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Introduction

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 up previous print clean
Next: Theory of PEF estimation Up: Guitton: Weighted PEF estimation Previous: Guitton: Weighted PEF estimation
Stanford Exploration Project
7/8/2003