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.
Herrmann, F. J., Moghaddam, P., & Stolk, C. C. (2008). Sparsity- and continuity-promoting seismic image recovery with curvelet frames. Applied and Computational Harmonic Analysis, 24(2), 150-173. https://doi.org/10.1016/j.acha.2007.06.007