Sparsity- and continuity-promoting seismic image recovery with curvelet frames

By Felix J. Herrmann, Peyman Poor Moghaddam, Christiaan C. Stolk. 2008

134 citations recorded by the source corpus.

A nonlinear singularity-preserving solution to seismic image recovery with sparseness and continuity constraints is proposed. We observe that curvelets, as a directional frame expansion, lead to sparsity of seismic images and exhibit invariance under the normal operator of the linearized imaging problem. Based on this observation we derive a method for stable recovery of the migration amplitudes from noisy data. The method corrects the amplitudes during a post-processing step after migration, such that the main additional cost is one application of the normal operator, i.e., a modeling followed by a migration. Asymptotically this normal operator corresponds to a pseudodifferential operator, for which a convenient diagonal approximation in the curvelet domain is derived, including a bound for its error and a method for the estimation of the diagonal from a compound operator consisting of discrete implementations for the scattering operator and its adjoint the migration operator. The solution is formulated as a nonlinear optimization problem where sparsity in the curvelet domain as well as continuity along the imaged reflectors are jointly promoted. To enhance sparsity, the l1-norm on the curvelet coefficients is minimized, while continuity is promoted by minimizing an anisotropic diffusion norm on the image. The performance of the recovery scheme is evaluated with a reverse-time ‘wave-equation’ migration code on synthetic datasets, including the complex SEG/EAGE AA′ salt model.

Subfields: Computer Vision and Pattern Recognition, Geophysics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/d91c23a5c9937c0541ec1aa77434bee52c884c46

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Citations in the corpus

The Curvelet Transform (2010, methodology): “Currently, a few researchers have applied the ICT method to compressed sensing in seismic data recovery [33], [34], [65], and remote sensing [44], [46].”
Fast randomized full-waveform inversion with compressive sensing (2012, background): “An important class of algorithms used to solve the FWI problem are the GN methods that involve the pseudo-inverse of the reduced Hessian given by the combined action of the Jacobian operator ∇F [mk”
Fast Discrete Curvelet Transforms (2006, background): “Section 3 introduces the main ideas underlying the USFFT-based and the wrapping-based digital implementations which are then detailed in sections 4 and 6, respectively.”