Introduction

Residual migration has proved useful both in imaging and in migration velocity analysis. (84) show that post-stack residual migration can improve the focusing of migrated sections. They also show that migration with a given velocity (v) is equivalent to migration with a reference velocity (v₀) followed by residual migration with a velocity (v_r) that can be expressed as a function of v₀ and v.

Residual migration has also been used in velocity analysis. (2; 3) discusses a residual migration operator in the prestack domain, and show that it can be posed as a function of a non-dimensional parameter ($\rho$), the ratio of the reference velocity to the updated velocity. (32; 35) defines a kinematic residual migration operator as a cascade of NMO and DMO, and shows that it is only a function of the non-dimensional parameter ($\rho$) defined by Al-Yahya. Finally, (102) defines a prestack residual migration operator in the $(\omega,k)$ domain, and shows that, as in the post-stack case, it depends on the reference (v₀) and the correct (v) migration velocities, but not on a residual velocity (v_r).

In this section, I review prestack Stolt residual migration, and show that it, too, can be formulated as a function of a non-dimensional parameter that is the ratio of the reference (v₀) and updated (v) velocities. Consequently, we can use Stolt residual migration in the prestack domain to obtain a better-focused image without making explicit assumptions about the velocity magnitude. Although, strictly speaking, the method is developed for constant velocity, numerical examples show that it can be used in an approximate way for images migrated with smoothly varying velocity v(x,y,z) which departs from the constant velocity assumption.

An alternative to the residual migration technique presented in this section is a suite of full depth migrations with velocities at a percentage change from a reference velocity. Although more accurate, such a technique is also more expensive and not much more useful than residual migration, except for a complicated geological model which violates the residual migration assumptions.

This method has direct application in wave-equation migration velocity analysis (wemva). In WEMVA, we invert for perturbations of the velocity model starting from perturbations of the seismic image. A quick residual migration technique is ideally suited for this task, since we are less interested in the accuracy of the perturbed images than in the direction of the change that needs to be applied to the velocity model in order to improve the image.