2011 02 RECORDER Wavelet Estimation

Mixed-phase wavelet estimation A case studySomanath Misra and Satinder ChopraArcis Corporation, Calgary, Alberta, Canada

Introduction

An accurate estimation of wavelet is crucial in the deconvo-lution of seismic data. As per the convolution model, therecorded seismic trace is the result of convolution of theearths unknown reflectivity series with the propagatingseismic source wavelet along with the additive noise. Thedeconvolution of the source wavelet from the recordedseismic traces provides useful estimates of the earthsunknown reflectivity and comes in handy as an aid to geolog-ical interpretation. This deconvolution process usuallyinvolves estimation of a wavelet, before it is removed bydigital filtering. Since the earths reflectivity and seismicnoise are both unknown, the wavelet estimation process isnot easy. The statistical methods estimate the wavelet usingthe statistical properties of the seismic data and are based oncertain mathematical assumptions. The most commonly usedmethod assumes that the wavelet is minimum phase and thatthe amplitude spectrum and the autocorrelation of thewavelet is the same as the amplitude spectrum and the auto-correlation of the seismic traces, within a scale factor, in thetime zone from where the wavelet is extracted. With theassumption that the wavelet is minimum phase, an estima-tion of the wavelet is done from the trace autocorrelation.This method always estimates a minimum phase wavelet andso is suitable for wavelet estimates from seismic dataacquired using explosive sources, only if their source signa-ture is purely minimum phase and that is retained throughduring processing. However, it is not applicable for esti-mating wavelets from sources giving mixed phase signature.For example, deconvolution of non-minimum phase seismicdata with a minimum phase wavelet will leave behind aspurious phase in the data. The reason for this is that theautocorrelation function and the power spectrum mentionedabove are second-order statistical measures. They contain nophase information and so cannot identify non-minimumphase signals. Also, these measures work well for Gaussianprobability distribution of amplitudes, and so will not yieldaccurate results for non-Gaussian or non-linear distributions.The non-Gaussianity in the seismic data could arise from thenon-minimum phase source signature, noise in the data likeswell noise, and a non-linear earth response. Consequently,higher-order statistics have been used for dealing with non-Gaussian distributions. These statistics, known as cumulants,and their associated Fourier transforms known as poly-spectra, not only reveal amplitude information, but also thephase information (Mendel, 1991). In this paper, we addressthis issue through the use of higher-order statistics such thatthe phase component in the data are more accurately esti-mated and removed.

Higher-order statistics for wavelet estimation

Cumulant, a higher-order statistical property, preserves thephase information of the wavelet under the assumption that

the reflectivity series is a non-Gaussian, stationary and statis-tically independent random process. The second-ordercumulant of a zero-mean process is just the autocorrelation,which as stated above, has no phase information. The third-order cumulant is a two dimensional correlation function. Fora Gaussian process, all cumulants above the second-order arezero, but are non-zero for non-Gaussian processes. Thus thesetwo statistics are not suitable for recovering a non-minimumphase from a convolution process such as the seismic trace.The fourth-order cumulant is a three-dimensional correlationfunction which contains information about phase. Just as theFourier transform of the autocorrelation function yields thepower spectrum, similarly, the trispectrum relates to thefourth-order cumulant via the 3D Fourier transformation.Lazear (1993) and Velis and Ulrych (1995) estimate the phaseof a wavelet by fourth-order cumulant matching wherein aninitial guess for the wavelet is iteratively updated until itsfourth order statistics match those of the seismic data.

Misra and Sacchi (2006) suggest the parameterization of theembedded mixed phase wavelet as a convolution of theminimum phase wavelet with an all-pass operator. The all-pass operator can further be parameterized as the ratio of amaximum phase time sequence and corresponding minimumphase time sequence with the necessary time delay required toenforce causality in the all-pass operator (Porsani and Ursin,1998). The denominator term in the paramerization of the all-pass operator is a minimum phase sequence whose length andcoefficients are unknown. As discussed in a later section, weoptimize for the unknown coefficients of the minimum phasesequence keeping the length as a constant parameter.

Seismic data is represented as the convolution of the reflec-tivity sequence with the unknown wavelet. The unknownwavelet showing mixed phase characteristics is further repre-sented as the convolution of minimum phase wavelet and theall-pass operator. Thus the seismic data is further representedin terms of two convolutions, namely convolution of reflec-tivity with the minimum phase wavelet which in turn isconvolved with the all-pass operator. Deconvolving the datawith the minimum phase wavelet increases the bandwidth ofthe output data and we subsequently refer to it as thewhitened data. The whitened data is thus represented as theconvolution of the underlying reflectivity series and the all-pass operator. Hence it is possible to make an estimation ofthe underlying reflectivity series by estimating the all-passoperator from the whitened data.

Development of the algorithm

The well known Barlett-Brillinger-Rosenblatt formula(Lazear 1993; Mendel 1991) links the fourth-order cumulantof the seismic trace with the fourth-order moment of theembedded wavelet. For non-Gaussian, statistically inde-pendent and identically distributed reflectivity series, the

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fourth-order cumulant of the seismic trace is equal to, within ascale factor, the fourth-order moment of the wavelet providedthat the noise distribution is Gaussian. The optimization proce-dure described in the following paragraph minimizes the costfunction given by the L2-norm between the fourth-order normal-ized trace cumulant and the fourth-order wavelet moment.

The optimization of the cost function thus involves computation ofthe normalized 4th order trace cumulant of the whitened data (dataobtained after deconvolution with a minimum phase wavelet) andthe normalized 4th order moment of the all-pass operator.

The shape of the cost function is unknown and may containseveral local minima. Local optimization methods based ongradient computation always proceed to the minimum nearest tothe chosen initial model. In the present optimization problemwhere the shape of the cost function is not known, a global opti-mization algorithm is a preferred choice. Simulated annealingalgorithm with a Metropolis acceptance/ rejection criterion(Misra, 2008) is adopted for the optimization of the cost function.The model parameters for the simulated annealing optimizationare the coefficients of the minimum phase sequence in the para-meterization of the all-pass operator. The all-pass operator foreach of the generated model is computed from equation bytaking the ratio of the corresponding maximum phase sequenceand the minimum phase sequence.

Application to post stack seismic data

In order to test the stability and reliability of the algorithm, themethod outlined above was applied to a seismic data volumefrom North America. For this volume, the data are further subdi-vided into smaller zones and for each of these zones a mixed-phase wavelet is estimated. Thedata in each of these subdivisionsare deconvolved using the esti-mated mixed-phase wavelet.Further, an average estimatedmixed-phase wavelet is obtainedby taking the mean of the indi-vidual mixed-phase wavelets. Themean mixed-phase wavelet is thenused to deconvolve the data. Theresults are correlated with the P-wave log curve.

Figure 1 shows the workflow forthe method outlined above.

Case study

A seismic data volume from aprovince in North America waspicked up for testing the algo-rithm outlined above. The datavolume had been processed usinga conventional processingsequence. Four different locationson the post-stack volume wereselected for the estimation of themixed-phase wavelets in the sametime interval (500ms) for each.

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Figure 2. (a) A segment of a seismic section around location 1. (b) The estimated minimum phase wavelet, (c) the estimatedall-pass operator and (d) the estimated mixed-phase wavelet for location 1, and (e) input seismic section in (a) after mixedphase wavelet deconvolution, Notice, the phase-corrected section exhibits higher frequency content than the input data asexpected and so exhibits much higher resolution.

Figure 1. Workflow for the methodology developed for deconvolution of mixed phasewavelets from seismic data.


34 CSEG RECORDER February 2011

Figure 2a shows a segment of aseismic section around location1. Figures 2b, c and d show theestimated minimum-phasewavelet, the estimated all-passoperator and the estimatedmixed-phase wavelet at the loca-tion 1. The minimum-phasewavelets are estimated from theaverage autocorrelation of theseismic traces by the Wiener-Levinson algorithm. The esti-mated minimum-phase waveletis deconvolved from the datawhich resulted in broadening ofthe bandwidth. Figures 2e showsthe phase-corrected data for thelocation 1. Figures 3, 4 and 5show a similar set of images forlocations 2, 3 and 4. Notice thatfor each set of images, the mixed-phase wavelet deconvolvedsections exhibit the highest levelof detail. Again, it would beadvisable to correlate the decon-volved sections with the P-wavelog curves to gain some confi-dence in ascertaining if theresolved reflections correlatewell and if they are authentic.Notice in Figure 6, the correlationof the section in Figure 6b isbetter than the one shown inFigure 6c.

Conclusions

Deconvolution of seismic datawith a minimum phase waveleteffectively removes the ampli-tude spectrum of the waveletfrom the data. However, in situa-tions where the minimum-phaseassumption about the wavelet isnot valid, the deconvolutionleaves behind a spurious phasecomponent in the data. Themethod adopted in estimatingand hence removing the spuriousphase involves estimation of thecoefficients of an all-pass oper-ator from the data that have beenwhitened by the deconvolution ofthe minimum-phase wavelet. Thewhitened data are used to opti-mize the cost function involvingthe 4th order normalized trace cumulant and the 4th ordermoment of the all-pass operators. The optimization procedureuses simulated annealing with the Metropolis acceptance/rejec-tion criterion. The estimated all-pass operator is subsequentlyconvolved with the earlier estimated minimum-phase wavelet to

estimate the mixed-phase wavelet in the data. The suggestedmethod is tested on a seismic data set belonging to a province inNorth America. The data set is subsequently deconvolved withthe estimated mixed-phase wavelets. Further, an average mixed-phase wavelet is computed from the individual mixed-phase wavelets. The data are then deconvolved with the average

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mixed-phase wavelet are corre-lated with the P-wave log dataand shows better correlation. R

Acknowledgements

We acknowledge Prof. MauricioSacchi for his help and sugges-tions during the development ofthe algorithm. We also acknowl-edge an anonymous provider ofthe data shown in and ArcisCorporation, Calgary, Canada forsupport of this work.

ReferencesLazear, G. D., 1993, Mixed-phase waveletestimation using fourth-order cumulants,Geophysics, 58, 1042-1051.

Mendel, J. M., 1991, Tutorial on higher-orderstatistics (spectra) in signal processing andsystem theory: theoretical results and someapplications, Proceedings of the IEEE, 79,278-305.

Misra, S. and Sacchi, M. D., 2006, Non-minimum phase wavelet estimation by non-linear optimization of all-pass operators,Geophysical Prospecting, 55, 223-234.

Misra, S., 2008, Global optimization withapplication to Geophysics, PhD thesis,University of Alberta, Canada.

Porsani, M. J., and Ursin, B., 1998,Mixed phase deconvolution, Geophysics, 63, 637-647.

Velis, D. R., and Ulrych, T. J., 1996, Simulated annealing wavelet estimation via fourth-order cumulant matching, Geophysics, 61, 1939-1948.

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Figure 6. Segment of a seismic section from (a) input data, and (b) data after phase correction with theaverage mixed phase wavelet. The inserted red curve is the P-wave log. Notice the higher level of correla-tion of the log curve with the section in (b).

Somanath Misra wasborn and raised in thecoastal province ofOrissa, India. Aftergraduating in Physicsmajor, Somanathjoined the IndianSchool of Mines for his

Masters degree in Applied Geophysics,which he successfully completed in 1991.Thereafter, Somanath preferred to jointhe exploration department in Orissa,India and worked there for about 10years. In 2003, he joined the SignalAnalysis and Imaging Group at theUniversity of Alberta and completed hisPh.D. in 2008. Since then he is working asa Reservoir Geophysicist in the reservoirservices group at Arcis Corporation,Calgary, Canada. He is a member ofCSEG and SEG.

Satinder Chopra:See March 2011RECORDER issuepage 5.

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2011 02 RECORDER Wavelet Estimation