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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