The fundamental idea of wave-equation migration velocity
analysis (WEMVA) is that we can establish a *linear* relation
between a small perturbation in the slowness field and the
corresponding perturbation in the image. Therefore, given an
image perturbation, we can invert for the slowness perturbation.

The main challenge of WEMVA is to construct a correct image
perturbation that can be inverted for slowness. This image
need not be fully accurate, but ought to provide the correct
*direction* and *magnitude* of the change.

In our early tests Sava and Biondi (2000), we construct the image perturbation using Prestack Stolt Residual Migration (PSRM) Sava (2000); Stolt (1996). In summary, this residual migration method provides updated images for new velocity maps that correspond to a fixed ratio of the new velocity with respect to the original velocity map.

The main disadvantage of building the image perturbation using PSRM is that, if the velocity ratio parameter () is too large, there is a good chance for the reference and the updated images to get out of phase. In other words, a large change in velocity violates the Born approximation. The end result is that the image perturbation computed by the forward operator and the one computed by residual migration are fundamentally different, and can have contradictory behaviors when using the Born WEMVA operator for inversion.

In previous reports, we have presented various attempts to solve this problem. All these attempts were related one way or another to the idea of scaling down the change in the map, thus obeying the Born approximation, followed by scaling up of the inverted slowness map. Although these approaches were successful in several examples, none of them addressed the fundamental problem: does the (scaled) image perturbation created by residual migration match the one obtained by the forward WEMVA operator? Furthermore, scaling and rescaling cannot lead to a robust method, since they involve ad hoc processes and since the inversion problem we are trying to address is already highly nonlinear.

In this paper, we present a new method that can be used to create image perturbations for WEMVA. The two main goals here are

- to create an image perturbation that is compatible with the one computed using the forward WEMVA operator, and
- to create the image perturbation directly from the background image, and therefore compliant with the Born approximation.

We begin with a discussion of the WEMVA scattering operator, and continue with the derivation of our new method. Next we show a complex synthetic example and provide a discussion of the future work and of the problems that are still unsolved.

9/18/2001