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Theory

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 ${\bf d}$ and a migration operator ${\bf L}$, we seek a model ${\bf m}$ such that
\begin{displaymath}
{\bf Lm = d}.\end{displaymath} (1)
This goal can be rewritten in the following form
\begin{displaymath}
{\bf 0 \approx r_d = Lm - d}\end{displaymath} (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 ${\bf m}$ in a least-squares sense, we want to minimize the objective function
\begin{displaymath}
f({\bf m})={\bf \Vert r_d\Vert}^2={\bf \Vert Lm-d\Vert}^2.\end{displaymath} (3)
The least-squares estimate of the model is given by  
 \begin{displaymath}
{\bf \hat{m} = (L'L)^{-1}L'd}\end{displaymath} (4)
where ${\bf L'L}$ is the Hessian of the transformation. My goal in this paper is to approximate the effects of the Hessian ${\bf L'L}$using non-stationary matching filters.



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