Next: Post-Stack Residual Migration Up: EXAMPLES Previous: Migration

## Post-Stack Time Migration

An interesting example of a stacking operator is the hyperbola summation used for time migration in the post-stack domain. In this case, the summation path is defined as
 (63)
where z denotes the vertical traveltime, x and y are the horizontal coordinates on the migrated and unmigrated sections respectively, and v stands for the effectively constant root-mean-square velocity Claerbout (1995b). The summation path for the reverse transformation (demigration) is found from solving equation (63) for z. It has the well-known elliptic form
 (64)
The Jacobian of transforming z to t is
 (65)
If the migration weighting function is defined by conventional downward continuation Schneider (1978), it takes the following form, which is equivalent to equation (44):
 (66)
The simple trigonometry of the reflected ray suggests that the cosine factor in formula (66) is equal to the simple ratio between the vertical traveltime z and the zero-offset reflected traveltime t:
 (67)
The equivalence of the Jacobian (65) and the cosine factor (67) has important interpretations in the theory of Stolt frequency-domain migration Chun and Jacewitz (1981); Levin (1986); Stolt (1978). According to equation (19), the weighting function of the adjoint operator is the ratio of (66) and (65):
 (68)
We can see that the cosine factor z/t disappears from the adjoint weighting. This is completely analogous to the known effect of dropping the Jacobian'' in Stolt migration Harlan (1983); Levin (1994). The product of the weighting functions for the time migration and its asymptotic inverse is defined according to formula (9) as
 (69)
Thus, the asymptotic inverse of the conventional time migration has the weighting function determined from equations (9) and (66) as
 (70)
The weighting functions of the asymptotic pseudo-unitary operators are obtained from formulas (34) and (35). They have the form
 (71) (72)
The square roots of the cosine factor appearing in formulas (71) and (72) correspond to the analogous terms in the pseudo-unitary Stolt migration proposed by Harlan and Sword 1986.

Next: Post-Stack Residual Migration Up: EXAMPLES Previous: Migration
Stanford Exploration Project
11/12/1997