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The first step is to consider non-stationary shaping filters.
Experience with missing data problems
Crawley et al. (1998); Crawley (1999b) has shown that
working with smoothly-varying non-stationary filters often gives
better results than working with filters that are stationary within
small patches.
With a non-stationary convolution filter, , the shaping
filter regression equations,

| |
(3) |

are massively underdetermined since there is a potentially unique
impulse response associated with every point in the dataspace
Rickett (1999). We need constraints to ensure the
filters vary-smoothly in some manner.
The simplest regularization scheme involves applying a generic
data-space roughening operator, , to the non-stationary
filter coefficients. can be a simple derivative operator,
for example. This leads to the set of equations,

| |
(4) |

| (5) |

By making the change of variables,
Fomel (1997b),
we get the following system of equations,
| |
(6) |

| (7) |

Equations (6) and (7) describe a
preconditioned linear system of equations, the solution to which
converges rapidly under an iterative conjugate-gradients solver.
In practice, I set , and keep the filters smooth by
restricting the number of iterations Crawley (1999a).

** Next:** Shaping filers on a
** Up:** Theory
** Previous:** Cross-correlation and shaping filters
Stanford Exploration Project

4/27/2000