Coherent noise attenuation using inverse problems and prediction-error filters

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. Nowadays, amplitude-preserving processing is a mandatory task for true-amplitude migration, AVO analysis or 4D interpretation; extracting the modelling part with approximate inverses is then risky. Inversion theory provides us with methods to compute a ‘good’ inverse that will honour the seismic data. Pioneering work by Tarantola (1987) has shown the usefulness of inversion for earthquake location and tomography. Since then inversion has been at the heart of many seismic processing breakthroughs, such as least-squares migration (Nemeth 1996), high-resolution radon transforms (Thorson & Claerbout 1985; Sacchi & Ulrych 1995) or projection filtering (Soubaras 1994; Abma & Claerbout 1995). 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 the 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 data components, or outliers which are better described by long-tailed probability density functions (PDFs) as opposed to short-tailed Gaussian PDFs, 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 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. A more ‘geophysical way’ of saying this is that the residual will not have a white spectrum. 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 out 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.