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Introduction

It is well known that most of our mathematical operators for seismic processing are not unitary. This means that for any linear transformation ${\bf L}$, ${\bf L'L \neq I}$ where (') stands for the adjoint and ${\bf I}$ is the identity matrix. Having non-unitary operators often results from approximations we make when building those operators. For example, we might not take the irregular and finite acquisition geometry of seismic surveys into account.

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.


next up previous print clean
Next: Theory Up: Guitton: Approximating the Hessian Previous: Guitton: Approximating the Hessian
Stanford Exploration Project
7/8/2003