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In this section, I show how the least-squares estimate of a migrated
image can be approximated using non-stationary matching
filters. In terms of cost, this approach is comparable to one and a half
iterations of a conjugate-gradient method (CG), the first ``half'' iteration being
the migration. The cost of estimating the non-stationary filters is
negligible compared to the total cost of migration.
First, given seismic data and a migration operator
, we seek a model such that

| |
(1) |

This goal can be rewritten in the following form
| |
(2) |

and is called the fitting goal. For migration, a model styling goal
(regularization) is necessary to compensate for irregular geometry artifacts
and uneven illumination Prucha et al. (1999). I omit this term in my derivations and focus
on the data fitting goal only.
By estimating in a least-squares sense, we want to minimize
the objective function
| |
(3) |

The least-squares estimate of the model is given by
| |
(4) |

where is the Hessian of the transformation. My goal in
this paper is to approximate the effects of the Hessian using non-stationary matching filters.

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Stanford Exploration Project

7/8/2003