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In seismic processing, we often have to estimate filters in order to,
for example, improve the result of tomography
Clapp (2001), interpolate data
Crawley et al. (1999); Spitz (1991) or perform signal-noise
separation Abma and Claerbout (1995); Soubaras (1994). The filter
estimation can be done in the time or frequency domain. In general,
we first start by estimating the filters and
then use them for a particular geophysical task. One general
assumption when we estimate filters is that the time series from which
we estimate the filters is wide-sense stationary.
Often with real data, however, this assumption is violated.
The non-stationarity can be related to a
different moveout or a different
amplitude behavior throughout the seismic record.
The filter estimation in the first case can
be improved by introducing patches. We then estimate one filter per
patch assuming that the data are locally stationary. The
boundary conditions for patching technology are rather difficult to
handle and a better solution is to use non-stationary filters
Crawley et al. (1999). The amplitude aspect of non-stationarity
is usually tackled before processing by applying, for example,
an Automatic Gain Control (AGC) on the data.
One problem with this approach is that it is a non-linear process that
can damage important amplitude information. In a setting where more attributes
are extracted from seismic data to derive rock properties,
treating amplitudes accurately becomes a major challenge in seismic processing.
In this paper, I show strategies for estimating filters when
amplitude problems exist with seismic data.
My claim is that (1) any weight on the data should be applied
as late as possible in the processing work-flow, preferably for
display purposes only and (2) if weights are needed, they should be
incorporated inside our processing scheme without letting any
footprint in the final image Claerbout (1992).
Inverse theory provides us with tools to handle
amplitude problems which are usually not used by the industry.
Theoretically, from a statistical view-point Tarantola (1987),
the residuals should be weighted during the filter estimation by the
inverse covariance matrix of the noise. This requirement might be
quite difficult to meet because we might not know the noise a-priori
and/or the covariance operator might be too expensive to compute.
In the context of filter estimation, I will assume that the covariance
operator is a diagonal weight. Here I investigate the filter estimation
problem for signal/noise separation. I will show with
2-D and 3-D field data that it is important to weight the residual during
the filter-estimation step to obtain the best noise attenuation
results.

In the first section following this introduction, I will describe the
theory of prediction-error filters (PEF) estimation when the data are
weighted and when the residuals are weighted. Then, I will show with
a 2-D example with stationary filters and a 3-D example with
non-stationary filters that a weighted-residual PEF estimation leads to the best
noise attenuation results, as predicted by inverse theory.

** Next:** Theory of PEF estimation
** Up:** Guitton: Weighted PEF estimation
** Previous:** Guitton: Weighted PEF estimation
Stanford Exploration Project

7/8/2003