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Conceptually, the first step of our method is to form the convolutional
matrices of both the estimated multiples and the estimated primaries
. In practice, these matrices are not explicitly formed but computed with
linear operators Claerbout and Fomel (2002). Next, we compute non-stationary filters in
micro-patches (that is, filters that act locally on overlapping two-dimensional
partitions of the data) to match the estimated multiples and the estimated
primaries, to the data containing both. We compute the filters by solving the
following least-squares inverse problem:
where and are the matching filters for the
multiples and
primaries respectively, is a parameter to balance the relative
importance of the two components of the fitting goal Guitton (2005),
is
the data (primaries and multiples), is a regularization
operator, (in our implementation a Laplacian operator), and is the
usual way to control how strong a regularization we want.
Once convergence is achieved, each filter is applied to its corresponding
convolutional matrix, and new estimates for *M* and *P* are computed:

| |
(5) |

| (6) |

These updated versions of *M* and *P* are then plugged into
equations 1 and and the process repeated until the
cross-talk has been eliminated or significantly attenuated.

** Next:** Examples with synthetic data
** Up:** Alvarez and Guitton: Adaptive
** Previous:** Introduction
Stanford Exploration Project

1/16/2007