The main goal of this paper is to clarify the theoretical origins of the WEMVA method, and to precisely show how it relates to the larger family of migration methods from which it is derived, the mixed-domain migration methods.
I begin with a review of mixed-domain migration. The two most general members of the family are known in the literature as the Fourier finite-difference (FFD) method, introduced by Ristow and Ruhl (1994), and the generalized screen (GSP) method de Hoop (1996). All the other methods in the family are just simplified cases, where we neglect some of the terms of the general relations. We can easily find many other methods known in the literature under various names, like: phase-shift Gazdag and Sguazzero (1984); Gazdag (1978), split-step Fourier Stoffa et al. (1990), local Born-Fourier or pseudo-screen Huang and Wu (1996), complexified pseudo-screen de Hoop and Wu (1996), extended local Born-Fourier Huang et al. (1999), a.s.o. Here, I present all these methods in a unified framework, with the goal of facilitating easy navigation through the relevant literature.
Next, I generalize the WEMVA operator based on the mixed-domain operators from which it is derived. It turns out that a Born linearization of either the FFD or the GSP relations, give general expressions for the WEMVA operator, from which we can obtain various special cases, as it is done for the associated mixed-domain migration methods.
Finally, I use a North Sea example to visualize the results of the WEMVA backprojection operator. I simulate small, localized perturbations on the seismic image, and show how they get backprojected in the slowness function using a particular choice of the WEMVA operator. The result is a bundle of ``fat'' rays, which can be correlated with the trajectories obtained for rays built using conventional ray tracing. This comparison enables us to easily visualize the band-limited character of wave-equation migration velocity analysis, and to gain insight into how this method operates in a real case.
Next section presents the general mixed-domain migration equations, followed by a discussion of the Born approximation and two sections dedicated to the various approximations encountered in the literature. In the end, I show how we can generalize the scattering and, implicitly, the WEMVA operators, and finish with the example on the North Sea dataset.