William S. Harlan

(Presented at a 1994 SEG Annual Meeting workshop on seismic amplitude preservation.)

We cannot process seismic data without changing seismic amplitudes, but we can determine whether processing affects an interpretation of amplitudes. If we adjust amplitudes in an invertible way (e.g., as a simple function of time, position, frequency, wavenumber, or dip), then we have lost no information that cannot be reconstructed. Moreover, if we understand how a particular wave phenomenon affects amplitudes, then we can remove these effects from the data to simplify the expression of more useful information. Variations in reflectivity versus angles of incidence (RVA) are very informative, while transmission and propagation effects are usually distracting.

To examine seismic amplitudes visually, we must reduce the dynamic range. Dynamic gain ideally removes only amplitude information that an interpreter can take for granted, without seeing it visually displayed. Gain over time corrects only partly for the geometric divergence of wavefronts. Ray modeling of this divergence should improve upon a constant-velocity correction; however, absorption and incoherent scattering weaken amplitudes over time even more, and these effects are less easily modeled.

A constant-Q model of absorption (Kjartansson, 1981) assumes individual frequencies decay exponentially in time. Integration over a finite bandwidth produces a decay dominated by a reciprocal power of time, with lesser exponential terms. J. Claerbout (1985) has noted that this effect, plus geometric divergence, explains why a second power of time balances very well the forty seismic profiles published by O. Yilmaz (1988). (Multiples, surface-noise, and the near field appear poorly balanced.) Dynamic gain can use a single analytic function to rescale all traces, with one or two parameters dependent upon the data.

Trace balancing should be handled separately from dynamic gain. We observe variations in source strengths, receiver receptivities, and near-surface impedances by their distortion of reflections. Surface-consistent trace balancing uses few degrees of freedom and yet removes most source and near-surface effects (ignoring radiation patterns). The constraint is so rigid that a change in the statistic of trace strength, such as norms or percentiles, appears not to change results.

Marine cables also require offset-dependent balancing, which more easily distort RVA information. Since the correction is stationary over time, and should be the same for all sources, the adjustment is easily inverted if necessary. Rapid changes in cable receptivity over offset can be decoupled and corrected independently of smoother changes, just as we decouple short and long-period static corrections.

Multi-trace processing of midpoint gathers requires increasing attention. For example, multiple suppression by generalized Radon transform weakens incoherent energy at far offsets. These linear filters suppress a larger range of dips at the far offset than at the near offset. Later statistical interpretations must acknowledge that the signal-tonoise ratio changes with offset. Dip filters affect incoherent noise consis tently over offset, but leave multiple energy at the near offsets. Multidimensional filters should use least-squares implementations (such as f -x) that invert only recorded, unmuted offsets. Discrete approximations of continuous analytic transforms (such as f -k filters) assume unrecorded offsets to be zero or to wrap around, and produce fatal edge effects.

As preprocessing mixes information from more traces, we must anticipate methods of amplitude interpretation. Many displays of RVA at tributes accentuate anomalous gradients with angle and de-emphasize background trends, such as a consistent weakening with angle. For example, fluid factor analysis and a geogain function (Gidlow et al., 1992) can remove background trends in the RVA gradients of waterfilled sands to emphasize the RVA of gassy reflections with anomalous Vs/Vp contrasts. Such an adjustment can depend on seismic rather than assumed rock properties. Preceding processes might be able to adjust amplitudes with offset and not affect the final displays.

Imaging inverts and removes only the effects of wave propagation that carry no useful interpretive information. For example, we do not want DMO (dip moveout or prestack partial migration) actually to convert constant-offset sections into zero-offset sections because RVA information must remain. DMO is often derived as prestack acoustic imaging in time, minus the effects of NMO and poststack migration. Unfortunately, we frequently refine and change our definition of full imaging.

Within Conoco, "full imaging" often avoids any separation of DMO from other imaging steps. A prestack cascaded migration by R.H. Stolt estimates reflectivities as a function of depth and reflection angle. A prestack depth migration by D.P. Wang includes amplitude adjustments for variations in aperture and velocity perturbations.

DMO still retains a useful role as preprocessing rather than as partial imaging. If we are working toward time displays of RVA, we apply DMO with linear space-time operators on constant-offset sections (mis named "Kirchhoff"). The kinematics of DMO remain the same. Rather than derive amplitude obliquity factors according to Born imaging, we choose amplitude scaling that can easily be reversed. Very simply, this constant-offset DMO should not affect the amplitude of an a unaliased monochromatic plane wave. The dip of the monochromatic plane wave changes, but not the amplitude. (Constant-velocity DMO is easy to adjust, but not DMO with depth-variable velocities or anisotropy.) Later programs can readjust amplitudes as a function of dip and frequency if necessary, or perform truer imaging. An implementation with antialiasing (Hale, 1991) may weaken high frequencies at high dips but should preserve unaliased frequencies at consistent strength.

Transmission effects that are not surface-consistent, such as defocusing by velocity anomalies, can greatly obscure RVA displays. A separate expanded abstract (Harlan, 1994) explains how to model and remove transmission perturbations of amplitudes.

Some RVA displays require partial stacking with variable folds of stack. If later processes are unaware of the fold of stack, then renormalization is crucial. The power ratio of coherent signal to incoherent noise ideally increases as the square root of stack fold. If we normalize for ideal signal, then less coherent energy will be weakened as the fold increases. Such statistical assumptions must be consciously made, and changed when appropriate.

The customers of a seismic processing shop naturally prefer reproducible processing sequences that allow consistent comparisons between many seismic sections. Such sequences often rely upon statistical am plitude balancing programs that highlight anomalous changes in amplitude with offset and de-emphasize consistent background changes in amplitude with angle. For example, amplitudes may be scaled locally ac cording to some statistic of neighboring samples. The original changes in amplitudes over offset are not recoverable. Statistical balancing often provides the best default correction of unusual amplitude distortions but also hides problems which reduce confidence in the results.

We learn more by adjusting amplitudes with one restrictive physical model at a time. Model parameters derived from the data should be diagnostic of the seriousness of each physical effect. Restrictive models minimize the accidental misinterpretation and removal of useful amplitude information.

Just as important, but ignored by this discussion, is phase preservation. Amplitude cannot be considered independently of phase and other spectral properties. Similar principles should apply: use restrictive models, allow invertible adjustments, and document the behavior.

Claerbout, J. F., 1985, Imaging the Earth's interior: Blackwell Scien- tific Publications.

Gidlow, P., Smith, G., and Vail, P., 1992, Hydrocarbon detection using fluid factor traces: a case history: Joint SEG/EAEG summer re- search workshop, How Useful is Amplitude-Versus-Offset Analysis?, Technical Program and Abstracts, 78-89.

Hale, D., 1991, A nonaliased integral method for dip moveout: Geo- physics, 56, no. 6, 795-805.

Harlan, W. S., 1994, Tomographic correction of transmission distortions in reflected seismic amplitudes: 64th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, SI2.2.

Kjartansson, E., 1981, Constant Q - wave propagation and attenuation in Toksoz, M. N., and Johnston, D. H., Eds., Seismic wave attenua- tion:: Soc. Expl. Geophys., 448-459.

Yilmaz, O., 1988, Seismic data processing: Soc. Expl. Geophys., Tulsa.

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