Wavelet Estimation and Blind Deconvolution of Realistic Synthetic Seismic Data by Log SpectralAveragingM. Tria1, M. van der Baan2, A. Larue1 and J.I. Mars1

1 LIS-ENSIEG, BP 46, 38402 Saint Martin d’Heres, France2 Dept. of Earth Sciences, University of Leeds

SUMMARY

Homomorphic wavelet estimation was a popular tool in the late 70s. Itconsists of taking the logarithm of traces transformed to the frequencydomain. This maps a convolution in the time domain to an addition inthe log(frequency domain). Wavelet estimation was however done onsingle traces. We improve on this concept by introducing a multi-traceapproach. In current marine acquisition the source wavelet is very uni-form across an entire survey. We assume that we have selected manytraces, randomly sampled from a large area, that are all characterizedby different reflectivity series. Averaging the log(spectra) then recov-ers the original source wavelet. The advantage of this approach is thatit does not make any assumptions on the phase of the wavelet. Afterwavelet estimation, a standard deconvolution algorithm can be used toimprove the resolution of the seismic data.

INTRODUCTION

The seismic trace d(t) is usually assumed to be the convolution of anemitted wavelet w(t) with the reflectivity r(t). Homomorphic decon-volution was introduced by Oppenheim and Schafer (Oppenheim etal., 1968) and has been used to deconvolve seismic records (Ulrych,1971). Otis & Smith (Otis et al., 1977) proposed a wavelet estima-tion method based on homomorphic transformation, that consists inaveraging the log spectra of several reflection records. This will av-erage out the spatially non-stationary Earth response while emphasingthe log spectrum of the emitted wavelet. We improve on their idea byintroducing a more robust phase reconstruction technique, while eval-uating the outcome on realistic synthetics. Log spectral averaging hastwo important advantages: (1) non-minimum phase assumptions aremade; (2) the standard white reflectivity assumption is eliminated.

HOMOMORPHIC SYSTEM AND WAVELET ESTIMATION

A commonly used homomorphic system T is the cascade of (a) theFourier transform of a time signal x(t), and (b) the complex logarithmof the spectrum X( f ). The previous steps applied to a seismic tracegive:

log[D( f )] = log[W ( f )]+ log[R( f )] (1)

The real (Re) and imaginary (Im) parts of (1) are expressed as:

Re : log |D( f )|= log |W ( f )|+ log |R( f )| (2)

Im : arg[D( f )] = arg[W ( f )]+ arg[R( f )] (3)

Now, let us consider sensors recording different reflections to providethe seismic profile. Given the stationarity assumption of the sourcewavelet, averaging the real and imaginary parts of log spectra givesthe two following relations:

1Ns

Ns

∑s=1

log(|Ds( f )|) = log(|W ( f )|)+1Ns

Ns

∑s=1

log(|Rs( f )|) (4)

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Ns

∑s=1

arg[Ds( f )] = arg[W ( f )]+1Ns

Ns

∑s=1

arg[Rs( f )] (5)

If for each frequency f the values are random variables that have identi-cal probability distributions with the same mean M and the same vari-ance , and are random variables distributed uniformly within a certaininterval , then due to the central limit theorem, we have:

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Ns

∑s=1

log |Rs( f )| Ns→+∞−→ K

1Ns

Ns

∑s=1

arg[Rs( f )] Ns→+∞−→ 0

In other words, log spectral averaging (LSA) recovers both the ampli-tude and phase spectrum of the unknown wavelet under these condi-tions. Recovery of the amplitude spectrum |W ( f ) is straightforwardbut accurately reconstructing the phase is more complex since thephase arg[D( f )] is always given as ”modulus (i.e., wrapped). We needto unwrap the phase before computing its average. Phase unwrappingis done refering to (Oppenheim et al., 1975; Tribolet, 1979). Errors inthe unwrapped phase are cumulative with frequency. We therefore ap-ply phase deramping before computing the average. Phase derampingconsists of ensuring that the phase derivative is a zero-mean function(Tribolet, 1979; Tria et al., 2007). This normalization is equivalent tosuppressing the linear component from the unwrapped phase. Figures1(a) and 1(b) illustrate the concept.

WAVELET ESTIMATION FROM REALISTIC SYNTHETIC DATA

In this part, we estimate the emitted wavelet from realistic syntheticnoiseless data, coming from Chevron Corporation. The dataset is aconvolutionary model assuming a stationary wavelet. The number ofsensors is Ns = 400 and the number of time samples is Nt = 560. A firstexample of dataset and their spectrum are shown in figures 2(a) and2(b): These data have been produced with a mixed-phase and narrow-bandwidth wavelet.

The LSA method assumes that the reflectivity is spatially nonstation-ary. To increase the reflectivity nonstationarity degree, we have builthighly uncorrelated traces from the data. The procedure consists inpicking randomly from the data, N′

s new traces of N′t time samples

(N′t < Nt ). The number of new traces we have chosen, is N′

s = 750 andthe number of time samples for each new trace is N′

t = 200. The abovetraces are rearranged in a new dataset that is used for estimating thewavelet by the LSA method. The new dataset is shown in figure 3.

Figure 4(a) illustrates the wavelet modulus estimate obtained by LSAmethod. From the previous estimate, we have calculated the DSP thatallows to incorporate a lowpass filtering on the data before phase un-wrapping is done. The cut-off frequency of the lowband filter is fixedat ∆dB =−30dB from the maximum of the DSP. Figure 4(b) illustratesthe wavelet phase estimate obtained by the above averaging method.Figure 4(c) shows the temporal wavelet estimate (solid line) resultingfrom the frequency wavelet estimation; the first arrival of one trace (indash line) is superimposed to the wavelet estimate, as a reference sig-nal. Their close similarity indicates robust wavelet phase estimation.

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Figure 5(a) shows the deconvolved data obtained after Wiener filteringusing the wavelet W ( f ) obtained by LSA. The spectrums of the de-convolved data are shown in figure 5(b) and look wider than those ofthe data in figure 2(b). Figures 6(a) to 6(j) compare the resolution ofthe data with the resolution obtained after data deconvolution, throughdifferent examples of image sections arbitrarly selected. The abovefigures underlines a resolution improvement in the deconvolved dataimage.

Figures 7(a) to 7(c) illustrate the wavelet estimation from a new datasetproduced from a zero-phase and very narrow bandwidth wavelet. Fig-ures 8(a) to 8(c) illustrate the wavelet estimation from a third datasetproduced from a mixed-phase and very narrow bandwidth wavelet.

CONCLUSION

Log Spectral Averaging is a useful tool for estimating the source wavelet.The method assumes that the wavelet is consistent throughout the en-tire dataset while only the local geology varies. An important advan-tage of Log Spectral Averaging is that it is not restricted to minimum-phase wavelets rendering blind deconvolution possible. Robust phaseestimation is a principal item in this technique. After wavelet esti-mation standard deconvolution methods can be used to enhance theresolution of the data. The method has the additional advantage thatthe wavelet estimate can be used as a quality control in the processingsequence.

ACKNOWLEDGMENTS

The authors thank BG, BP, Chevron, the DTI and Shell for financialsupport of the project Blind Identification of Seismic Signals (BLISS),and Chevron for creating and permission to use the synthetic dataset.

(a)

(b)

Figure 1: Unwrapping then deramping the data phase: (a) Phase un-wrapped by Oppenheim’s method and corresponding linear compo-nent, (b) Data phase obtained by suppressing the linear component

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(b)

Figure 2: Realistic synthetic data: (a) data, (b) spectrum.

Figure 3: Highly uncorrelated traces used for wavelet estimation byLSA.

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Figure 4: Wavelet estimation by LSA: (a) modulus, (b) phase, (c) tem-poral wavelet (mixed-phase and narrow-bandwith wavelet).

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Figure 5: Deconvolved data: (a) deconvolved data, (b) spectrum.

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Figure 7: Realistic synthetic data: (a) data, (b) wavelet modulus, (c)temporal wavelet (zero-phase and very narrow-bandwith wavelet).

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Figure 8: Realistic synthetic data: (a) data, (b) wavelet modulus, (c)temporal wavelet (mixed-phase and very narrow-bandwith wavelet).

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2007 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Oppenheim, A. V., and R. W. Schafer, 1975, Digital signal processing: Prentice-Hall. Oppenheim, A. V., R. W. Schafer, and T. G. Stockham, 1968, Nonlinear filtering of multiplied and convolved signals:

Proceedings of IEEE, 65, 1264–1291. Otis, R. M., and R. B. Smith, 1977, Homomorphic deconvolution by log spectral averaging: Geophysics, 42, 1146–1157. Tria, M., M. van der Baan, A. Larue, and J. I. Mars, 2007, Wavelet estimation in homomorphic domain by spectral averaging for

deconvolution of seismic data: Proceedings of Physics in Signal and Image Processing. Tribolet, J. M., 1979, Seismic applications of homorphic signal processing: Prentice-Hall. Ulrych, T. J., 1971, Application of homomorphic deconvolution to seismology: Geophysics, 36, 650–660.

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