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

TitleSparsity- and continuity-promoting seismic image recovery with curvelet frames
Publication TypeJournal Article
Year of Publication2008
AuthorsFelix J. Herrmann, Peyman P. Moghaddam, Christiaan C. Stolk
JournalApplied and Computational Harmonic Analysis
Volume24
Number2
Page150-173
Month03
Keywordscurvelet transform, Imaging, Processing, SLIM
Abstract

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 ap- plication 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 $ell_1$-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 time-reversed ’wave-equation’ migration code on synthetic datasets, including the complex SEG/EAGE AA salt model.

URLhttps://www.slim.eos.ubc.ca/Publications/Public/Journals/ACHA/2008/herrmann2008ACHAsac/herrmann2008ACHAsac.pdf
DOI10.1016/j.acha.2007.06.007
Citation Keyherrmann2008ACHAsac