Next: Differential Geometrical Spreading for Up: Brown: Pegleg amplitudes Previous: Introduction

# Snell Resampling Removes AVO/Attenuation Differences

If we are modeling seabed pegleg multiples, Figure illustrates the fact that (ignoring the seabed reflection) in a v(z) medium, there exists a single offset xp such that a pegleg with offset x and primary with offset xp are physically invariant with respect to AVO behavior and anelastic attenuation (water is assumed perfectly elastic). Ottolini (1982) introduced the concept of Snell Traces'' - a resampling of multi-offset reflection data along curves of constant time dip, or stepout''. I adopt a similar line of reasoning to infer xp as a function of x.

 schem-snell Figure 1 A primary and pegleg multiple with the same emergence angle () and midpoint (y). Note different offsets (x and xp) and a shift () in reflection point.

Since the pegleg multiple and primary in Figure have the same emergence angle, , the stepout of the two events is the same at x and xp. This fact is the basis of the derivation in Appendix A, which obtains the following result for xp as a function of x.
 (1) (2)
is the two-way traveltime of a -order pegleg multiple in the top layer. Equation (1) defines a time-variable compression of the offset axis. In constant velocity, Veff=V, and equation (1) reduces to the radial trace resampling used by Taner (1980) for long-period deconvolution of peglegs. Figure demonstrates the Snell resampling on the first- and second-order pegleg multiples of a synthetic dataset.

Graphically (Figure ), we infer that the shift in midpoint, , of the reflection points of the primary and pegleg is
 (3)
As a function of time, decreases asymptotically to zero from a maximum of x/4 at the seabed. The deeper the reflector, the smaller becomes.

Oftentimes, non-seabed pegleg multiples (e.g. top of salt) are strong enough to merit modeling. In a v(z) earth, the results derived in this section are equally valid, with one exception: attenuation. In this case, the effects of attenuation are encoded - in a possibly non-linear way - in the effective reflection coefficient that we estimate in a subsequent section.