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## Approximating the Hessian

In equation (4), I define as the migrated image after a first migration such that
 (5)
In equation (5), and are unknown. Since I am looking for an approximate of the Hessian, I need to find two known images that are related by the same expression as in equation (5). This can be easily achieved by remodeling the data from with
 (6)
and remigrating them with as follows:
 (7)
Notice a similarity between equations (5) and (7) except that in equation (7), only is unknown. Notice that has a mathematical significance: it is a vector of the Krylov subspace for the model . Now, I assume that we can write the inverse Hessian as a linear operator such that
 (8)
and
 (9)
Equation (9) can be approximated as a fitting goal for a matching filter estimation problem where is the convolution matrix with a bank of non-stationary filters Rickett et al. (2001). This choice is rather arbitrary but reflects the general idea that the Hessian is a transform operator between two similar images. My hope is not to perfectly'' represent the Hessian, but to improve the migrated image at a lower cost than least-squares migration. In addition in equations (8) and (9), the deconvolution process becomes a convolution, which makes it much more stable and easy to apply. Hence, I can rewrite equation (9) such that the matrix becomes a vector and becomes a convolution matrix Robinson and Treitel (1980):
 (10)
The goal now is to minimize the residual
 (11)
in a least-squares sense. Because we have many unknown filter coefficients in , I introduce a regularization term that penalizes differences between filters as follows:
 (12)
where is the Helix derivative Claerbout (1998). The objective function for equation (12) becomes
 (13)
where is a constant. The least-squares inverse is thus given by
 (14)
Once is estimated, the final image is obtained by computing
 (15)
where (*) is the convolution operator.

Therefore, I propose computing first a migrated image , then computing a migrated image (equation (7)), and finally estimating a bank of non-stationary matching filters , e.g., equation (12). The final improved image is obtained by applying the matching filters to the first image , e.g., equation (8). In the next section, I illustrate this idea with the Marmousi dataset. I show that an image similar to the least-squares migration image can be effectively obtained.

Next: Migration results Up: Theory Previous: Theory
Stanford Exploration Project
7/8/2003