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GEOPHYSICS, VOL. 74, NO. 6 �NOVEMBER-DECEMBER 2009�; P. V123–V132, 12 FIGS., 1 TABLE.10.1190/1.3245216

ive-dimensional interpolation: Recovering from acquisition constraints

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ABSTRACT

Although 3D seismic data are being acquired in larger vol-umes than ever before, the spatial sampling of these volumesis not always adequate for certain seismic processes. This isespecially true of marine and land wide-azimuth acquisi-tions, leading to the development of multidimensional datainterpolation techniques. Simultaneous interpolation in allfive seismic data dimensions �inline, crossline, offset, azi-muth, and frequency� has great utility in predicting missingdata with correct amplitude and phase variations. Althoughthere are many techniques that can be implemented in five di-mensions, this study focused on sparse Fourier reconstruc-tion. The success of Fourier interpolation methods dependslargely on two factors: �1� having efficient Fourier transformoperators that permit the use of large multidimensional datawindows and �2� constraining the spatial spectrum along di-mensions where seismic amplitudes change slowly so thatthe sparseness and band limitation assumptions remain valid.Fourier reconstruction can be performed when enforcing asparseness constraint on the 4D spatial spectrum obtainedfrom frequency slices of five-dimensional windows. Binningspatial positions into a fine 4D grid facilitates the use of theFFT, which helps on the convergence of the inversion algo-rithm. This improves the results and computational efficien-cy. The 5D interpolation can successfully interpolate sparsedata, improve AVO analysis, and reduce migration artifacts.Target geometries for optimal interpolation and regulariza-tion of land data can be classified in terms of whether theypreserve the original data and whether they are designed toachieve surface or subsurface consistency.

INTRODUCTION

All current 3D seismic acquisition geometries have poor sam-ling along at least one dimension. This affects migration quality,hich is based on the principle of constructive and destructive inter-

Manuscript received by the Editor 12August 2008; revised manuscript rec1CGGVeritas, Calgary,Alberta, Canada. E-mail: [email protected] Society of Exploration Geophysicists.All rights reserved.

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erence of data and thus is sensitive to irregular and coarse samplingAbma et al., 2007�.Analysis of amplitude variations with offset andzimuth �AVO, AVAz�, which we want to observe in the migratedomain, are also affected by the presence of gaps and undersam-ling.

There are many different approaches to tackling this problem. Thenly perfect solution is to acquire well-sampled data; all other ap-roaches deal with the symptoms of the problem rather than theroblem itself, and there is no guarantee that they can adequatelyolve it. However, given that, in the real world, we usually cannot goack to the field and fix the actual problem, we need to address thisssue using the processing tools at our disposal.

Most seismic algorithms implicitly apply some sort of interpola-ion because they assume correctly sampled data. Typically, missingamples are assumed to be zero or similar to neighboring values. Thedvantage of using a separate interpolation algorithm is that more in-elligent assumptions can be made by using a priori information. Forxample, sinc interpolation uses the constraint that there is no energyt frequencies above Nyquist. This is more reasonable than assum-ng that the unrecorded data are zeros. Interpolation algorithms canhen be viewed as methods to precondition the data with intelligentonstraints.

Interpolation of wide-azimuth land data presents many challeng-s, some quite different from those of interpolating narrow-azimutharine data sets. The most familiar interpolation algorithms have

een developed for marine streamer surveys. Marine data are usual-y well sampled in the inline direction and coarsely sampled in therossline direction. Many algorithms based on Fourier interpolationre quite successful at infilling the crossline direction, even in theresence of aliasing and complex structure �Schonewille et al., 2003;u et al., 2005; Abma and Kabir, 2006; Poole and Herrmann, 2007;wartjes and Sacchi, 2007�. Land data interpolation brings addition-l complications because of noise, topography, and the wide-azi-uth nature of the data. In particular, the azimuth distribution re-

uires interpolation to use information from all spatial dimensions athe same time because sampling along any particular subset of theour spatial dimensions is usually very poor.

Multidimensional interpolation algorithms have become feasibleven for five dimensions �Trad et al., 2005�. This capability raises

May 2009; published online 25 November 2009.

SEG license or copyright; see Terms of Use at http://segdl.org/

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ew possibilities but also brings new challenges and questions. Theeneral principle is the same: Missing data are assumed to have aimilar nature to data recorded in their neighborhood, but the termneighborhood” can have different meanings in multiple dimen-ions. An additional complication for wide-azimuth data interpola-ion in five dimensions is that these data are always very irregularnd sparse in at least two of the four spatial dimensions because ofcquisition and processing costs.

Interpolation implementations have two different aspects: theeneral interpolation strategy �choice of spatial dimensions, win-ow size, and target geometry� and the mathematical engine used toredict the new traces from some kind of model. A discussion ofhese two aspects follows.

INTERPOLATION STRATEGIES

Interpolation methods differ in complexity, assumptions, and op-rator size. Local methods �e.g., short-length prediction filters� useimple models �usually linear events� to represent the data in smallindows. Therefore, they tend to be robust, fast, adaptable, and easy

o implement. Their shortcoming is an inability to interpolate largeaps because the local information they need does not exist �therere no data around the trace to interpolate�.

Global methods use all of the data simultaneously �up to some ap-rture limit defined by the physics of the problem� and models withany degrees of freedom because they cannot assume simple data

vents at a large scale. They are slower, less adaptable, and harder tomplement. However, they can, at least in theory, interpolate largeaps by using information supplied from distant data. Most practicalethods fall between these two extremes; but the sparser the sam-

ling, the larger the operator size needs to be. If the geology is com-lex, some methods with a large operator can smear geologic fea-ures and decrease resolution.Asafe choice is to work with global in-erpolation methods that behave like local interpolators when localnformation is available.

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igure 1. Synthetic data: comparison of 5D versus 3D interpolation.a� Synthetic data, one shot �window�. �b�After removing every sec-nd line. �c� Interpolation in 5D �inline/crossline/offset/azimuth/fre-uency�. �d� Interpolation in 3D �receiver.x, receiver.y, frequency�.races are sorted according to receiver numbers.


A related distinction is the number of dimensions that the algo-ithm can handle simultaneously. Usually, the time dimension is wellampled, so only spatial dimensions need be interpolated. AlthoughD seismic data have four spatial dimensions, many traditionalethods use data along one spatial dimension only. If the method is

ascaded through the different dimensions, the order of these opera-ions becomes extremely important. However, interpolation ofparse wide-azimuth data is more likely to succeed in a full 5D spaceecause often at every point there is at least one spatial directionlong which seismic amplitudes change slowly. Information alonghis direction helps to constrain the problem along the other dimen-ions where events are harder to predict.

Also, seismic amplitude variations are smoother in five dimen-ions than they are in any projection into a lower dimensional space.o see why, consider an analogy: Imagine the shadow of an airplaneying over a mountain range. The shadow of the airplane is a com-lex path even if the airplane goes in a simple trajectory. Interpola-ion of the airplane flight path is much more difficult on the 2D sur-ace �shadow� than in the original 3D space. A similar argument cane made about seismic wavefield variations in the full 5D space.

My approach to interpolation is to work with large operators in 5Dindows. In practice, the data window size is often constrained by

he processing system capabilities, particularly when using clustersn busy computing networks. I normally apply windows of 30�30ines, 1000-m offsets, and all azimuths. Larger windows are occa-ionally required to deal with very sparse data. The spatial dimen-ions in these windows are chosen so that the data look as simple asossible along each dimension. After extensive testing in differentomains �shot, receiver, cross spreads, and common-offset vectoromains�, I have chosen the inline-crossline-azimuth-offset-fre-uency �i.e., midpoint, offset, and azimuth� with NMO-correctedata for the following reasons:

� These are the dimensions where amplitude variations are mostimportant �structure, AVO, and AVAz�. Interpolation is alwaysan approximation of the truth, and that approximation is betteralong the dimensions where the algorithm is applied.

� AVO and AVAz are usually slow �after NMO�; therefore, datahave limited bandwidth in the Fourier spectra along these di-mensions. The azimuth dimension also has the advantage of be-ing cyclic in nature, making it particularly fit for discrete Fouri-er transform representation.

� The interval between samples in the inline crossline dimen-sions �i.e., midpoints� is on the order of the common-midpoint�CMP� bin size. In the shot or receiver domain, the samplingcan be as coarse as shot/receiver line sampling �several CMPbins�.

Figure 1 shows a simple synthetic experiment to demonstrate thedvantage of 5D interpolation over 3D interpolation. The originalraces from an orthogonal survey were replaced by synthetic seismicvents while preserving the original recording geometry of the trac-s. The distance between receiver lines was 500 m, with 12 receiverines per shot. Every second receiver line of this synthetic data setas removed, simulating a 1000-m line interval, and then predictedith Fourier reconstruction.In the first case, I interpolate on a shot-by-shot basis �three dimen-

ions�, and in the second case in the inline-crossline-offset-azimuth-requency domain �five dimensions�. It is evident in Figure 1 that thelgorithm can reproduce all data complexity when using five inter-


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olation dimensions, but it is unable to repeat this using only threeimensions. Because the algorithm is exactly the same, this examplehows the importance of the additional information supplied by thextra dimensions for Fourier interpolation.

The actual location of the newly created traces is an important is-ue for interpolation. I can distinguish six cases, of which only fourre used for land data wide-azimuth surveys:

� Preserving original data �interpolation�

a� Decrease shot and receiver interval �decrease bin size�.b� Decrease shot and receiver line interval �increase offset

and azimuth sampling�.c� Make shot and receiver line interval equal to shot and re-

ceiver interval �fully sampled�. This is a particular case of1b.

� Replacing data totally with predicted traces �regularization�

a� Target geometry regular on shot and receiver locations�surface consistency�.

b� Target geometry regular on CMP, offset, and azimuth �sub-surface consistency�.

c� Target geometry regular on surface and in subsurface.

Possibilities 1a, 1b, 2a, and 2b each have important applicationssee Table 1�.Adapting to the acquired data by adding new shots and

able 1. Types of land data interpolation and main benefits. Tbased on use from 2005 to 2008). The font style in the bottom

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erging surveys withifferent bin size and/oresign �2D and 3D, parallelnd orthogonal, etc.� � �

ncrease resolution forteep dips �relax antialiaslters during migration� �mprove CIGs �AVO,VAz, velocity analysis� �D applications �matchingime-lapse surveys� � �

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eceivers following the original design �types 1a and 1b� has the ad-antage that original data can be preserved and interpolation is wellonstrained. Preserving the original data is generally safer than re-lacing all of the acquisition with interpolated data, particularly foromplex noisy data from structured areas in the presence of topogra-hy. This approach works well for Kirchhoff time and depth migra-ion. By adding new shots and receivers, the subsurface samplingan be improved according to well-understood acquisition conceptse.g., Cordsen et al., 2000�.

Type 2a, surface-consistent interpolation with perfectly regularhot and receiver lines, is useful for wave equation migration, inter-olation of very irregular surveys, and time-lapse applications. Typeb, subsurface-consistent uniform coverage of offsets and azimuthsor each CMP, is desirable for migration in general. However, thisesign implies a large number of shots and receivers with nonuni-orm shot and receiver fold. This is a problem for ray-tracing meth-ds and any kind of shot or receiver processing. Therefore, its appli-ation seems to be limited to time migration and, because of the largeize of the resulting data sets, for small surveys. Probably, it can alsoe applied well to common-offset Gaussian beam migration.

Finally, types 1c and 2c, complete coverage of shots and receiv-rs, are desirable for all seismic processing, but the resulting largeize of the data makes it impractical.

Any of these interpolation types can be used for infilling acquisi-ion gaps. A modification of type 2b from polar to Cartesian coordi-ates can be used to produce common-offset vector gathers. Types

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c, 2b, and 2c are fully implemented and have been used in internalests but have not yet been used in production projects. Notice thatne case missing in the table is to replace all data with predictionsnto a given geometry. This is the situation in 4D time lapse, where its usual to interpolate the monitor locations to match the baseline.

For typical wide-azimuth land data surveys in a complex environ-ent, the safest choice seems to be surface-consistent interpolation

1a and 1b�. This allows one to preserve the original data untouchednd to apply careful quality control �QC� to the new traces. Some QCarameters can be added to the headers, making it possible to discardew traces with high risk or low confidence after the interpolation.here are several possible quality parameters. Two QC parameters

hat complement each other and are often useful are �1� the distancelong the four spatial dimensions between the new and the originalraces and �2� the ratio of original to interpolated traces.

For the 5D configuration discussed in this paper, a meaningful cal-ulation of the first parameter requires a weighted average of the dis-ance along inline, crossline, offset, and azimuth. The weights de-end on the structural complexity, residual moveout, and anisotropy.he second parameter refers to the ratio of the number of original

races to the number of sampling points on the 4D spatial grid used inhe numerical algorithm. This ratio is usually much smaller than theatio of input to output traces for a given area.

INTERPOLATION ENGINE

The second major component of the interpolation problem is thehoice of a mathematical algorithm to predict new information giv-n a set of recorded traces. One method with the flexibility to adapt tohe requirements for multidimensional global interpolation is mini-

um weighted norm interpolation �MWNI� �Liu and Sacchi, 2004�,hich extends the work from Sacchi and Ulrych �1996� to multipleimensions. MWNI is a constrained inversion algorithm. The actualata d are the result of a sampling matrix T acting on an unknownully sampled data set m �m and d are containers for multidimen-ional data, and T is a mapping between these two containers�.

The unknown �interpolated� data are constrained to have the sameultidimensional spectrum as the original data. Enforcing this con-

traint requires a multidimensional Fourier transform, which is theost expensive part of the algorithm. To solve for the unknown data,cost function is defined for every frequency slice and is minimizedsing standard optimization techniques. The cost function J is de-ned, frequency by frequency, as

J� �d�Tm�2��m�W, �1�

here � �2 indicates an �2-norm and � �w indicates an �2-weightedorm calculated as

�m�W�mHFn�1�pk��2Fnm . �2�

n equation 2, Fn is the multidimensional Fourier transform, with nndicating the number of spatial dimensions of the data, mH theranspose conjugate of the model m, and pk the multidimensionalpectrum of the unknown data.

The multidimensional vector pk contains weight factors that givereedom to the model to be large where it needs to be large. They cane obtained by bootstrapping from the previous temporal frequencyn a manner similar to that done for Radon transforms �Herrmann etl., 2000�. These weights are defined in the �-k domain, where � ishe temporal frequency and k is the wavenumber vector along each


patial dimension. They link the frequency slices, making the fre-uency axis behave as the fifth interpolation dimension, althoughrequencies are not really interpolated.

The model m is in the �-x domain �x is a vector representing allpatial directions�. If kmax is the maximum wavenumber on each di-ension for the maximum dip of the data, then the case of pk�1 for�kmax and pk�0 for k � kmax corresponds to sinc interpolation.he variable � in equation 1 is a hyperparameter that controls thealance between fitting the data and enforcing sparseness on thepectrum. This parameter is eliminated by changing the cost func-ion in equation 1 to the standard form and using the residuals to de-ne the number of iterations �Trad et al., 2003�. The actual geophysi-al meaning of the spatial dimensions is irrelevant to the algorithm.owever, for the method to work well, at least one of these dimen-

ions must have a sparse spectrum or a band limited spectrum.The multidimensional spectrum can be calculated using discrete

ourier transforms �DFTs� that exactly honor space locations or theast Fourier transforms �FFTs� that require binning the data into arid with exactly one trace per position. In practice, I define m to be aegular supersampled 4D grid that contains many more traces thanhe target geometry. This allows us to use FFTs but forces us to binhe data during the interpolation.

The bin intervals along the spatial dimensions are kept small tovoid smearing and data distortion. The binning errors along the in-ine/crossline directions can be made negligible by subdividingMP bins into subbins if necessary, but CMP grid bin size usually isdequate. The binning errors along offset and azimuth dimensionsre kept small by applying NMO and static corrections before inter-olation. However, data with significant residual NMO and strongnisotropy require small bin intervals along offset and azimuth.arge bins reduce computation time and improve numerical stabilityut reduce precision. There is a trade-off between precision and nu-erical stability that requires careful parameterization and imple-entation. A good rule of thumb for land surveys is to use, as offset

in interval, a fraction of the receiver group interval �e.g., 12 or 1

4 �, de-reasing from near to far offsets and geologic complexity. Azimuthntervals are usually chosen in the range between 20° and 45°, de-reasing with offset and anisotropy.

DFTs can also be used for the spectrum with the advantage thathey do not require binning. The problem with DFTs is computation-l cost. For N variables, a 1D FFT requires computation time propor-ional to N log N, but DFT requires a computation time proportionalo N2. This constraint makes the cost in two spatial dimensions pro-ortional to N4 and four spatial dimensions proportional to N8. Al-hough numerical tricks such as nonuniform FFTs �Duijndam andchonewille, 1999� can improve these numbers dramatically, a 4DFT algorithm is quite expensive in terms of computer time and haseen unfeasible for production demands until now. Very recently,his has become possible �Gordon Poole, personal communication,009�, although it demands large computer resources.

There are many differences between working with FFTs or DFTs.n the negative side, working with FFTs has the potential to distortata because of the binning. However, binning spatial coordinates isften applied in seismic processing, even by methods that can usexact spatial coordinates. For example, when working on common-ffset volumes, a binning along offset is applied. On the positiveide, when working with FFTs, the results improve because the in-reased speed of the iterations permits us to obtain a solution close tohe one that would have been obtained after full convergence.

Furthermore, the nature of the system of equations solved at every


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requency changes, depending on whether we use regular sampling,rregular sampling, or regular sampling as a result of binning. To un-erstand why, let us incorporate the sparseness constraint into theperator by transforming equation 1 from the general form to thetandard form �Hansen, 1998�. By defining a new model u�k,

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hich is m after transforming to the �-k domain and inverse weight-ng with pk, equation 1 becomes

J� �d�TFn�1pku�k�2��u�k�2. �4�

The weighted norm �� w� now becomes an �2-norm, and the oper-tor absorbs the spectral weights. This allows us to include theparseness constraint into the operator, i.e., to modify the basis func-ions of the transformation to include the sparseness constraint �Tradt al., 2003�. The mapping between d and the new model u is nowerformed by the operator:

L�TFn�1pk. �5�

olving this equation for u�k requires solving the following systemf equations:

�pkHFnTHTFn

�1pk��I�u�k�pkHFnTHdH, �6�

here I is the identity matrix and the super index H means conjugateranspose.

Because of the large size of the system of equations in our prob-em, on the order of 105 equations, the final full solution u�k is neverchieved. Instead, an approximate solution is obtained by using anterative algorithm and running only a few iterations. Components of�k that have a weak mapping through operator L �such as low-am-litude spectral components� can be resolved with this limited num-er of iterations only if the system of equation 6 has good conver-ence. This convergence improves as the operator L�TFn

�1pk be-omes closer to orthogonal, i.e., as

LHL→I . �7�

The operator pk is usually a diagonal operator; therefore, conver-ence depends mainly on the two operators Fn

�1 and T, which in turnepend on the spatial axes and the missing samples, respectively.

The operator Fn�1 maps the 4D spatial wavenumber k to the inter-

olated 4D spatial axis x. If x and k are perfectly regular, thennFn

�1�I. If, in addition, there are no missing traces, the left side ofquation 6 is diagonal and the system converges in one iteration. Theavenumber axes k �one axis per spatial dimension� can always beade regular, but the axes x depend on the input data. Binning the in-

ut data makes x regular.The sampling operator T that connects interpolated data to ac-

uired data depends on the missing traces. It is orthogonal whenhere are no missing traces. Binning the data without decreasingampling interval does not affect T but can introduce data distortion.

aking bin intervals small to avoid data distortion introduces nonor-hogonality into the system of equations 6, making convergence

ore difficult.As we see from this analysis, there is a trade-off between nonor-

hogonality on Fn�1 and T. Moreover, there is a trade-off between

inning �and data distortion� on one side and convergence of the sys-em of equations on the other. Precisely honoring spatial coordinateslows down convergence because of the nonorthogonality of Fn

�1.lternatively, F�1 can be made regular by fine binning of x, practi-
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ally without loss of precision, but T becomes nonorthogonal. In-reasing bin interval does not affect Fn

�1 and decreases nonorthogo-ality on T but introduces data distortion.

Figure 2 illustrates the effects of sampling in the matrix distribu-ion for the left side of the system of equation 6. Let us consider twoifferent cases: coarse regular sampling �left column� and irregularampling �right column�. The matrix distribution for these two casess shown when applying three different methods: coarse binning,rue locations, and fine binning.

The first row, Figure 2a and b, shows the structure of the system ofquations when coarse binning is used �which allows the use ofFTs�. The system of equations is quite sparse, with most elementslong the main diagonal; therefore, the optimization converges veryuickly. In the decimation case on the left �Figure 2a�, the secondaryeaks produced by operator aliasing are as strong as the nonaliasomponent. In practice, they can be taken care of by filters and boot-trapping weights from low to high frequency.

The second row, Figure 2c and d, shows the same for irregularampling �true spatial locations�. The system of equations is fullyopulated because the irregularly sampled Fourier transform intro-uces cross-terms between the model elements �the basis functionsre nonorthogonal� and convergence is slower �Figure 2c and d�. Op-rator aliasing, on the other hand, becomes less strong �Figure 2c�.he third row, Figure 2e and f, shows the same for fine binning. Theystem of equations becomes almost fully populated again, but inhis case not because of F as before but because of T. The multidi-ensional case is more difficult to visualize, but the same ideas ap-

ly. In that case, the nonaliased directions help to constrain the solu-ion and attenuate the effect of aliasing.

In my experience, if the bin size is not made too small, the largeomputational advantage of FFT algorithms over DFTs is more ben-ficial than the consequent increase on nonorthogonality on T. Thiss possible when working along spatial dimensions where the dataook simple. In this case, the method can preserve localized ampli-ude variations better than inversion using irregularly sampled spa-ial locations because it is possible to iterate more and to obtain a so-

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igure 2. Matrix distributions for the left side of the system of equa-ions 6 for irregularly sampled data in two different cases, coarseampling and gaps �columns�, using coarse binning, true locations,nd fine binning �rows�. �a� Coarse binning on decimated data. �b�oarse binning on data with gaps. �c� True locations on decimatedata. �d� True locations on data with gaps. �e� Fine binning on deci-ated data. �f� Fine binning on data with gaps. The color represents

mplitudes in absolute numbers, with dark blue representing zeros.


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ution closer to the one obtained with full convergence. This high fi-elity for localized events makes the algorithm very useful for landata, where amplitude changes very quickly. At the same time, itakes the method less useful in removing noise.Pseudorandom noise �noise that looks random but has a coherent

ource� can be propagated into the new traces, becoming locally co-erent and therefore very difficult to remove. Although this is a dis-dvantage, it is important to realize that interpolation and noise at-enuation have different and sometimes conflicting goals. Noise at-enuation should predict only signal and should filter out noncoher-nt events. Interpolation should predict all of the data, even if thevents are very weak or badly sampled. Undersampled events canook noncoherent; therefore, their preservation depends on the algo-ithm not being too selective in terms of coherence. Although simul-aneous interpolation and noise attenuation is a very desirable goal,etter chances of success are achieved by applying noise attenuationnd interpolation iteratively in sequence rather than in a single pass.

On the other hand, there are many advantages in using exact posi-ions to eliminate aliasing and difficulties of binning for complextructure. The first aspect is balanced by working in the full 5D spacef the data. The second can be addressed in most cases by using smallinning intervals.

Some problems appear often, however, when binning long offsetsn structured data because of rapid amplitude variations caused bynisotropy and residual moveout. This is a problem for land andcean-bottom �OBC� data, where far offsets usually have poor sam-ling because of the rectangular shape of shot patches. Also, this is-ue makes binning more difficult for marine data where residual mo-eout can be very significant at long offsets. A possible solution is tose larger bins along inline and crossline for far offsets, taking ad-

a)

b)

igure 3. Foothills survey: orthogonal geometry. Shots are locatedlong vertical lines; receivers are located along horizontal lines. Col-r represents fold. �a� Before interpolation. �b� After interpolationshot and receiver line spacing decreased by a factor of two�.


antage of the fact that the Fresnel zone increases in size with offset.practical combination would be a hybrid method where binning is

pplied for near and middle offsets and exact locations are appliedn long offsets.

A complete discussion of the topic is beyond the scope of this pa-er. The comments above are intended to point out the effect of sam-ling in the system of equation 6 and the impact this has in predictingocalized amplitude variations in the data.

APPLICATIONS AND DATA EXAMPLES

Applications for land data interpolation usually involve increas-ng inline and crossline sampling �decreasing bin size� and/or in-reasing offset and azimuth sampling �increasing fold�. This classi-cation is too broad, however, because there are many possible ways

o increase the sampling, just as there are many possible geometryesigns. Table 1 shows several applications classified according tohe six types defined earlier. All of these cases have been used inractice, but only a few of them are often required in productionrojects. In this section, we review examples for the most commonases:

increasing offset and azimuth sampling by decreasing shot andreceiver line interval to improve migration �type 1b�increasing offset and azimuth sampling for better velocity, AVO,andAVAz estimates after migration �type 1b�increasing inline and crossline sampling to improve imaging ofsteep reflectors by relaxing antialias filters in migration algo-rithms �type 1a�increasing inline and crossline sampling for changing natural binsize, as in merging surveys acquired with different geometries�type 1a�infilling missing shots and receivers in acquisition gaps �type 1bin this example�

ncreasing offset and azimuth sampling for imaging

The first example shows the benefits of interpolation for aniso-ropic 3D depth migration in a Canadian Foothills area with signifi-ant structure, topography, and noise. These surveys often can bene-t from interpolation because usually they have shot and receiver

ines acquired quite far apart because of high acquisition costs on to-ographic areas. Foothills acquisitions on structurally complex ar-as, however, are difficult to interpolate because small imperfectionsn static corrections affect coherence in the space domain. Also,hese data often are severely affected by ground roll noise, which

akes interpolation difficult — particularly for shallow structures.Figure 3a shows an orthogonal geometry �vertical lines are shots,

orizontal lines are receivers�. CMPs are overlain on the shot/receiv-r locations, with their color indicating the fold. Figure 3b shows thearget geometry that contains all of the same shots and receivers as inigure 3a, along with the new interpolated shot and receiver lines.otice that these new lines follow the geometry of the original lines,ermitting us to preserve all original data because original shots andeceivers do not need to be moved.

By halving shot and receiver line intervals, the CMP fold increas-s by a factor of four, giving a better sampling of offsets and azi-uths. This can be seen in Figure 4a and b, which shows the offset/

zimuth distribution for a group of CMPs before and after interpola-ion. The increased sampling benefits migration because imaging al-


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orithms rely on interference to form the correct image and thereforeequire a proper sampling �at least two samples per cycle� to workorrectly.

These benefits can be observed in the final stacked image, but theyre more obvious in common-image gathers �CIGs�. Figure 5 com-ares the CIGs with and without interpolation. The better continuityf the events will certainly bring improvements to the results of gath-r processing, especially for AVAz and AVO analysis �which areery sensitive to acquisition footprint� and automated processesuch as tomography based on residual curvature analysis. Figure 6hows the image stack from 0 to 1000-m offsets. The continuity ofvents has been improved over nearly all of the section.

ncreasing offset-azimuth sampling for AVO

Prestack migration of the seismic data prior to performing AVOnversion has been advocated for more than 10 years �Mosher et al.,996�. However, poor sampling typical of land seismic acquisitionakes practical implementation of this concept quite difficult.ownton et al. �2008� demonstrate that these problems can be ad-ressed by performing interpolation prior to prestack migration, re-ulting in betterAVO estimates. These workers performed a series ofomparisons of processing flows forAVO, including the 5D interpo-ation method presented in this paper. By calculating the correlationetween AVO estimates to well-log information for the Viking For-ation in Alberta, they concluded that the combination of interpola-

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igure 6. Foothills survey: migrated stack section, 0–1000 m �a�ithout interpolation, and �b� with interpolation before migration.

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igure 4. Foothills survey: offset/azimuth distribution in an area ofhe �a� original and �b� interpolated surveys.

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igure 5. Foothills survey: migrated gathers �3D anisotropic depthigration� from �a� original data and �b� with interpolation beforeigration.


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ion and prestack time migration provided the best AVO estimates,chieving a correlation increase from 0.39 for migration without in-erpolation to 0.57 for migration after interpolation. This improve-

ent can be taken as evidence of amplitude preservation during in-erpolation.

Figure 7 shows CIGs from prestack time migration with and with-ut interpolation before migration. The migration applied in thisase was type 1b, decreasing shot and receiver lines by half and in-reasing fold by four times. Hunt et al. �2008� give a complete de-cription of the experiment.

ncreasing inline-crossline sampling for steep dips

This example, also described in Gray et al. �2006�, shows the ben-fits of reducing the bin size �increasing inline-crossline sampling�efore migration rather than afterward. The land data set in this ex-mple was acquired over a structured area in Thailand using an or-hogonal layout. The objective of the interpolation was to obtain

ore information on steep dips by including moderate- to high-fre-uency energy that the migration antialias filter removed from theriginal, more coarsely sampled data. For this purpose, the shot

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igure 7. Foothills II. CIGs from prestack time migration �a� withoutnterpolation and �b� with interpolation before migration.

igure 8. Thailand: shot locations after interpolation for an orthogo-al geometry. Red dots are original shots; blue dots are new �interpo-ated� shots. The two large gaps are 1000–1500 m in diameter �be-ore interpolation�.


pacing along lines was halved to reduce the bin size from 12.550 to 12.5�25 m. Figure 8 shows the shot locations after inter-

olation. The red dots indicate the locations of the original shots, andhe blue dots indicate the locations of the new shots.

As a comparison, a prestack time migration stack was producedsing the original acquired data; then the stack was interpolated, ashown in Figure 9a. In Figure 9b, the prestack data were interpolatedefore migration using 5D interpolation. The prestack interpolationroduced a data set input for migration that was better sampled thanhe noninterpolated data set. This allowed the migration to operateith greater fidelity on the steep-dip events — in this case, applying

ewer antialiasing constraints. The prestack interpolation did not addnformation to the data, but it did allow the migration to make betterse of the information that was already in the data, allowing it to pro-uce an image with greater structural detail.

ncreasing inline-crossline sampling for survey merging

Often, surveys acquired with different natural bin sizes need to beerged into a common grid. If a survey is gridded into a bin size

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igure 9. Thailand: prestack time migration stacks. �a� Interpolationerformed after stacking the migrated images. �b� Interpolation per-ormed before migration. The improved imaging of the steep-dipvent in the center of the section is evident in �b�. �Data courtesy ofTT Exploration and Production�


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maller than it was designed for, the CMP coverage becomes veryoor, affecting further interpretation even after migration.Asolutions to use prestack interpolation to reduce the natural bin size to matchhe merge grid. This can be achieved by increasing sampling in thenline-crossline domain or, alternatively, by using the surface-con-istent approach to decrease the distance between shots and receiv-rs along lines.

Trad et al. �2008� show a case history from the Surmont bitumenroject in northern Alberta. In this area, nine surveys had to beerged into a common 10�10-m CMP grid. Of the nine surveys in

he project, one was acquired with a natural bin size of 15�30 m,iving poor coverage when binned in the 10�10-m CMP grid usedor the merge. Furthermore, this survey was the only one in theroject with a parallel design �the other surveys were acquired withn orthogonal geometry�. By adding new shots and receivers usinghe method presented in this paper, the coarser survey was trans-ormed from a parallel geometry with 10�30-m bin size to an or-hogonal survey of 10�10-m bin size and twice the original fold.he original data were fully preserved and the numbers of shots and

eceivers were each increased by three, so the final size was nineimes the original size. The interpolation allowed this survey to

erge with the other surveys in the Surmont area, avoiding the needor reshooting.

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igure 10. Surmont: comparison of a CMP �a� before and �b� after in-erpolation. Empty traces have been added to the CMP before inter-olation to match the traces obtained after interpolation. The CMPinb� was created by using information from many other CMPs �nothown in the figure�.


Figure 10a and b shows one CMP before and after interpolation.igure 11a shows a time slice from the stack of original data in the0�10-m grid. Figure 11b shows the same time slice from the stackf the interpolated data.

nfilling large gaps

It is common for 3D acquisitions to have large gaps with missinghots or receivers because of inaccessibility in some areas �lakes,ills, population, etc.�. Although it usually is impossible to infillarge gaps completely, decreasing their size has a large impact on mi-ration results. The following example shows the infilling of a largeap produced by an open-pit coal mine in an area with structured ge-logy. This obstacle prevented shots and receivers from being de-loyed at this location during the 3D acquisition. New shots and re-eivers were added on the border of the gap. Time migration of the

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igure 11. Surmont: time slice comparison from �a� the stack of theriginal data and �b� a time slice from the stack of the interpolatedata.


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riginal seismic data produced the image in Figure 12a. After inter-olation, time migration produced the image in Figure 12b. The in-erpolated image shows an anticline underneath the open-pit minehat was confirmed by 2D seismic and well logs acquired before thexistence of the mine.

DISCUSSION

Interpolation fills missing traces by incorporating information inhe data using a priori assumptions. This provides, for standard pro-esses, information that is already in the data but is not accessibleithout these constraints. On the negative side, results from interpo-

ated data sometimes can be worse than results without interpola-ion. This can happen because some processes such as stacking

ight work better for zero traces than for wrong traces. In addition,nterpolation can add spurious information in a coherent manner, aroblem that stacking is unable to fix. Interpolation must be appliedery carefully to ensure this does not happen.

Several factors play against interpolation because it is by naturen ill-conditioned problem. Not only do unrecorded samples have toe estimated, but they also must be located in a manner consistentith the rest of the data, for example, at the proper elevations withroper static information. Usually, geometries that can benefit fromnterpolation do not lend themselves to good noise attenuation. Ac-urate interpolation becomes more difficult as the structure becomesore complex, as gaps become larger or sampling poorer, and as the

ignal-to-noise ratio gets lower. Therefore, careful QC is necessaryo select interpolated data according to some quality criteria. A use-ul criterion is the minimum distance between a trace and its originaleighbors, but many other QC parameters can be estimated andaved into headers.After interpolation, these parameters can be usedogether to decide if a new trace is acceptable.

CONCLUSIONS

Wide-azimuth geometries often are undersampled along one orore dimensions, and interpolation is a very useful tool to precondi-

ion the data for prestack processes such as migration, AVO, andVAz. I have discussed a 5D interpolation technique to create newhots and receivers for 3D land seismic data that honors amplitudeariations along inline, crossline, offset, and azimuth. Although notntended to replace acquiring adequate data for processing, this tool

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igure 12. Coal mine: comparison of time migration images �a�ithout and �b� with interpolation for a 3D survey acquired on top oflarge gap.

s useful for overcoming acquisition constraints and for obtaining


enefits of tighter acquisition sampling patterns, higher fold, and/ormaller bin size.

By working in five dimensions, this interpolation method can in-rease sampling in surveys that are problematic for lower dimen-ional interpolators. The technique might be applied to overcome ac-uisition constraints at a fraction of field acquisition costs, mergeata sets with different bin sizes, and eliminate differences caused bycquisition, avoiding the need to reshoot surveys. Benefits includeore reliable prestack processes: velocity analysis, prestack migra-

ion, AVO and AVAz analyses, reduction of migration artifacts, andmproved imaging of steep dips.

ACKNOWLEDGMENTS

I would like to thank CGGVeritas for permission to publish thisaper and CGGVeritas Library Canada, PTT Exploration and Pro-uction, PetroCanada, ConocoPhillips Canada Ltd., and Total E&Panada Ltd. for data examples. Special thanks are owed to severalolleagues who helped produce the interpolation examples and whorovided useful ideas and discussions on interpolation over theears. In particular, my thanks to Bin Liu and Mauricio Sacchi,hose work constitutes the cornerstone of this method.

REFERENCES

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ownton, J., B. Durrani, L. Hunt, S. Hadley, and M. Hadley, 2008, 5D inter-polation, PSTM and AVO inversion for land seismic data: 70th AnnualConference and Technical Exhibition, EAGE, ExtendedAbstracts, G029.

uijndam, A. J. W., and M. A. Schonewille, 1999, Nonuniform fast Fouriertransform: Geophysics, 64, 539–551.

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ansen, P., 1998, Rank-deficient and discrete ill-posed problems: Numericalaspects of linear inversion: Society for Industrial and Applied Mathemat-ics.

errmann, P., T. Mojesky, M. Magesan, and P. Hugonnet, 2000, De-aliased,high-resolution Radon transforms: 70th Annual International Meeting,SEG, ExpandedAbstracts, 1953–1957.

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iu, B., and M. D. Sacchi, 2004, Minimum weighted norm interpolation ofseismic records: Geophysics, 69, 1560–1568.osher, C. C., T. H. Keho, A. B. Weglein, and D. J. Foster, 1996, The impactof migration onAVO: Geophysics, 61, 1603–1615.

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rad, D., J. Deere, and S. Cheadle, 2005, Understanding land data interpola-tion: 75th Annual International Meeting, SEG, Expanded Abstracts,2158–2161.

rad, D., M. Hall, and M. Cotra, 2008, Merging surveys with multidimen-sional interpolation: CSPG CSEG CWLS Conference, Expanded Ab-stracts, 172–176.

rad, D., T. Ulrych, and M. Sacchi, 2003, Latest views of the sparse Radontransform: Geophysics, 68, 386–399.

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