Parabolic dictionary learning for seismic wavefield reconstruction across the streamers

Abstract

Dictionary learning (DL) methods are effective tools to automatically find a sparse representation of a data set. They train a set of basis vectors on the data to capture the morphology of the redundant signals. The basis vectors are called atoms, and the set is referred to as the dictionary. This dictionary can be used to represent the data in a sparse manner with a linear combination of a few of its atoms. In conventional DL, the atoms are unstructured and are only numerically defined over a grid that has the same sampling as the data. Consequently, the atoms are unknown away from this sampling grid, and a sparse representation of the data in the dictionary domain is not sufficient information to interpolate the data. To overcome this limitation, we have developed a DL method called parabolic DL, in which each learned atom is constrained to represent an elementary waveform that has a constant amplitude along a parabolic traveltime moveout. The parabolic structure is consistent with the physics inherent to the seismic wavefield and can be used to easily interpolate or extrapolate the atoms. Hence, we have developed a parabolic DL-based process to interpolate and regularize seismic data. Briefly, it consists of learning a parabolic dictionary from the data, finding a sparse representation of the data in the dictionary domain, interpolating the dictionary atoms over the desired grid, and, finally, taking the sparse representation of the data in the interpolated dictionary domain. We examine three characteristics of this method, i.e., the parabolic structure, the sparsity promotion, and the adaptation to the data, and we conclude that they strengthen robustness to noise and to aliasing and that they increase the accuracy of the interpolation. For both synthetic and field data sets, we have successful seismic wavefield reconstructions across the streamers for typical 3D acquisition geometries.

Cross-streamer wavefield reconstruction through wavelet domain learning

“…In this case, one regards the reconstruction problem as a matrix rank reduction problem. Rather than relying on predefined transformations or matrix decomposition, some approximation methods represent the data as a sparse linear combination of an overcomplete and learned dictionary [82,95]. DL-based methods can actually be viewed as special cases of the sparse approximation methods, but instead of optimizing each component separately, i.e., the dictionary including its sparse vectors, both components are instead jointly optimized [16].…”

Dual‐sensor wavefield separation in a compressed domain using parabolic dictionary learning

“…The parabolic structure was used by Turquais et al. (2018) for interpolation purposes, because it is consistent with the physics of wavefield propagation (Hoecht et al., 2009; Hubral et al., 1992; Zhang et al., 2001). The PDL problem may be mathematically expressed as follows: $$$$\begin{eqnarray} &&\mathop {{\rm{min}}}\limits_{\left\{ {{{\bf{x}}_i}} \right\}_{i{\rm{\;}} = {\rm{\;}}1}^M,\left\{ {{{\bf{d}}_k}} \right\}_{k{\rm{\;}} = {\rm{\;}}1}^K} \mathop \sum \limits_{i\; = \;1}^M \parallel {{\bf{x}}_i}{\parallel _0}{\rm{\;subject\;to}} \\ &&\left\{ \def\eqcellsep{&}\begin{array}{l} {\parallel {{\bf{y}}_i} - \mathop \sum \limits_{k{\rm{\;}} = {\rm{\;}}1}^K {{\bf{d}}_k}{{\bf{x}}_i}{\parallel _2} \le \epsilon,\;i\; = \;1, \ldots ,M\;}\\[2pt] {\;{{\bf{d}}_k}\;\left( {t,o_k^{{\rm{ref}}}} \right) = \;{{\bf{d}}_k}\left( {t + {c_k}\Delta {{\bf{o}}^2} + {\rm{\;}}{s_k}\Delta {\bf{o}},\;o_k^{{\rm{ref}}} + \Delta {\bf{o}}} \right),\;\forall \left( {t,...$$…”

“…The PDL (Turquais et al, 2018) is a modification of the conventional DL method where a geometrical structure is imposed to the atoms while learning them. The parabolic structure was used by Turquais et al (2018) for interpolation purposes, because it is consistent with the physics of wavefield propagation (Hoecht et al, 2009;Hubral et al, 1992;Zhang et al, 2001). The PDL problem may be mathematically expressed as follows:…”

Seismic data interpolation using deep learning with generative adversarial networks

“…Domain-transform methods assume the linearity of events if the basis functions of the selected transform are linear, and predictive-filter methods make assumptions about the local linearity of seismic events Yu et al (2015); Oliveira et al (2018). In addition to the methods described above, some authors have also worked on rank-reduction algorithms Chen et al (2016Chen et al ( , 2019 and dictionary learning methods Yu et al (2015); Hou et al (2018); Turquais et al (2018) for seismic data interpolation and regularization.…”