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Signature after predictive deconvolutionJohn F. Parrish*, Periseis Company

Summary

Conventional predictive deconvolution is very good forsuppressing normal incidence water bottom reverberations.

However, the output signature can vary significantly withthe value selected for the prediction distance (lag).Generalizing predictive deconvolution with relative entropydeconvolution concepts can provide consistent

dereverberation filters for lags shorter than the length of thewavelet kernel.

Introduction

Classic papers describing the behavior of waterreverberations (Backus, 1959) and the calculation of

predictive deconvolution filters (Peacock and Treitel, 1969)have provided rules of thumb for conventional seismicdeconvolution processing. These rules were invaluable in

shortening field wavelets enough to allow structuralinterpretation of the subsurface.

However, concepts of seismic deconvolution processing

have evolved. Merely shortening the interpretation waveletis no longer enough. In order to interpret rock properties, itis necessary to know the interpretation wavelet shape andto maintain its amplitude and phase spectrum throughout an

entire seismic volume.

Theory

Burg (1975 & 1967) clearly enunciated the maximumentropy formulation for seismic deconvolution. Thisspiking or whitening version of deconvolution is veryeffective but does not always provide a known outputsignature. In addition, the filter shape can be very sensitive

to the selected value of the noise level constraint. Burgsstudent, John E. Shore (1979, 1980, and 1981) generalized

maximum entropy into a consistent relative entropyformulation for spectral estimation and zero-lagdeconvolution (see Parrish, 1997 and 1999).

Predictive deconvolution can also be generalized byrelative entropy concepts. The result is a consistent outputsignature for interpretation that is independent of the valueselected for the prediction distance or lag.

Example

In order to illustrate these concepts, a synthetic digital

seismogram has been constructed entirely from minimumphase components, sampled at 2 milliseconds. A finiteimpulse response (FIR) instrument is simulated by three

pairs of (magnitude = 0.99) z-domain zeros at 155, 185 and250 Hz. The dashed line in Figure 1 shows the resulting,minimum phase, time signature response of this syntheticFIR instrument.

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Figure 1: FIR instrument and kernel signatures.

NOTE: actual instrument and hydrophone responses werenot used in this example because they have infinite impulse

responses (IIR). However, with appropriate spectral energyconstraints and with a sufficiently long (designature)compensation filter, it would be possible to convert an

actual seismic instruments IIR into a close approximationof a specified FIR digital response like the one chosen forthis example.

Source and receiver ghosts are synthesized with sea surfacereflection coefficients of 0.99 and two-way travel times of

8 ms. The solid line in Figure 1 shows the combined finiteimpulse response of both ghosts convolved with theinstrument (as if recorded with an ideal SEG standard

polarity hydrophone). This FIR wavelet kernel for thesynthetic seismogram has exactly 15 non-zero samples

between 0 ms and 28 ms. With a 14 ms shift, it would besuitable as a nearly zero phase interpretation wavelet.

The earth model has a sea bottom with a two way traveltime of 80 ms and a reflection coefficient of 0.3. This will

SEG Int'l Exposition and 72nd Annual Meeting * Salt Lake City, Utah * October 6-11, 2002

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Signature after predictive deconvolution

generate a normal incidence water bottom multiplesequence with the time and amplitude series shown in the

first and second columns respectively of Table 1. At normalincidence, each subsurface reflection will be convolved

with the reverberation series shown in column 3 of Table 1.

In order to avoid any significant interference from the

water bottom multiple in this example, the syntheticseismogram contains only a single isolated reflection at atwo way travel time of 1 second with a positive reference

amplitude of 1.0. A 2 seconds long, synthetic seismogramrecord (compensated perfectly for spherical spreading) wasconstructed from these various components. The time

interval from 1000 ms to 1400 ms is displayed in Figure 2.

Such a long field signature could obscure even a structuralinterpretation of an earth seismogram. However, asdiscussed earlier, a prediction operator should be able to

predict the (IIR) reverberation sequence. Figure 3 showsthe 190 ms (160 ms plus a lag of 30 ms) prediction error

filter for a noise level constraint of 0.001 (0.1%). Thispredictive deconvolution filter is manifestly a bandpassed

version of Backuss (1959) dereverberation operator forthis synthetic seismogram.

Figure 4 shows the filtered output between 1000 ms and

1400 ms. Not only is the reverberation suppressed, but thereflections signature (the interpretation wavelet) is anundistorted copy of the models FIR wavelet kernel. The

prediction error filters in this example are nearly exact for

lags between 30 and 50 ms. In practice, the filtered output,including the reflections signature, is very good for lags upto and including 80 ms.

For long or unknown field wavelets, Peacock and Treitel

(1969) suggested choosing a lag that corresponds to the

second zero crossing of the autocorrelation function. Inthis example, the second zero crossing occurs between 14ms and 16 ms. Figure 5 shows the prediction error filter for

a lag of 14 ms. The filter is distorted significantly byincluding a portion of the autocorrelation of the FIRwavelet kernel. Backuss dereverberation operator is nolonger recognizable.

Time Multiple Reverberation

0 0.00000 1.0000080 0.30000 -0.60000

160 -0.09000 0.27000

240 0.02700 -0.10800320 -0.00810 0.04050400 0.00243 -0.01458480 -0.00073 0.00510

560 0.00022 -0.00175640 -0.00007 0.00059720 0.00002 -0.00020800 -0.00001 0.00006

880 0.00000 -0.00002960 -0.00000 0.00001

Table 1: Multiple and reverberation time and

amplitude series.

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Figure 2: Synthetic seismogram.

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Figure 3: Predictive deconvolution filter for lag = 30 ms.

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Figure 4: Dereverberated output for lag = 30 ms.


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Nevertheless, the filtered output in Figure 6 shows that thewater bottom reverberation has indeed been suppressed.Unfortunately, the reflections signature is distorted and no

longer nearly zero-phase. In addition, the signature ringsand extends beyond the lag time. This phase-rotated

interpretation wavelet could be adequate for most structuralinterpretations but it might distort the interpretation of rock

properties and well log synthetics.

The distortion of the dereverberation operator for short lags

can be avoided by reformulating predictive deconvolution.Figure 7 shows a relative entropy predictive deconvolution

filter for a lag of 14 ms. The resulting filter is very close tothe classic predictive deconvolution filter for a lag of 30 ms

shown in Figure 3. Backuss dereverberation operator isrecognizable. The filtered output (not displayed) isindistinguishable from that shown in Figure 4.

Conclusions

Classic papers describing the behavior of water

reverberations (Backus, 1959) and the calculation ofpredictive deconvolution filters (Peacock and Treitel, 1969)

can provide rules of thumb for conventional seismicdeconvolution processing. By utilizing an exampleseismogram synthesized with a finite impulse response,wavelet kernel, these rules can be refined:

1. Predictive deconvolution can suppress reverberationsas long as the lag is less than or equal to the minimumtime of the water bottom reverberation sequence.

2. An isolated reflections signature is not distorted by

predictive deconvolution, as long as the lag is largerthan the length of the wavelet kernel.

3. The dereverberation filter changes shape as soon asthe lagged interval includes a significant portion of the

wavelet kernels autocorrelation.4. Placing the lag at the second zero crossing is a

reasonable compromise but the reflections signaturewill be distorted and it will extend beyond the lag.

Generalizing predictive deconvolution with relative entropydeconvolution (Parrish, 1997 & 1999) concepts can provideconsistent dereverberation filters for lags shorter than the

length of the wavelet kernel. Actual field signatures can becompensated to any convenient wavelet shape, including

those with infinite impulse responses, before applying arelative entropy predictive deconvolution.

References

Backus, M. M., 1959, Water reverberationstheir natureand elimination: Geophysics, v. 24, p. 233-261.

Figure 5: Predictive deconvolution filter for lag = 14 ms.

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Figure 6: Dereverberated output for lag = 14 ms.

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Figure 7: Relative entropy predictive deconvolution filter for

lag = 14 ms.


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Burg, J. P., 1975, Maximum entropy spectral analysis,Ph.D. dissertation, Stanford University, Stanford, CA.

(University Microfilms No. 75-25, 499)

Burg, J. P., 1967, Maximum entropy spectral analysis,Society of Exploration Geophysicists InternationalExposition and 37thAnnual Meeting.

Parrish, John F., 1997, Relative entropy spectrumdeconvolution, Society of Exploration GeophysicistsInternational Exposition and 67thAnnual Meeting, Dallas.

Parrish, John F., 1999, Applying minimum relative entropyspectrum deconvolution, Society of ExplorationGeophysicists International Exposition and 69th AnnualMeeting, Houston.

Peacock, K. L., and Treitel, S., 1969, Predictivedeconvolution: theory and practice: Geophysics, v. 34, p.155-169.

Shore, John E., 1979, Minimum cross-entropy spectralanalysis, NRL-MR 3921, Naval Research Laboratory,Washington, D. C., Jan.

Shore, J. E. and Johnson, R. W., 1980, Axiomaticderivation of the principle of maximum entropy and the

principle of minimum cross-entropy, IEEE Trans. Inform.Theory, IT-26, 26-37, Jan.

Shore, J. E., 1981, Minimum cross-entropy spectralanalysis, IEEE Trans. Acous. Speech Signal Processing,ASSP-29, 230-237, Apr.

Shore, John E. and Johnson, Rodney W., 1983, Propertiesof cross-entropy minimization, IEEE Trans. Inform.Theory, IT-27, 472-482, July 1981. See also: commentsand corrections, IEEE Trans. Inform. Theory, IT-29, Nov.

Acknowledgments

Thank you Wulf Massell and EPIC Geophysical forallowing me to investigate and test some of these concepts

on actual seismic records.


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SEG-2002-2194