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# Introduction

In seismology, we can express many of the processing steps as linear operators. These operators perform a mapping of one domain, usually a model of the earth parameterized in terms of velocity, reflectivity, into another domain, usually seismic data sorted into CMP or shot gathers. This mapping is called modeling because it models the seismic data. Usually we desire the opposite of modeling, i.e, given the seismic data, we want to retrieve earth parameters that will help us to understand the physical properties of the subsurface. In many cases the adjoint of a modeling operator is used to estimate the model. For some operators like the Fourier transform the adjoint is the exact inverse, for others, the vast majority, the adjoint is not the true inverse but rather an approximation of the inverse.

Nowadays amplitude-preserving processing is an absolutely necessary task for migration, amplitude versus angle (AVA) analysis or 4D interpretation; undoing the modeling part with approximate inverses is then dangerous. Inversion theory provides us with methods to compute a ``good'' inverse that will honor the seismic data. Pioneering work by Tarantola (1987) shown the usefulness of inversion for earthquake location and tomography. Ever since, inversion has been at the heart of many seismic processing breakthroughs like least-squares migration Nemeth (1996), high-resolution radon transforms Sacchi and Ulrych (1995); Thorson and Claerbout (1985) or projection filtering Abma and Claerbout (1995); Soubaras (1994). A very popular method of inversion is the least-squares approach, which can be related to a Bayesian estimation of the model parameters.

It is well understood that inversion in a least-squares sense is very sensitive to the noise level present in the data. By noise I mean abnormally large or small residual components, or outliers which are better described by long-tailed probability density functions (PDF) as opposed to short-tailed gaussian PDF (Chapter ), and coherent noise that the seismic operator is unable to model. The noise will spoil any analysis based on the result of the inversion and affect the amplitude recovery of the input data. From a more statistical point of view if the data residual, which measures the quality of the data fitting, is corrupted by outliers or coherent noise in the data, it will not have independent and identically distributed (IID) components. In other words, the residual will not have a white spectrum.

In this Chapter, two methods that yield IID data residuals are presented. One method approximates the inverse data covariance operator with prediction-error filters (PEFs), which amounts to a filtering (or weighting) of the data residual. The other method introduces a noise modeling operator that separates signal and coherent noise. I prove that this method amounts to a filtering of the data residual with projection filters.

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Stanford Exploration Project
5/5/2005