.Int. 209, no. 1, 358–372, doi: 10.1093/gji/ggx021.

Stehly, L., M. Campillo, B. Froment, and R. L. Weaver (2008). Reconstruct-ing Green’s function by correlation of the coda of the correlation (C3)of ambient seismic noise, J. Geophys. Res. 113, no. B11306, doi:10.1029/2008JB005693.

Stehly, L., M. Campillo, and N. M. Shapiro (2006). A study of the seismicnoise from its long-range correlation properties, J. Geophys. Res. 111,no. B10306, doi: 10.1029/2005JB004237.

Tsai, V. C. (2011). Understanding the amplitudes of noise correlationmeasurements, J. Geophys. Res. 116, no. B09311, doi: 10.1029/2011JB008483.

Viens, L., A. Laurendeau, L. F. Bonilla, and N. M. Shapiro (2014). Broad-band acceleration time histories synthesis by coupling low-frequencyambient seismic field and high frequency stochastic modeling, Geo-phys. J. Int. 199, 1784–1797.

Wapenaar, K., J. van der Neu, and E. Ruigrok (2008). Passive seismic inter-ferometry by multidimensional deconvolution, Geophysics 73, no. 6,A51–A56, doi: 10.1190/1.2976118.

Yao, H., R. D. Van der Hilst, and M. V. de Hoop (2006). Surface-wave arraytomography in SE Tibet from ambient seismic noise and two-stationanalysis—I. Phase velocity maps, Geophys. J. Int. 166, 732–744.

Zhang, J., and X. Yang (2013). Extracting the surface wave attenuation fromseismic noise using correlation of the coda of correlation, J. Geophys.Res. 118, 2191–2205, doi: 10.1002/jgrb.50186.

Department of GeophysicsStanford UniversityStanford, California 94305yixiao2@stanford.edu

(Y.S., G.C.B.)

Department of Earth and Planetary SciencesHarvard UniversityCambridge, Massachusetts 02138

(M.A.D.)

Manuscript received 17 February 2017;Published Online 26 September 2017

10 Y. Sheng, M. A. Denolle, and G. C. Beroza

BSSA Early Edition

http://dx.doi.org/10.1029/2008GL034428http://dx.doi.org/10.1029/2008JB006067http://dx.doi.org/10.1029/JB095iB06p08593http://dx.doi.org/10.1029/JB095iB06p08593http://dx.doi.org/10.1111/j.1365-246X.2011.05263.Xhttp://dx.doi.org/10.1016/j.epsl.2015.01.016http://dx.doi.org/10.1016/j.epsl.2015.01.016http://dx.doi.org/10.1093/gji/ggx021http://dx.doi.org/10.1029/2008JB005693http://dx.doi.org/10.1029/2005JB004237http://dx.doi.org/10.1029/2011JB008483http://dx.doi.org/10.1029/2011JB008483http://dx.doi.org/10.1190/1.2976118http://dx.doi.org/10.1002/jgrb.50186