For migration, different
approaches exist to correct for these defects of our
operators. Bleistein (1987), based on earlier
works from Beylkin (1985), derived an inverse operator
for Kirchhoff migration. A similar path is followed by
Thierry et al. (1999) with the addition of non-linear
inversion with approximated Hessian. Alternatively, least-squares
migration with regularization has proven efficient with incomplete surface data, e.g.,
Nemeth et al. (1999) and illumination problems due to complex
structures, e.g., Kuehl and Sacchi (2001); Prucha et al. (1999).
Hu et al. (2001) introduce a deconvolution operator that
approximates the inverse od the Hessian in the least-squares
estimate of the migrated image. However, this method assumes
a *v*(*z*) medium which means that the deconvolution operators are horizontally invariant.
Recently, Rickett (2001) proposed estimating weighting functions
from reference images to compensate illumination effects for
finite-frequency depth migration. This method corrects for amplitude
effects only and requires some smoothing that can be rather difficult
to estimate.

In this paper, I propose a new strategy for approximating the inverse of the Hessian. This approach aims to estimate a bank of non-stationary matching filters Rickett et al. (2001) between two migrated images that theoretically embed the effects of the Hessian. This approach is implemented after migration and is relatively cheap to apply. It can be applied on images migrated at zero-offset or in the angle domain. I illustrate this method with the Marmousi dataset. I demonstrate that this approach can effectively recover amplitudes similar to the one obtained with least-squares inversion. In addition, these filtered images have less artifacts than the least-squares result without regularization and at a much reduced cost.

7/8/2003