SVD enhanced seismic interferometry for traveltime estimates between microquakesG abriela Melo & Alison MalcolmE arth Resources Laboratory - E arth, Atmospheric, and Planetary Sciences Department - MIT

SUM M ARY

In general, Green’s functions obtained with seismic interfer-ometry are only estimates of the true Green’s function, intro-ducing uncertainties to the information recovered from them.However, there are still many cases in which the source-receiver geometries are suitable for seismic interferometry,usually allowing the recovery of kinematic information. Herewe show how to use the singular value decomposition to re-enforce the accuracy of traveltimes obtained from interfero-metric Green’s functions. We apply the combination of seis-mic interferometry and the singular value decomposition to ob-tain physically accurate inter-event traveltimes for microquakepairs at a geothermal reservoir. With a synthetic example, weshow that the P-wave phase and coda-wave energy informa-tion are closer to correct with the singular value decomposi-tion than without. These traveltimes could be used for velocitytomography and event location algorithms to obtain more ac-curate event locations and locally accurate velocity models.

INTRODUCTION

To recover the full Green’s function (G F) between two re-ceivers using seismic interferometry (SI) requires that thesetwo receivers be surrounded by a surface of both monopole anddipole sources. E ven if one is interested in obtaining only ac-curate estimates of traveltimes, this still requires full (monopole)source coverage, a condition which is rarely met in practice.A s a result, we generally recover only estimates of the trueG F, which introduces uncertainties to the information recov-ered from these empirical (interferometric) Green’s functions,that here we refer to as E G Fs. E ven though most practical sce-narios are not ideal, there are many cases in which the source-receiver geometry is suitable for SI, allowing for the recov-ery of at least some information (generally kinematic) fromthe E G F. Snieder (2004) showed that the sources that give themain contribution to the E G Fs are the ones located along theray path between the two receivers, and those in the Fresnelzone around these sources. This conclusion comes from themethod of stationary phase; the sources along the raypath arethe sources for which the phase of the receiver crosscorrela-tions is stationary. From this argument, assuming full sourcecoverage, energy emanated by sources outside the Fresnel zoneshould cancel. Thus, as long as we have sources and receiverson stationary paths, we have a good chance of being able toapply SI to recover traveltime information.

In order to improve the accuracy of E G Fs for cases with gaps inthe source coverage, Melo et al. (2010) introduce a method inwhich they use the singular value decomposition (SV D) (seee.g. Golub and van Loan (1996)) to enhance E G Fs. SV D isa numerical technique commonly used in seismic data pro-

cessing (see e.g. U lrych et al. (1988); Sacchi et al. (1998)),to increase the signal to noise ratio and filter linear events.Melo et al. (2010) show that E G Fs obtained by stacking lower-rank approximations of crosscorrelograms, obtained throughSV D, have enhanced physical arrivals that are not properlyrecovered using the standard stacking in SI. The idea behindthis technique is that the stationary and non-stationary energyin the crosscorrelograms generally have different signaturesthat, under favorable circumstances, can be separated throughSV D. Stationary energy in the crosscorrelogram is character-ized by linearity, coherency, low wavenumber, and thus nearlyin-phase energy along the source dimension. Non-stationaryenergy by contrast is characterized by non-linearity, incoherency,high wavenumber, and out-of-phase events along the sourcedimension. The relationship between frequency (or source-wavenumber in our case) and singular values is key to un-derstanding why SV D is able to separate stationary and non-stationary energy. Hansen et al. (2006) explain this relation-ship - large singular values correspond to low frequencies andsmall singular values correspond to high frequencies. A s theycorrespond to low-frequencies, large singular values are as-sociated with events that are in phase in the crosscorrelogram,coming from stationary sources whose energy contribute to theE G F. In the context of waveguides, for instance, Philippe et al.(2008) exploit this connection between singular values and fre-quency for characterization of targets. They show that the firstsingular value associated with a given target is proportional tothe backscattering form function of the target, and that the sec-ond singular value is proportional to the second derivative ofthe angular form function. They then use SV D to extract thebackscattered frequency signature of a target in a waveguide.

Here the goal is to use the SI-SV D method to obtain accuratetraveltime estimates between a pair of microquakes. Curtiset al. (2009) show that, by reciprocity, while most applicationsof SI estimate the G F between two receivers surrounded bysources, it is also possible to use SI to estimate the G F be-tween a pair of sources. The sources in our particular casewill be microquakes recorded at a geothermal reservoir. Someof the challenges in working with microseismic data are gapsin the receiver coverage, uncertainties in the source locations,and low signal-to-noise ratio. These challenges will generallyresult in errors in the E G F. To alleviate some of these prob-lems, Melo and Malcolm (2011) use the SV D technique forestimating the E G Fs. Their acoustic modeling results indicatethat, in some cases where microseismic data is highly contam-inated by noise, SV D gives an accurate estimate of the E G Feven when the standard technique fails to converge. They alsosuggest that, in general, the phase of direct wave and the codaenergy are closer to correct with SV D than without. F inally,they demonstrate that as well as stability with respect to noise,this method is also stable with respect to the aperture of thesource coverage, which is important because errors in the lo-cation of microquakes can be significant.

SVD enhanced interferometric traveltimes

A n example application for the SI-SV D based traveltimes cal-culated here is for use in algorithms that jointly perform to-mography and event location, such as the double-differences(D D) techniques (Waldhauser and E llsworth, 2000; Zhang andThurber, 2003). These algorithms generally use both abso-lute and relative arrival time information, leading to improvedvelocity models and more accurate source locations. One ad-vantage of using the SI-SV D based traveltimes is that they cor-respond to the physical traveltimes between microquakes, andnot just relative times. This should allow for more accuraterelative event locations. A s with all interferometric methods,this can only be done for pairs of events for which there is areceiver along the stationary path.

In the next section, we briefly review the SI-SV D techniquewith a synthetic example. Following that, we show some pre-liminary inter-event traveltime estimates for a real dataset con-sisting of microseismic events from a geothermal reservoir.Our results suggest that for cases where usual stacking over re-ceivers is enough to obtain traveltime estimates, the traveltimesobtained through SV D can be used to confirm the accuracy oftraveltimes obtained with SI alone. When the estimates are ac-curate both SV D-SI and SI give similar results; when there areno stationary events the E G Fs will be quite different indicat-ing errors. A lso, our modeling results indicate that traveltimesmay also be accurately estimated through SV D when regularcrosscorrelation stacking fails. In addition, these synthetic re-sults suggest that the energy of the coda-wave in the E G F ob-tained through SV D may be closer to the reference coda-wavethan for the case without SV D.

REVIE W OF SI-SVD M ETHOD

We define a crosscorrelogram as the matrix, C , where eachrow is the crosscorrelation of the signals recorded at two re-ceivers from a single source. Thus, the vertical dimension ofC is the source dimension and the horizontal is time. Next,we decompose the crosscorrelogram using SV D. The SV D de-composition of the crosscorrelogram matrix is, C = U !V t ,where U and V are the left and right singular vector matrices,and is the diagonal matrix whose elements are the singularvalues of C . F igure 1 shows the standard method of obtaininga lower-rank approximation C ! = U ! V t of the crosscor-relogram by selecting only the largest singular values of theSV D decomposition of C . Stacking the rows of C gives thestandard E G F, G, and stacking the rows of the approximationC ! gives the modified E G F, Gj , where j is the rank of C ! (thenumber of singular values retained). In the synthetic exam-ple that follows, we compare both E G Fs, Gj and G, with thereference G F, Gref , and explain how Gj can decrease uncer-tainties in G, or substitute for it completely when G is poorlyestimated.

This synthetic example approximately mimics an idealized sourceand receiver geometry for downhole monitoring of microseis-mic activity in a geothermal reservoir. We use a single bore-hole with 35 receivers and estimate the G F between two mi-croquakes, as shown in figure 2. The reference and interfer-ometric G Fs (E G Fs) shown here are all normalized to make

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up for the lack of dipole sources. The idea is to obtain anE G F between a pair of microquakes instead of a pair of re-ceivers, as done in Curtis et al. (2009) with earthquakes at aglobal scale. The medium is weakly scattering with a con-stant background velocity and density, and noise was added tothe modeled dataset. The additive weakly-correlated noise andrandom scattering we use here are realizations of a Gaussianrandom field with prescribed correlation lengths along givendirections. For this synthetic case, we focus our observationson the phase of the direct wave and the energy in the coda. Weadded enough noise to completely obscure the direct wave anddistort the waveform of the coda in the E G F and show howSV D improves both of these measures.

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F igure 2: On the top, the source-receiver geometry with 35receivers (blue triangles) and two microquakes (red stars). Theoriginal (left) and rank-1 (right) crosscorrelograms are in themiddle. On the bottom, the black curves are the E G Fs, G andGj , and the red curve is the reference G F, Gref . Contrary toG, in G1 the random noise is reduced to the level of the codaand the arrival is clear.

In the original crosscorrelogram in figure 2, there is enoughnon-stationary energy from receivers outside the Fresnel zoneto create high-amplitude fluctuations that hide the direct wavein G. We choose the rank-1 approximation here because Gref

consists of only one direct wave and thus there is only one zoneof stationary-phase energy in the crosscorrelogram. In figure 2,

SVD enhanced interferometric traveltimes

G appears strongly contaminated by noise and neither the di-rect arrival nor the coda wave are visible. A lso, fluctuationsand random noise are strongly attenuated in G1, revealing notonly the direct arrival but also reducing the noise to be close tothe coda wave level. The phase differences between G and G1

and Gref are 0.2850 and 0.0015 s, respectively. The relativeerrors in the L2-norm of the coda are 148% for G and 11%for G1. We see that SV D eliminates most of the noise in thecoda wave, as well as the fluctuations before the direct wave,illustrating its stability with respect to noise.

GEOTHERM A L FIE LD DATA

We now apply the SI-SV D method to a real microseismic datasetfrom a geothermal field. F igure 3 shows 2D projections of thedistribution of a set of microquakes, red dots, and the receivers,blue triangles. The microquake locations are preliminary esti-mates obtained with a rough 1D approximation of the real ve-locity model. The source-receiver geometry is similar to thesynthetic example presented above. We estimate P-wave trav-eltimes between pairs of microquakes, with the goal of ulti-mately using these traveltimes as inputs for velocity tomogra-phy and microquake location code, to obtain a high-resolutionvelocity model and accurate event locations in the microquakesnest area.

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F igure 3: Two-dimensional projections of microquake (reddots) and receiver (blue triangle) locations in the standardXY -, XZ-, and Y Z-planes. The receiver and the pair ofevents in black are in a near-stationary configuration.

In figure 3, we see a nest of microquakes that seems to comefrom directly below one of the receivers (marked in black),which, we refer to as the stationary receiver Rs (note that Rs,in general, is not exactly but near a stationary location withrespect to a microquake pair). We now choose pairs of micro-quakes (e.g., the two microquakes marked in black in figure 3)that are nearly on a stationary path with Rs. For simplicity, wechoose the direction of the stationary path to be parallel to theZ-axis (black line coming off of Rs in figure 3). We choose

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F igure 4: The original (left) and rank-1 (right) crosscorrelo-grams are on the top. On the bottom, the black curves are theE G Fs, G and Gj , and the red curve is reference G F, Gref , andthe vertical green line indicates the traveltime based on the P-wave arrival. G and G1 agree well, indicating that there isenough stationary energy in the crosscorrelograms, suggestingthat the kinematic information is reliable.

many pairs near this stationary path and for each pair we es-timate the P-wave traveltimes in four different ways: i) sincethe microquakes and Rs are nearly along a stationary path,we subtract the P-wave arrival times picked in the recordeddata; ii) standard SI (crosscorrelation and stack of the vertical-component signal for all receivers); iii) SI-SV D method (stackof lower-rank crosscorrelogram over the receivers); iv) cross-correlation of vertical component for Rs only. For measure-ments ii), iii), and iv), the traveltime will be the peak of the(stacked) crosscorrelations. If the microquakes and Rs arealong an exactly stationary path, then all four estimates shouldbe equal. F igure 4 shows original and rank-1 crosscorrelo-grams and the respective crosscorrelogram stacks for a micro-quake pair. In this case, the signal-to-noise ratio is good andboth G and G1 are similar, suggesting that the kinematic in-formation is reliable. This is often the case for the waveformsof the events in figure 3, but it is not the case for the entiredataset. Note that the crosscorrelation stacks (E G Fs) here donot actually converge to the true G F.

Since the pair of microquakes and Rs are nearly along a sta-tionary path, the true P-wave traveltime between any two eventsshould approach the difference between the traveltimes of theP-waves recorded at Rs, which is our measurement i). Thus,we use these traveltimes as the reference traveltimes. Now,we investigate the correlation between the three traveltime es-timates - ii), iii), and iv) - and the reference one - i). F ig-ure 5 shows cross-plots for these traveltimes. E ven thoughall three cases show some deviated points (the ones fartheraway from the trend line), the estimates obtained via SI and SI-SV D correlate better with the P-wave traveltimes than the es-timates obtained using signal from Rs only. This suggests thatinter-event traveltime estimates for this dataset based solely oncrosscorrelation at one station may have large errors. Thus, in

SVD enhanced interferometric traveltimes

this case, the stacking step in estimates ii) and iii) is importantfor the covergence of the traveltimes.

We can think of a few ways to classify a given traveltime es-timate as accurate enough or not. One could remove the de-viated points of one and/or another of the different estimates(blue, red, and black in figure 5) or, alternatively, of all ofthem for a more conservative approach. Since here we arealso investigating the use of SV D in SI, we remove the scatter-ing only from red estimates in figure 5. Removing the sevenred deviated points in figure 5 we obtain figure 6. A s men-tioned above, the signal-to-noise ratio in this data set is gener-ally good, leading to relatively sharp crosscorrelations in gen-eral. However, the three deviated data points in figure 6 areexceptions. F igure 7 shows the SI results for one of the bluedeviated points in figure 6. Similar to the synthetic example,in this case the crosscorrelograms are fairly different (as arethe respective E G Fs) indicating a high level of incoherency inthe crosscorrelograms. However, G1 still peaks at a reason-able time shift, as seen in figure 6, which indicates that theresult based on SV D for this case is reasonable. It is impor-tant to note that we do not expect estimantes ii), iii), and iv)to coincide perfectly with the traveltimes based on the P-wavepicks (but we expect it to be close) for a few reasons. F irst,

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F igure 5: Cross-plots showing the correlation between the es-timated and reference traveltimes for P-waves. The green lineis the slope-1 line that crosses the origin. The first two es-timates, obtained through SI, have less scattering than usingsignal from Rs only.

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F igure 6: Similar to figure 5 but the red deviated points in theSV D estimates were removed.

the sources and receivers are not located on exactly stationarypaths. A s said above, the microquake locations here were ob-tained with a rough velocity model so there are likely errors inthe event locations, which lead to uncertainties in whether thesources are actually located near stationary paths. However,the SI-SV D method is stable with respect to these location er-

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F igure 7: The original (left) and rank-1 (right) crosscorrelo-grams are on the top. On the bottom, the black curves are theE G Fs, G and Gj , and the red curve is reference G F, Gref , andthe vertical green line indicates the traveltime based on the P-wave arrival. G and G1 are different, indicating a high levelof incoherency in the crosscorrelograms. However, G1 peaksat a reasonable time shift as measured by its relationship to thedifference time pick figure 6.

rors as long as the events are within the Fresnel zone. Second,low signal-to-noise ratios are also a source of errors. We didnot exclude data with low signal-to-noise ratio from our anal-ysis because we expected SI-SV D to be able to give the cor-rect peak on the E G F in some of these cases, as is indeed thecase as mentioned above. G iven these uncertainties, we thecorrelation between SI-SV D and P-wave based traveltimes isgenerally good, making the method promising for improvingestimated locations and velocities.

Conclusions and future work

We have shown that physically accurate inter-events traveltimeestimates are obtained, for microquake pairs of a geothermalreservoir, through a combination of SV D and SI. We haveillustrated this both with synthetic and field data examples.These traveltimes can be used in velocity tomography and mi-croquake location algorithms, to obtain locally high-resolutionvelocity models and accurate event locations in the area of themicroquake nest. While the SI-SV D technique in general maylead to more accurate estimates of inter-events traveltimes, itcan only be applied to sets of sources and receivers in near-stationary paths.

Acknowledgments

We would like to thank O leg Poliannikov and M ichael Fehlerfor great suggestions during our discussions. This work is sup-ported by grants from the Department of Energy (D E-F G36-08G O18197) and the founding members consortium at EarthResources Laboratory (E R L).

SVD enhanced interferometric traveltimes

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