shear-wave imaging from traﬃc noise using seismic ... · shear waves constructed by seismic...

CWP-688

Shear-wave imaging from traffic noise using seismicinterferometry by cross-coherence

Nori Nakata1,2, Roel Snieder2, Takeshi Tsuji1, Ken Larner2,

& Toshifumi Matsuoka11Department of Urban Management, Kyoto University2Center for Wave Phenomena, Colorado School of Mines

ABSTRACTWe apply the ”cross-coherence method” to seismic interferometry of traffic

noise, which originated from roads and railways, to retrieve both body waves andsurface waves. Our preferred algorithm in the presence of highly variable andstrong additive random noise uses cross-coherence, which uses normalizationby the spectral amplitude of each of the traces, rather than cross-correlation ordeconvolution. This normalization suppresses the influence of additive noise andovercomes problems resulting from amplitude variations among input traces.By using only the phase information and ignoring amplitude information, themethod effectively removes the source signature from the extracted response andyields a stable structural reconstruction even in the presence of strong noise.This algorithm is particularly effective where the relative amplitude amongthe original traces is highly variable from trace to trace. We use the extractedreflected shear waves from the traffic noise data to construct a stacked andmigrated image, and use the extracted surface waves (Love waves) to estimatethe shear velocity as a function of depth. This shear-velocity profile agrees wellwith the interval velocity obtained from the normal moveout of the reflectedshear waves constructed by seismic interferometry. These results are useful fora wide range of situations applicable in both geophysics and civil engineering.

Key words: noise, passive seismic, shear wave, near surface, coherence

1 INTRODUCTION

The main purpose of seismic interferometry is to con-struct a Green’s function between two geophones, hy-drophones, or accelerometers through data processingof signals generated by earthquakes, microtremors, cul-tural noise, or artificial seismic sources. Green’s functionextraction can be derived from normal modes (Lobkisand Weaver, 2001), representation theorems (Wape-naar, 2004; Wapenaar and Fokkema, 2006), principleof time reversal (Roux and Fink, 2003), and stationaryphase analysis (Snieder et al., 2006). After such process-ing, one geophone serves as a (virtual) source for wavesrecorded by other receivers, which leads to a pseudoshot gather for many receivers, without using an activesource.

Although the first application of seismic interferom-

etry was based on cross-coherence (Aki, 1957), the firstapplied algorithm in seismic interferometry that foundwide application is based on cross-correlation (Claer-bout, 1968; Wapenaar, 2003; Bakulin and Calvert, 2004;Schuster et al., 2004). Another proposed algorithm isbased on deconvolution. In this method, the source sig-nal is removed by means of spectral division. The mathe-matical theory of deconvolution interferometry has beenderived by (Vasconcelos and Snieder, 2008a), and themethod has been applied to field data (Snieder andSafak, 2006; Vasconcelos and Snieder, 2008b; Vascon-celos et al., 2008). A multidimensional deconvolutionmethod has been formulated for seismic interferometry(Wapenaar et al., 2008; @warning Citation ‘ 2008wape-naar’ on page 139 undefined).

The various methods each have both advantages

140 N. Nakata, et al.

and disadvantages (Table 1 of Snieder et al., 2009).Cross-correlation, for example, is stable but needs es-timation of the power spectrum of the noise source, anddeconvolution is potentially unstable and needing regu-larization but does not require estimation of the sourcespectrum. We should choose the method that best suitsthe data; to date, however, it has not been clear howthese different methods behave when applied to datacontaminated with highly variable and strong additivenoise.

In this study we analyze the use of cross-coherence.This approach, used in seismology and engineering (e.g.,Aki, 1957; Bendat and Piersol, 2000; Chavez-Garıaand Luzon, 2005), calculates the cross-correlation oftraces normalized by their spectral amplitudes in thefrequency-domain. Thus, the method uses the phaseof each trace, ignoring amplitude information, for sup-pressing the influence of additive noise and handlingirregular input amplitudes. (Bensen et al., 2007) showexamples of normalization techniques applied in seismicinterferometry.

Extracting surface waves from cross-corrrelation ofambient noise is by now an established technique (e.g.,Campillo and Paul, 2003; Shapiro et al., 2005). In con-trast, the extraction of body waves has been proven tobe much more difficult. Extracting body waves by cross-correlation has, however, been accomplished in somestudies. Examples include extraction of P waves usingdistributed sources (e.g., Roux et al., 2005; Hohl andMateeva, 2006; Draganov et al., 2007; Gerstoft et al.,2008; Draganov et al., 2009; Zhang et al., 2009) and ofS waves using localized noise sources (e.g., O’Connell,2007; Miyazawa et al., 2008). This paper presents dataprocessing of field data dominated by strong and highlyvariable traffic noise as well as incoherent additive noise.We first present the basic equations of cross-correlation,deconvolution, and cross-coherence interferometry. Wefurther demonstrate the merits of cross-coherence in-terferometry applied to traffic noise data for retrievalof surface waves and reflected shear waves. Becausethe interferometry is based on the transverse horizon-tal component, the extracted surface waves consist ofLove waves. Finally, we explain why the cross-coherenceis particularly suitable to extract the approximatedGreen’s function from this type of noisy data.

2 EQUATIONS OF INTERFEROMETRY

Consider the wavefield u(r, s) excited at s and receivedat r. In this work we use a frequency-domain formu-lation for all data processing. Ignoring additive noise,the wavefield can be described as the multiplication ofa source wavelet and a Green’s function:

u(r, s) = W (s)G(r, s), (1)

where W (s) is the source wavelet and G(r, s) is theGreen’s function.

2.1 Cross-correlation and deconvolution

Let us first review the cross-correlation and deconvo-lution methods (Snieder et al., 2006; Vasconcelos andSnieder, 2008a) in order to compare them with the cross-coherence method. The cross-correlation of wavefieldsrecorded at locations rA and rB is

CAB =u(rA, s)u∗(rB , s)

=|W (s)|2G(rA, s)G∗(rB , s), (2)

where the asterisk denotes complex conjugate. Integrat-ing this equation over a closed surface ∂V , which is apart of the earth’s free surface and an arbitrarily shapedsurface at depth that includes all sources, gives

I

∂V

CABds =˙

|W (s)|2¸

I

∂V

G(rA, s)G∗(rB , s)ds, (3)

where˙

|W (s)|2¸

is the average of the power spec-tra for the source wavelets. Because the integralH

∂VG(rA, s)G

∗(rB , s)ds in equation 3 approximatesG(rA, rB) (Wapenaar and Fokkema, 2006), this givesthe approximated Green’s function between the two re-ceivers.

In contrast, the basic equation of deconvolution in-terferometry is

DAB =u(rA, s)

u(rB , s)=G(rA, s)

G(rB , s)=G(rA, s)G

∗(rB , s)

|G(rB , s)|2.

(4)

Deconvolution removes the influence of the sourcewavelet W (s). Because of the absolute value in the de-nominator, the phase of DAB is determined by the nu-merator G(rA, s)G

∗(rB , s) in the last term of equation4, hence the deconvolution gives the same phase as thecross-correlation method (equation 2). The deconvolu-tion method can also be used to extract the impulseresponse (Vasconcelos and Snieder, 2008a) when inte-grating over sources located on a closed surface ∂V . Be-cause of the spectral division, the result is independentof the source signature. The method thus can deal withdata generated by long and complicated source signals.

2.2 Cross-coherence

The cross-coherence HAB is defined in the frequency-domain as

HAB =u(rA, s)u

∗(rB , s)

|u(rA, s)||u(rB , s)|. (5)

The numerator of equation 5 is the same as the productin the expression for cross-correlation (equation 2), andthe denominator is the product of the amplitude spec-tra of the waveforms. This equation indicates that whilethe phase information is used, the amplitude informa-tion is discarded. Because the amplitude is, in practice,prone to inaccuracies, e.g., as a result of difference ofsensitivity among receivers or variable orientation of re-ceivers, use of this equation is expected to retrieve more

Swave imaging from traffic noise 141

Road

Road

Road

Survey line (blue)

Railway

Railway

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RoadRailway Railway Road

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tim

e[s

]

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(a) location of survey line (b) observed data (c) powerspectrum

Figure 1. (a) Location of the survey line for observing trafficnoise at Gunma, Japan: the line parallels a river and crossessome roads and train lines. (b) Observed noise record. The

receiver number increases from south to north. (c) Powerspectrum of the data in panel (b).

robust information than do either cross-correlation ordeconvolution. We can rewrite equation 5 as

HAB =G(rA, s)G

∗(rB , s)

|G(rA, s)||G(rB , s)|(6)

because cross-coherence cancels the source wavelet termW (s) by division, as does deconvolution. Integratingequation 6 over a closed surface ∂V containing allsources gives

I

∂V

HABds =

I

∂V

G(rA, s)G∗(rB , s)

|G(rA, s)||G(rB , s)|ds; (7)

hence it provides the phase of the approximated Green’sfunction between two receivers, but the amplitude is notpreserved in this cross-coherence approach. To clarifycharacteristics of each approach, (Nakata, 2010) showsTaylor expansions of equations 2, 4, and 5 for small-amplitude scattered waves.

When rA = rB in equations 5-7, the right-handside is equal to 1, which corresponds to the Dirac deltafunction δ(t) in the time-domain. This means that thefield extracted by cross-coherence satisfies a so-calledclamped boundary condition at rB ; the same boundarycondition occurs in deconvolution interferometry (Vas-concelos and Snieder, 2008a; Snieder et al., 2009) butnot in cross-correlation interferometry.

3 FIELD DATA PROCESSING

3.1 Data acquisition and pseudo shot gathers

We apply the cross-coherence method to traffic noisedata acquired in Gunma, Japan. An aerial photographof the observation site is shown in Figure 1a. The surveyline (blue line in Figure 1a) is quasi-linear, parallelingthe river; several roads and railways were crossed byor parallel to the line (shown by solid arrows in Fig-ure 1a). In order to retrieve body waves correctly, itis necessary to have sources at the stationary phase

depth

horizontal

Figure 2. Stationary points of a body wave propagatingbetween two receivers (triangles) from multiples reflected

from different interfaces. The white triangle denotes a re-ceiver that acts as a pseudo source. The wave radiated bythis pseudo source is reflected by a particular reflector andthen is recorded by a receiver marked with the black triangle.

Gray circles denote stationary source locations for multiplesthat first reflect off layers on the left, and that then prop-agate between the receivers marked by triangles. If a noise

source (e.g. the white star) is close to one of the stationarypoints, we obtain the reflected wave.

points from every combination of receivers and reflec-tors (Snieder et al., 2006; Forghani and Snieder, 2010).It might appear that the sparsity of roads and rail-ways does not provide an adequate illumination, but, asshown in Figure 2, the number of stationary points in-creases dramatically when one considers multiples thatare reflected from different reflectors. The length of thesurvey line is about 2180 m, with single-horizontal-component geophones oriented orthogonal to the sur-vey line at 10-m intervals aimed at obtaining subsurfacestructure from shear-wave data. The data were storedin 1200 time windows of 30-s duration, at a 4-ms sam-pling interval. Figure 1b shows an example of a noiserecord along the entire line. Higher levels of traffic noiseoriginated from roads and railways that cross the sur-vey line at receiver numbers 20, 50, 180, and 200. Thesource wavelets for the traffic noise had wide-rangingand complex frequency spectra because much traffic rancontinuously with differing characteristics attributableto varying speed or weight of the vehicles. Frequencyanalysis reveals that the most energetic part of trafficnoise is in the range 12-16 Hz (Figure 1c).

Figures 3a-3c compare the pseudo shot gathers de-rived from cross-correlation, deconvolution, and cross-coherence. Because we used transverse geophones, theseshot gathers are dominated by shear waves. For eachof the three different operations, the data acquired atreceiver point 60 is used as the reference trace. The in-terferometric data from the 1200 records are stacked;and no other filter is applied to the records to constructthese interferometry profiles. In the cross-correlation re-sult (Figure 3a), ringing noise is dominant because ofthe periodic characteristic of the source wavelets of thetraffic noise, and amplitude levels are particularly highat positions near the traffic noise sources (receiver num-


receiver number50 100 150 200

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receiver number80 100

tim

e[s

]

0

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0.5

1.5

20 40 60

(d) detail of cross-coherence

Figure 3. Virtual source gathers generated by (a) cross-correlation, (b) deconvolution, and (c) cross-coherence. The

pseudo source point of these gathers is at receiver number60. We applied no filter to these displayed data. (d) Detailshowing hyperbolic events of panel (c). The main hyperbolicevents are highlighted in transparent yellow. A bandpass fil-

ter and f-k filter have been applied to the data in panel (d).The data gaps in panel (d) are caused by the removal ofincoherent traces.

bers 20, 50, 180, and 200). While the ringing noise issuppressed in the deconvolution shown in Figure 3b,the signal-to-noise ratio is low, and large local ampli-tude variations remain. Of these methods, for reasonsexplained below, cross-coherence (Figure 3c) gives thebest results in terms of both signal-to-noise ratio andtrace balance. This virtual source record exhibits re-flected shear waves with hyperbolic moveouts of theevents particularly around 0.4 and 1.3 s (highlighted inFigure 3d). Figure 3d shows events with non-hyperbolicmoveouts that do not correspond to a virtual source atreceiver number 60. These are likely to be artifacts of animperfect location distribution of noise sources. Theseevents are, however, suppressed by the normal moveout(NMO) correction and the common midpoint (CMP)stack.

3.2 Reflection profiles from body waves

We perform seismic reflection data processing using theCMP stack method and apply time migration to thedata sets from each of the three methods. CMPs arenumbered by location of the receivers along the acqui-sition line, consistent with the numbering in Figure 1a.After generating pseudo shot gathers by each interfer-

0.0

1.25

1.0

0.75

0.5

0.25

1.5

0 500 00 0 500500500

time[s]

offset[m]

vnmo= 500m/s vnmo= 680m/s vnmo= 880m/soriginal

(a) CMP gathers

vnmo=500m/soriginal

0.3

0.5

0 250

offset[m]

time[s]

vnmo=680m/s vnmo=880m/s

0 250 0 2500 250

0.4

(b) CMP gathers focused on one moveout event

Figure 4. Constant-velocity NMO-corrected CMP gathersat the CMP number 60. (a) Leftmost panel is the originalgather, and the other three panels are gathers corrected usingthe NMO velocity shown below each panel. The highlighted

areas show the time intervals within which the RMS velocitybest flattens the local reflection. (b) Detailed views of theCMP gathers highlighting the reflection event at about 0.4

s.

ometry method, we apply identical steps of bandpassfiltering (5-35 Hz), f-k filtering (to reject surface waveswith velocities outside the range 250-2500 m/s), NMOcorrection by the root-mean-square (RMS) velocity ob-tained from cross-coherence, CMP stack, time migra-tion, and depth conversion. We determine the stack-ing velocity by performing constant-velocity stacks onNMO-corrected CMP gathers (Yilmaz et al., 2001; Stuc-chi and Mazzotti, 2009).

Figure 4 displays a CMP gather at the CMP num-ber 60 corrected with three different constant values ofmoveout velocity; highlighted areas in Figure 4a showthe time intervals used for determining the stacking ve-locity. As shown in Figure 4b, the reflection arrival ataround 0.43 s is flattened for a RMS velocity between680-880 m/s. Given the noise level in this figure, it isdifficult to estimate the stacking velocity with great ac-curacy. The panels in Figure 4b and the correspond-ing range of RMS velocity (680-880 m/s) suggest anuncertainty of perhaps 100 m/s. The interval velocitystructure obtained from the RMS velocity function esti-mated by cross-coherence interferometry is shown withthe black line in Figure 5d. Figures 6a-6c show themigrated depth sections using all pseudo shot recordsderived through cross-correlation, deconvolution, and


de

pth

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(d) active source

Figure 6. Subsurface structure obtained by CMP stack, time migration, and depth conversion using reflected waves obtainedby (a) cross-correlation, (b) deconvolution, and (c) cross-coherence interferometry. Panel (d) is generated from active-source

data. A bandpass filter between 5 and 35 Hz has been applied to all the data before imaging.

cross-coherence, respectively. We applied the same shearwave stacking velocity function to the recorded active-shot data.

In the image obtained from cross-correlation (Fig-ure 6a), anomalously strong waves dominate the image.These are caused by strong vibrations generated by thecrossing traffic. This result agrees with that of (Hohland Mateeva, 2006) whose image is also contaminatedby local strong waves induced by rig activity. The imagein figure 6a displays a marked periodicity, this is due tothe narrow-band character of the noise sources (trucksand trains). As shown in figure 3a, the cross-correlationdoes not compensate for the narrow-band properties ofthe noise sources. The image retrieved by deconvolu-tion (Figure 6b) is also noisy, although the amplitudeis less variable from one location to another. The imagein figure 6b is very noisy and incoherent. This is dueto the fact that the virtual source gathers obtained bydeconvolution interferometry (e.g., figure 3b) show fewcoherent arrivals. Cross-coherence interferometry givesby far the clearest image of the three methods (Figure6c). Because cross-coherence interferometry flattens thepower spectrum via the normalization in frequency do-main, this type of interferometry gives an image thatcontains a much larger range of spatial wavenumbersthan the image obtained from cross-correlation (Figure6a).

For comparison, we show in Figure 6d a conven-

tional stacked and migrated reflection seismic sectionusing 224 transverse active seismic sources, which are atapproximately 10-m intervals along the receiver line andrecorded by transverse-component geophones along thesame line. Since for practical reasons it is not possibleto deploy sources close to the receiver line in the activeshot experiments, the structure shallower than about 50m is not imaged well in Figure 6d. In contrast, shallowreflections are evident in the seismic section of Figure 6cobtained from cross-coherence interferometry. Althoughwe apply the same bandpass filter, the images in Fig-ures 6c and 6d have different depth resolution becauseof the different frequency content of the sources. Imagesfrom reflected shear waves are usually nosier than P-wave images, and the images obtained from traffic noise(Figure 6c) and from active shots (Figure 6d) are bothcontaminated with noise. Yet both images show coher-ent layered structures with a region of large reflectivitybetween receivers 80 and 140.

Delineation of structures shallower than 300 min Figure 6c is useful not only for geophysical explo-ration, e.g., static corrections and near-surface tomog-raphy, but also for ground-motion prediction by seismicmonitoring in earthquake disaster prevention and base-ment surveys in civil-engineering applications. Althoughtrace-to-trace amplitudes are not preserved in the cross-coherence method, the method can still be used for thedelineation of underground structures.

144 N. Nakata, et al.receiver number

50 100 150 200

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e[s

]

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(a) pseudo shot gather

0 12.5 25 37.50

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frequency[Hz]

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e ve

loci

ty[m

/s]

(b) phase velocity

0 12.5 25 37.50

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frequency[Hz]

phas

e ve

loci

ty[m

/s]

observedinversion

(c) dispersion curve

0 200 400 600 800 1000 1200 14000

50

100

150

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250

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S velocity[m/s]

de

pth

[m]

from body waves

from surface waves

uncertainty bounds

(d) interval velocity

Figure 5. (a) Pseudo shot gather obtained by cross-coherence interferometry for a virtual source at receiver num-ber 190. (b) Frequency-dependent phase velocity, (c) disper-

sion curve overlaying the phase velocity plot (b), and (d)interval velocity computed by inversion of the data in panel(c). In panel (c), the blue ×’s show the picking points for

the fundamental mode of the surface wave, and the red lineis the dispersion curve obtained from inversion. The blackline in panel (d) is the interval velocity function obtainedfrom the RMS velocity profile used in the CMP stack of the

reflected transverse waves, the red line is the estimated shear-wave velocity, and the blue dashed lines show an estimate ofthe uncertainty resulting from the uncertainly in the phasevelocity picks in panel (c).

3.3 S-wave velocity from surface waves

We apply the cross-coherence interferometry tech-nique to shear-wave velocity estimation from the re-trieved Love waves. The virtual shot record from cross-coherence interferometry (Figure 5a) clearly displayssurface waves. Figures 5b and 5c show the frequency-dependent phase velocity and dispersion curve esti-mated from the pseudo shot record at receiver number190.

We pick the phase velocity of the fundamental modeLove wave (blue crosses in Figure 5c) and use these mea-surements to invert for the shear velocity as a functionof depth using a genetic algorithm (Saito and Kabasawa,1993; Yamanaka and Ishida, 1995; Hayashi, 2008). Theinverted shear velocity model is shown by the red linein Figure 5c. The shear-wave interval-velocity distribu-tion down to around 300 m obtained from the phasevelocity measurements is shown by the red line in Fig-ure 5d. This shear-velocity profile agrees well with theinterval-velocity profile obtained from NMO correctionof the reflected shear waves (black line in Figure 5d).The shallow part of the shear-velocity profile from the

surface-wave analysis resolves several layers, with grad-ually changing velocity. This shear-wave interval veloc-ity, obtained by cross-coherence interferometry of trafficnoise, is useful for estimating and monitoring ground soilstrength.

4 DISCUSSION - ERROR PROPAGATION

In this section, we compare the statistical properties ofcross-coherence with those of cross-correlation and de-convolution, and show theoretically why cross-coherenceis preferable in the data applications shown in this work.

4.1 Correction of amplitude variation amongtraces

Consider the processing of data whose amplitudes varytrace by trace as a result of variations in source strengthand differences in positioning or sensitivity of receivers.Ideally, the sensitivity is the same for all receivers, butin practice this is not the case because of variations inground coupling and local topography. The equations ofthe three methods are

CAα =u(rA, s)u∗(rα, s) (8)

DAα =u(rA, s)u

∗(rα, s)

|u(rα, s)|2(9)

DαA =u(rα,s)u

∗(rA, s)

|u(rA, s)|2(10)

HAα =u(rA, s)u

∗(rα, s)

|u(rA, s)||u(rα, s)|, (11)

where two receivers are at rA, and rα. Deconvolution in-terferometry is asymmetric in that the amplitude of theextracted signal changes when we exchange the pseudosource and receiver points: this method thus has twodifferent forms. Consider the case where receiver rArecords the average amplitude of all receivers and rαrecords an anomalously large amplitude; the recordedmotion at receiver rA is u(rA, s) = G(rA, s), and the mo-tion recorded at rα is u(rα, s) = RG(rα, s) with R� 1,an amplification factor. The amplitude of the signals ex-tracted with cross-correlation and cross-coherence doesnot change by exchanging α and A.

Let us compare the amplitudes among equations 8- 11. In equation 8, CAα = RG(rA, s)G

∗(rα, s), and theamplitude of CAα is thus amplified by a factor R. Simi-larly, the amplitudes of DAα and DαA are multiplied by1/R and R, respectively. As mentioned above, the am-plitudes of CAα, DAα, and DαA differ from the averageamplitude. Accordingly, in an analysis based on cross-correlation or deconvolution that includes the anoma-lous receiver α, the amplitude of the extracted responseis unbalanced, thus requiring the additional task of re-moving these variations. The amplitude of HAα is 1,independent of R. That is, cross-coherence removes theinfluence of amplitude variations and achieves a stable


amplitude without separate processing to normalize theamplitude of traces constructed by interferometry.

In the deconvolution approach here, some level ofwhite noise has to be added to prevent numerical in-stability. If we choose a regularization parameter thatis too large, the regularized deconvolution reduces tocross-correlation. If, however, the regularization param-eter is too small, the amplitude variation between tracesremains large. In cross-coherence this problem does notarise because the numerator and denominator are bothsmall when the spectral amplitude is small.

4.2 The influence of additive noise

When the data are contaminated by additive randomnoise N(r) with zero mean, the wavefield u(rA, s) in-cludes a noise term N(rA),

u(rA, s) = W (s)G(rA, s) +N(rA). (12)

For simplicity, let us set the source signature |W (s)| = 1.This additive noise might be caused by microtremors,electric noise in the equipment, and human activities.Henceforth, we abbreviate G(rA, s) as GA, N(rA, s) asNA, and W (s) as W . In Appendix A, we insert equation12 into equations 2, 4, and 5 and expand in the smallquantity |N |/|G| < 1.

To investigate the influence of additive noise, wetake the ensemble average to estimate the mean andvariance of equations A5 - A7 (Appendix A). Becausethe ensemble average of random noise is assumed tovanish, the ensemble average of the convolution of Gand N also vanishes. As shown in the equations A5 -A7, the ensemble average values of the cross-correlation,deconvolution, and cross-coherence in the presence ofadditive noise are thus given by

〈CAB〉 =GAG∗B (13)

〈DAB〉 =GAG

∗B

|GB |2(14)

〈HAB〉 =

„

1 − 1

4

|NA|2

|GA|2− 1

4

|NB |2

|GB |2

«

GAG∗B

|GA||GB |, (15)

in which ”〈〉” indicates an ensemble average. Their vari-ances are

σ2C =|GB |2σ2

NA+ |GA|2σ2

NB(16)

σ2D =

1

|GB |2σ2NA

+|GA|2

|GB |4σ2NB

(17)

σ2H =

1

2|GA|2σ2NA

+1

2|GB |2σ2NB

. (18)

Here σN denotes standard deviation of the additivenoise. Note that the noise does not bias the ensembleaverage of the cross-correlation and the deconvolution(equations 13 and 14), but according to expression 15it does lead to a bias in the cross-coherence. Therefore,when we stack many times to mimic an ensemble av-erage, the influence of noise remains as a bias in cross-coherence. Since, however, the cross-coherence does not

(1500, 900)

x

z

(3500, 900)

2500 m

(500, 500) (4500, 500)

50 m

1500 m/s

2000 m/s

Figure 7. Two horizontal constant-density layers model,with an interface at 2500-m depth. The velocities of layers are1500 m/s and 2000 m/s, respectively. Two receivers, whichare shown with triangles, are positioned at 900-m depth and

at lateral positions x = 1500 m and 3500 m. Dashed andthick arrows denote created direct and reflected wavefield byinterferometry, respectively. The sources, which are repre-sented by stars, are distributed in a horizontal line at 500-m

depth, ranging from x = 500 m to 4500 m, in increments of50 m.

preserve amplitude even in the absence of noise, themultiplicative bias in equation 15 is of little concern inpractice.

It is difficult to compare the variances in equations16, 17, and 18 because they express the variance in dif-ferent quantities (cross-correlation, deconvolution, andcross-coherence, respectively). Instead, we consider thenormalized standard deviations:

σC|CAB |

=

s

σ2NA

|GA|2+

σ2NB

|GB |2(19)

σD|DAB |

=

s

σ2NA

|GA|2+

σ2NB

|GB |2(20)

σH|HAB |

=

s

σ2NA

2|GA|2+

σ2NB

2|GB |2. (21)

From equations 19 - 21, the normalized standard devi-ations are related as follows:

σH|HAB |

=1√2

σC|CAB |

=1√2

σD|DAB |

. (22)

As shown in equation 22, the relative uncertainty in theof cross-coherence is about 70% of that of the othermethods.

In summary, additive random noise causes an in-consequential bias in the cross-coherence, but the rela-tive statistical uncertainty in the cross-coherence is re-duced by a factor 1/

√2 (≈ 70%) compared with that of

cross-correlation and deconvolution. Thus, in additionto treating the problem of anomalous trace amplitudes,cross-coherence is more stable in the presence of noise.

We study the influence of additive random noiseby a numerical example of synthetic data generatedby a two-dimensional acoustic finite-difference time-


1.0

3.0

2.0

time[s]

20 51015 1signal-to-noise ratio


1.0

3.0

2.0

time[s]


(b) deconvolution

1.0

3.0

2.0

time[s]


(c) cross-coherence

Figure 8. The influence of random noise added to the simulation data before applying (a) cross-correlation, (b) deconvolution,and (c) cross-coherence interferometry. In each figure, the signal-to-noise ratio varies between traces. No noise is added to theleftmost trace. The second trace from the left has a signal-to-noise ratio of 20, the signal-to-noise ratio is decreased by one for

each successive trace. The signal-to-noise ratio of the rightmost trace is 1.

domain method with a model consisting of two hori-zontal constant-density layers (Figure 7). The virtualsource sections, obtained from the noise-contaminatedtraces and shown in Figure 8, display the direct arrival,which is represented with the dashed arrow in Figure7 at 1.3 s, and the reflected wave, which is depictedby the thick arrow in Figure 7 at 2.5 s. The leftmosttrace in each plot is noise-free so the signal-to-noise ra-tio is infinite. The signal-to-noise ratio (calculated fromthe maximum amplitude of direct arrival) of the sec-ond trace from the left is 20, which means that weadded 5% random noise, while that of the third traceis 19, and that of the fourth trace is 18. The amountof noise gradually increases until the rightmost tracewhose signal-to-noise ratio is 1. Figures 8a, 8b, and 8cshow the wavefields retrieved from cross-correlation, de-convolution, and cross-coherence, respectively. For lowsignal-to-noise ratios, the direct and reflected waves areburied in ambient noise in both the cross-correlation anddeconvolution results (Figures 8a and 8b). In contrast,cross-coherence interferometry (Figure 8c) reduces theinfluence of ambient noise.

5 CONCLUSION

In our study, cross-coherence interferometry providedthe clearest pseudo shot gathers generated from highlyvariable and strong traffic noise and retrieved both re-flected shear waves and Love waves. Since we usedrecordings of the transverse motion, this procedureyielded virtual source gathers for shear waves. The im-print of the source signature and amplitude variationsbetween receivers is suppressed in the virtual sourcegathers obtained from cross-coherence. They provideshear-wave images obtained by migrating virtual sourcedata that agree to a large extent with those obtainedwith active sources. Moreover, the images obtained byactive sources lack the shallow structures seen in the

image obtained from the cross-coherence of traffic noisebecause the active sources were placed at a distancefrom the survey line. The virtual source sections ob-tained from cross-coherence exhibit both surface wavesand body waves; we used the surface-wave data to carryout dispersion measurements of the fundamental modeLove waves and obtained a shear-wave velocity pro-file that agrees with the interval-velocity function cal-culated by the stacking velocity profile. Correspondingwith early studies, it is easier to extract surface wavesthan body waves.

Compared with the standard deviation of the vir-tual source sections obtained from cross-correlation anddeconvolution, the relative statistical uncertainty incross-coherence is 70% as large. In contrast to cross-correlation and deconvolution, additive noise leads toa multiplicative bias in virtual source signals based oncross-coherence. Since the cross-coherence method doesnot conserve trace-to-trace amplitude, this multiplica-tive bias is of little concern. Because of the normaliza-tion employed, the method overcomes amplitude vari-ations among traces. In any case, because the ampli-tude information is lost, cross-coherence interferometryis inappropriate for data analysis that exploits ampli-tude information, such as the measurement of reflec-tion coefficients, amplitude versus offset, and attenua-tion. Cross-coherence is particularly suitable for datathat are noisy, vary in amplitude among traces, or havelong and complex source wavelets. The primary target ofcross-coherence interferometry here is the estimation ofshear-wave velocity from surface waves and the shapeof subsurface structures obtained from reflected bodywaves. By using the transverse-component of the groundmotion for cross-coherence, we obtain a shear-velocityprofile and a shear-wave image of the subsurface. Thisinformation is useful for various applications such asstatic corrections, near-surface tomography, ground mo-tion prediction for earthquake disaster prevention, mon-


itoring the ground soil strength, and basement surveysfor civil engineering.

6 ACKNOWLEDGMENTS

We are grateful to Kyosuke Onishi of Akita University,Toshiyuki Kurahashi of Public Works Research Insti-tute, and Takao Aizawa of Suncoh Consultants for theacquisition of the seismic data and Tatsunori Ikeda ofKyoto University for the technical support in surfacewave analysis. We also thank Akihisa Takahashi of JGI,Inc., Shohei Minato of Kyoto University, the editor, andthree anonymous reviewers for suggestions, corrections,and discussions. Norimitsu Nakata is grateful for sup-port from the Japan Society for the Promotion of Sci-ence (JSPS: 22-5857).

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APPENDIX A: ERROR PROPAGATION

By substituting equation 12 into equations 2, 4, and 5, and assuming that |W (s)| = 1, we obtain the followingexpressions for cross-correlation, deconvolution, and cross-coherence using wavefields that include random noise withzero mean:

CAB =(WGA +NA)(W ∗G∗B +N∗

B) (A1)

DAB =WGA +NAWGB +NB

(A2)

HAB =(WGA +NA)(W ∗G∗

B +N∗B)

|WGA +NA||WGB +NB |. (A3)

Below, we assume that the additive noise at different locations is uncorrelated; hence˙

|NA|2¸

= σ2NA,

˙

|NB |2¸

= σ2NB

, 〈NAN∗B〉 = 0.

When ψ represents the phase,

N = |N |eiψ; (A4)

then under the assumption that the amplitude and phase are uncorrelated, the ensemble average of N2 is

˙

N2¸

=D

|N |2e2iψE

=˙

|N |2¸

D

e2iψE

.

We assume that the phase has a uniform distribution, thus˙

e2iψ¸

= 0; hence˙

N2A

¸

=˙

N2B

¸

= 0.

We further assume that the level of additive noise is small (|N |/|G| < 1) and expand equations A1 - A3 in |N |/|G|.Ignoring noise terms higher than second-order in |N |/|G|, gives

CAB =GAG∗B +W ∗NAG

∗B +WGAN

∗B +NAN

∗B (A5)

DAB =GAG

∗B

|GB |2+W ∗NAG

∗B

|GB |2− W ∗GAG

∗BNBG

∗B

|GB |4− (W ∗)2NAG

∗BNBG

∗B

|GB |4+

(W ∗)2GAG∗BNBG

∗BNBG

∗B

|GB |6(A6)

HAB =

„

1 − 1

4

|NA|2

|GA|2− 1

4

|NB |2

|GB |2

«

GAG∗B

|GA||GB |

+1

2

W ∗NAG∗B

|GA||GB |+

1

2

WGAN∗B

|GA||GB |+

1

4

NAN∗B

|GA||GB |

− 1

2

WGAG∗BGAN

∗A

|GA|3|GB |− 1

2

W ∗GAG∗BNBG

∗B

|GA||GB |3− 1

8

(W ∗)2NAG∗ANAG

∗B

|GA|3|GB |− 1

8

W 2GAN∗BGBN

∗B

|GA||GB |3

− 1

4

W 2GAN∗AGAN

∗B

|GA|3|GB |− 1

4

(W ∗)2NAG∗BNBG

∗B

|GA||GB |3

+3

8

W 2GAG∗BGAN

∗AGAN

∗A

|GA|5|GB |+

3

8

(W ∗)2GAG∗BNBG

∗BNBG

∗B

|GA||GB |5+

1

4

GAG∗BGAN

∗ANBG

∗B

|GA|3|GB |3. (A7)

Taking expectation values gives the mean and variance of equations 13 - 18.

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