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Discussion

The preceding section shows that we can formulate prestack Stolt residual migration without directly assuming the new velocity to which we residually migrate the data. Instead, we can select a ratio of the current velocity to the updated one ($\rho=v_0/v$). This conclusion is true in the 3D case (myresmig-3d-pr and myresmig-3d-po), the 2D case (myresmig-2d-pr and myresmig-2d-po), and the case of 3D common-azimuth data (myresmig-ca.

(102) comments that, in the post-stack case, the frequency after time migration can be related to the frequency of the original data and the difference of the squares of the two velocities, before and after residual migration (stoltold). However, he also shows that such a conclusion is no longer true in the prestack case. If we reformulate Stolt residual migration as a function of the ratio of two velocities, we can apply the process to images which have been depth-migrated with arbitrary velocity models. This is technically possible because the equations do not operate with velocity differences, but with velocity ratios. The residual migration transforms using a scaled version of the original velocity field. In this formulation, prestack Stolt residual migration is a constant velocity ratio method, and not a constant velocity one. It is important to understand that this approach is an approximation and it may not work in regions of extreme complexity and large velocity contrasts. Furthermore, the method is correct only from a kinematic point of view, and does not incorporate amplitude corrections.

An extension of this method can go beyond constant velocity ratios. Since it is a fast, Stolt-type technique, we can run a large number of different residual migrations at different velocity ratios and then pick the values of the ratio parameter $\rho(x,y,z)$ that give the best image at every location in space. This method is therefore a good companion to wave-equation migration velocity analysis (wemva), where the goal is to obtain improved images regardless of the procedure we use.


next up previous print clean
Next: Conclusions Up: Prestack residual migration Previous: Examples
Stanford Exploration Project
11/4/2004