Adaptive multiple subtraction: Unification and comparison of matching filters based on the ℓq-norm and statistical independence

Abstract

An adaptive multiple subtraction step is necessary for almost all methods that predict seismic multiple reflected waves. We aim at giving a better understanding of matching filters based on l q -norms and on statistical independence. We found that the formulation of all of these techniques can be gathered in a mutual framework by introducing a space-time operator, called the primary enhancer, acting on the estimated primaries. The differences between the considered matching filters become more intuitive because this operator behaves as a simple amplitude compressor. In this perspective, all the methods tend to uncorrelate the predicted multiples and the enhanced estimated primaries. The study of these matching-filter methods can be narrowed to the study of the primary enhancer operator because it is the only difference. Moreover, we have emphasized the role of using adjacent traces or windowing approaches in terms of statistics, and we show that an adequate windowing strategy may overbear the choice of the objective function. Indeed, our analysis showed that setting a good windowing strategy may be more important than changing the classical leastsquares adaptation criterion to other approaches based on l q -norm minimization or independent component analysis.

A Marchenko equation for acoustic inverse source problems

“…Here, we assume that the PA signals and the artifacts are not or only weakly interfering, such that the removal of events decreases the norm of the traces. This type of adaptive filters is often applied in seismic data processing 39,40 to overcome the limitations of inaccurate amplitudes, due to the many factors that have been outlined earlier in this section. To find the filters, we aim to solve the following minimization problem:…”

“…Apart from the most conventional applications, which are based on the ' 2 -norm, 39 various studies in the geophysical literature have been devoted to the minimization of ' 1 and other norms. 40 Note that a 1 ðx F ; tÞ and a 2 ðx F ; tÞ can be chosen space-and time-dependent. To facilitate this dependence without offering too much freedom to the filters, it is common to make use of sliding windows or non-stationary regression.…”

Application of multi-objective optimization to blind source separation

“…Several practical situations are formulated as BSS or BSE problems, e.g. in audio signal processing (Choi & Cichocki, 1997;Xue et al, 2009), chemical data analysis (Duarte et al, 2014b,a), geophysical signal processing (Takahata et al, 2012;Batany et al, 2016) and manufacturing process analysis (Gandini et al, 2011). Recent works also applied BSS techniques in face recognition problems (Bhowmik et al, 2019).…”