GEOPHYSICS, VOL. 68, NO. 2 (MARCH-APRIL 2003); P. 733–744, 12 FIGS.10.1190/1.1567243

Seismic reflection data interpolation with differentialoffset and shot continuation

Sergey Fomel∗

ABSTRACT

I propose a finite-difference offset-continuation filterfor interpolating seismic reflection data. The filter is con-structed from the offset-continuation differential equa-tion and is applied on frequency slices in the log-stretchfrequency domain. Synthetic and real data tests demon-strate that the proposed method succeeds in structurallycomplex situations where more simplistic approachesfail.

INTRODUCTION

Data interpolation is one of the most important problemsof seismic data processing. In 2D exploration, the interpola-tion problem arises because of missing near and far offsets,spatial aliasing, and occasional bad traces. In 3D exploration,the importance of this problem increases dramatically because3D acquisition almost never provides a complete regular cov-erage in both midpoint and offset coordinates (Biondi, 2003).Data regularization in three dimensions can solve the prob-lem of Kirchhoff migration artifacts (Gardner and Canning,1994), prepare the data for wave-equation common-azimuthimaging (Biondi and Palacharla, 1996), or provide the spatialcoverage required for 3D multiple elimination (van Dedemand Verschuur, 1998).

Claerbout (1992, 2003) formulated the general principle ofmissing-data interpolation: “A method for restoring missingdata is to ensure that the restored data, after specified filtering,has minimum energy.” How can one specify an appropriate fil-tering for a given interpolation problem? Smooth surfaces areconveniently interpolated with Laplacian filters (Briggs, 1974).Steering filters help us interpolate data with predefined dipfields (Clapp et al., 1998). Prediction-error filters in the time-space or frequency-space domains successfully interpolate datacomposed of distinctive plane waves (Spitz, 1991; Claerbout,1999). Local plane waves are handled with plane-wave de-

Published on Geophysic Online October 1, 2002. Manuscript received by the Editor December 6, 2000; revised manuscript received September 16,2002.∗Bureau of Economic Geology, University of Texas at Austin, University Station, Box X, Austin, Texas 78713-8972. E-mail: sergey.fomel@beg.utexas.edu.c© 2003 Society of Exploration Geophysicists. All rights reserved.

struction filters (Fomel, 2002). Because prestack seismic data isnot stationary in the offset direction, nonstationary prediction-error filters need to be estimated, which leads to an accurate butrelatively expensive method with many adjustable parameters(Crawley et al., 1999).

A simple model for reflection seismic data is a set of hy-perbolic events on a common midpoint gather. The simplestfilter for this model is the first derivative in the offset directionapplied after the normal moveout (NMO) correction. Goingone step beyond this simple approximation requires taking thedip moveout (DMO) effect into account (Deregowski, 1986).The DMO effect is fully incorporated in the offset continuationdifferential equation (Fomel, 1994, 2003).

Offset continuation is a process of seismic data transforma-tion between different offsets (Deregowski and Rocca, 1981;Bolondi et al., 1982; Salvador and Savelli, 1982). Differenttypes of DMO operators (Hale, 1991) can be regarded as con-tinuation to zero offset and derived as solutions of an initial-value problem with the revised offset-continuation equation(Fomel, 2003). Within a constant-velocity assumption, thisequation not only provides correct traveltimes on the con-tinued sections, but also correctly transforms the correspond-ing wave amplitudes (Fomel et al., 1996; Fomel and Bleistein,2001). Integral offset continuation operators have been de-rived independently by Chemingui and Biondi (1994), Bagainiand Spagnolini (1996), and Stovas and Fomel (1996). The 3Danalog is known as azimuth moveout (AMO) (Biondi et al.,1998). In the shot-record domain, integral offset continua-tion transforms to shot continuation (Schwab, 1993; Bagainiand Spagnolini, 1993; Spagnolini and Opreni, 1996). Inte-gral continuation operators can be applied directly for miss-ing data interpolation and regularization (Bagaini et al., 1994;Mazzucchelli and Rocca, 1999). However, they don’t behavewell for continuation at small distances in the offset spacebecause of limited integration apertures and, therefore, arenot well suited for interpolating neighboring records. Addi-tionally, as all integral (Kirchhoff-type) operators, they sufferfrom irregularities in the input geometry. The latter problem

733

734 Fomel

is addressed by accurate but expensive inversion to commonoffset (Chemingui, 1999; Chemingui and Biondi, 2002).

In this paper, I propose an application of offset continu-ation in the form of a finite-difference filter for Claerbout’s(1992, 2003) method of missing data interpolation. The filter isdesigned in the log-stretch frequency domain, where each fre-quency slice can be interpolated independently. Small filter sizeand easy parallelization among different frequencies assure ahigh efficiency of the proposed approach. Although the offsetcontinuation filter lacks the predictive power of nonstationaryprediction-error filters, it is much simpler to handle and servesas a good a priori guess of an interpolative filter for seismic re-flection data. I first test the proposed method by interpolatingrandomly missing traces in a constant-velocity synthetic dataset. Next, I apply offset continuation and the related shot con-tinuation field to a real data example from the North Sea. Usinga pair of offset continuation filters operating in two orthogo-nal directions, I successfully regularize a 3D marine data set.These tests demonstrate that offset continuation can performwell in complex structural situations where more simplisticapproaches fail.

OFFSET CONTINUATION

A particularly efficient implementation of offset continua-tion results from a log-stretch transform of the time coordinate(Bolondi et al., 1982), followed by a Fourier transform of thestretched time axis. After these transforms, the offset continu-ation equation from (Fomel, 2003) takes the form

h

(∂2 P̃

∂y2− ∂

2 P̃

∂h2

)− iÄ

∂ P̃

∂h= 0, (1)

where Ä is the corresponding frequency, h is the half-offset,y is the midpoint, and P̃(y, h, Ä) is the transformed data. Asin other f − x methods, equation (1) can be applied indepen-dently and in parallel on different frequency slices.

We can construct an effective offset-continuation finite-difference filter by studying first the problem of wave ex-trapolation between neighboring offsets. In the frequency-wavenumber domain, the extrapolation operator is defined bysolving the initial-value problem on equation (1). The solutiontakes the following form (Fomel, 2003):

ˆ̂P (h2) = ˆ̂P (h1)Zλ(kh2)/Zλ(kh1), (2)

where λ= (1+ iÄ)/2, and Zλ is the special function defined as

Zλ(x) = 0(1− λ)(

x

2

)λJ−λ(x) = 0 F1

(; 1− λ;−x2

4

)=∞∑

n=0

(−1)n

n!0(1− λ)

0(n+ 1− λ)

(x

2

)2n

, (3)

where 0 is the gamma function, J−λ is the Bessel function, and0 F1 is the confluent hypergeometric limit function (Petkovseket al., 1996). The wavenumber k in equation (2) correspondsto the midpoint y in the original data domain. In the high-frequency asymptotics, operator (2) takes the form

ˆ̂P (h2) = ˆ̂P (h1)F(2kh2/Ä)/F(2kh1/Ä)

× exp[iÄ9(2kh2/Ä− 2kh1/Ä)], (4)

where

F(ε) =√

1+√1+ ε2

2√

1+ ε2exp

(1−√1+ ε2

2

), (5)

and

9(ε) = 12

(1−

√1+ ε2 + ln

(1+√1+ ε2

2

)). (6)

Returning to the original domain, we can approximate thecontinuation operator with a finite-difference filter with theZ-transform

P̂h+1(Zy) = P̂h(Zy)G1(Zy)G2(Zy)

. (7)

The coefficients of the filters G1(Zy) and G2(Zy) are found byfitting the Taylor series coefficients of the filter response aroundthe zero wavenumber. In the simplest case of three-point filters[an analogous technique applied to the case of wavefield depthextrapolation with the wave equation would lead to the famous45◦ implicit finite-difference operator (Claerbout, 1985)], thisprocedure uses four Taylor series coefficients and leads to thefollowing expressions:

G1(Zy) = 1− 1− c1(Ä)h22 + c2(Ä)h2

1

6

+ 1− c1(Ä)h22 + c2(Ä)h2

1

12

(Zy + Z−1

y

), (8)

G2(Zy) = 1− 1− c1(Ä)h21 + c2(Ä)h2

2

6

+ 1− c1(Ä)h21 + c2(Ä)h2

2

12

(Zy + Z−1

y

), (9)

where

c1(Ä) = 3(Ä2 + 9− 4iÄ)Ä2(3+ iÄ)

and

c2(Ä) = 3(Ä2 − 27− 8iÄ)Ä2(3+ iÄ)

.

Figure 1 compares the phase characteristic of the finite-difference extrapolators (7) with the phase characteristics ofthe exact operator (2) and the asymptotic operator (4). Thematch between different phases is poor for very low frequen-cies (left plot in Figure 1) but sufficiently accurate for fre-quencies in the typical bandwidth of seismic data (right plot inFigure 1).

Figure 2 compares impulse responses of the inverse DMOoperator constructed by the asymptotic Ä-k operator withthose constructed by finite-difference offset continuation. Ne-glecting subtle phase inaccuracies at large dips, the two imageslook similar, which provides an experimental evidence of theaccuracy of the proposed finite-difference scheme.

When applied on the offset-midpoint plane of an individualfrequency slice, the 1D implicit filter (7) transforms to a 2Dexplicit filter with the 2D Z-transform

G(Zy, Zh) = G1(Zy)− ZhG2(Zy). (10)

Convolution with filter (10) is the regularization operator thatI propose to use for interpolating prestack seismic data.

Interpolation with Offset Continuation 735

APPLICATION

I start numerical testing of the proposed regularization firston a constant-velocity synthetic data set, where all the assump-tions behind the offset continuation equation are valid.

Constant-velocity synthetic data set

A sinusoidal reflector shown in Figure 3 creates complicatedreflection data, shown in Figures 4 and 5. To set up a test forregularization by offset continuation, I removed 90% of ran-domly selected shot gathers from the input data. The synclineparts of the reflector lead to traveltime triplications at large off-sets. A mixture of different dips from the triplications wouldmake it extremely difficult to interpolate the data in individualcommon-offset gathers, such as those shown in Figure 4. Theplots of time slices after NMO (Figure 5) clearly show that thedata are also nonstationary in the offset direction. Therefore,a simple offset interpolation scheme is also doomed.

Figure 6 shows the reconstruction process on individual fre-quency slices. Despite the complex and nonstationary charac-ter of the reflection events in the frequency domain, the offset

FIG. 1. Phase of the implicit offset-continuation operators in comparison with the exact solution. The offsetincrement is assumed to be equal to the midpoint spacing. The left plot corresponds to Ä = 1, the right plot toÄ = 10.

FIG. 2. Inverse DMO impulse responses computed by the Fourier method (left) and by finite-difference offsetcontinuation (right). The offset is 1 km.

continuation equation is able to accurately reconstruct themfrom the decimated data.

Figure 7 shows the result of interpolation after the data aretransformed back to the time domain. The offset continuation

FIG. 3. Reflector model for the constant-velocity test.

736 Fomel

result (right plots in Figure 7) reconstructs the ideal data (leftplots in Figure 4) very accurately even in the complex triplica-tion zones, while the result of simple offset interpolation (leftplots in Figure 7) fails as expected. The simple interpolationscheme applied the offset derivative ∂/∂h in place of the offsetcontinuation equation, and thus did not take into account themovement of the events across different midpoints.

The constant-velocity test results allow us to conclude that,when all the assumptions of the offset continuation theory aremet, it provides a powerful method of data regularization.

Being encouraged by the synthetic results, I proceed to a 3Dreal data test.

FIG. 4. Prestack common-offset gathers for the constant-velocity test. Left: ideal data (after NMO). Right: inputdata (90% of shot gathers removed). Top, center, and bottom plots correspond to different offsets.

3-D data regularization with the offset continuation equation

Three-dimensional differential offset continuation amountsto applying two differential filters, operating on the inline andcrossline projections of the offset and midpoint coordinates.The corresponding system of differential equations has theform

h1

(∂2 P̃

∂y21

− ∂2 P̃

∂h21

)− iÄ

∂ P̃

∂h1= 0;

h2

(∂2 P̃

∂y22

− ∂2 P̃

∂h22

)− iÄ

∂ P̃

∂h2= 0,

(11)

Interpolation with Offset Continuation 737

where y1 and y2 correspond to the inline and crossline midpointcoordinates, respectively, and h1 and h2 correspond to the inlineand crossline offsets, respectively. The projection approach isjustified in the theory of azimuth moveout (Fomel and Biondi,1995; Biondi et al., 1998).

The result of a 3-D data regularization test is shown inFigure 8. The input data is a subset of a 3-D marine data setfrom the North Sea, complicated by salt dome reflections anddiffractions. The same data set was used previously for test-ing azimuth moveout (Biondi et al., 1998). I used neighboringoffsets in the inline and crossline directions and the differen-

FIG. 5. Time slices of the prestack data for the constant-velocity test. Left: ideal data (after NMO). Right: inputdata (90% of random gathers removed). Top, center, and bottom plots correspond to time slices at 0.3, 0.4, and0.5 s.

tial 3-D offset continuation to reconstruct the empty traces ina selected midpoint cube. Although the reconstruction is notentirely accurate, it successfully fulfills the following goals:

1) The input traces are well hidden in the interpolation re-sult. It is impossible to distinguish between input and in-terpolated traces.

2) The main structural features are restored without us-ing any assumptions about structural continuity in themidpoint domain. Only the physical offset continuity isused.

738 Fomel

Analogously to integral azimuth moveout operator (Biondiet al., 1998), differential offset continuation can be applied in3-D for regularizing seismic data prior to prestack imaging.

In the next section, I return to the 2-D case to consider theimportant problem of shot gather interpolation.

SHOT CONTINUATION

Missing or undersampled shot records are a common exam-ple of data irregularity (Crawley, 2000). The offset continuationapproach can be easily modified to work in the shot record do-

FIG. 6. Interpolation in frequency slices. Left: input data (90% of the shot gathers removed). Right: interpolationoutput. Top, bottom, and middle plots correspond to different frequencies. The real parts of the complex-valueddata are shown.

main. With the change of variables s= y− h, where s is theshot location, the frequency-domain equation (1) transformsto the equation

h

(2∂2 P̃

∂s∂h− ∂

2 P̃

∂h2

)− iÄ

(∂ P̃

∂h− ∂ P̃

∂s

)= 0. (12)

Unlike equation (1), which is second order in the propagationvariable h, equation (12) contains only first-order derivativesin s. We can formally write its solution for the initial conditions

Interpolation with Offset Continuation 739

at s= s1 in the form of a phase-shift operator:

ˆ̂P (s2) = ˆ̂P (s1) exp[ikh(s2 − s1)

khh−Ä2khh−Ä

], (13)

where the wavenumber kh corresponds to the half-offset h.Equation (13) is in the mixed offset-wavenumber domain and,therefore, not directly applicable in practice. However, we canuse it as an intermediate step in designing a finite-differenceshot-continuation filter. Analogously to the cases of plane-

FIG. 7. Interpolation in common-offset gathers. Left: output of simple offset interpolation. Right: output ofoffset continuation interpolation. Compare with Figure 4. Top, center, and bottom plots correspond to differentcommon-offset gathers.

wave destruction (Fomel, 2002) and offset continuation, shotcontinuation leads us to the rational filter

P̂s+1(Zh) = P̂s(Zh)S(Zh)

S̄(1/Zh). (14)

The filter is nonstationary because the coefficients of S(Zh) de-pend on the half-offset h. We can find them by the Taylor expan-sion of the phase-shift equation (13) around zero wavenumberkh. For the case of the half-offset sampling equal to the shotsampling, the simplest three-point filter is constructed with

740 Fomel

three terms of the Taylor expansion. It takes the form

S(Zh) = −(

112+ i

h

2Ä

)Z−1

h +(

23− i

Ä2 + 12h2

12Äh

)+(

512+ i

Ä2 + 18h2

12Äh

)Zh. (15)

Let us consider the problem of doubling the shot density.If we use two neighboring shot records to find the missing

FIG. 8. Three-dimensional data regularization test. Top: input data, the result of binning in a 50 × 50 m offsetwindow. Bottom: regularization output. Data from neighboring offset bins in the inline and crossline directionswere used to reconstruct missing traces.

record between them, the problem reduces to the least-squaressystem [

SS̄

]ps ≈

[S̄ps−1

Sps+1

], (16)

where S denotes convolution with the numerator of equation(14), S̄ denotes convolution with the corresponding denomina-tor, ps−1 and ps+1 represent the known shot gathers, and ps rep-resents the gather that we want to estimate. The least-squares

Interpolation with Offset Continuation 741

solution of system (16) takes the form

ps = (ST S+ S̄T S̄)−1(

ST S̄ps−1 + S̄T Sps+1

). (17)

If we choose the three-point filter (15) to construct the oper-ators S and S̄, then the inverted matrix in equation (17) willhave five nonzero diagonals. It can be efficiently inverted witha direct banded matrix solver using the LDLT decomposition(Golub and Van Loan, 1996). Since the matrix does not dependon the shot location, we can perform the decomposition oncefor every frequency so that only a triangular matrix inversionwill be needed for interpolating each new shot. This leads toan extremely efficient algorithm for interpolating intermediateshot records.

Sometimes, two neighboring shot gathers do not fully con-strain the intermediate shot. In order to add an additionalconstraint, I include a regularization term in equation (17),

FIG. 9. Top: Two synthetic shot gathers used for the shot interpolation experiment. An NMO correction has beenapplied. Bottom: simple average of the two shot gathers (left) and its prediction error (right).

as follows:

ps = (ST S+ S̄T S̄+ ε2AT A)−1(ST S̄ps−1 + S̄T Sps+1),

(18)

where A represents convolution with a three-point prediction-error filter (PEF), and ε is a scaling coefficient. The appro-priate PEF can be estimated from ps−1 and ps+1 using Burg’salgorithm (Burg, 1972, 1975; Claerbout, 1976). A three-pointfilter does not break the five-diagonal structure of the invertedmatrix. The PEF regularization attempts to preserve offset dipspectrum in the underconstrained parts of the estimated shotgather.

Figure 10 shows the result of a shot interpolation experimentusing the constant-velocity synthetic from Figure 4. In this ex-periment, I removed one of the shot gathers from the originalNMO-corrected data and interpolated it back using equation(18). Subtracting the true shot gather from the reconstructed

742 Fomel

one shows a very insignificant error, which is further reducedby using the PEF regularization (right plots in Figure 10). Thetwo neighboring shot gathers used in this experiment are shownin the top plots of Figure 9. For comparison, the bottom plotsin Figure 9 show the simple average of the two shot gathersand its corresponding prediction error. As expected, the erroris significantly larger than the error of shot continuation. Aninterpolation scheme based on local dips in the shot directionwould probably achieve a better result, but it is significantlymore expensive than the shot continuation scheme introducedabove.

A similar experiment with real data from a North Sea ma-rine data set is reported in Figure 12. I removed and recon-structed a shot gather from the two neighboring gathers shownin Figure 11. The lower parts of the gathers are complicated bysalt dome reflections and diffractions with conflicting dips. Thesimple average of the two input shot gathers (bottom plotsin Figure 12) works reasonably well for nearly flat reflection

FIG. 10. Synthetic shot interpolation results. Left: interpolated shot gathers. Right: prediction errors (the dif-ferences between interpolated and true shot gathers), plotted on the same scale. Top: without regularization.Bottom: with PEF regularization.

events but fails to predict the position of the back-scattereddiffractions events. The shot continuation method works wellfor both types of events (top plots in Figure 12). There is somesmall and random residual error, possibly caused by local am-plitude variations.

Analogously to the case of offset continuation, it is possibleto extend the shot continuation method to three dimensions.A simple modification of the proposed technique would alsoallow us to use more than two shot gathers in the input or toextrapolate missing shot gathers at the end of survey lines.

CONCLUSIONS

Differential offset continuation provides a valuable toolfor interpolation and regularization of seismic data. Start-ing from analytical frequency-domain solutions of the offset-continuation differential equation, I have designed accuratefinite-difference filters for implementing offset continuation as

Interpolation with Offset Continuation 743

FIG. 11. Two field marine shot gathers used for the shot interpolation experiment. An NMO correction has beenapplied.

FIG. 12. Field-data shot-interpolation results. Top: interpolated shot gather (left) and its prediction error (right).Bottom: simple average of the two input shot gathers (left) and its prediction error (right).

744 Fomel

a local convolutional operator. A similar technique works forshot continuation across different shot gathers. Missing dataare efficiently interpolated by an iterative least-squares opti-mization. The differential filters have an optimally small size,which assures high efficiency.

Differential offset continuation serves as a bridge betweenintegral and convolutional approaches to data interpolation. Itshares the theoretical grounds with the integral approach butis applied in a manner similar to that of prediction-error filtersin the convolutional approach.

Tests with synthetic and real data demonstrate that the pro-posed interpolation method can succeed in complex structuralsituations where more simplistic methods fail.

ACKNOWLEDGMENTS

The financial support for this work was provided by the spon-sors of the Stanford Exploration Project (SEP).

The 3-D North Sea data set was released to SEP by Conocoand its partners, BP and Mobil. For the shot continuation test,I used another North Sea data set, released to SEP by ElfAquitaine.

I thank Jon Claerbout and Biondo Biondi for helpful dis-cussions about the practical application of differential offsetcontinuation.

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