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Morphological component analysis (MCA) is a powerful tool used in image processing to separate different geometrical components (cartoons and textures, curves and points etc.). MCA is based on the observation that many complex signals may not be sparsely represented using only one dictionary/transform, however can have sparse representation by combining several over-complete dictionaries/transforms. In this paper we propose seislet-based MCA for seismic data processing. MCA algorithm is reformulated in the shaping-regularization framework. Successful seislet-based MCA depends on reliable slope estimation of seismic events, which is done by plane-wave destruction (PWD) filters. An exponential shrinkage operator unifies many existing thresholding operators and is adopted in scale-dependent shaping regularization to promote sparsity. Numerical examples demonstrate a superior performance of the proposed exponential shrinkage operator and the potential of seislet-based MCA in application to trace interpolation and multiple removal. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approximate shrinkage operator in the iterative shrinkage–thresholding (IST) algorithm to interpolate the nonuniformly sampled seismic traces. The key to designing the approximate shrinkage operator is the use of Taylor series. The redundant Fourier transform is employed to enhance the reconstruction performance, inspired by the theory of spectral compressive sensing. Numerical experiments using 3D real data and 5D synthetic data demonstrate the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Interpolating the missing traces of regularly or irregularly sampled seismic record is an exceedingly important issue in the geophysical community. Many modern acquisition and reconstruction methods are designed to exploit the transform domain sparsity of the few randomly recorded but informative seismic data using thresholding techniques. In this paper, to regularize randomly sampled seismic data, we introduce two accelerated, analysis-based two-step interpolation algorithms, the analysis-based FISTA (fast iterative shrinkage-thresholding algorithm) and the FPOCS (fast projection onto convex sets) algorithm from the IST (iterative shrinkage-thresholding) algorithm and the POCS (projection onto convex sets) algorithm. A MATLAB package is developed for the implementation of these thresholding-related interpolation methods. Based on this package, we compare the reconstruction performance of these algorithms, using synthetic and real seismic data. Combined with several thresholding strategies, the accelerated convergence of the proposed methods is also highlighted. [Copyright &y& Elsevier]

Abstract: An exceedingly important inverse problem in the geophysical community is the interpolation of the seismic data, which are usually nonuniformly recorded from the wave field by the receivers. Researchers have proposed many useful methods to regularize the seismic data. Recently, sparseness-constrained seismic data interpolation has attracted much interest of geophysicists due to the surprisingly convincing results obtained. In this article, a new derivation of the projection onto convex sets (POCS) interpolation algorithm is presented from the well known iterative shrinkage–thresholding (IST) algorithm, following the line of sparsity. The curvelet transform is introduced into the POCS method to characterize the local features of seismic data. In contrast to soft thresholding in IST, hard thresholding is advocated in this curvelet-based POCS interpolation to enhance the sparse representation of seismic data. The effectiveness and the validity of our method are demonstrated by the example studies on the synthetic and real marine seismic data. [Copyright &y& Elsevier]