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Coherent noise filtering and modeling in practice

The core of the filtering method is the PEF ${\bf A}$ that destroys the coherent noise present in the residual. In order to accommodate the possible complexity in space and time of the undesirable events, these filters can have any dimensions thanks to the helical boundary conditions Claerbout (1998); Mersereau and Dudgeon (1974). In addition, these filters can be made non-stationary in any directions Claerbout and Fomel (2002); Crawley (2000); Margrave (1998). Chapters [*] and [*] will illustrate these possibilities.

If a noise model is known in advance, a PEF can be estimated from it and used directly for the inversion of ${\bf m}$ in equation ([*]). If a noise model is not known in advance, I propose the following algorithm based on the iterative inversion of ${\bf m}$:

Set ${\bf A=I}$ and ${\bf m=0}$.
Iterate until the data residual ${\bf r_d}$ contains mostly coherent noise.
Estimate a new PEF ${\bf A}$ from the residual.
Set ${\bf m=0}$ and iterate.
Inspect the residual. If white then stop else reestimate the PEF ${\bf A}$ from the unweighted residual ${\bf Lm - d}$ and go to (4).
This nonlinear process tries to refine the PEF estimation by bootstrapping in order to obtain the best possible coherent noise filter ${\bf A}$ Clapp (2003). It is also based on the assumption that the coherent noise will stay in the residual after a certain number of iterations. Therefore, the residual needs to be closely watched at each iteration, which requires a certain level of human interpretation.