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# Introduction

When perturbations of velocity models are small, residual migration may be considerably more efficient than full migration. In a classic paper, Rothman et al. (1985) described residual time 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 certain 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. He calculated the residual migration operators by solving a pair of nonlinear equations numerically. In SEP-65 (Zhang, 1990a), I derived analytical formulas for calculating the kinematic operators of residual profile migration. However, these results are limited to the cases in which migration velocities are constant.

In my last report (Zhang, 1990b), I described a conceptual algorithm for calculating residual depth migration operators. The algorithm is generally applicable to both pre- and post-stack images. In principle, it can handle general variable-slowness models. This paper explains in detail the procedure for implementing this algorithm and demonstrates the results of the algorithm with a synthetic example. The next section reviews the key equations I derived in the last report. The sections that follow explain the algorithm step by step, including how to calculate the partial derivatives of traveltimes and surface horizontal slownesses with an upwind finite-difference method and how to extrapolate the operators starting from the known initial conditions at the surface. The procedure for applying the operators to migrated images is briefly described. Finally, the report includes a synthetic example of residual depth migration of a post-stack image.

Next: REVIEW Up: Zhang: Residual depth migration Previous: Zhang: Residual depth migration
Stanford Exploration Project
12/18/1997