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The number of coefficients to estimate is usually much greater
than the number of data points. This makes the problem very under-determined.
A solution is to introduce more equations in equation (30) as follows:
| |
(32) |

The second term in equation (32) improves the conditioning
of our problem and is called regularization. In the filter
estimation problem, it is reasonable that penalizes strong variations between filter
coefficients. Hence, is usually a gradient or a Laplacian.
Crawley (2000) proposes smoothing the filter coefficients along radial
directions. This proposal is valid for shot or common mid point
gathers only where constant dips are roughly aligned along radial
segments. For instance, if is a gradient operator
we have
| |
(33) |

with the identity matrix.
Equation (32) can be solved for in a
least-squares sense. We then want to minimize the objective function
| |
(34) |

which gives for the least-squares estimate
| |
(35) |

Because of the number of unknowns and of the sparseness of the
problem, we use a conjugate gradient method to estimate our PEFs.

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Stanford Exploration Project

7/8/2003