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Seismic noise attenuation is an important part of the processing workflow before a migrated image can be obtained and interpreted. This thesis tackles the noise problem with inversion to preserve any signal present in the data. In particular, spiky and coherent noise are attenuated by looking at the statistics (i.e., PDF and pattern) of the undesirable events.

For spiky events, Chapter [*] introduces the Huber norm. The Huber norm behaves like the $\ell^1$ norm for large residuals and like the $\ell^2$ norm for small residuals. Because it is continuous and differentiable everywhere, the Huber norm is minimized with a quasi-Newton method called L-BFGS. This technique maintains most of the convergence properties of the standard BFGS method for non-linear problems while keeping the memory requirements very low. This last point is particularly important for seismic processing where the volume of data can be quite large.

In this thesis, the combination Huber norm/L-BFGS solver proves being very versatile. For instance, in Chapter [*], velocity scans are derived from data with bad traces and noise bursts. In Chapter [*], small geological features at the bottom of the Sea of Galilee are unraveled from extremely noisy data. In Chapter [*], multiples and primaries are better separated by estimating matching filters with the Huber norm. This list of applications is not exhaustive. It is my belief that many more geophysical problems could benefit from using the Huber norm with the L-BFGS solver.

For coherent noise attenuation, I introduce in Chapter [*] two strategies based on the observation that coherent noise generally results from an approximate modeling of the seismic data. These techniques achieve an important goal of least-squares inversion: to have IID residuals. One strategy approximates the inverse data covariance operator with multidimensional prediction-error filters (PEFs) by weighting the data residual. The other strategy incorporates a coherent noise modeling term inside the inversion. Compared to the weighting approach, the modeling technique yields the smallest residual with the best convergence properties. One advantage of the weighting approach with PEFs, however, is its ability to separate non-stationary noise and signal according to their pattern. This pattern-based approach is very effective at separating primaries and multiples as illustrated in Chapters [*] and [*]. In particular, I demonstrate that the pattern-based approach with 3-D PEFs is more robust to modeling uncertainty than adaptive subtraction, an industry standard for this task.

Therefore, the main contributions of this thesis are four fold:

It presents a particular implementation of a robust solver for the inversion of data contaminated with spiky noise. This implementation is based on the Huber norm and the L-BFGS method (Chapters [*] and [*]).
It develops processing techniques for coherent noise attenuation based on the need to have IID residuals (Chapters [*] and [*]).
It establishes multidimensional prediction-error filters with helical boundary conditions as an extremely flexible tool for noise filtering and/or modeling (Chapters [*], [*], and [*]).
It highlights the limitations of the standard adaptive subtraction technique for the multiple attenuation problem by introducing two viable alternatives: (1) matching filters estimation with the Huber norm (Chapter [*]), and (2) multiple subtraction with a pattern-based approach (Chapters [*] and [*]).
One important problem omitted in this thesis that directly links to my work, however, is how to build stable inverse non-stationary PEFs. In practice, patching solves it but in a very crude manner. More elegant solutions to that problem should be developed. Valcarce et al. (2000) give conditions for building stable inverse filters, but their approach seems to be difficult to apply in practice. Beylkin (1995) proposes transforming infinite impulse response filters (IIR) into finite response filters (FIR). This approach could impact our way of doing non-stationary deconvolution. If stable non-stationary PEFs can be computed, then the modeling approach, which has very good signal/noise separation properties, could be also used for non-stationary coherent noise removal. There, inverse non-stationary PEFs would play the role of modeling operators, as illustrated in Chapter [*] for the stationary case.

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