In seismic, 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 modelling because it models the seismic data. Usually we desire the opposite of modelling, i.e. given some data, we want to retrieve the model. In many cases the adjoint of the modelling 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. In this paper I show how the residual can be whitened when coherent noise is present in the data. Outliers and noise-burst problems are not addressed here. They can be winnowed our by applying, iteratively, a locally re-weighted regression (Wang, White & Pratt 2000). In the first section I review some basics of inverse theory. Then in the following section I introduce two inversion methods that yield white residuals. The first method proposes approximating the noise covariance operators with prediction-error filters (PEFs). The second method handles the coherent noise by introducing a noise modelling operator within the inversion. These methods are tested with field data.
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