The dream of an explorationist is to be able to carry on AVO or AVA attributes analysis in areas with poor illumination. But the quality of the images that a state of the art migration can produce is not good enough for that purpose. One way to improve the image is to use an inversion formalism introduced by Tarantola (1987) to solve geophysical imaging problems. This procedure computes an image by convolving the migration result with the inverse of the Hessian matrix.
When the dimensions of the problem get large, the explicit calculation of the Hessian matrix and its inverse becomes unfeasible. That is why Valenciano and Biondi (2004) and Valenciano et al. (2006) proposed the following approximations: (1) to compute the one way wave equation Green functions from the surface to the target (or vice versa), to reduce the size of the problem; (2) to compute the Hessian, exploiting its sparse structure; and (3) to compute the inverse image following an iterative inversion scheme. The last item renders unnecessary an explicit computation of the inverse of the Hessian matrix. For efficiency reasons the Green's functions necessary to compute the Hessian are computed by means of a one-way wave-equation extrapolator.
In this paper I study the impact of the one-way wave-equation approximation to the wave propagation in: migration, modeling, and wave-equation inversion. I illustrate the differences between two-way and one-way data modeling, and the migration of two-way and one-way modeled data.
I also show how the approximations used to compute the Hessian have an impact on the recovery of the medium AVO and AVA response. This is done by comparing the inversion of the two-way modeled TRIP data, a one-way modeled data (equivalent to the use of the full Hessian), and a one-way modeled data using an approximated Hessian.