Figure illustrates the fact that 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, assuming perfect elasticity in the top layer (often water), to anelastic attenuation. Since the pegleg multiple and primary in Figure have the same emergence angle, , the time dip, or ``stepout'' of the two events is the same at x and x_{p}. In Appendix , I derive the following expression for x_{p}, where , , and were defined in Section :
schem-snell
Figure 4 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. |
(21) |
Figure illustrates application of Snell Resampling to a synthetic CMP gather. From the Figure, we see that Snell Resampling is an important vehicle for the exploitation of the additional information contained in the multiples. Notice how energy from the multiples is spread into the coverage gaps at near offsets and at 1.0 km offset. Snell Resampling moves energy from the multiples to the offset corresponding to its reflection angle at the target.
Graphically (Figure ), we may infer that the shift in midpoint, , of the reflection points of the primary and first-order pegleg is:
(22) |
For peglegs arising from sub-seabed reflectors, the assumption of perfect elasticity for the multiple bounce breaks down. To some extent, however, the additional attenuation suffered by the multiple can, to first order, be treated as a decrease in reflection coefficient.