Waveform tomography based on local image correlationsFrancesco Perrone∗ and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

Common velocity analysis is based on the invariance ofmigrated images with respect to the experiment index(shot number, plane-wave take-off angle, etc.) or exten-sion parameters (reflection angle, correlation lags in ex-tended images, etc.). All the information available, i.e.the entire survey, is used for assessing the quality of themodel used for imaging. This approach is effective butforces a clear separation between imaging and velocitymodel building. Here, we ask a complementary ques-tion: how much information about the velocity modelis contained in a minimum number of images? Startingfrom an alternative statement of the semblance prin-ciple, we propose a measure of velocity error based onlocal correlations of pairs of migrated images. We designan objective function and implement a local, gradient-based optimization scheme to reconstruct the velocitymodel. Our methodology is “full-wave” because it isnot based on a linearization of the imaging operator (incontrast with linearized wave-equation migration veloc-ity analysis techniques). The gradient of the objectivefunction is computed using the adjoint-state method.

INTRODUCTION

Seismic imaging includes estimation of both the posi-tion of the structures responsible for the recorded dataand a model that describes the propagation in the sub-surface. The two problems are related since a modelis necessary to infer the position of the reflectors. Therecorded wavefields are extrapolated in the model bysolving a wave equation and crosscorrelated with a syn-thetic source wavefield simulated in the same model(Claerbout, 1985). Under a single scattering approxi-mation, reflectors are located where the source and re-ceiver wavefields match in time and space. If the veloc-ity model is inaccurate, the reflectors are positioned atincorrect locations.

Wave-equation tomography is a family of techniquesthat estimate the velocity model from finite bandwidthsignals. The inversion is usually formulated as an opti-mization problem, where the correct velocity minimizesan objective function that measures the inconsistencybetween simulated and observed data. Full-waveforminversion (FWI) (Tarantola, 1984; Pratt, 1999; Sirgueand Pratt, 2004) addresses the estimation problem inthe data space and measures the mismatch between ob-servations and simulated data. FWI aims to reconstructthe exact model that generates the recorded data. Bymatching both traveltime and amplitude information,FWI allows one to achieve high-resolution results (Sir-

gue et al., 2010). Nonetheless, it needs a source es-timate, the physics of wave propagation must be cor-rectly modelled and a good parametrization (for exam-ple, impedance vs. velocity contrasts) is crucial (Kellyet al., 2010). Moreover, an accurate initial model is keyto avoid cycle skipping and convergence to the globalminumum of the objective function. Migration velocityanalysis (MVA) (Fowler, 1985; Faye and Jeannot, 1986;Al-Yahya, 1989; Chavent and Jacewitz, 1995; Biondiand Sava, 1999; Sava et al., 2005; Albertin et al., 2006)defines the objective function in the image space andis based on the semblance principle (Al-Yahya, 1989).Groups of experiments and the associated images areanalyzed. If the velocity model is correct, the imagesfrom different experiments must be consistent since theEarth is assumed stationary on the time scale of the seis-mic experiments. MVA leads to smooth objective func-tions and well-behaved optimization problems (Symes,1991; Symes and Carazzone, 1991), and it is less sensi-tive to the initial model than FWI. On the other hand,because we lose the information about the data ampli-tudes, the estimated model shows lower resolution thanthe FWI result. MVA measures either the invariance ofthe migrated images in an auxiliary dimension (reflec-tion angle, shot, etc.) (Al-Yahya, 1989; Sava and Fomel,2003; Xie and Yang, 2008) or focusing in an extendedspace (Rickett and Sava, 2002; Symes, 2008; Sava andVasconcelos, 2009). A large number of shots and goodillumination are necessary to pick moveout curves or as-sess focusing. Moreover, because of the high memoryrequirement for storing the information from each ex-periment, only a subset of the image is considered inthe evaluation of the objective function. For example,common-image gathers (CIGs) (Rickett and Sava, 2002;Yang and Sava, 2011) or common-image point (CIPs)(Sava and Vasconcelos, 2009) are computed at fixed lat-eral positions or picked image points along reflectors,respectively, and both methods require migration of allthe shots that illuminate the location where the CIGs/-CIPs are constructed.

We propose an objective function that operates in theimage space and does not need CIGs. We consider smallgroups of images from neighboring experiments and for-mulate the semblance principle in the shot-image do-main, instead of the extended space at selected CIGs.We use the morphologic relationship between imagesfrom neighboring experiments to define a measure of rel-ative shifts in the image space. This approach reducesthe memory requirements and allows us to include allimage points in the velocity analysis step.

Waveform tomography based on local image correlations

(a) (b) (c)

(d) (e) (f)

Figure 1: Images of a flat reflector obtained using a velocity model that is (a) too high, (b) correct, and (c) too low.For each model we compute the sensitivity kernel associated with the highlighted box. Panels (d), (e), and (f) showthe sensitivity kernels for the corresponding models.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 2: Local image correlations ((a), (b), and (c)),penalty operators ((d), (e), and (f)), and penalized localcorrelations ((g), (h), and (i)) for the box highlightedin Figure 1 and the three models (velocity too high,correct, and too low, respectively).

THEORY

MVA is based on the semblance principle (Al-Yahya,1989): a single Earth model generates the recorded data;then, if the velocity model is correct, different exper-iments must produce consistent images of the reflec-tors. The similarity between migrated images is usuallyassessed through the analysis of common-image gath-ers (CIGs), which can be constructed in different do-mains (reflection angle, ray parameter, sub-surface off-set, etc.) (Sava and Fomel, 2003; Shen and Symes, 2008;Xie and Yang, 2008), or extended images (EIs) (Rick-ett and Sava, 2002; Symes, 2008; Sava and Vasconcelos,2009). These approached require migration of a largenumber of shots in order to measure moveout in the

CIGs or focusing in EIs. Alternatively, we can measurethe similarity of pairs of images by computing local cor-relations at each image point. We define the objectivefunction

J (m) =1

2

�

i

��

λ

P (x,λ) ci (x,λ) �2x,λ, (1)

where ci (x,λ) =�

w(x)Ri+1

�ξ − λ

2

�Ri

�ξ + λ

2

�dξ is the

local correlation of Ri and Ri+1 and P is a penalty oper-ator that highlights features that are related to velocityerrors. The correlations are computed in local windowsw (x), which allow us to consider small subsets of thedata. The index i spans the shots and m denotes themodel. In this work, we define the objective function us-ing pairs of shots, but we use the entire images obtainedfrom them, instead of analyzing only sparsely selectedCIGs or CIPs. The reduced input data leads to lowermemory requirements with respect to other techniquesbased on common image-gathers or extended images. Apossible alternative to measure inconsistencies betweenmigrated images is the energy of the difference betweenthem. Plessix (2006) uses the image difference as a reg-ularization term for full-waveform inversion and not asa stand-alone objective function. The difference opera-tor is a high-pass filter, enhances noise, and is sensitiveto amplitude patterns. Images from different shots donot have identical amplitudes at a given spatial locationbecause the wavefields experience different propagationpaths. The bigger the separation between the shots, thehigher the amplitude mismatch. The difference objec-tive function considers this discrepancy a velocity errorindicator and thus this objective function is intrinsicallybiased. Local correlations measure phase shifts betweentwo images. Correlation is more robust because ampli-tudes are not directly used for measuring velocity errors.Nonetheless, the size of the correlation window must besuch that it captures the relative shift between the twoimages. Local correlation in the image space is analo-gous to data correlation for FWI, as proposed by vanLeeuwen and Mulder (2008, 2010).

Waveform tomography based on local image correlations

We restate the semblance principle as follows: if the ve-locity model is correct, two images from nearby experi-ments constructively interfere along the direction of thereflectors (Perrone and Sava, 2012). This is equivalentto having no moveout in shot-domain CIGs (Xie andYang, 2008). Figure 1 shows the migrated images of ahorizontal reflector with three velocity models (too high(a), correct (b), and too low (c)) and the sensitivity ker-nels ((d), (e), and (f)) for the local window highlightedby the box. The sign of the sensitivity kernel is consis-tent with the velocity error. Figure 2 shows the localcorrelations, penalty operators, and penalized correla-tions for the box in Figure 1. The similarity betweentwo images is measured by the inconsistency betweenthe local correlation and the dip estimated from oneimage. If the velocity model is correct, the maximum ofthe correlation lies along the reflector slope (the centralcolumn of Figure 2); the penalized correlation is an oddfunction, and the stack over correlation lags is practi-cally zero, i.e., the two image are aligned and the modelis correct. If the velocity is incorrect (the first and thirdcolumn of Figure 2), we observe a deviation that de-pends on the sign and extent of the velocity error. Thisdeviation is measured as a mismatch between the ori-entation of the penalty operator (constructed from theestimated dip in the image) and of the the local corre-lation. The penalized correlation are not odd functionsand their mean value measures the relative shift betweenthe two images.

The penalty operator P is constructed from the esti-mated dip field (which depends on the model parametersas well). The dependence of P on the wavefield u can beaccounted for by expressing the dip as a function of theimage through gradient square tensors (van Vliet andVerbeek, 1995). This relationship is highly nonlinearand a thorough study of ∂P/∂u and its impact on theadjoint-state calculations is subject of further research.

SYNTHETIC EXAMPLES

We test our methodology on a synthetic example (Fig-ure 3). Figures 3(a) and 3(c) show the initial model andthe migrated image, respectively. The inaccuracy of thevelocity model can be assessed from the shot-domainCIGs Figure 3(e): the depth of the reflectors changes asa function of experiment, i.e., the semblance principleis not satisfied. After 14 steepest-descent iterations, werecover the model in Figure 3(b). The layers can nowbe identified, and the image shows better positioning ofthe interfaces (Figure 3(d)). Figure 3(f) shows align-ment of the partial images, which indicates an accuratemodel. The simplicity of the structures in the modelhelps the inversion because the definition of the penaltyoperators at each image point is unequivocal. We candefine a dip vector at each image point; the pinch-outsof the sincline (x = 2 km and x = 8 km) do not in-fluence the computed gradient. For more complicatedmodels with conflicting dips and complicated geologic

features (where there is no clear definition of the dipfield), a more sophisticated design of the penalty opera-tor is necessary. The eigenvalues and eigenvectors of thegradient square tensors (van Vliet and Verbeek, 1995)can be used to define ellipses that, in turn, may offera structure oriented criterion for the definition of thepenalty operators.

We run a second test using the Marmousi model. Wesimulate 78 shots with a finite-difference single-scatteringmodeling code and absorbing boundary conditions onthe 4 sides of the model. The spacing between the shotsis 0.08 km, and the first source is at x = 0.96 km; re-ceivers are placed 0.008 km apart on the surface. Theinitial model is a heavily smoothed and scaled versionof the original Marmousi slowness model. Figures 4(a),4(c), and 4(e) show the initial model, migrated image,and shot-domain CIGs extracted every 1 km from x =0 km, respectively. Conflicting dips are not yet handledcorrectly by our methodology because they make thedefinition of the penalty operators ambiguous. For thisreason, we restrict the inversion to the first 1.5 km indepth, where the reflectors are more coherent.

The migrated image (Figure 4(c)) presents crossing in-terfaces and defocused fault planes; the shot-domainCIGs (Figure 4(e)) show variation of the depth of reflec-tors as a function of experiment, which indicates modelinaccuracy. After 14 steepest-descent iterations, we ob-tain the results in Figures 4(b), 4(d), and 4(f). The in-terfaces are moved toward the correct position, and weobserve better focusing of the fault planes. The shot-domain CIGs in Figure 4(f) show flatter events, whichindicate a more kinematically accurate model.

CONCLUSIONS

We present an approach to waveform tomography basedon the semblance principle in the shot-image domain.We define an objective function using appropriately pe-nalized local image correlations that measure the rela-tive shift between two images obtained from nearby ex-periments; the penalty operator enhances the mismatchbetween the structural information (the orientation ofthe reflector) and the shift between two images. Thegradient of the objective function with respect to themodel is computed with the adjoint-state method, whichmakes our technique a close relative to the more con-ventional full waveform inversion. Inversion on a simplesynthetic and the Marmousi model shows the ability ofour strategy to correct model errors.

ACKNOWLEDGMENTS

We would like to thank eni E&P for the permission topublish this work. The reproducible numerical examplesin this paper use the Madagascar open-source softwarepackage freely available from http://www.reproducibility.org.

Waveform tomography based on local image correlations

(a) (b)

(c) (d)

(e) (f)

Figure 3: (a) Initial slowness model and (b) slowness model after 14 iterations of waveform tomography; (c) intialimage and (d) migrated image with the updated model; (e) intial shot-domain CIGs and (f) shot-domain CIGs withthe updated model. Observe the improved positioning of the reflectors after the inversion. The curvature of the CIGsis corrected by the inversion algorithm.

(a) (b)

(c) (d)

(e) (f)

Figure 4: (a) Initial slowness model and (b) slowness model after 14 iterations of waveform tomography; (c) intialimage and (d) migrated image with the updated model; (e) intial shot-domain CIGs and (f) shot-domain CIGs withthe updated model.

Waveform tomography based on local image correlations

REFERENCES

Al-Yahya, K., 1989, Velocity analysis by iterative profilemigration: Geophysics, 54, P. 718–729.

Albertin, U., P. Sava, J. Etgen, and M. Maharramov,2006, Adjoint wave-equation velocity analysis: Pre-sented at the 74th Ann. Internat. Mtg., SEG.

Biondi, B., and P. Sava, 1999, Wave-equation migrationvelocity analysis: Presented at the 69th Ann. Inter-nat. Mtg., SEG.

Chavent, G., and C. A. Jacewitz, 1995, Determinationof background velocities by multiple migration fitting:Geophysics, 60, 476–490.

Claerbout, J. F., 1985, Imaging the earth’s interior:Blackwell Publishing.

Faye, J.-P., and J.-P. Jeannot, 1986, Prestack migrationvelocity analysis from focusing depth imaging: Pre-sented at the 56th Ann. Internat. Mtg., Soc. of Expl.Geophys.

Fowler, P., 1985, Migration velocity analysis by opti-mization: Technical Report 44, Stanford ExplorationProject.

Kelly, S., J. R. Martinez, B. tsimelzon, and S. Crawley,2010, A comparison of inversion results for two full-waveform methods that utilize the lowest frequenciesin dual-sensor recordings: Presented at the 80th Ann.Internat. Mtg., Soc. of Expl. Geophys.

Perrone, F., and P. Sava, 2012, Wavefield tomographybased on local image correlation: Presented at the72th Ann. Internat. Mtg., European Association ofGeoscientists and Engineers.

Plessix, R.-E., 2006, A review of the adjoint-statemethod for computing the gradient of a functionalwith geophysical applications: Geophys. J. Int., 167,495–503.

Pratt, R. G., 1999, Seismic waveform inversion in thefrequency domain, part 1: Theory and verification ina physical scale model: Geophysics, 64, 888–901.

Rickett, J., and P. Sava, 2002, Offset and angle-domaincommon image-point gathers for shot-profile migra-tion: Geophysics, 67, 883–889.

Sava, P., B. Biondi, and J. Etgen, 2005, Wave-equationmigration velocity analysis by focusing diffractionsand reflections: Geophysics, 70, U19–U27.

Sava, P., and S. Fomel, 2003, Angle-domain common-image gathers by wavefield continuation methods:Geophysics, 68, 1065–1074.

Sava, P., and I. Vasconcelos, 2009, Extended common-image-point gathers for wave-equation migration:Presented at the 71th Ann. Internat. Mtg., EuropeanAssociation of Geoscientists and Engineers.

Shen, P., and W. W. Symes, 2008, Automatic velocityanalysis via shot profile migration: Geophysics, 73,VE49–VE59.

Sirgue, L., O. I. Barkved, J. Dellinger, J. Etgen, U. Al-bertin, and J. H. Kommedal, 2010, Full waveform in-version: the next leap forward in imaging at valhall:First Break, 28, 65–70.

Sirgue, L., and R. Pratt, 2004, Efficient waveform inver-

sion and imaging: A strategy for selecting temporalfrequencies: Geophysics, 69, 231–248.

Symes, W. W., 1991, Layered velocity inversion: Amodel problem from reflection seismology problemfrom reflection seismology: SIAM J. on Math. Anal-ysis, 22, pp. 680–716.

——–, 2008, Migration velocity analysis and waveforminversion: Geophysical Prospecting, 56, 765–790.

Symes, W. W., and J. J. Carazzone, 1991, Velocity in-version by differential semblance optimization: Geo-physics, 56, 654–663.

Tarantola, A., 1984, Inversion of seismic reflection datain the acoustic approximation: Geophysics, 49, 71–92.

van Leeuwen, T., and W. A. Mulder, 2008, Veloc-ity analysis based on data correlation: GeophysicalProspecting, 56, 791 – 803.

——–, 2010, A correlation-based misfit criterion forwave-equation traveltime tomography: Geophys. J.Int., 182, 1383–1394.

van Vliet, L. J., and P. W. Verbeek, 1995, Estimatorsfor orientation and anisotropy in digitalized images:Proceedings of the first annual conference of the Ad-vanced School for Computing and Imaging, ASCI’95,442–450.

Xie, X. B., and H. Yang, 2008, The finite-frequencysensitivity kernel for migration residual moveout andits applications in migration velocity analysis: Geo-physics, 73, S241–S249.

Yang, T., and P. Sava, 2011, Wave-equation migrationvelocity analysis with time-shift imaging: Geophysi-cal Prospecting, 59, 635–650.