Introduction

In a classic paper, Rothman et al., (1985) described residual zero-offset migration providing a method for changing the migration velocity of a zero-offset section without remigrating the data from scratch. Residual zero-offset migration (also known as cascaded migration) proved to be very useful. Larner et al., (1986) used residual migration to enhance the accuracy of 15 degree migrations extending them to higher dips. Beasley et al., (1988) use cascaded migration to extend Stolt (constant velocity f-k) migration to variable velocity media. Residual migration seemed so useful, that one would hope the concept of residual migration would extend to prestack migration. It might be useful for reducing sensitivity to the velocity model used for prestack migration because errors in the migration could be corrected by doing residual migration. Residual migration might also be useful for migration velocity analysis because many migrations could be obtained for little cost. It might also be possible to do prestack depth migration by first performing inexpensive constant-velocity migration, followed by a residual migration to account for v(x,z).

Because of the potential rewards, Fowler and Al-Yahya (1986) tried to formulate residual prestack migration. They concluded that it is possible to construct an operator that converts constant-offset or shot-profile data, migrated with one velocity, to data migrated at another velocity. However, the operator is not equivalent to applying constant-offset migration or shot profile migration with some other velocity which is the case for zero-offset residual migration. Because the operator they obtained was not a ``migration" in the classical sense, it seemed that residual prestack migration would not be as useful as residual zero-offset migration.

My velocity analysis project (Etgen, 1988, 1989) requires the ability to find changes in migrated images due to perturbations in the velocity model used for prestack depth migration. In the past, approximated residual migration to derive residual moveout corrections that can be applied to migrated constant-offset sections. The residual migration operators of this paper supercede the previous residual moveout operators. To obtain an inexpensive residual migration operator I make two approximations. First, I only use the stationary phase path of the full residual migration operator. Second, I build the operators only for the constant-velocity case. The residual migration is only a residual time migration; ray bending is ignored in the construction of the migration operator.

Subtracting the kinematics of residual zero-offset migration from the kinematics of residual constant-offset migration, leaves an operator that performs only residual DMO and residual NMO. This converts migrated constant-offset sections to zero-offset sections for a change in the migration velocity. Note that if the velocity doesn't change, migrated constant-offset sections are already converted to migrated zero-offset sections. Residual constant-offset NMO+DMO is useful for velocity analysis. It provides a velocity analysis method analogous to stacking velocity analysis, that works in arbitrary structure and after prestack depth migration.