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

 Title Sparsity- and continuity-promoting seismic image recovery with curvelet frames Publication Type Journal Article Year of Publication 2008 Authors Felix J. Herrmann, Peyman P. Moghaddam, Christiaan C. Stolk Journal Applied and Computational Harmonic Analysis Volume 24 Number 2 Page 150-173 Month 03 Keywords curvelet 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. URL https://www.slim.eos.ubc.ca/Publications/Public/Journals/ACHA/2008/herrmann2008ACHAsac/herrmann2008ACHAsac.pdf DOI 10.1016/j.acha.2007.06.007 Citation Key herrmann2008ACHAsac