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 x_{p} such that a pegleg with offset x and primary with offset x_{p} 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 x_{p} 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 x_{p}) 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 x_{p}. This fact is the basis of the derivation in Appendix A, which obtains the following result for x_{p} as a function of x.
(1) | ||
(2) |
Graphically (Figure ), we infer that the shift in midpoint, , of the reflection points of the primary and pegleg is
(3) |
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.