exploiting the crossterms of the virtual rayleigh-wave green … · 2018. 12. 18. · exploiting...

Exploiting the crossterms of the virtual Rayleigh-wave Green tensor for surface-wave inversionK. van Wijk∗, Department of Geosciences, Boise State UniversityH. Douma, ION Geophysical/ GXT Imaging SolutionsD. Mikesell and M. Haney, Department of Geosciences, Boise State University

SUMMARY

Surface-wave tomography from the correlations of ambientnoise records has provided a powerful tool to investigate theEarth’s subsurface with new surface-wave information. Ex-ploration surveys can benefit from inversion of these virtualsurface waves, as well. Most commonly, correlations are per-formed on the vertical components of the wavefields, but herewe use the crossterms of multicomponent data and show vir-tual surface wave recordings are less sensitive to artifacts fromsignal not in-line with the two stations. We illustrate this witha subsection of the Coronation data set and the Batholiths tem-porary seismic deployment, showing estimates of the Rayleighwave and the consequent phase-velocity dispersion curve.

INTRODUCTION

Being able to estimate the impulse response between seismicstations from crosscorrelating noise wavefields has added anew dimension to surface-wave inversion for the Earth’s litho-sphere. Phase- and group-velocity dispersion curves betweendistributed station pairs are inverted for 3D velocity structure(e.g., Sabra et al., 2005; Shapiro et al., 2005; Lin et al., 2008;Ekstrom et al., 2009). Ideally, station pairs are surrounded byambient seismic noise sources (mostly thought to be associ-ated with the oceans). The exploration seismology communityis benefiting from parallel developments by inverting surfacewaves for near-surface structure (Haney and Douma, 2010;Gouedard et al., 2011). Here we aim to optimize surface-waveinformation via seismic interferometry on combinations of thevertical and radial components of the wavefield.

The elastic Green tensor can then be found by summing cross-correlations of the different components(i, j) of the wavefield(Wapenaar and Fokkema, 2006):

Gi j(x,x′, t)+Gi j(x,x′,−t) ∝I

SuS

i (x, t)⋆uSj (x

′, t)dS, (1)

whereGi j(x,x′, t) is the Green tensor with componentj at lo-cationx from a source in directioni atx′. uS

i (x, t) anduSj (x

′, t)

are the components of the measured wavefield atx andx′ froma source on contourS. The ⋆ indicates the crosscorrelationoperator.

A vertically heterogeneous earth has an anti-symmetry betweenthe horizontal component of the Rayleigh wave from a verti-cal force source, and the vertical component of the Rayleighwave from a horizontal force source (equation 7.147 of Akiand Richards, 1980):

Grz(x,x′, t) = −Grz(x′,x, t) = −Gzr(x,x′, t), (2)

xx’

source

receiver

source

receiver

anti−symmetry

Figure 1: The antisymmetry of the crossterms of the Rayleigh-wave Green tensor canbe seen from the receiver for the blueexperiment pointing inward, while the red receiver is point-ing away from its source. This means these wavefields are ofopposite sign.

where subscriptr stands for radial andz for vertical. For aretrograde elliptical Rayleigh wave,Grz is 90 degrees phase-delayed with respect toGzz and a 180 degrees phase-delayedwith respect toGzr. In graphical form, the antisymmetry isdisplayed in Figure 1.

After obtaining estimates of the Rayleigh-wave Green tensorvia equation 1, we compareGzz to Gzr − Grz. The Hilberttransform (e.g., page 20 of Claerbout, 1985) equalizes the phasebetweenGzz and the difference of the cross terms:

Gc(x,x′, t) = Hˆ

Gzr(x,x′, t)−Grz(x,x′, t)˜

. (3)

Crosscorrelations of multicomponent data from the Corona-tion data set illustrate the value of using these crossterms. Inaddition, the Batholiths passive seismic experiment (Calkinset al., 2010) provides estimates of the crossterms of the GreentensorGc that are more robust thanGzz in the presence of seis-mic signal not in-line with the stations.

CORONATION DATA

Figure 2 is a map of sources and receivers for a subset ofthe 80 km2 Coronation field data set, where sources are 1 kgof dynamite buried at 18 m and the receivers are three com-ponents (Calvert et al., 2005). The horizontal receivers areoriented such that there is an East-West (radial) and a North-South (transverse) component. Surface-wave inversion between(virtual) source-receiver pairs can be used for near-surface char-acterization (Haney and Douma, 2010; Gouedard et al., 2011).In the following we present robust virtual source-receiver pairswith the aim to increase resolution in velocity models fromsurface-wave inversion.

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Exploiting the Crossterms of Rayleigh Waves

476 476.5 477 477.5 478 478.55773

5773.5

5774

5774.5

5775

5775.5

Easting (km)

Nor

thin

g (k

m)

Figure 2: A map of the subset of sources and receivers of theCoronation data set. Blue symbols aresources, and red dia-monds are receivers. The green symbol is the location of thevirtual source used in this abstract.

The top panel in Figure 3 is the virtual shot record from 1-5 Hzobtained from seismic interferometry applied to the verticalcomponent data. The bottom panel is the difference betweenthe crossterms of the Green tensor estimated via seismic inter-ferometry.

The tenth receiver in the shotrecords is depicted in Figure 4 toemphasize key differences betweenGzz andGc: the latter re-sult lacks the strong artifact neart = 0, and the Rayleigh wavehas a larger signal-to-noise ratio thanGzz. For more insightinto the crossterms, Figure 5 shows the antisymmetry of thesecrossterms. Amplitudes of the two crossterms are not equal.This may be due to a violation of the lateral homogeneity un-der the virtual source-receiver pair, or a difference in couplingbetween stations. Nevertheless, the difference of these termsclearly benefits from constructive interference.

We have shown that we can use a subset of the sources to in-crease the number of (virtual) source-receiver pairs that cango into a surface-wave inversion scheme via seismic interfer-ometry. This number may be further grown by obtaining theGreen tensor between sources, appealing to reciprocity (Cur-tis et al., 2009). Ultimately, the goal is to improve resolutionin the surface-wave inversion scheme. The next section is onestep closer to this goal, as we illustrate a more coherent disper-sion curve for the crossterm results.

BATHOLITHS DATA

In passive seismic interferometry, the spatial integration in equa-tion 1 is replaced with summation overk time sections of thewavefieldu, aiming to capture surface-wave signal from sourcesaround the stations at(x,x′):

Gi j(x,x′, t) ≈X

k

`

u j(x′, t)⋆ui(x, t)´

k . (4)

Vertical component (i= j = z) Rayleigh wave estimates aremost commonly used in ambient noise tomography, but anuneven source distribution and contamination by wave modesother than Rayleigh waves can lead to artifacts in the estimated

476.8 477 477.2 477.4 477.6 477.8 478

0

1

2

3

4

5

6

Receiver Location (km)

Tim

e (s

)

476.8 477 477.2 477.4 477.6 477.8 478

0

1

2

3

4

5

6

Receiver Location (km)

Tim

e (s

)

Figure 3: Estimated Green functions forGzz (top) andGc (bot-tom). For the smallerstation spacings inGzz, an artifact ispresent att ≈ 0 s. Gc lacks most of this artifact, and appearsmore coherent at greater offsets.

1 2 3 4 5 6

−1

−0.5

0

0.5

1

x 10−7

Time (s)

Am

plitu

de (

a.u.

)

G

zzG

c

Figure 4: A comparison betweenGzz (red) andGc (blue) forthe receiver10 stations East of the virtual source.




1 2 3 4 5 6

−6

−4

−2

0

2

4

6

x 10−8

Time (s)

Am

plitu

de (

a.u.

)

G

zrG

rz

Figure 5: A comparison betweenGzr (blue) andGrz (red) forthe receiver 10 stations to the East of the virtual source.

−131˚ −130˚ −129˚ −128˚ −127˚53˚

54˚

55˚

56˚

50 km

BN01

BN04

BN06

BN08BN10

BN12

BN13A

BN16BN18

BN22

BN23

BN07

BN09

BN13BN14BN15

BN17

BN19BN20

BN21

−131˚ −130˚ −129˚ −128˚ −127˚53˚

54˚

55˚

56˚

Figure 6: Map of the active stations in August 2006 of theNorth line of the Batholithsexperiment. Red squares on theregional inset are BN01 and BN23.

Green functions from cross correlation. Wapenaar et al. (2011)show in a numerical example how multi-dimensional deconvo-lution can suppress unwanted signal, but we propose to exploitthe crossterms of the Green tensor estimated from crosscorre-lation (van Wijk et al., 2011).

The azimuth from station BN01 to BN23 from the north lineof the Batholiths experiment is 29 degrees from North in theclockwise direction (Figure 6). We rotate the horizontal com-ponents of the wavefield recordings to a generally radial (r)and transverse (t) component, band-pass filter (0.1 - 1 Hz),and sign-bit the data. We then crosscorrelate combinationsof the vertical and radial components of the wavefield fromstation BN01 with those of all 20 active stations accordingto equation 4. The Green tensor estimate is the sum of non-overlapping, ten-minute crosscorrelations from August 1, 200600:00:00 (HH:MM:SS) to August 4, 2006 00:00:00. We cor-rect the amplitudes for geometrical spreading.

Figure 7: Estimated Green functions forGzz (blue) andGc

(red). For the smallerstation spacings inGzz, an artifact att ≈ 0 s interferes with the Rayleigh wave.

van Wijk et al. (2011) show a dominant source area to the West,with a secondary region to the Southwest, both with slow-nesses typical for Rayleigh waves. The dominant energy fromthe West propagates obliquely to the array, and is the source forthe out-of-line energy observed close to zero lag time in thevertical component (blue) crosscorrelations in Figure 7. Thecrossterm combinationGc (red) is also shown in Figure 7. TheGc estimates lack the artifact near zero time lag attributed toout-of-line sources to the West. Biased velocity and/or ampli-tude information in the estimate of the Rayleigh wave couldbe erroneously attributed to attenuation and anisotropy (alsodiscussed in Harmon et al., 2010).

To illustrate the improvement ofGc compared toGz, we com-pute the phase-velocity dispersion curve – commonly used toinvert for velocity structure. We compute phase-velocity dis-persion curves using the Full-Offset Dispersion Imaging tech-nique (Park, 2011), which includes spectral whitening in thephase-velocity transformation. Figure 8 shows more coher-ence inGc than the dispersion curve obtained fromGzz.

FUTURE WORK: SPAC

An alternative way to obtain surface wave dispersion informa-tion is the spatial autocorrelation method (SPAC), where thespatial distribution of the crosscorrelated wavefields are fit byBessel functions, dependent on the velocity as a function offrequency (Aki, 1957). (Haney et al., 2011) show that our re-sults for the off-diagonal components of the Rayleigh wavecorrelation coefficient matrices, combined with Aki’s results




Figure 8: Phase velocity dispersion curve for the estimatedRayleigh-wave Green function.The dispersion curve for thecross termsGc (top) has greater coherence across the spectrumthan the result for the vertical component of the wavefieldGzz

(bottom).

for the main diagonal, can be summarized as follows:

φ R(r,ω) =

2

4

φ RZZ φ R

ZR φ RZT

φ RRZ φ R

RR φ RRT

φ RT Z φ R

T R φ RT T

3

5 = PR(ω)×

2

6

4

J0(ωrcR

) −R J1(|ω|rcR

) 0

R J1(|ω|rcR

) R2

2 [J0(ωrcR

)− J2(ωrcR

)] 0

0 0 R2

2 [J0(ωrcR

)+ J2(ωrcR

)]

3

7

5

.

(5)

For completeness, here is theresult for Love waves:

φ L(r,ω) =

2

4

φ LZZ φ L

ZR φ LZT

φ LRZ φ L

RR φ LRT

φ LT Z φ L

T R φ LT T

3

5 = PL(ω)×

2

4

0 0 00 1

2 [J0(ωrcL

)+ J2(ωrcL

)] 00 0 1

2 [J0(ωrcL

)− J2(ωrcL

)]

3

5 , (6)

whereω is angular frequency,r is radial distance,cR is Rayleighwave velocity,cL is Love wave velocity,R is the ratio of thehorizontal-to-vertical motion of the Rayleigh waves,PR is thepower spectrum of the Rayleigh waves,PL is the power spec-trum of the Love waves, andJ0, J1, andJ2 are Bessel functionsof the zeroth, first, and second orders, respectively.

CONCLUSIONS

Correlations of multicomponent wavefields from the Corona-tion seismic experiment provide estimates of the Rayleigh-wave Green tensor. These estimates can serve in surface-waveinversion for near-surface velocity models. Taking advantageof anti-symmetry, the difference between the crossterms pro-vides a superior estimate compared to the estimate from cross-correlations of the vertical components only. The crossterm re-sult is especially robust in the presence of out-of-line Rayleighwaves. Improvements in signal-to-noise ratio and suppressionof artifacts in the estimated Rayleigh waves allow us to includemore (virtual) source-receiver pairs in surface-wave inversionfor velocity models of the near surface.

ACKNOWLEDGMENTS

We thank Vera Schulte-Pelkum and Joshua Stachnik for thework on the Batholiths data set.



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