Wavelet estimation for surface multiple attenuation using a simulatedannealing algorithm

Paulo M. Carvalho, Petrobras, Brazil; and Arthur B. Weglein*, SchlumbergerCambridge Research, UK SP2.4

SUMMARY THE SURFACE MULTIPLE ATTENUATION METHOD

examples. Encouraging

The inverse scattering surface multiple attenuation

resu Its were

method (Carvalho et. al.,

obtained

1991 and Carvalho, 1992) requires

thatdemonstrate the robustness of the process under real data

knowledge of the source wavelet. The method itself providesa way to estimate the wavelet. The estimation process usesan optimization algorithm. A global search procedure, basedon simulated annealing, is used to perform the optimization.Simulated annealing has a random component that allows itto search for the global minimun of a function containingmany local minima. In this work, the surface multipleattenuation method and the wavelet estimation procedure aredescribed and illustrated with synthetic and real data

The surface multiple attenuation method may be writtenin the following form (see, for example, Carvalho, 1992):

order. is just the pre-stack seismic data with source andreceiver array and ghost effects removed. The higher order

D = D, + + . . . + with

D,(m) = O(w) , = 2 , (2)where the data without surface multiples, is obtained bythe sum of a series of terms, 1 being the highest ordermultiple existing in the input data. For example, the serieswith five terms is free of all multiples from first to fourth

conditions.

INTRODUCTION

The inverse scattering series aproach to surfacemultiple attenuation does not rely on periodicity or differentialmoveout, nor does it require a model of subsurface structureor velocities. However, the effectiveness of this approachdepends on the satisfaction of certain prerequisites, forexample, an estimated source wavelet, compensation formissing near traces, and source and receiver deghosting.Experience indicates that estimating the source wavelet is aparticularly important precondition. This experience has beenshared by others, e.g., Verschuur et al. (1988), Fokkema andVan den Berg (1990) and Dragoset (1992), using similar orclosely related surface multiple attenuation methods.

The idea is to use the sensitivity of the multipleattenuation method to source signature to turn the situationaround, that is, when the multiples have been attenuated thewavelet has been well estimated. A criteria is needed tojudge when multiple suppression is working. The criteria isbased on the assumption that when multiples are removedfrom the data, there is less energy than when multiples arepresent. Verschuur et. al. (1989) first introduced this criteriain association with their surface multiple suppressionrnethod. In their work, a gradient aproach was used to searchfor the minimum of energy. In this work the same criteria,that is, object function, is used, but instead of the gradientapproach, a simulated annealing algorithm (Kirkpatricet. al., 1983) is used to search for the global minimum.

SA is an optimization method that has the ability tosearch for the global minimum of a function in the presenceof many local minima. It derives its name from an analogywith the thermodynamic process of liquid crystallization. Aparameter called temperature, together with a coolingschedule, guides the search in the model space. Inthermodynamics, slow cooling leads to a low energy finalstate and a well formed crystal which, in the analogy, meansthe global minimum; quick cooling leads to a high energyfinal state and a disordered polycrystalline solid whichcorresponds to a local minimum. There is no theoreticalguaranty that this process leads to the optimal solution but ithas experienced success in different applications. In seismicprocessing the first use of SA was made by Rothman (1986)for the problem of residual statics. Sen and Stoffa (1991)applied it for inverting elastic parameters from seismic data.

terms remove the multiples, order by order. They areobtained by the recursive equation (2). and aretwo-dimensional matrices corresponding to frequency slicesof three-dimensional data cubes where the other twodimensions correspond to the positions of the source and thereceiver of each particular trace. 0 is a matrix operatorobtained from D1 by applying a frequency dependent obliquityfactor, an extrapolation from the depths of sources andreceivers to the free surface datum and the waveletdeconvolution. In certain applications, when dealing withsmall to moderate angles of propagation, which roughlycorresponds to short offsets compared to depth of reflectors,angle dependent effects such as the removal of array andghost efects from and the obliquity factor applied to 0 maybe neglected. In this case the extrapolation should not beapplied to the operator 0. Since is the input data with nomodification, 0 is just the deconvolved version of Whenthe obliquity and other factors are neglected the wavelet tobe used in the deconvolution is an effective waveletcontaining the frequency dependence of these factors.

The problem of the wavelet arises in the series ofequation (1) when one does not matte the deconvolution ordoes it with a wrong wavelet. In this case Dn will have thewavelet, or a residual wavelet, to the power and themultiple attenuation that should result from the sum of theterms will fail.

WAVELET ESTIMATION AND SIMULATED ANNEALING

Here we will describe both the use of the SA method,and the complete strategy for wavelet estimation.

The process requires an initial estimate for the waveletwhich can be obtained by smoothing and averaging theamplitude spectra of selected traces. The initial phase can bemade equal to zero. Any other reasonable first guess maybe used, but it is important that the bandwidth matchesaproximately with the one of the actual wavelet. This firstguess is used in the deconvolution that generates theoperator 0, so that the estimation process accomodates aresidual wavelet.

Another crucial point is the problem of the scale factorthat is always present in real data. This factor may be verysignificant and it propagates in powers throughout the termsof the series. The estimation process is divided into twomajor steps, both using SA algorithm for optimization and

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2 Wavelet estimation for multiple attenuationperformed in the frequency domain. In the first step theoptimization is applied to three parameters: one is theamplitude scale factor and the other two are a time shift anda phase rotation. Remembering that an amplitude spectrumhas already been given to the operator 0, the first step is thencomplete and a rough version of the final wavelet is available.The second step refines this estimation by allowing theamplitude and the phase spectra to vary more freely, butconstrained by the first estimate. In this step a user suppliednumber of frequencies are used to parametrize both spectra.

The first example is from a synthetic data set. Figure 1has two primary reflections and its surface related multiples.Figure 2.a shows the result of the method after the first stepin which the scale factor was applied to the smoothed andaveraged amplitude spectrum of the input data and thecorresponding time shift and phase rotation applied to thephase spectrum. Figure 2.b shows the result after the secondstep in which the amplitude and the phase spectra estimationwere refined. All seismograms have the same variable gainapplied. Figure 3 shows the amplitude spectra after bothsteps and of the exact wavelet. Figure 4 is the same but forthe phase spectra and Figure 5 shows the wavelets in thetime domain.

The exact wavelet is a Gabor signal -- a cosine functionwith side lobes attenuated -- with a dominant frequency of 40Hz. Its phase spectrum has a linear componentcorresponding to a 8 ms time shift plus a constant factorcorresponding to 90 degrees of phase rotation. There is alsoa scale factor of the order of IO? The previous figures showthat the spectral average provides a reasonable first guess.After the first step of the estimation the wavelets roughlymatch and the multiple attenuation is reasonable. The scalefactor found was 972, the time shift 7.5 ms and the phaserotation 99.7 degrees. After the second step an improvementachieved in the wavelet estimation resulted in improvedmultiple attenuation. In this step the amplitude spectrum wasallowed to vary freely from the first step estimation to a shapecloser to the exact one. A parametrization of 20 frequenciesequally spaced was used for the amplitude. The phasespectrum was parametrized in the same way used in the firststep and a 7.6 ms time shift plus a 97.4 degrees of constantphase was found.

Figures 6.a and 6.b show the energy of the data at eachtemperature level for the first and second steps of theoptimization process, respectively. In the first steps 45 levelswere used with 45 moves per level. The best wavelet wasfound in the last level although a reasonably stable state wasreached after level 33. In the second step 140 temperaturelevels were used with 140 moves per level. The best estimatecorresponds to level 128 but a stable state was reachedaround level 65. The need for more levels with more movesin the second step is due to the fact that in this case there aremore parameters to be optmized so there are more points inthe parameter space to be visited. Also, since the secondstep is the final one, it is pushed toward lower levels oftemperature to assure that the estimate goes as close aspossible to the optimal point.

Figure 7 shows a shot from marine real data and Figure8 shows the same shot after the final step of the waveletestimation. Figure 9 shows the amplitude spectrum of thebest wavelet. The phase spectrum was again estimated as astraight line in both steps leading to a final time shift of 3.4ms and a phase rotation of 233.8 degrees. Figure 10 showsthe wavelet. It should be mentioned that it still has the efectsof source and receiver ghosts plus a 90 degrees phaserotation that should be corrected together with the applicationof the obliquity factor. In this real data example no angledependent correction was applied. All of the variables thatcontrol the optimization process such as initial temperatures,decay rates and number of temperature levels, were thesame in both examples.

DISCUSSION AND CONCLUSSIONSThe synthetic data example shows that although

providing a successful multiple attenuation the estimatedwavelet does not exactly match the true one. This fact maybe explained by one or a combination of the following causes.The first one is that the process has ended at a localminimum, close to the global minimum. Another explanationis that it has stopped at the same region of the globalminimum but slightly displaced from it; in this case a gradientmethod could bring the estimation to the minimum. A thirdpossibility is that the optimization had no understanding tosearch for lower energy states either due to numericalproblems when computing the summation or because ofnoise that was introduced in the process.

The real data example shows a relativelly good result,the multiple energy being strongly reduced. The applicationof the multiple attenuation method to real data raises someproblems that can be only approximately solved and that canexplain, at least in part, why the multiples were notcompletely attenuated. First there is the need to fill withtraces the gap from the nearest trace in the data to zerooffset. This was done by applying NMOcorrection to neartraces, then repeating them and applying inverse NMO withtheir new offsets. Another problem is that the methodrequires 2D structures and the line to be shot along the dipdirection. Deviations from this geometry causes incorrectprediction of the multiples. The amplitude of the data shouldalso correspond to 2D geometrical s reading. This wasaproximately corrected by applying a

factor to the inputdata where t is the time of the sample. It should also bementioned that none of the angle dependent correctionsdescribed before were applied to the data.

As an overall conclusion it can be said that the use of aSA algorithm in the wavelet estimation process for surfacemultiple attenuation is feasible and provides good results.The method is particulary effective for finding the scale factorin the data even when it is very large. It should be mentionedhere that earlier attempts with gradient optimization methodsfailed in this case. The use of a gradient method associatedwith SA is a point to be studied and may improve the finalestimation.

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Wavelet estimation for multiple attenuation 3

ACKNOWlEDGEMENTS

The authors would like to acknowledge Petrobras andSchlumberger for support and permission to publish thiswork.

REFERENCES

Carvalho, P.M, Weglein, A.B., and Stolt, R.H, 1991, Examplesof a nonlinear inversion method based on the T matrix ofscattering theory: Application to multiple suppression: 61stAnn. Internat. Mtg., Soc. Expl. Geophys., Expanded abstracts.

Carvalho, P.M., 1992, Metodo de de relacionadas corn a superficie livre em linear de dados sismicos: Tese de doutorado,

Universidade Federal da Bahia.Dragoset, W.H., 1992, Surface multiple attenuation: Thecnicalreport, Western Geophysical.

Fokkema, J.T., and Van den Berg, P.M., 1990, Removal ofsurface-related wave phenomena: the marine case: 60th Ann.Internat. Mtg., Soc. Expl. Geophys., Expanded abstracts.

Kirkpatric, S., Gelatt, C.D., Jr., and Vecchi, M.P., 1983,Optimization by simullated annealing: Science, 220, 671-680.

Rothman, D.H., 1986, Automatic estimation of large residualstatics correction: Geophysics, 51, 332-346.

Sen, M.K., Stoffa, P.L., 1991, Nonlinear one-dimensionalseismic waveform inversion using simulated annealing:Geophysics, 56, 1624-I 638.Verschuur, D.J., Herrmann, P., Kinneging, N.A., Wapenaar,C.P.A., and Berkhout, A.J., 1988, Elimination of surface relatedmultiple reflected and converted waves: 58th Ann. Internat.Mtg., Soc. Expl. Geophys., Expanded abstracts.Verschuur, D.J., Berkhout, A.J., and Wapenaar, C.P.A., 1989,Wavelet estimation by prestack multiple elimination: 59thAnn. Internat. Mtg., Soc. Expl. Geophys., Expanded abstracts.

Figure 1: Seismogram of synthetic data containing twoprimaries reflections and its surface related multiples.

Figure 3: Amp0 Frequency (Hz)litude spectra of the wave

step (dashed line) and the second step ((estimation process was applied to data from Figure 1, showntogheter with the exact spectrum (solid line).

125ets after the firstotted line) of the

Figure 4:Frequency (Hz) 125

Phase spectra corresponding to the amplitudespectra shown in Figure 3.

Figure 5: The wavelets corresponding to the spectra shownon Figures 3 and 4.

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Wavelet estimation for multiple attenuation

Figure 6: Energy of the data computed during the waveletestimation process at each temperature level: (a) for the firststep and (5) for the second step.

Figure 7: Seismogram of a shot from real marine data.

Figure 8: Seismogram showing the result of the multipleattenuation method applied to the data of Figure 7 with thewavelet estimated after the two steps of the process.

Figure 9: Amplitude spectrum of the wavelet used to obtainthe result shown in Figure 8.

Figure 1O: Wavelet corresponding to the amplitude spectrumshown in Figure 9 and linear phase components described inthe text.

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