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G EOP HYSIC S, VOL. 69, NO. 6 (NOVEMBE R-DE CEMB ER 2004); P. 15601568, 9 FIG S.10.1190/1.1836829

Minimum weighted norm interpolation of seismic records

Bin Liu1 and Mauricio D. Sacchi1

ABSTRACT

In seismic data processing, we o ften need to interpo-

late and extrapolate data at missing spatial locations. The

reconstruction problem can be posed as an inverse prob-lem where, from inadeq uate and incomplete data , we at-

tempt to reconstruct the seismic wavefield at locations

where measurements were not acquired.

We propose a wavefield reconstruction scheme for

spatially band-limited signals. The method entails solv-

ing an inverse problem where a wavenumber-domain

regularization term is included. The regularization term

constrains the solution to be spatially band-limited and

imposes a prior spectral shape. The numerical algorithm

is quite effi cient since the method of conjugate gra dients

in conjunction with fast matrixvector multiplications,

implemented via the fast Fourier transform (FFT), is

adopted. The algorithm can be used to perform multi-

dimensional reconstruction in a ny spatial do main.

INTRODUCTION

The seismic da ta reconstr uction problem arises in many pro-

cessing steps that require regular sampling. Different methods

have been proposed: for example, prediction error filtering

interpolation (Spitz, 1991; Claerbout, 1992), wave equation-

based interpolation (Ronen, 1987), and Fourier reconstruc-

tion (Sacchi and U lrych, 1996; Cary, 1997; Hindriks et al.,

1997; Sa cchi et a l., 1998; D uijndam et al., 1999; Zw artjes and

D uijndam, 2000). A mong those methods, Fourier-based re-

construction starts by posing the interpolation/extrapolation

problem a s an inverse problem where, from inadequate and

incomplete data, one attempts to recover the discrete Fourier

transform of the seismic wavefi eld.Inverse problems are known to be ill posed and require

regularization to obtain unique and stable solutions. Criteria

to choose a suitable regularization strategy in the context of

interpolation and extrapolation are discussed by several re-

Manuscript received b y t he E ditor January 22, 2003; revised manuscript received M ay 16, 2004.1U niversity of A lberta, Institute for G eophysical R esearch and D epartment of P hysics, E dmonton, Alberta, C anada, T6G 2J1. E -mail: binliu@

phys.ualberta.ca; [email protected] 2004 Society of Exploration G eophysicists. All rights reserved.

searchers (Cabrera and Pa rks, 1991; Sa cchi and U lrych, 1996;

H indriks et a l., 1997; Sa cchi et a l., 1998; D uijndam et al., 1999;

Zw artjes and D uijndam, 2000). For example, minimum norm

spectral regularization ca n be used when w e assume that seis-

mic data are band-limited in the spatial wavenumber domain

(D uijndam et a l., 1999). Similarly, a regulariza tion der ived us-

ingthe Cauchy criterioncan be used to obtain a high-resolution

(sparse) discrete Fourier transform that can synthesize the data

at new spatia l positions (Sacchi and U lrych, 1996; Sacchi et al.,

1998; Z wa rtjes and D uijndam, 2000). The sparse spectrum a s-

sumption is appropriate for da ta tha t consist of a superposition

of a few plane wa ves (Sacchi and U lrych, 1996). Pro cessing

the input data in windows is often necessary w hen assuming a

sparsespectrummodel. How ever, an interpolation schemethat

operates on small windows might not be optimal for multidi-

mensional data reconstruction in the presence of large portions

of missing data .

In this paper we introduce a minimum weighted norm inter-

polation (MWNI) a lgorithm to perform multidimensional re-

construction o f seismic wa vefields. In pa rticular, w e minimize

a wavenumber weighted norm that lets us incorporate a priorspectral signature of the unknown wa vefield. The procedure is

an extension of the a daptive frequency-domain w eighted norm

scheme proposed by Cabrera and Parks (1991) to extrapolate

time series. O ur work adapts the Ca brera a nd P arks (1991)

method to the seismic data reconstruction problem. We also

extend the problem to the multidimensional case. In a ddition,

we a void direct inversion methods and opt for a more effi cient

optimization scheme based on t he method of conjugate gra di-

ents with preconditioning.

Numerical examples with synthetic and field data demon-

strate the merits of the proposed interpolation scheme.

INTERPOLATION OF BAND-LIMITED DATA

Basic definitions

We start our a nalysis with a 1D interpolation problem. The

extension to higher dimensions is proposed in the next section.

By 1D interpolation, we understand interpolation in the fx

doma in along the spatial dimension x . In other words, a seismic

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Minimum Weighted Norm Interpolation 1561

gather in the tx domain is first transformed to the frequency

domain and then interpolation is carried out along the spatial

dimension x for each temporal frequency f.

We denote xas thelength-Mvector of data sampled on a reg-ular grid x1,x2,x3, . . . ,xM. The observations are given by the

elements of the vector y

=[xn(1),xn(2),xn(3), . . . ,xn(N)]

T, where

the set N={n(1), n(2), n(3), . . . , n(N)} indicates the po sitionof the known samples or observations. We now defi ne the sam-

pling ma trix T with elements Ti,j = n(i),j , where indicatesthe Kronecker operator.I t is quitesimpleto show that thecom-

plete data and the observations are connected by the following

linear system:

y= Tx. (1)For example, let us assume the complete data set consists of

M= 5 consecutive samples, or x= [x1,x2,x3,x4,x5]T; the ob-servations (available da ta) are given by samples at positions

N={2,3, 5}, or y= [x2,x3,x5]T. Then equa tion 1 becomes

x2

x3

x5

= 0 1 0 0 0

0 0 1 0 0

0 0 0 0 1

x1

x2

x3

x4

x5

. (2)

Note that the sampling operator T has the following property:

TTT = IN, (3)where IN denotes the NN identity matrix. I n addition,TTT= IM. We also define the discrete Fourier transfo rm (DFT)and the inverse discrete Fourier transform (ID FT) respectively,

as follows:

Xk =1M

M

m=1

xmei2(m 1)(k 1) /M,

k= 1, . . . ,M, (4)

xm =1M

Mk=1

Xkei2(m 1)(k1) /M,

m = 1, . . . ,M. (5)We use the following compact nota tion for the D FTa nd ID FT,

respectively:

X = Fx, (6)x= FHX, (7)

where the superscript Hdenotes the complex conjugate trans-

pose. Notice tha t F is the DFT unitary mat rix whose inverse is

given by F1 =FH.

Minimumweighted norminversion of thesamplingoperator

The signal reconstruction or interpolation problem given by

equa tion 1enta ilssolving an underdetermined system of equa -

tions (more unknowns than observations). It is clear that the

problem does not have a unique solution. In general, one way

of solving this type of problem is by restricting the class of

solutions through providing suitable prior informa tion.

Let us continue the analysis by saying that among all the

possible solutions, w e seek a solution t hat minimizes a mo del

norm. In the ab sence of errors, the inversion can be reduced to

solving the following constrained minimization problem:

Minimize x2WSubject t o Tx= y,

where .W indicates a weighted norm. Following C abreraand Pa rks (1991), we select the follow ing wavenumber-domain

norm:

x2W =kK

XkXkP2k

, (8)

where P2k are spectral domain weights with support a nd shape

similar to those of t he signal to interpolate. The set of indexes

K indicates the region of spectral support of the signal. It isunderstood that Pk= 0 f o r kK. The coefficient Pk representsthe spectral power at wavenumber index k.

We now introd uce a diagona l matrix with elements given

by

k =P2k , k K0, k K . (9)

Similarly, we define the pseudoinverse of the diagonal matrix

, as the matrix with elements given by

k =

P2k , k K

0, k K . (10)

The wa venumber-doma in norm ca n now be expressed a s

x2W = XH X. (11)After combining equations 7 and 8, we arrive at the following

expression:

x2W = xH FHFx= xH Qx, (12)

where Q=FHF is a circulant matrix (Strang, 1986). Sim-ilarly, we define the circulant matrix Q=FHF. Notice thatboth Q and Q are band-limiting operators. In other words,they annihilate a ny spectral component kK.

The minimum norm solution is found by minimizing the fol-

lowing cost function:

J= bT(Tx y) + x2W.

In the above equation, b denotes the vector of Lagrange mul-tipliers. Minimizing J with respect to x subject to Tx

y

=0

leads to the fo llowing solution:

x= QTT(TQTT)1y. (13)In the previous derivation we a ssume that t he matrix TQTT isinvertible. If this is not the case, the inverse can be replaced by

the Moore-Penrose pseudoinverse (Cabrera and Parks, 1991).

The above solution is designated as MWNI. We reserve the

name minimum norm interpolation (MNI) for the case where

Q is a band-pass filter with spectral weights P2k =1, kK. I n


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1562 Liu and Sacchi

other words, we constraint the solution to the class of band-

limited signals with spectral components in kK, and we makeno a ttempt to impose a prior spectral shape.

Let us consider the case when Q is an a ll-pass filtering ma-trix with D FTcoefficients k=1 f o r k=1, . . . ,M. In this case,

=I, and after invoking the orthono rmality of the DFT oper-

ator we obtain the following expression:

x= TT(TTT)1y= TTy. (14)In the toy example provided by equa tion 2, the minimum norm

solution becomes

x= TTy=

0

x2

x3

0

x5

. (15)

In other w ords, missing samples are fi lled in with zeros.

Inversionof T in thepresenceof noise

When the observations contain additive noise ra ther than

trying to fit exactly all of the observations, we attempt to fi t the

observations in the least-squares sense. In this case we min-

imize a cost function that combines a data misfit function in

conjunction with the mo del norm:

J= Tx y2 + 2x2W, (16)where 2 ist he trade-off parameter of the problem. Notice that

minimizing J is equivalent to finding the least-squares solution

of the fo llowing overdetermined system of equa tions:

TWx

y

0 , (17)

where, according to our previous definitions, the matrix of

weights W is given by

W= 1/2 F. (18)U nfortunately, the augmented matrix of the problem is rank

deficient; therefore, equation 17 does not have a unique solu-

tion. The latter can be solved by choosing, among a ll possible

least-squares solutions, the one w ith the minimum E uclidean

norm. This can be done with the aid of the singular value de-

composition(SVD ) of the augmented matrix.A lternatively, we

can use the method of conjugate gra dients. For rank-deficient

problems, the solution to w hich the conjugate gra dient method

converges depends upon the initial approximation ad opted. If

the initial approximation is chosen to be x=0, then the con-jugate gradient converges to t he minimum-norm least-squa res

solution (Hestenes, 1975). One advantage of using the con-

jugate gradient method is that the computational cost of the

algorithm heavily depends on matrixvector multiplications.

These operations can be performed efficiently using the fast

Fourier transform (FFT).

In our numerical implementation, equation 17 is modified

with the following change of variab le: z=Wx. The augmented

system becomes TW

I

z

y

0

. (19)

The trade-off parameter can be set to

=0, and the number

of iterations in the conjugate gradient method plays the roleof regularization para meter (H ansen, 1998). We end up solv-

ing TWzy and stopping the algorithm when a maximumnumber of iterations is reached or a desired misfit is achieved:

Txy< tolerance, tolerance =103 105. We find tha t theconjugate gradient method often converges in less than 15 it-

erations.

At this point a few comments are in order. The transition

from equation 17 to equation 19 is only valid for a full-rank

matrix W. The fact t hat we a re solving for a band-limited solu-tion (kK), however, permitsus to claim that solving equation19 is equivalent to solving equation 17 (Nichols, 1997).

It is important to clarify that the proposed algorithm dif-

fers from the one proposed by Sacchi a nd U lrych (1996) to

invert the coefficient of the D FT using sparseness constraints.

First of all, our algorithm does not a ssume a sparse distributionof spectral amplitudes. The latter is only valid for estimating

the D FT of a process that consists of a fi nite number of spec-

tral lines (Sacchi et al. 1998). To be more specific, the high-

resolution Fourier transform (H R FT) proposed by Sa cchi and

U lrych (1996) utilizes a C auchy regularization criterion of the

form

xc =k

ln

1+ Xk X

k

2c

, (20)

where c is the scale parameter of the Cauchy norm.

Notice the difference between the a bove norm (equa tion 20)

and the norm utilized in this paper (equation 8). It is very im-

portant to stress that the C auchy criterion w as proposed a s a

means of estimating sparse (high-resolution) spectral estima-tors for waveforms that can be approximated by plane waves.

In this case, a sparse spectrum is the appropriate model for

data that consist of a superposition of a few plane wa ves. In

this paper, however, we propose a more general norm that is

capable of handling nonsparse spectral models. The new ap-

proach is particularly relevant for multidimensional seismic

data, where the common assumption of a local superposition

of a few plane w aves is suboptimal. Windowing can be used to

validate the no nsparse spectral model. How ever, we prefer an

alternative procedure wheresparsenessis not invoked.It istrue

that both MWNI a nd H RFT lead to very similar algorithms.

However, in HRFT the amplitude of the Fourier transform

|Xk|2 plays the role of a data-dependent diagonal regulariza-tion matrix[see equation 19in Sacchi and Ulrych(1996)]. In the

MWNI formulation, on the other hand, the matrix of weights

is derived from the power spectrum of the data using a non-

para metric spectral estimato r. In the next section w e propose

a procedure to estimate the power spectrum of the unknown

data.

It is also important to clarify that the present work does

not attempt to invert the nonuniform D FT (Hindriks et al.,

1997). Our implementation utilizes FFTs; therefore, an impor-

tant gain in efficiency is achieved when interpolating dat a tha t

depend on more than one spatial dimension.


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Adaptiveestimation of theweightingoperator

To obt ain the ma trix of weights W, in practice one shouldknow the power spectrum of the complete data x. U nfortu-nately, the complete da ta x are the unknowns of our problem.The latter can be overcome by defining an iterative scheme

to bootstrap the spectral weights from the dat a. O ur numeri-cal implementation uses the smooth periodogram of the data

(B ingham et al., 1967):

P2k =L

l=L wl|Xkl |2, k K,0, k K , (21)

where wl is a smoothing window of length 2L +1. We initializethe algorithm with the band-limiting operator with spectral

weights P2k = 1, kK; we solve for x and use the solution torecompute P2k using equation 21.

Alternatively, it is possible to adopt a noniterative strat-

egy similar to the one proposed by Herrmann et al. (2000)

to compute the high-resolution parabolic R adon transform.

The metho d is well document ed in Hugo nnet et a l. (2001). The

power spectrum P

2

k required to interpolate spatial data at atemporal frequency f can be estimated from the a lready in-

terpolated da ta at frequency ff. Such a scheme is ofteneffective in dealing with situations where the data exhibit a

mild degree of spatial aliasing at high frequencies.

In particular, in situations with aliasing produced by non-

conflicting dips, the weighting operator computed from the

nonaliased low frequencies attenuates the aliasing that might

arise at high frequencies. Clea rly, the a ssumption at the time of

adopting such scheme is that the power spectrum of the data

at frequency f f is similar in shape to the pow er spectrumof the data at frequency f. This assumption is often valid when

f is small. This is achieved, in general, by padding t he da ta

with zeros before applying the Fourier transform.

1D reconstruction examples

Reconstruction along one spatial coordinate is illustrated

with a synthetic shot ga ther. Figure 1a shows a complete shot

gather with a smallamount of random noise. The syntheticdata

were modeled with a ray-tracing algorithm for laterally invari-

ant media; the a mplitude va riation with offset (AVO) effect

is added using Shueys equation (Shuey, 1987). A total of 18

traces were removed from the original data , including some

near-offset traces (Figure 1b). The incomplete data set is used

as input for o ur reconstruction a lgorithm. The dat a set is fi rst

transformed to the temporal frequency domain. The recon-

struction is then performed along the spatial coordinate (re-

ceiver position) for each t emporal freq uency. Figure 1c shows

the reconstruction using the MWNI algorithm. The modified

periodogram (equation 21) is used to iteratively estimate the

matrix of weights. The reconstruction error is portrayed in

Figure 1d.

For comparison, we also tried to reconstruct the data using

the HR FT algorithm (Sacchi and Ulrych, 1996) and the

minimum norm interpolation (MNI) algorithm. Note that in

the HRFT example we used all of the traces that compose

the synthetic shot gather (no attempt a t data windowing was

made). The MNI a lgorithm had diffi culties when interpolating

large gaps. The MWNI and H R FTa lgorithmsb oth mana ged to

retrieve comparable interpolation results. H owever, numerical

experiments have shown that the HRFT tends to produce

spectral models that a re too sparse and tends to produce large

interpolation errors when dealing with data that do not fi t the

sparse spectral model (seismic events with curvature in tx).

We also compare the reconstructed power spectrum at the

temporal frequency component f= 15.6Hz for the MWNI,HR FT, and MNI methods. Figure2a showsthe power spectrum

of t he reconstructed da ta using the MWNI metho d. Figure 2b

portrays the power spectrum of the reconstructed data us-

ing the HRFT approach. The spectrum of the reconstructed

dat a using the MNI method is portra yed in Figure 2c. Finally,

the power spectrum of the true (complete) data is displayed

in Figure 2d. Notice the good agreement of the spectral sig-

natures of the interpolated and original data in Figures 2a

and 2d. The spectrum obtained using the HRFT approach

(Figure 2b) isb etter tha n the spectrum obta ined using the MNI

method (Figure 2c;) however, as we have already mentioned,

Figure 1. (a) Original synthetic shot gather.(b) Incomplete shotgather ob tained by removing 18 traces from the complete shotgather in (a). (c) Reconstruction using the MWNI algorithm.(d) R econstruction error after interpolation with the MWNImethod. (e) Reconstruction using the HR FTa pproach. (f) Re-construction error after interpolation with the HRFTmethod.(g) Reconstruction using MNI. (h) Reconstruction error afterinterpolation with the MNI method. Error panels (d), (f), and(h) are multiplied by two to b etter depict differences.


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1564 Liu and Sacchi

interpolation with the HRFTapproach tends to produce spec-

tral estimates that are too sparse.

I t i s i mp ort an t t o st r ess t h at f or t h e M NI m et h od we

have used a frequency-dependent bandwidth. The maximum

wavenumber at frequency f is estimated using the formula

kmax

=f/vmin , where vmin is the minimum appa rent velocity of

the da ta (D uijndam et al. 1999).The interpolation of a real marine shot gather using the

MWNI method is portrayed in Figure 3. Figure 3a shows the

marine shot gather befo re interpolation. The interpolated dat a

at twice the original sample rate is portrayed in Figure 3b.

In this example, we have d etermined the spectral weights us-

ing the noniterative scheme described above. The noniterative

scheme is much faster tha n the iterative a pproach. Therefore,

the no niterative scheme is utilized in a ll of t he remaining ex-

amples. Our tests show minimal difference between t hese two

approaches.

2D SPATIAL INTERPOLATION

The 1D interpolation a lgorithm proposed in the previous

section can be extended to tw o dimensions by using Kronekerproducts (D avis, 1979; Ja in and R anga nath, 1981). First, w e

denote the lexicographic ordering of the elements of the 2D

Mu Mv complete data matrix along two arbitrary spatial di-mensions u and v as the vector x. Similarly, the observations(obtained after binning the data) can also be organized in a

data vector y. Again, we can relate the complete data in theregular grid to the observations with a simple mapping of the

form Tx=y. The band-limiting operato r W is now defined interms of the following operat ions:

W= 1/2(Fu Fv), (22)

Figure 2. Power spectra at frequency component f=15.6 H zfor data shown in Figure 1. (a) Power spectrum of the recon-structed data using the MWNI method. (b) P ower spectrum ofthe reconstructed da ta using the H RFT approach. (c) Powerspectrum of the reconstructed da ta using the MNI method.(d) Power spectrum of the original (complete) data.

where Fu a nd Fv denote 1D DFTs along dimensions u a nd v,respectively. The K roneker product Fu Fv isthe 2D DFTma-trix operating on the vectorized dat a. Similarly, the 2D po wer

spectrum of the d ata (in vectorized form) is distributed a long

the diagonal of . With these new definitions, we can easily

extend the conjugate gradient algorithm discussed in a pre-

ceding section to the 2D case. It is clear tha t this schemecan be extended to interpolate three or more spatial variables

simultaneously.

Interpolation in source-receivercoordinates

The effectiveness of the 2D MWNI method is first demon-

strated using the Marmo usid ata set. The spatial coordinates to

interpolate a re source and receiver positions. It is important to

stress that similar results could be obtained by interpolating in

midpoint-offset coordinates. The input data are a subset of the

Marmo usid ata set that consists of 24 shots with 96 receiver po-

sitions per shot. The original shots and receivers were sampled

every 25 m. We simulat e a survey with shot an d receiver inter-

vals of 75 m. In other words, 8 shots with 36 receivers per shot

were extracted from the data and input to our reconstruction

algorithm.

Figure 4 shows the shot-receiver distribution of the observa-

tions and positions to reconstruct. The original da ta, the input

data, the reconstructed data, and the reconstruction error for

three shots in the survey (source positions: 3075, 3100, and

Figure 3. (a) Incomplete data from a real marine shot gather.(b) Interpolated data using the MWNI method.


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Figure 4. Source and receiver location ma p for a subset of theMarmo usi model.

Figure 5. (a) Three shots extracted from t he Ma rmousi data set. (b) D ecimated shots. (c) Reconstructed dat a using the 2D MWNIalgorithm. (d) Error panel.

3125m ) are sho wn in Figures 5ad, respectively. The fkspec-

tra of the original, decimated, and reconstructed shot gathers

at 3125 m are shown in Figures 6ac, respectively.

3D poststack seismic datareconstruction

We a lso illustrat e the reconstruction of a real 3D poststackdata cube using the 2D MWNI algorithm. In this case the in-

terpolation is carried out along the inline and crossline co-

ordinates. Figure 7a show s a complete 3D po ststack da ta cube

that consists of 51 inlines and 31 crosslines. The decimated

poststack data cube (Figure 7b) is obta ined by removing every

second tra ce along bo th the inline an d crossline directions. The

incomplete data cube isused as input to the MWNI reconstruc-

tion a lgorithm. Figure 7c shows the cube after reconstruction.

D etailed panels showing the true complete da ta, the recon-

structed data, and the reconstruction error for inline 39 and

crossline 19 are provided in Figures 8 and 9, respectively. No-

tice that the proposed interpolation has also attenuated the

rando m noise. The degree of noise attenuat ion versus fidelity


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1566 Liu and Sacchi

of the reconstruction is regulated by the number of iterations

of the conjugate gra dient solver.

CONCLUSIONS

In thispa per, we have formulated a band-limited dat a recon-

struction a lgorithm that can incorporate prior spectral weightsto control the bandwidth and the spectral shape of the recon-

structed data . The MWNI method ha s been shown to perform

better than standard MNI when dealing with large data gaps.

Figure 6. (a) The fk spectrum of the original shot gather at3125 m. (b) The fk spectrum of the same shot gather afterdecimat ion. (c) The fkspectrum of the same shot ga ther afterinterpolation.

We have also discussed the differences between MWNI and

dat a reconstruction via H R FT (Sacchi and U lrych, 1996). The

MWNI method avoids the sparse spectrum assumption; this is

an important advantage when processing seismic data that do

not sat isfy the sparse spectrum model. It is clear that the sparse

spectrum assumption is only valid for da ta that consist of a few

Figure 7. (a) A complete 3D poststack data cube. (b) D eci-mated cube; every second line is removed. (c) Reconstructedcube using the MWNI metho d.


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plane waves (linear events in tx). This is a valid assumption

when reconstructing data in small windows. However, an in-

terpolation scheme that operates on small windows might not

be optimal for data reconstruction in the presence of large

gaps.

In the presence of a dditive noise, a least-squares minimum

weighted norm solution can be computed efficiently using the

method of conjugate gradients. It is important to stresstha t the

computational cost of the conjugate gradient method heavily

depends on matrixvector multiplications. These operations

can be implemented effi ciently using the FFT. Ad ditional effi -

ciency can be o btained by t runcating the number of conjugate

gra dient iterat ions. As pointed out by H ansen (1998), the num-

ber of iterations plays a role similar to a trade-off parameter.

Figure 8. (a) Original data along inline 39 (line A, Figure 7b). (b) Reconstructed data using theMWNI method. (c) R econstruction error.

Figure 9. (a) O riginal d ata along crossline 19 (line B, Figure 7b). (b) R econstructed d ata using theMWNI method. (c) R econstruction error.


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1568 Liu and Sacchi

Co nsequently, by truncating the number o f iterations, additive

noise can be a ttenuated.The computational cost of the metho d

makes it at tractive for multidimensional interpolat ion.

Fourier interpolation methods can handle crossing events in

as much as the data a re not aliased. Although we did not show

numerical results highlighting the interpolations with conflict-

ing dips, our metho d ma y not be ab le to ha ndle multiple cross-ing events if the data are aliased. This is significant because

seismic data interpolation often involves spatia lly aliased data

and multiple crossing events.

ACKNOWLEDGMENTS

The Signal Ana lysis and I maging G roup at the U niversity of

Alberta would like to acknowledge financial support from the

following companies: G eo-X Ltd., Encana Ltd., Veritas G eo-

services, and the Schlumberger Foundation. This research has

also been supported by the Na tural Sciences and E ngineering

R esearch Council of Canada and the Alberta D epartment of

Energy.

We a lso a ppreciate the va luable comments and suggestions

from the reviewerso f Geophysicsand Assistant E ditor Yonghe

Sun.

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