CWP-599

Seismic imaging with Wigner distribution functions

Paul Sava1 and Oleg Poliannikov21 Center for Wave Phenomena, Colorado School of Mines2 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology

ABSTRACTThe fidelity of depth seismic imaging depends on the accuracy of the velocity mod-els used for wavefield reconstruction. Models can be decomposed in two componentscorresponding to large scale and small scale variations. In practice, the large scalevelocity model component can be estimated with high accuracy using repeated migra-tion/tomography cycles, but the small scale component cannot. When the Earth hassignificant small-scale velocity components, wavefield reconstruction does not com-pletely describe the recorded data and migrated images are perturbed by artifacts.There are two possible ways to address this problem: improve wavefield reconstructionby estimating more accurate velocity models and image using conventional techniques(e.g. wavefield cross-correlation), or reconstruct wavefields with conventional methodsusing the known background velocity model, but improve the imaging condition toalleviate the artifacts caused by the imprecise reconstruction, which is what we suggestin this paper.We describe the unknown component of the velocity model as a random function withlocal spatial correlations. Imaging data perturbed by such random variations is char-acterized by statistical instability, i.e. various wavefield components image at wronglocations that depend on the actual realization of the random model. Statistical stabil-ity can be achieved by pre-processing the reconstructed wavefields prior to the imag-ing condition. We employ Wigner distribution functions to attenuate the random noisepresent in the reconstructed wavefields, parametrized as a function of image coordi-nates. Wavefield filtering using Wigner distribution functions and conventional imag-ing can be lumped-together into a new form of imaging condition which we call an“interferometric imaging condition” due to its similarity to concepts from recent workon interferometry. The interferometric imaging condition can be formulated both forzero-offset and for multi-offset data, leading to robust and efficient imaging proceduresthat are effective in attenuating imaging artifacts due to unknown velocity models.

Key words: wave-equation, imaging, Wigner distribution functions

1 INTRODUCTION

Seismic imaging in complex media requires accurate knowl-edge of the medium velocity. Assuming single scattering(Born approximation), imaging requires propagation of therecorded wavefields from the acquisition surface, followed bythe application of an imaging condition highlighting locationswhere backscattering occurs, i.e. where reflectors are present.Typically, this is achieved with simple image processing tech-niques, e.g. cross-correlation of wavefields reconstructed fromsources and receivers.

The main requirement for good-quality imaging is ac-

curate knowledge of the velocity model. Errors in the modelused for imaging lead to inaccurate reconstruction of the seis-mic wavefields and to distortions of the migrated images. Inany realistic seismic field experiment the velocity model isnever known exactly. Migration velocity analysis estimateslarge scale approximations of the model, but some fine scalevariations always remain elusive. For example, when geol-ogy includes complicated stratigraphic structures or complexsalt/carbonate bodies, the rapid velocity variations on the scaleof the seismic wavelength and smaller cannot be estimatedcorrectly by kinematic methods. Therefore, even if the broadkinematics of the seismic wavefields are reconstructed cor-

152 P. Sava & O. Poliannikov

rectly, the extrapolated wavefields also contain phase and am-plitude distortions that lead to image artifacts obstructing theimage of the geologic structure under consideration. While itis certainly true that even the recovery of a long-wave back-ground may prove to be a challenge in some circumstances, wedo not attempt to address that issue in this paper. Instead, weconcentrate solely on the problem of dealing with the effect ofa small scale random variations not estimated by conventionalmethods.

There are two ways in which we can approach thisproblem. The first option is to improve our velocity analysismethods to estimate the small-scale variations in the model.Such techniques take advantage of all information containedin seismic wavefields and are not limited to kinematic in-formation of selected events picked from the data. Exam-ples of techniques in this category are waveform inversion(Tarantola, 1987; Pratt & Worthington, 1990; Pratt, 1990; Sir-gue & Pratt, 2004), wave-equation tomography (Woodward,1992) or wave-equation migration velocity analysis (Sava &Biondi, 2004a,b; Shen et al., 2005). A more accurate veloc-ity model allows for more accurate wavefield reconstruction.Then, wavefields can be used for imaging using conventionalprocedures, e.g. cross-correlation. The second option is to con-centrate on the imaging condition, rather than concentrate on“perfect” wavefield reconstruction. Assuming that the large-scale component of the velocity models is known (e.g. by it-erative migration/tomography cycles), we can design imagingconditions that are not sensitive to small inaccuracies of thereconstructed wavefields. Imaging artifacts can be reduced atthe imaging condition step, despite the fact that the wavefieldsincorporate small kinematic errors due to velocity fluctuations.

Of course, the two options are complementary to eachother, and both can contribute to imaging accuracy. In this pa-per, we concentrate on the second approach. For purposes oftheoretical analysis, it is convenient to model the small-scalevelocity fluctuations as random but spatially correlated varia-tions superimposed on a known velocity. We assume that weknow the background model, but that we do not know the ran-dom fluctuations. The goal is to design an imaging conditionthat alleviates artifacts caused by those random fluctuations.Conventional imaging consists of cross-correlations of extrap-olated source and receiver wavefields at image locations. Sincewavefield extrapolation is performed using an approximationof the true model, the wavefields contain random time delays,or equivalently random phases, which lead to imaging arti-facts.

One way of mitigating the effects of the random modelon the quality of the resulting image is to use techniques basedon acoustic time reversal (Fink, 1999). Under certain assump-tions, a signal sent through a random medium, recorded by areceiver array, time reversed and sent back through the samemedium, refocuses at the source location in a statistically sta-ble fashion. Statistical stability means that the refocusing prop-erties (i.e. image quality) are independent of the actual realiza-tion of the random medium (Papanicolaou et al., 2004; Fouqueet al., 2005).

We investigate an alternative way of increasing imag-

ing statistical stability. Instead of imaging the reconstructedwavefields directly, we first apply a transformation based onWigner distribution functions (Wigner, 1932) to the recon-structed wavefields. We consider a special case of the Wignerdistribution function (WDF) which has the property that it at-tenuates random fluctuations from the wavefields after extrap-olation with conventional techniques. The idea for this methodis borrowed from image processing where WDFs are used forfiltering of random noise. Here, we apply WDFs to the recon-structed wavefields, prior to the imaging condition. This is incontrast to data filtering prior to wavefield reconstruction or toimage filtering after the application of an imaging condition.

Our procedure closely resembles conventional imagingprocedures where wavefields are extrapolated in the imagevolume and then cross-correlated in time at every image lo-cation. Our method uses WDFs defined in three-dimensionalwindows around image locations which makes it both robustand efficient. From an implementation and computational costpoint of view, our technique is similar to conventional imag-ing, but its statistical properties are improved. Although con-ceptually separate, we can lump-together the WDF transfor-mation and conventional imaging into a new form of imagingcondition which resembles interferometric techniques (Papan-icolaou et al., 2004; Fouque et al., 2005). Therefore, we usethe name interferometric imaging condition for our techniqueto contrast it with the conventional imaging condition.

A related method discussed in the literature is knownunder the name of coherent interferometric imaging (Borceaet al., 2006a,b,c). This method uses similar local cross-correlations and averaging, but unlike our method, itparametrizes reconstructed wavefields as a function of receivercoordinates. Thus, the coherent interferometric imaging func-tional requires separate wavefield reconstruction from everyreceiver position, which makes this technique prohibitively ex-pensive and probably unusable in practice on large-scale seis-mic imaging projects. In contrast, the imaging technique ad-vocated in this paper achieves similar statistical stability prop-erties as coherent interferometric imaging, but at an affordablecomputational cost since we apply wavefield reconstructiononly once for all receiver locations corresponding to a givenseismic experiment, typically a “shot”.

2 IMAGING CONDITIONS

2. 1 Conventional imaging condition

Let D (x, t) be the data recorded at time t at receivers locatedat coordinates x for a seismic experiment with buried sources(Figure 1(a)) also known as an exploding reflector seismic ex-periment (Loewenthal et al., 1976). A conventional imagingprocedure for this type of data consists of two steps (Claer-bout, 1985): wavefield reconstruction at image coordinates yfrom data recorded at receiver coordinates x, followed by animaging condition taking the reconstructed wavefield at timet = 0 as the seismic image.

Mathematically, we can represent the wavefield V recon-structed at coordinates y from data D recorded at coordinates

WDF imaging 153

y

x

sources

receivers

(a)

sourcex

yscatterers

receivers

(b)

Figure 1. Zero-offset seismic experiment sketch (a) and multi-offset seismic experiment sketch (b). Coordinates x = {x, y} characterize receiverpositions on the surface and coordinates y = {x, y, z} characterize reflector positions in the subsurface.

x as a temporal convolution of the recorded trace with theGreen’s function G connecting the two points:

V (x,y, t) = D (x, t) ∗t G (x,y, t) . (1)

The total wavefield U reconstructed at y from all receiversis the superposition of the wavefields V reconstructed fromindividual traces

U (y, t) =

Zx

dx V (x,y, t) , (2)

where the integral over x spans the entire receiver space. Asstated, the imaging condition extracts the image R (y) fromthe wavefield U (y, t) at time t = 0, i.e.

R (y) = U (y, t = 0) . (3)

The Green’s function used in the procedure described byequation 1 can be implemented in different ways. For our pur-poses, the actual method used for computing Green’s functionsis not relevant. Any procedure can be used, although differentprocedures will be appropriate in different situations, with dif-ferent cost of implementation. We assume that a satisfactoryprocedure exists and is appropriate for the respective velocitymodels used for the simulations. In our examples, we computeGreen’s functions with time-domain finite-difference solutionsto the acoustic wave-equation, similar to reverse-time migra-tion (Baysal et al., 1983).

Consider the velocity model depicted in Figure 2(a) andthe imaging target depicted in Figure 2(b). We assume that the

model with random fluctuations (Figure 3(b)) represents thereal subsurface velocity and use this model to simulate data.We consider the background model (Figure 3(a)) to representthe migration velocity and use this model to migrate the datasimulated in the random model. We consider one source lo-cated in the subsurface at coordinates z = 8 km and x =13.5 km, and receivers located close to the top of the model atdiscrete horizontal positions and depth z = 0.0762 km. Fig-ure 3(c)-3(d) show snapshots of the simulated wavefields at alater time. The panels on the left correspond to modeling in thebackground model, while the panels on the right correspond tomodeling in the random model.

Figure 3(f) shows the data recorded on the surface. Thedirect wavefield arrival from the seismic source is easily iden-tified in the data, although the wavefronts are distorted by therandom perturbations in the medium. For comparison, Fig-ure 3(e) shows data simulated in the background model, whichdo not show random fluctuations.

Conventional imaging using the procedure describedabove implicitly states that data generated in the randommodel, Figure 3(b), are processed as if they were generatedin the background model, Figure 3(a). Thus, the random phasevariations in the data are not properly compensated during theimaging procedure causing artifacts in the image. Figure 6(b)shows the image obtained by migrating data from Figure 3(f)using the model from Figure 3(a). For comparison, Figure 6(a)shows the image obtained by migrating the data from Fig-ure 3(e) using the same model from Figure 3(a).

Ignoring aperture effects, the artifacts observed in the im-

154 P. Sava & O. Poliannikov

(a) (b)

Figure 2. Velocity model (a) and imaging target located around z = 8 km (b). The shaded area is imaged in Figures 6(a)-6(d).

ages are caused only by the fact that the velocity models usedfor modeling and migration are not the same. Small artifactscaused by truncation of the data on the acquisition surfacecan also be observed, but those artifacts are well-known (i.e.truncation butterflies) and are not the subject of our analysis.In this example, the wavefield reconstruction procedure is thesame for both modeling and migration (i.e. time-domain finite-difference solution to the acoustic wave-equation), thus it isnot causing artifacts in the image. We can conclude that themigration artifacts are simply due to the phase errors betweenthe Green’s functions used for modeling (with the random ve-locity) and the Green’s functions used for migration (with thebackground velocity). The main challenge for imaging in me-dia with random variations is to design procedures that atten-uate the random phase delays introduced in the recorded databy the unknown variations of the medium without damagingthe real reflections present in the data.

The random phase fluctuations observed in recorded data(Figure 3(f)) are preserved during wavefield reconstruction us-ing the background velocity model. We can observe the ran-domness in the extrapolated wavefields in two ways, by recon-structing wavefields using individual data traces separately, orby reconstructing wavefields using all data traces at once.

The first option is to reconstruct the seismic wavefield atall image locations y from individual receiver positions on thesurface. Of course, this is not conventionally done in reverse-time imaging, but we describe this concept just for illustra-tion purposes. Figure 4(a) shows the wavefield reconstructedseparately from individual data traces depicted in Figure 3(f)using the background model depicted in Figure 3(b). In Fig-ure 4(a), the horizontal axis corresponds to receiver positionson the surface, i.e. coordinates x, and the vertical axis repre-sents time. According to the notations used in this paper, Fig-ure 4(a) shows the wavefield V (x,y, t) reconstructed to a par-ticular image coordinate y from separate traces located on thesurface at coordinates x. A similar plot can be constructed forall other image locations. Ideally, the reconstructed wavefieldshould line-up at time t = 0, but this is not what we observe inthis figure, indicating that the input data contain random phase

delays that are not compensated during wavefield reconstruc-tion using the background velocity.

The second option is to reconstruct the seismic wavefieldat all image locations y from all receiver positions on the sur-face at once. This is a conventional procedure for reverse-timeimaging. Figure 5(a) shows the wavefield reconstructed fromall data traces depicted in Figure 3(f) using the backgroundmodel depicted in Figure 3(b). In Figure 5(a), the vertical andhorizontal axes correspond to depth and horizontal positionsaround the source, i.e. coordinates y, and the third cube axisrepresents time. According to the notations used in this paper,Figure 5(a) corresponds to wavefield U (y, t) reconstructedfrom data at all receiver coordinates x to image coordinates y.The reconstructed wavefield does not focus completely at theimage coordinate and time t = 0 indicating that the input datacontains random phase delays that are not compensated duringwavefield reconstruction using the background velocity.

2. 2 Wigner distribution functions

One possible way to address the problem of random fluctua-tions in reconstructed wavefields is to use Wigner distributionfunctions (Wigner, 1932) to pre-process the wavefields prior tothe application of the imaging condition. Appendix C providesa brief introduction for readers unfamiliar with Wigner distri-bution functions. More details about this topic are presentedby Cohen (1995).

Wigner distribution functions (WDF) are bi-linear repre-sentations of multi-dimensional signals defined in phase space,i.e. they depend simultaneously on position-wavenumber (y−k) and time-frequency (t−ω). Wigner (1932) developed theseconcepts in the context of quantum physics as probabilityfunctions for the simultaneous description of coordinates andmomenta of a given wave function. WDFs were introduced tosignal processing by Ville (1948) and have since found manyapplications in signal and image processing, speech recogni-tion, optics, etc.

A variation of WDFs, called pseudo Wigner distribu-tion functions are constructed using small windows local-ized in space and/or time (Appendix C). Pseudo WDFs are

WDF imaging 155

(a) (b)

(c) (d)

(e) (f)

Figure 3. Seismic snapshots of acoustic wavefields simulated in the background velocity model (a)-(c), and in he random velocity model (b)-(d).Data recorded at the surface from simulation in the background velocity model (e) and from the simulation in the random velocity model (f).

156 P. Sava & O. Poliannikov

simple transformations with efficient application to multi-dimensional signals. In this paper, we apply the pseudo WDFtransformation to multi-dimensional seismic wavefields ob-tained by reconstruction from recorded seismic data. We usepseudo WDFs for decomposition and filtering of extrapolatedspace-time signals as a function of their local wavenumber-frequency. In particular, pseudo WDFs can filter reconstructedwavefields to retain their coherent components by removinghigh-frequency noise associated with random fluctuations inthe wavefields due to random fluctuations in the model.

The idea for our method is simple: instead of imagingthe reconstructed wavefields directly, we first filter them usingpseudo WDFs to attenuate the random phase noise, and thenproceed to imaging using a conventional or an extended imag-ing conditions. Wavefield filtering occurs during the applica-tion of the zero-frequency end-member of the pseudo WDFtransformation, which reduces the random character of thefield. For the rest of the paper, we use the abbreviation WDFto denote this special case of pseudo Wigner distribution func-tions, and not its general form.

As we described earlier, we can distinguish two options.The first option is to use wavefield parametrization as a func-tion of data coordinates x. In this case, we can write the pseudoWDF of the reconstructed wavefield V (x,y, t) as

Vx (x,y, t) =

Z|th|≤T

dth

Z|xh|≤X

dxh

V

„x− xh

2,y, t− th

2

«V

„x +

xh

2,y, t +

th

2

«, (4)

where xh and th are variables spanning space and time in-tervals of total extent X and T , respectively. For 3D sur-face acquisition geometry, the 2D variable xh is defined onthe acquisition surface. The second option is to use wavefieldparametrization as a function of image coordinates y. In thiscase, we can write the pseudo WDF of the reconstructed wave-field U (y, t) as

Wy (y, t) =

Z|th|≤T

dth

Z|yh|≤Y

dyh

U

„y − yh

2, t− th

2

«U

„y +

yh

2, t +

th

2

«, (5)

where yh and th are variables spanning space and time in-tervals of total extent Y and T , respectively. For 3D surfaceacquisition geometry, the 3D variable yh is defined around im-age positions.

For the examples used in this section, we employ 41 gridpoints for the interval X centered around a particular receiverposition, 5 × 5 grid points for the interval Y centered arounda particular image point, and 21 grid points for the interval Tcentered around a particular time. These parameters are notnecessarily optimal for the transformation, since they charac-terize the local WDF windows and depend on the specific im-plementation of the pseudo WDF transformation. The maincriterion used for selecting the size of the space-time win-dow for the pseudo WDF transformation is that of avoiding

cross-talk between nearby events, e.g. reflections. Finding theoptimal size of this window is an important consideration forour method, although its complete treatment falls outside thescope of the current paper and we leave it for future research.Preliminary results on optimal window selection are discussedby Borcea et al. (2006a).

Figure 4(b) depicts the results of applying the pseudoWDF transformation to the reconstructed wavefield in Fig-ure 4(a). For the case of modeling in the random model and re-construction in the background model, the pseudo WDF atten-uates the random character of the wavefield significantly, Fig-ure 4(b). The random character of the reconstructed wavefieldis reduced and the main events cluster more closely aroundtime t = 0. Similarly, Figure 5(b) depicts the results of apply-ing the pseudo WDF transformation to the reconstructed wave-fields in Figure 5(a). For the case of modeling in the randommodel and reconstruction in the background model, the pseudoWDF also attenuates the random character of the wavefieldsignificantly, Figure 5(b). The random character of the recon-structed wavefields is also reduced and the main events focusat the correct image location at time t = 0.

2. 3 Zero-offset interferometric imaging condition

After filtering the reconstructed wavefields with pseudoWDFs, we can perform imaging with normal procedures. Forthe case of wavefields parametrized as a function of data coor-dinates, we obtain the total wavefield at image coordinates bysumming over receiver coordinates x

Wx (y, t) =

Zx

dx Vx (x,y, t) , (6)

followed by a conventional imaging condition extracting timet = 0 from the pseudo WDF of the reconstructed wavefields:

Rx (y) = Wx (y, t = 0) . (7)

The image obtained with this imaging procedure is shownin Figure 6(c). As expected, the artifacts caused by the un-known random fluctuations in the model are reduced, leavinga cleaner image of the source.

Similarly, for the case of wavefields parametrized as afunction of image coordinates, we obtain the image by appli-cation of the conventional imaging condition extracting timet = 0 from the pseudo WDF of the reconstructed wavefield:

Ry (y) = Wy (y, t = 0) . (8)

The image obtained with this imaging procedure is shown inFigure 6(d). As in the preceding case, the artifacts caused bythe unknown random fluctuations in the model are reduced,producing a cleaner image of the source, comparable with theone in Figure 6(c).

2. 4 Multi-offset interferometric imaging condition

The imaging procedure in equations 5-8 can be generalized forimaging prestack (multi-offset) data (Figure 1(b)). The con-

WDF imaging 157

(a) (b)

Figure 4. Reconstructed seismic wavefield as a function of data coordinates (a) and its pseudo Wigner distribution function (b) computed as afunction of data coordinates x and time t. The wavefield is reconstructed using the background model from data simulated in the random model.

ventional imaging procedure for this type of data consists oftwo steps (Claerbout, 1985): wavefield simulation from thesource location to the image coordinates y and wavefield re-construction at image coordinates y from data recorded at re-ceiver coordinates x, followed by an imaging condition eval-uating the match between the simulated and reconstructedwavefields.

Let US (y, t) be the source wavefield constructed fromthe location of the seismic source and UR (y, t) the receiverwavefield reconstructed from the receiver locations. A con-ventional imaging procedure produces a seismic image as thezero-lag of the time cross-correlation between the source andreceiver wavefields. Mathematically, we can represent this op-eration as

R (y) =

Zt

dt US (y, t) UR (y, t) , (9)

where R (y) represents the seismic image for a particular seis-mic experiment at coordinates y. When multiple seismic ex-periments are processed, a complete image is obtained by sum-mation of the images constructed for individual experiments.The actual reconstruction methods used to produce the wave-fields US (y, t) and UR (y, t) are irrelevant for the present dis-cussion. As in the zero-offset/exploding reflector case, we usetime-domain finite-difference solutions to the acoustic wave-equation, but any other reconstruction technique can be ap-plied without changing the imaging approach.

When imaging in random media, the data recorded at thesurface incorporates phase delays caused by the velocity varia-tions encountered while waves propagate in the subsurface. Ina typical seismic experiment, random phase delays accumulateboth on the way from the source to the reflectors, as well as onthe way from the reflectors to the receivers. Therefore, the re-ceiver wavefield reconstructed using the background velocitymodel is characterized by random fluctuations, similar to theones seen for wavefields reconstructed in the zero-offset situa-tion. In contrast, the source wavefield is simulated in the back-ground medium from a known source position and, therefore,it is not affected by random fluctuations. However, the zero-lagof the cross-correlations between the source wavefields (with-out random fluctuations) and the receiver wavefield (with ran-dom fluctuations), still generates image artifacts similar to theones encountered in the zero-offset case.

Statistically stable imaging using pseudo WDFs can be

obtained in this case, too. What we need to do is attenuate thephase errors in the reconstructed receiver wavefield and thenapply a conventional imaging condition. Therefore, a multi-offset interferometric imaging condition can be formulated as

R (y) =

Zt

dt US (y, t) WR (y, t) , (10)

where WR (y, t) represents the pseudo WDF of the receiverwavefield UR (y, t) which can be constructed, in principle, ei-ther with parametrization relative to data coordinates, accord-ing to equations 4-6, or relative to image coordinates, accord-ing to equation 5. Of course, our choice is to use image-spaceparametrization for computational efficiency reasons.

2. 5 Discussion

The strategies described in the preceding section have notablesimilarities and differences. The imaging procedures 4-6-7 and5-8 are similar in that they employ wavefields reconstructedfrom the surface data in similar ways. Neither method uses thesurface recorded data directly, but they use wavefields recon-structed from those data as boundary conditions to numericalsolutions of the acoustic wave-equation. The actual wavefieldreconstruction procedure is identical in both cases.

The techniques are different because imaging with equa-tions 4-6-7 employs independent wavefield reconstructionfrom receiver locations x to image locations y. In practice, thisrequires separately solving the acoustic wave-equation, e.g. bytime-domain finite-differences, from all receiver locations onthe surface. Such computational effort is often prohibitive inpractice. In contrast, imaging with equations 5-8 is similar toconventional imaging because it requires only one wavefieldreconstruction using all recorded data at once, i.e. only onesolution to the acoustic wave-equation, similar to conventionalshot-record migration.

The techniques 4-6-7 and 5-8 are similar in that they bothemploy noise suppression using pseudo Wigner distributionfunctions. However, the methods are parametrized differently,the former relative to data coordinates with 2D local space av-eraging and the later relative to image coordinates with 3Dlocal space averaging.

The imaging functionals presented in this paper are de-scribed as functions of space coordinates, x or y, and time,

158 P. Sava & O. Poliannikov

(a) (b)

Figure 5. Reconstructed seismic wavefield as a function of image coordinates (a) and its pseudo Wigner distribution function (b) computed as afunction of image coordinates y and time t.The wavefield is reconstructed using the background model from data modeled in the random model.

t. As suggested in Appendix C, pseudo WDFs can be imple-mented either in time or frequency, so potentially the imagingconditions discussed in this paper can also be implemented inthe frequency-domain. However, we restrict our attention inthis paper to the time-domain implementation and leave thefrequency-domain implementation subject to future study.

Equations 4-6-7 can be collected into the zero-offsetimaging functional

RCINT (y) =

Zt

dt δ (t)

Zx

dx

Z|th|≤T

dth

Z|xh|≤X

dxh

V

„x− xh

2,y, t− th

2

«V

„x +

xh

2,y, t +

th

2

«, (11)

where the temporal δ function implements the zero time imag-ing condition. A similar form can be written for the multi-offset case. Equation 11 corresponds to the time-domain ver-sion of the coherent interferometric functional proposed byBorcea et al. (2006a,b,c). Consistent with the preceding dis-cussion, the cost required to implement this imaging func-tional is often prohibitive for practical application to seismicimaging problems.

3 STATISTICAL STABILITY

The interferometric imaging condition described in the pre-ceding section is used to reduce imaging artifacts by attenuat-ing the incoherent energy corresponding to velocity errors, asillustrated in Figures 6(b) and 6(d). The random model usedfor this example corresponds to the weak fluctuation regime,as explained in Appendix A (characteristic wavelength of sim-

ilar scale with the random fluctuations in the medium and fluc-tuations with small magnitude).

By statistical instability we mean that images obtainedfor different realizations of random models with the identicalstatistics are different. Figures 7(a)-7(c) illustrate data mod-eled for different realizations of the random model in Fig-ure 3(b). The general kinematics of the data are the same,but subtle differences exist between the various datasets due tothe random model variations. Migration using a conventionalimaging condition leads to the images in Figures 7(d)-7(f)which also show variations from one realization to another.In contrast, Figures 7(g)-7(i) show images obtained by the in-terferometric imaging condition in equations 5-8, which aremore similar to one-another since many of the artifacts havebeen attenuated.

In typical seismic imaging problems, we cannot ensurethat random velocity fluctuations are small (e.g. σ ≤ 5%). It isdesirable that imaging remains statistically stable even in caseswhen velocity varies with larger magnitude. We investigate thestatistical properties of the imaging functional in equations 5-8 using numerical experiments similar to the one used earlier.We describe the random noise present in the velocity modelsusing the following parameters explained in Appendix A: seis-mic spatial wavelength λ = 76.2 m, wavelet central frequencyω = 20 Hz, random fluctuations parameters: ra = 0.0762,rc = 0.0762, α = 2, and random noise magnitude σ be-tween 15% and 45%. This numerical experiment simulates asituation that mixes the theoretical regimes explained in Ap-pendix B: random model fluctuations of comparable scale withthe seismic wavelength lead to destruction of the wavefronts,as suggested by the “weak fluctuations” regime; large magni-tude of the random noise leads to diffusion of the wavefronts,

WDF imaging 159

(a) (b) (c) (d)

Figure 6. Images produced by the conventional imaging condition using the data simulated in the background model (a) and using the datasimulated in the random model (b). Images produced from data simulated in the random model using the interferometric imaging condition withparametrization as a function of data coordinates (c) and as a function of image coordinates (d).

as suggested by the “diffusion approximation” regime. Thiscombination of parameters could be regarded as a worst-case-scenario from a theoretical standpoint.

Figures 8(a)-8(c) show data simulated in models similarto the one depicted in Figure 3(b), but where the random noisecomponent is described by σ = 15, 30, 45%, respectively. Asexpected, the wavefronts recorded at the surface are increas-ingly distorted to the point where some of the later arrival arenot even visible in the data.

Migration using a conventional imaging condition leadsto the images in Figures 8(d)-8(f). As expected, the imagesshow stronger artifacts due to the larger defocusing caused bythe unknown random fluctuations in the model. However, mi-gration using the interferometric imaging condition leads tothe images in Figures 8(g)-8(i). Artifacts are significantly re-duced and the images are much better focused.

4 MULTI-OFFSET IMAGING EXAMPLES

There are many potential applications for this interferometricimaging functional. One application we illustrate in this paperis imaging of complex stratigraphy through a medium char-acterized by unknown random variations. In this situation, ac-curate imaging using conventional methods requires velocitymodels that incorporate the small scale (random, as we viewthem) velocity variations. However, practical migration veloc-ity analysis does not produce models of this level of accuracy,but approximates them with smooth, large-scale fluctuationsone order of magnitude larger than that of the typical seis-mic wavelength. Here, we study the impact of the unknown(random) component of the velocity model on the images andwhether interferometric imaging increases the statistical sta-bility of the image.

For all our examples, we extrapolate wavefields usingtime-domain finite-differences both for modeling and for mi-gration. Thus, we simulate a reverse-time imaging procedure,although the theoretical results derived in this paper applyequally well to other wavefield reconstruction techniques, e.g.downward continuation, Kirchhoff integral methods, etc. Theparameters used in our examples, explained in Appendix A,are: seismic spatial wavelength λ = 76.2 m, wavelet cen-

tral frequency ω = 20 Hz, random fluctuations parameters:ra = 76.2 m, rc = 76.2 m, α = 2, and random noise magni-tude σ = 20%.

Consider the model depicted in Figures 10(a)-10(d). Asin the preceding example, the left panels depict the knownsmooth velocity v0, and the right panels depict the model withrandom variations. The imaging target is represented by theoblique lines, Figure 9(b), located around z = 8 km, whichsimulates a cross-section of a stratigraphic model.

We model data with a random velocity model and imageusing the smooth model. Figures 10(a)-10(d) show wavefieldsnapshots in the two models for different propagation times,one before the source wavefields interact with the target reflec-tors and one after this interaction. The propagating waves areaffected the the random fluctuations in the model both beforeand after their interaction with the reflectors. Figures 10(e) and10(f) show the corresponding recorded data on the acquisitionsurface located at z = λ, where λ represents the wavelengthof the source pulse.

Migration with a conventional imaging condition of thedata simulated in the background model using the same veloc-ity produces the image in Figure 11(a). The targets are wellimaged, although the image also shows artifacts due to trun-cation of the data on the acquisition surface. In contrast, mi-gration with the conventional imaging condition of the datasimulated in the random model using the background velocityproduces the image in Figure 11(b). This image is distorted bythe random variations in the model that are not accounted forin the background migration velocity. The targets are harder todiscern since they overlap with many truncation and defocus-ing artifacts caused by the inaccurate migration velocity.

Finally, Figure 11(c) shows the migrated image using theinterferometric imaging condition applied to the wavefields re-constructed in the background model from the data simulatedin the random model. Many of the artifacts caused by the inac-curate velocity model are suppressed and the imaging targetsare more clearly visible and easier to interpret. Furthermore,the general patterns of amplitude variation along the imagedreflectors are similar between Figures 11(b) and 11(c).

We note that the reflectors are not as well imaged as theones obtained when the velocity is perfectly known. This is be-cause the interferometric imaging condition described in this

160 P. Sava & O. Poliannikov

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 7. Illustration of statistical stability for the interferometric imaging condition in presence of random model variations. Data modeled usingvelocity with random variations of magnitude σ = 30%, for different realizations of the noise model n. Images obtained by conventional imaging(d)-(f) and images obtained by interferometric imaging (g)-(i).

WDF imaging 161

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 8. Illustration of interferometric imaging condition robustness in presence of random model variations. Data modeled using velocity withrandom variations of magnitudes σ = 15% (a), σ = 30% (b), and σ = 45% (c). Images obtained by conventional imaging (d)-(f) and imagesobtained by interferometric imaging (g)-(i).

162 P. Sava & O. Poliannikov

paper does not correct kinematic errors due to inaccurate ve-locity. It only acts on the extrapolated wavefields to reducewavefield incoherency and add statistical stability to the imag-ing process. Further extensions to the interferometric imag-ing condition can improve focusing and enhance the imagesby correcting wavefields prior to imaging. However, this topicfalls outside the scope of this paper and we do not elaborateon it further.

5 CONCLUSIONS

We extend the conventional seismic imaging condition basedon wavefield cross-correlations to achieve statistical stabilityfor models with rapid, small-scale velocity variation. We as-sume that the random velocity variations on a scale compa-rable with the seismic wavelength are modeled by correlatedGaussian distributions. Our proposed interferometric imag-ing condition achieves statistical stability by applying conven-tional imaging to the Wigner distribution functions of the re-constructed seismic wavefields. The interferometric imagingcondition is a natural extension of the cross-correlation imag-ing condition and adds minimally to the cost of migration. Themain characteristic of the method is that it operates on extrap-olated wavefields at image positions (thus the name interfero-metric imaging condition), in contrast with costlier alternativeapproaches using interferometry parametrized as a function ofreceiver coordinates.

6 ACKNOWLEDGMENTS

We acknowledge Geophysics Associate Editor John Etgen fornumerous comments and suggestions which have significantlyimproved the manuscript. This work is partially supported bythe sponsors of the Consortium Project at the Center for WavePhenomena at Colorado School of Mines.

REFERENCES

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Borcea, L., Papanicolaou, G., and Tsogka, C., 2006a, Adap-tive interferometric imaging in clutter and optimal illumina-tion: Inverse Problems, 22, 1405–1436.

——– 2006b, Coherent interferometric imaging in clutter:Geopysics, 71, no. 4, SI165–SI175.

——– 2006c, Coherent interferometry in finely layered ran-dom media: SIAM Journal on Multiscale Modeling andSimulation, 5, no. 1, 62–83.

Claerbout, J. F., 1985, Imaging the Earth’s interior: , Black-well Scientific Publications.

Cohen, L., 1995, Time Frequency Analysis: , Prentice HallSignal Processing Series.

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Papanicolaou, G., Ryzhik, L., and Solna, K., 2004, Statisticalstability in time reversal: SIAM J. Appl. Math., 64, 1133–1155.

Pratt, R. G., and Worthington, M. H., 1990, Inverse the-ory applied to multi-source cross-hole tomography. part 1:Acoustic wave-equation method: Geophys. Prosp., 38, no.03, 287–310.

Pratt, R. G., 1990, Inverse theory applied to multi-sourcecross-hole tomography. part 2: Elastic wave-equationmethod: Geophys. Prosp., 38, no. 03, 311–312.

Sava, P., and Biondi, B., 2004a, Wave-equation migration ve-locity analysis - I: Theory: Geophysical Prospecting, 52,593–606.

——– 2004b, Wave-equation migration velocity analysis -II: Subsalt imaging examples: Geophysical Prospecting, 52,607–623.

Shen, P., Symes, W., Morton, S., and Calandra, H., 2005, Dif-ferential semblance velocity analysis via shot profile migra-tion: 74th Annual International Meeting, SEG, ExpandedAbstracts, 2249–2253.

Sirgue, L., and Pratt, R., 2004, Efficient waveform inversionand imaging: A strategy for selecting temporal frequencies:Geophysics, 69, no. 1, 231–248.

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7 APPENDIX A: NOISE MODEL

Consider a medium whose behavior is completely defined bythe acoustic velocity, i.e. assume that the density ρ (x, y, z) =ρ0 is constant and the velocity v (x, y, z) fluctuates around ahomogenized value v0 (x, y, z) according to the relation

1

v2 (x, y, z)=

1 + σm (x, y, z)

v20 (x, y, z)

, (A-1)

where the parameter m characterizes the type of random fluc-tuations present in the velocity model, and σ denotes theirstrength.

Consider the covariance orientation vectors

a = (ax, ay, az)> ∈ R3 (A-2)

b = (bx, by, bz)> ∈ R3 (A-3)

c = (cx, cy, cz)> ∈ R3 (A-4)

WDF imaging 163

(a) (b)

Figure 9. Velocity model (a) and imaging target located around z = 8 km (b). The model consists of a smooth version of the Sigsbee2A velocity,in order to avoid backscattering during reverse-time migration. The shaded area is imaged in Figures 11(a)-11(c).

defining a coordinate system of arbitrary orientation in space.Let ra, rb, rc > 0 be the covariance range parameters in thedirections of a,b and c, respectively.

We define a covariance function

cov (x, y, z) = exp [−lα (x, y, z)] , (A-5)

where α ∈ [0, 2] is a distribution shape parameter and

l (x, y, z) =

s„a · rra

«2

+

„b · rrb

«2

+

„c · rrc

«2

(A-6)

is the distance from a point at coordinates r = (x, y, z) to theorigin in the coordinate system defined by {raa, rbb, rcc}.

Given the IID Gaussian noise field n (x, y, z), we obtainthe random noise m (x, y, z) according to the relation

m (x, y, z) = F−1hpdcov (kx, ky, kz) bn (kx, ky, kz)

i,

(A-7)where kx, ky, kz are wavenumbers associated with the spatialcoordinates x, y, z, respectively. Here,

dcov = F [cov] (A-8)bn = F [n] (A-9)

are Fourier transforms of the covariance function cov and thenoise n, F [·] denotes Fourier transform, and F−1[·] denotesinverse Fourier transform. The parameter α controls the visualpattern of the field, and a,b, c, ra, rb, rc control the size andorientation of a typical random inhomogeneity.

8 APPENDIX B: WAVE PROPAGATION AND SCALEREGIMES

Acoustic waves characterized by pressure p (x, y, z, t) propa-gate according to the second order acoustic wave-equation forconstant density

∂2p

∂t2= v2∇2p + Fλ(t) , (B-1)

where Fλ (t) is a wavelet of characteristic wavelength λ.Given the parameters l (size of inhomogeneities), λ

(wavelength size), L (propagation distance) and σ (noisestrength), we can define several propagation regimes.

The weak fluctuations regime characterized by waveswith wavelength of size comparable to that of typical inhomo-geneities propagating over a medium with small fluctuations toa distance of many wavelengths. This regime is characterizedby negligible back scattering, and the randomness impacts thepropagating waves through forward multipathing. The relevantlength parameters are related by

l ∼ λ � L , (B-2)

and the noise strength is assumed small

σ � 1 . (B-3)

The diffusion approximation regime characterized bywaves with wavelength much larger than that of typical inho-mogeneities propagate over a medium with strong fluctuationsto a distance of many wavelengths. This regime is character-ized by traveling waves that are statistically stable but diffusewith time. Back propagation of such waves in a medium with-out random fluctuations results in loss of resolution. The rele-vant length parameters are related by

l � λ � L , (B-4)

and the noise strength is not assumed small

σ ∼ 1 . (B-5)

9 APPENDIX C: WIGNER DISTRIBUTIONFUNCTIONS

Consider the complex signal u (t) which depends on timet. By definition, its Wigner distribution function (WDF) is(Wigner, 1932):

W (t, ω) =1

2π

Zu∗

„t− th

2

«u

„t +

th

2

«e−i ω thdth ,

(C-1)where ω denotes temporal frequency, th denotes the relativetime shift of the considered signal relative to a reference time tand the sign ∗ denotes complex conjugation of complex signal

164 P. Sava & O. Poliannikov

(a) (b)

(c) (d)

(e) (f)

Figure 10. Seismic snapshots of acoustic wavefields simulated in the background velocity model (a)-(c), and in the random velocity model (b)-(d).Data recorded at the surface from the simulation in the background velocity model (e) and from the simulation in the random velocity model (f).

WDF imaging 165

(a)

(b)

(c)

Figure 11. Image produced using the conventional imaging condition from data simulated in the background model (a) and from data simulated inthe random model (b). Image produced using the interferometric imaging condition from data simulated in the random model (c).

s. The same WDF can be obtained in terms of the spectrumU (ω) of the signal u (t):

W (t, ω) =1

2π

ZU∗

“ω +

ωh

2

”U

“ω − ωh

2

”e−i t ωhdωh ,

(C-2)where ωh denotes the relative frequency shift of the consideredspectrum relative to a reference frequency ω. The time inte-gral in equation C-1 or the frequency integral in equation C-2spans the entire domain of time and frequency, respectively.When the interval is limited to a region around the reference

value, the transformation is known as pseudo Wigner distribu-tion function.

A special subset of the transformation equation C-1 cor-responds to zero temporal frequency. For input signal u (t), weobtain the output Wigner distribution function W (t) as

W (t) =1

2π

Zu∗

„t− th

2

«u

„t +

th

2

«dth . (C-3)

The WDF transformation can be generalized to multi-dimensional signals of space and time. For example, for 2D

166 P. Sava & O. Poliannikov

(a) (b) (c)

Figure C-1. Random velocity model (a), wavefield snapshot simulated in this model by acoustic finite-differences (b), and its 2D pseudo Wignerdistribution function (c).

real signals function of space, u (x, y), the zero-wavenumberpseudo WDF can be formulated as

W (x, y) =1

4π2

Z|xh|≤X

Z|yh|≤Y

u“x− xh

2, y − yh

2

”× u

“x +

xh

2, y +

yh

2

”dxhdyh , (C-4)

where xh and yh denote relative shift of the signal s relativeto positions x and y, respectively. In this particular form, thepseudo WDF transformation has the property that it filters theinput of random fluctuations preserving in the output imagethe spatially coherent components in a noise-free background.

For illustration, consider the model depicted in Fig-ure 1(a). This model consists of a smoothly-varying back-ground with 25% random fluctuations. The acoustic seismicwavefield corresponding to a source located in the middle ofthe model is depicted in Figure 1(b). This wavefield snap-shot can be considered as the random “image”. The applica-tion of the 2D pseudo WDF transformation to images shownin Figure 1(b) produces the image shown in Figure 1(c). Wecan make three observations on this image: first, the randomnoise is strongly attenuated; second, the output wavelet is dif-ferent from the input wavelet, as a result of the bi-linear natureof the pseudo WDF transformations; third, the transformationis isotropic, i.e. it operates identically in all directions. Thepseudo WDF applied to this image uses 11× 11 grid points inthe vertical and horizontal directions. As indicated in the bodyof the paper, we do not discuss here the optimal selection of theWDF window. Further details of Wigner distribution functionsand related transformations are discussed by Cohen (1995).