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GEOPHYSICS, VOL. 74, NO. 3 �MAY-JUNE 2009�; P. V43–V48, 6 FIGS.10.1190/1.3085643

tacking seismic data using local correlation

uochang Liu1, Sergey Fomel2, Long Jin3, and Xiaohong Chen4

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ABSTRACT

Stacking plays an important role in improving signal-to-noise ratio and imaging quality of seismic data. However, forlow-fold-coverage seismic profiles, the result of convention-al stacking is not always satisfactory. To address this prob-lem, we have developed a method of stacking in which weuse local correlation as a weight for stacking common-mid-point gathers after NMO processing or common-image-pointgathers after prestack migration. Application of the methodto synthetic and field data showed that stacking using localcorrelation can be more effective in suppressing randomnoise and artifacts than other stacking methods.

INTRODUCTION

Stacking as one of the three crucial techniques �deconvolution,tacking, and migration� plays an important role in improving sig-al-to-noise ratio �S/R� in seismic data processing �Yilmaz, 2001�.onventional stacking, which is performed by averaging an NMO-orrected data set or migrated data set, is optimal only when noiseomponents in all traces are uncorrelated, normally distributed, sta-ionary, and of equal magnitude �Mayne, 1962; Neelamani et al.,006�. Therefore, different stacking technologies have been pro-osed, along with improvements in optimizing weights of seismicraces.

Robinson �1970� proposes using an S/N-based weighted stack tourther minimize noise. Using crosscorrelation of seismic traces andormalized crosscorrelation processing, Chang et al. �1996� propos-s preserved frequency stacking. Schoenberger �1996� proposes op-imum weighted stack for multiple suppression, with weight deter-

ined by solving a set of optimization equations. Neelamani et al.2006� propose a stack-and-denoise method called SAD, which ex-loits the structure of seismic signals to obtain an enhanced stack.

Manuscript received by the Editor 19 May 2008; revised manuscript receiv1China University of Petroleum, State Key Laboratory of Petroleum Resou

f Econonic Geology,Austin, Texas, U.S.A. E-mail: guochang.liu@yahoo.co2The University of Texas atAustin, Bureau of Economic Geology,Austin,3The University of Texas atAustin, Institute for Geophysics,Austin, Texas4China University of Petroleum, State Key Laboratory of Petroleum Resou2009 Society of Exploration Geophysicists.All rights reserved.

V43

hang and Xu �2006� present a high-order correlative weightedtacking technique on the basis of wavelet transformation and high-rder statistics. By estimating the probability distribution of noise,rickett �2007� applies a maximum-likelihood estimator to stacking.o eliminate artifacts in angle-domain common-image gathersCIGs� caused by sparsely sampled wavefields, Tang �2007� pre-ents a selective stacking approach that applies local smoothing ofnvelope function to achieve the weighting function. Rashed �2008�roposes smart stacking, which is based on optimizing seismic am-litudes of the stacked signal by excluding harmful samples from thetack and applying larger weight to the central part of the sampleopulation.

The global correlation coefficient can measure the similarity ofwo signals, but it is not a local attribute. Not only does the sliding-indow global-correlation approach need many parameters to be

pecified, but this approach cannot smoothly characterize thin layersell. Fomel �2007a� uses shaping regularization, which controls lo-

ality and smoothness to define local correlation �Fomel, 2007b�.ocal correlation is applied to multicomponent seismic image regis-

ration �Fomel et al., 2005; Fomel, 2007b� and time-lapse image reg-stration �Fomel and Jin, 2007�.

In this paper, we present a new stacking method using local corre-ation. This method applies time-dependent smooth weights �whichre taken as local correlation coefficients between reference tracesnd prestack traces�, stacks the common-midpoint �CMP� gather,nd effectively discards parts of the data that least contribute totacked reflection signals. Using synthetic and field data examples,e show that, compared with other stacking methods, this method

an greatly improve the S/N and suppress artifacts.

METHODOLOGY

eview of local correlation

The global uncentered correlation coefficient between two dis-rete signals ai and bi can be defined as the functional

tober 2008; published online 27 March 2009.d Prospecting, Beijing, China and The University of Texas atAustin, Bureau

.S.A. E-mail: sergey.fomel@beg.utexas.edu.. E-mail: jinlong@ig.utexas.edu.d Prospecting, Beijing, China. E-mail: chenxh@cup.edu.cn.

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here N is the length of a signal. The global correlation in equation 1upplies only one number for the whole signal. For measuring theimilarity between two signals locally, one can define the sliding-indow correlation coefficient

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here w is window length.Fomel �2007b� proposes the local correlation attribute that identi-

es local changes in signal similarity in a more elegant way. In a lin-ar algebra notation, the correlation coefficient in equation 1 can beepresented as a product of two least-squares inverses � 1 and � 2:

� 2 � � 1� 2, �3�

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� 2 � arg min� 2

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here a and b are vector notions for ai and bi. Let A and B be two di-gonal operators composed of the elements of a and b. Localizingquations 4 and 5 amounts to adding regularization to inversion. Us-ng shaping regularization �Fomel, 2007a�, scalars � 1 and � 2 turnnto vectors c1 and c2, defined as

c1 � ��2I � S�ATA � �2I���1SATb , �6�

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igure 1. �a� Zero-phase Ricker wavelet. �b� Reflection coefficient. ���,� � � N�0,10�6�. �d� Noisy signal with N��,� � � N�0,0.07�.orrelation. �f� Local correlation.

c2 � ��2I � S�BTB � �2I���1SBTa , �7�

here � scaling controls relative scaling of operators A and B andhere S is a shaping operator such as Gaussian smoothing with an

djustable radius. The componentwise product of vectors c1 and c2

efines the local correlation measure. Local correlation is a measuref the similarity between two signals.

An iterative, conjugate-gradient inversion for computing the in-erse operators can be applied in equations 6 and 7. Interestingly, theutput of the first iteration is equivalent to the algorithm of fast localrosscorrelation proposed by Hale �2006�.

tacking using local correlation

The problem of combining a collection of seismic traces into aingle trace is commonly referred to as stacking in seismic data pro-essing. This process is used to attenuate random noise and simulta-eously amplify the coherent signal in the gather. Typically, the de-ired stacked trace is estimated by averaging traces from the CMPather �Mayne, 1962�:

aj�t� �1

N�i�1

N

ai,j�t�, j � �1,2,3, . . . ,M� , �8�

here N is the fold of the stack and ai,j�t� is the sample value on tracefrom the jth CMP gather at two-way time t. Such a technique pro-ides the optimal unbiased estimate of aj�t�. Robinson �1970� pro-oses weighted stacking of seismic data:

aj�t� �1

�i�1

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wi,j · ai,j�t�, j � �1,2,3, . . . ,M� ,

�9�

here wi,j denotes the weight of the ith trace from the jth CMP gath-r, which is determined by noise variances wi,j � 1/� i,j

2 . However, its difficult to estimate noise variances reliably in practice. Neela-

mani et al. �2006� use an iterative algorithmcalled “leave me out” �LMO� to estimate noisevariances from data. The desired signal is as-sumed to be flat with constant amplitude acrossall the traces within a gather in the LMO method.

For using time-dependent smooth weights inthe stacking process, we choose the local correla-tion coefficient from the previous section asweights to stack seismic data. We find that localcorrelation better characterizes local similaritybetween prestack and reference traces than thesliding-window method.

Consider the two noisy signals with Gaussianrandom noise but different noise levels in Figure1c and d. The signals are derived from addingnoise on convolution of the Ricker wavelet �Fig-ure 1a� with synthetic reflectivity �Figure 1b�.The distribution of noise in Figure 1c is N��,� �� N�0,10�6�, where � and � are mean and vari-ance of noise, respectively. The distribution ofnoise in Figure 1d is N��,� � � N�0,0.07�. Thesliding-window correlation and local correlation

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Stacking using local correlation V45

etween Figure 1c and d are shown in Figure 1e and f, respectively.ote that local correlation coefficients �Figure 1f� are smooth andetter distinguish the thin layer, represented by the first two reflec-ivities in Figure 1b. In application to stacking, the prestack trace isnalogous to Figure 1d with larger variance noise, and the referencerace is analogous to Figure 1c with smaller variance noise.

To implement seismic data stacking using local correlation, werst apply the equal-weight stack to the NMO-corrected CMPgather

o obtain the reference trace. Then we compute the local correlationoefficients between the reference trace and the NMO-correctedMP gather and apply soft thresholding �Donoho, 1995� to all localorrelation coefficients. Finally, we apply the weighted stack to theMP gather using local correlation to get the final stacked trace.Mathematically, stacking using the local correlation approachodifies equation 9 to

aj�t� �1

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N

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�10�

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0, � i,j � � , �11�

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N wi,j�t� is the sum ofeights of the jth CMP gather, Hj�t� is the number of samples withi,j�t� ·ai,j�t��0 for a given two-way time, and � i,j�t� is the local cor-

elation between the ith prestack trace from jth gather and the refer-nce trace. The local correlation � i,j�t� can be computed using equa-ions 6 and 7. The reference trace is derived from averaging all theraces of one CMP gather. Here we assume that the equal-weighttacked trace is close to the desired trace. Becausehe weights are a function of two-way traveltimend offset, recovery scalar Kj has the same valueor the same CMP gather. Meanwhile, the sam-les with wi,j�t� ·ai,j�t� � 0 at a given two-wayime are assumed to be full noise or null valueuch as muting parts; we therefore use Hj�t� tocale the stacked trace.

Changes occurring between equation 9 andquations 10 and 11 result from time-dependentmooth weights for the stack and application ofhresholding to the weights. All local correlationoefficients below a specified threshold are dis-arded, and the rest, with values above the thresh-ld, are included. We thus stack only those partsf prestack data whose similarity to the referencerace is comparatively large. Equations 10 and 11mplicitly estimate the noise variance by analyz-ng local crosscorrelations between prestackrace and the reference trace. This operation cane understood as a nonlinear filter that enhanceshe coherency of events. We perform this opera-ion for all gathers using this method to improvehe stack profile.

When applied after angle-domain migration,ormalization provided by soft thresholding is

0

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Figure 2. Simin local-corre�S/N � 8.4�.�S/N � 10.2�

nalogous to true-amplitude illumination compensation from the so-alled Beylkin determinant �Albertin et al., 1999; Audebert et al.,005�. Local correlation normalizes the image by the number oftrongly illuminated angles in angle-domain CIGs.

In the following, we discuss the distinctions between seismictacking using local correlation and other methods. Our method cre-tes time-dependent smooth weights without depending on slidingindows, as compared to other weighted stacking methods such as

tatistically optimal stacking �Robinson, 1970; Neelamani et al.,006� and the semblance method �Yilmaz, 2001�. In contrast tomart stacking, proposed by Rashed �2008� and based on sign differ-nce between sample point and the alpha-trimmed mean to removerequency distortions, our method applies waveform similarity be-ween prestack trace and mean to make the stacking selective.

EXAMPLES

To illustrate the proposed method using synthetic and field data,e apply our approach to three examples. The first example is a sim-le case involving a fivefold prestack gather �Figure 2a� with a time-hifted-upward trace, which might be distortion by poor static cor-ection. The peak of the signal in this gather is one. We add Gaussianandom noise with distribution N��,� � � N�0,0.01� on the fiveraces. The result of an equal-weight stack is shown in Figure 2c. Thepside wing in Figure 2c is distorted because of the first time-shiftrace. Then we use three weighted stacking methods to stack the fiveraces.

Figure 1d and e illustrates results of smart stacking �Rashed,008� and LMO-based weighted stacking in which the weights areomputed by the LMO method �Rabinson, 1970; Neelamani et al.,006�. Figure 1f shows the result of stacking using local correlationith weights �Figure 2b� determined by the similarity between therestack trace �Figure 2a� and the reference �Figure 2c�. Because theaveform in the first trace in Figure 2a is most likely noise or arti-

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c) d) e) f)

king test with fivefold gather. �a� Prestack gather. �b� Weights usedeighted stacking. �c� Conventional equal-weight stacking method

art stacking method �S/N � 9.2�. �e� LMO-based weighted methodcal-correlation weighted stacking �S/N � 13.5�.

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V46 Liu et al.

act, it is reasonable that the weight in the stack procedure is lower.se of local correlation as weights of prestack traces lets us select

hose portions, which are more similar to the reference trace to con-ribute to the stack.

Comparing the three methods, one can find that smart stackingnd LMO-based weighted stacking can remove upside wing distor-ion cleanly, but stacking using local correlation removes more ran-om noise than the other two methods and meanwhile corrects up-ide wing distortion.

0

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igure 3. �a� One CMP gather from synthetic data set. �b� NMO-corrult of conventional equal-weight stacking. �d� Result of LMO-basede� Result of local-correlation weighted stacking.

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igure 4. Comparison among three stacking methods including all srs. �a� Conventional equal-weight stacking �S/N � 7.1�. �b� LMtacking �S/N � 9.6�. �c� Local-correlation weighted stacking �S/N

To judge the effect of denoising quantitatively between differentethods, we apply equal-weight stacking on the last four tracesithout any noise to get the exact desired stacked trace dj�t�, which

an be regarded as a signal trace. The S/N of the jth CMP can there-ore be estimated as

S/N j � 10 log10 �t

�dj2�t��

�t

�dj�t� � aj�t��2� , �12�

where aj�t� is the stacked trace from differentstacking methods. In the first simple example�Figure 2�, the S/N of equal-weight stacking is8.4 dB and the other three weighted methods are,respectively, 9.2, 10.2, and 13.5 dB. Stacking us-ing local correlation can improve S/N greatly.

The second example is a 2D synthetic modelthat includes four reflectors. Synthetic data aregenerated with Kirchhoff modeling. The peak ofthe data set is one and Gaussian random noisewith distribution N��,� � � N�0,0.05� is added.We show the results of stacking one CMP gather�Figure 3a� by three methods in Figure 3c-e.Compared to other methods, our method is themost effective in denoising.

The stacked profile of all CMP gathers isshown in Figure 4. We use equation 12 to com-pute the S/N of the stacked profile �Figure 4�. TheS/Ns of three methods are 7.1, 9.6, 10.9 dB, re-spectively. Noise is attenuated more effectively inthe stacking result using local correlation �Figure4c�.

The third example involves a historic 2D linefrom the Gulf of Mexico �Claerbout, 2005�. Thestacked sections, using three different methods,are shown in Figure 5. Figure 6 shows the localcorrelation cube between prestack and referencetraces. Similar cubes have been used in multi-component seismic image registration �Fomel etal., 2005; Fomel, 2007b� and time-lapse imageregistration �Fomel and Jin, 2007�. For syntheticdata, the exact desired stacked section can be cal-culated by stacking prestack traces without anynoise. But for field data, the S/N is difficult to esti-mate using equation 12. We therefore use singularvalue decomposition �SVD� �Andrews andPatterson, 1976� to evaluate different stackingmethods. The SVD of stacked section metrixgives

M � UVT. �13�

The diagonal elements � r of are the singularvalues of M. The S/N can be estimated as �Freireand Ulrych, 1988; Peterson and DeGroat, 1988;Grion and Mazzotti, 1998�

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Stacking using local correlation V47

S/N � 10 log10� 12 �

1

R � 1 �r�2

R

� r2

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R � 1 �r�2

R

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here R is the number of all singular values. The S/Ns of stackedections resulting from three stacking methods are, respectively,7.4, 29.2, and 33.9 dB. Comparing Figure 5a-c, we can find alsohat random noise is attenuated and coherent reflections are en-anced better using local correlation �e.g., 0.5–1.5-s range�.

CONCLUSIONS

We have developed a new method of stacking NMO-corrected origrated seismic data using local correlation. We substitute local

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Time(s)

CMP (km) CMP (km)a) b) c)

igure 5. Results of �a� conventional equal-weight stacking �S/Nased weighted stacking �S/N � 29.2� and �c� local-correlationS/N � 33.9�.

igure 6. Local correlation cube of the field-data example.

correlation for the weight value in statisticallyweighted stacking and then use soft thresholdingto make the stacking selective. Because weightsare derived from the input data, our method canbe regarded as a nonlinear filter. Synthetic andfield data examples confirm that our approach canbe significantly more effective than other weight-ed stacking methods in improving S/N and sup-pressing distortions resulting from prestack pro-cessing. Seismic stacking using local correlationcan give a poor result if the quality of the refer-ence trace is very poor. Because the coherencyenhancement from local correlation is not basedon physics, one should use this approach withcaution when aiming to preserve physicallymeaningful amplitudes.

ACKNOWLEDGMENTS

We would like to thank Yang Liu and Jingye Lifor inspiring discussions and three reviewersfor helpful suggestions. G. Liu would like tothank the China Scholarship Council �grant2007U44003� for partial support of this work. X.Chen’s research was supported partially by theNational Basic Research Program of China �973program, grant 2007CB209606� and by the Na-tional High Technology Research and Develop-ment Program of China �863 program, grant2006AA09A102-09�. Publication was autho-rized by the Director, Bureau of Economic Geol-ogy, The University of Texas atAustin.

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(km)

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� 27.4weigh

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V48 Liu et al.

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