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The goal of adaptive subtraction is as follows: given a time series
and a desired time series , we seek a filter
that minimizes the difference between and
where * is convolution. We can rewrite this definition
in the fitting goal
| |
(1) |

where represents the convolution with the time series
. We can minimize this fitting goal in a least-squares sense
leading to the objective function
| |
(2) |

where (') is the transpose. The minimum energy solution is given by
| |
(3) |

where is the least-squares estimate of .This approach is very popular but has some intrinsic limitations.
In particular is by construction orthogonal
to the residual . In the multiple attenuation
problem is the data, the multiple model and
the estimated primaries. If both signal and
noise are correlated, the separation will suffer because of
the orthogonality principle.
From now on I will refer to this method as the ``standard approach''.

In the next section I propose improving the adaptive subtraction
scheme. This improvement leads to an unbiased matched-filter
estimation when both signal and noise are correlated.

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** Up:** Improving adaptive subtraction
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Stanford Exploration Project

6/7/2002