Next: Examples Up: Fomel & Prucha: Angle-gather Previous: Traveltime considerations

# Amplitude considerations

One simple approach to amplitude weighting for angle-gather migration is based again on Cheops' pyramid considerations. Stacking along the pyramid in the data space is a double integration in midpoint and offset coordinates. Angle-gather migration implies the change of coordinates from to . The change of coordinates leads to weighting the integrand by the following Jacobian transformation:
 (11)
Substituting formulas (5) and (6) into equation (11) gives us the following analytical expression for the Jacobian weighting:
 (12)
Weighting (12) should be applied in addition to the weighting used in common-offset migration. By analyzing formula (12), we can see that the weight increases with the reflector depth and peaks where the angles and approach condition (10).

The Jacobian weighting approach, however, does not provide physically meaningful amplitudes, when migrated angle gathers are considered individually. In order to obtain a physically meaningful amplitude, we can turn to the asymptotic theory of true-amplitude migration Goldin (1992); Schleicher et al. (1993); Tygel et al. (1994). The true-amplitude weighting provides an asymptotic high-frequency amplitude proportional to the reflection coefficient, with the wave propagation (geometric spreading) effects removed. The generic true-amplitude weighting formula Fomel (1996b) transforms in the case of 2-D angle-gather time migration to the form:
 (13)
where Ls and Lr are the ray lengths from the reflector point to the source and the receiver respectively. After some heavy algebra, the true-amplitude expression takes the form
 (14)
Under the constant-velocity assumption and in high-frequency asymptotic, this weighting produces an output, proportional to the reflection coefficient, when applied for creating an angle gather with the reflection angle . Despite the strong assumptions behind this approach, it might be useful in practice for post-migration amplitude-versus-angle studies. Unlike the conventional common-offset migration, the angle-gather approach produces the output directly in reflection angle coordinates. One can use the generic true-amplitude theory Fomel (1996b) for extending formula (14) to the 3-D and 2.5-D cases.

Next: Examples Up: Fomel & Prucha: Angle-gather Previous: Traveltime considerations
Stanford Exploration Project
4/20/1999