Asymptotic pseudounitary stacking operators

1. Common-shot migration

In the case of common-shot migration, we can simplify equation (46) with the help of Gritsenko's formula (35) to the form

$$\widehat{w}_{CS}(r;z,x) = {1\over{\left(2 \pi\right)^{m/2}}} ... ...ht)^{m/2}}} {{\cos{\alpha(r)}} \over {v(r)}} {{R(s,x)} \over {R(r,x)}} \;,$$ (48)

where the angle $\alpha(r)$ is measured between the reflected ray and the normal to the observation surface at the reflector point $r$ . Formula (48) coincides with the analogous result of Keho and Beydoun (1988), derived directly from Claerbout's imaging principle (Claerbout, 1970). An alternative derivation is given by Goldin (1987). Docherty (1991) points out a remarkable correspondence between this formula and the classic results of Born scattering inversion (Bleistein, 1987).

For common-shot migration, pseudo-unitary weighting coincides with the weighting of datuming and corresponds to the downward continuation of the receivers.