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# Introduction

When perturbations of a velocity model are small, residual migration may be considerably more efficient than full migration. In a classic paper, Rothman et al., (1985) described residual migration of zero-offset sections. Their analysis shows that residual migration is efficient for two reasons: (1) the residual migration operator is sparse; and (2) it can be economically computed. Fowler and Al-Yahya (1986) made an effort to extend these results to prestack migration. They concluded that it is possible to construct a residual prestack migration operator. However, the operator is not equivalent to a full prestack migration operator with some velocity. This conclusion seems to imply that residual prestack migration operators are difficult to compute even when migration velocities are constant.

Using a kinematic approach, Etgen (1989) formulated residual prestack migration of constant offset sections that are initially migrated with a constant velocity. In his method, the residual migration operators are calculated by solving a pair of nonlinear equations numerically. Examples with synthetic and field data show that the operators work despite troubles near their triplications. In my previous report (Zhang, 1990), I showed that the kinematic operators of residual profile migration can be efficiently computed with analytical formulas, provided the migration velocities are constant. I also proposed to calculate the operators with a finite-difference scheme when migration velocities are not constant.

In this paper, I present an algorithm for calculating residual migration operators that transform an image migrated with one velocity model to an image migrated with another velocity model. This algorithm is generally applicable to both post-stack and pre-stack migration, and to both common-shot and constant-offset geometries. In principle, it handles general velocity models. Because it uses finite-difference techniques, the algorithm is potentially efficient. To test the accuracy of the algorithm, I calculate residual profile migration operators, and compare them with the analytical solutions. The results show that the errors in position are small fractions of the spatial-sampling intervals.

I will start with an analysis of the computational costs of full prestack migration and residual prestack migration, and explain why calculating operators is an important task in residual migration. Then, I will derive two kinematic relations, and convert them into partial differential equations, which is the key step of the formulation. I will briefly describe the method I use to solve these differential equations. Finally, examples show the accuracy of the algorithm.

Next: RESIDUAL MIGRATION OPERATOR Up: Zhang: Residual migration Previous: Zhang: Residual migration
Stanford Exploration Project
1/13/1998