Next: Migration results
In equation (4), I define as the migrated
image after a first migration such that
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
and remigrating them with as follows:
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
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):
The goal now is to minimize the residual
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:
where is the Helix derivative Claerbout (1998).
The objective function for equation (12) becomes
where is a constant.
The least-squares inverse is thus given by
Once is estimated, the final image is obtained by
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
Stanford Exploration Project