Seismic multiple removal with a primal-dual proximal algorithm

Abstract

Both random and structured perturbations affect seismic data. Their removal, to unveil meaningful geophysical information, requires additional priors. Seismic multiples are one form of structured perturbations related to wave-field bouncing. In this paper, we model these undesired signals through a timevarying filtering process accounting for inaccuracies in amplitude, time-shift and average frequency of available templates. We recast the problem of jointly estimating the filters and the signal of interest (primary) in a new convex variational formulation, allowing the incorporation of knowledge about the noise statistics. By making some physically plausible assumptions about the slow time variations of the filters, and by adopting a potential promoting the sparsity of the primary in a wavelet frame, we design a primal-dual algorithm which yields good performance in the provided simulation examples.

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

“…, and the a priori information available on the filters and the primary are expressed through hard constraints modeled by nonempty closed convex sets C and D. One of the potential advantages of such a constrained formulation is that it facilitates the choice of the related parameters with respect to the regularized approach which was investigated in some of our previous works [21,22] (this point will be detailed later on). We will now turn our attention to the choice of Ψ, C and D.…”

“…This work improves upon [21] by taking into account several multiple templates. Part of it was briefly presented in [22], by incorporating an additional noise into the generic model, and by introducing alternative norms in multiple selection objective criteria.…”

A constrained-based optimization approach for seismic data recovery problems

“…In [7,15], the authors incorporate plausible knowledge via additional metrics. Prior multiple templates are supplemented with Gaussian noise assumptions, wavelet-domain sparsity, smooth variations and energy concentration criteria.…”

“…by appropriately defining R ∈ R N ×Q , where Q = N P with P = J−1 j=0 P j and h ∈ R Q [15]. On the one hand, the matrix R contains the J templates for every time index n and tap index p. On the other hand, the vector h is similarly defined as the concatenation of all (unknown) time-varying filter impulse responses.…”

“…As mentioned earlier, the filters are assumed to be time-varying. However, real case study showed that those filters have smooth variations along time index n. To ensure that this a priori characteristic is satisfied for the estimated filters, we propose to introduce the following upper bound on the impulse response variations [15]:…”