A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

A year ago we talked about a technique for Adaptive multiple subtraction with wavelet-based complex unary Wiener filters. The field of application is seismic signal processing. The fast and simple design was heuristic (helping discovery, stimulating interest as a means of furthering investigation.based on experimentation), based on an appropriate combination of "a sparsifying transform" and a closed-form one-tap, sliding-window adaptive filter. To make it more pragmatic (based on observation and real-world models), an alternative approach uses proximal algorithms to incorporate sparsity priors, either on data in redundant frame transforms and in the short-support filter design. Here is the preprint: A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal and the version published by IEEE Transactions on Signal Processing.

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured ``noises''. As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by  wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations,  based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames.  The designed primal-dual algorithm solves a  constrained  minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates  significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field data.

While applied here to seismic signal, the concept is heavily related to pattern matching in images, echo cancellation in audio or voice signals, exemplar search in speech.