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

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 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 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 reverse-time 'wave-equation' migration code on synthetic datasets, including the complex SEG/EAGE AA salt model.

Curvelet‐based migration preconditioning

“…First, we correct for the order of the normal operator by introducing a left preconditioning consisting of a fractional time integration. This first level is consistent with earlier work reported by Herrmann et al (2008) and Symes (2008). The next level of preconditioning is made off a simple diagonal scaling in the physical domain to compensate for spherical spreading of seismic waves.…”

“…contributions from Claerbout and Nichols (1994); Rickett (2003); Guitton (2004), and more recently from Herrmann et al (2008) and Symes (2008). These methods vary in degree of sophistication with regard to the estimation of the diagonal through migrated-image to remigratedimage matching and in the way the scaling is applied-i.e., by division in the physical or via sparsity promotion in the curvelet domain as reported in Herrmann et al (2008). Amplitudes are restored during all these methods by applying the scaling as a post-processing step after migration.…”

Adaptive curvelet-domain primary-multiple separation

“…Although distinct, our approach is similar to recent work in migration-amplitude recovery, in which scaling methods with smoothness constraints have been proposed ͑Guitton, 2004; Symes, 2008͒. This paper builds explicitly on a curvelet-based approach to this problem introduced by Herrmann et al ͑2008a͒.…”

Sparse seismic imaging using variable projection

“…Sparse regularization has proven to be an indispensable tool in many areas, including inverse problems [1] and compressive sensing [2,3]. If a signal y is known to have a sparse or compressible (quickly decaying) representation y = Sx, this information can be used to formulate optimization problems of the form…”

“…In more general inverse problems, these guarantees have not been found; it is therefore appropriate to consider (1) as a regularization approach to the least squares problem. For example, in the seismic setting, where (1) has been particularly useful [4], A is a linearized Born-scattering operator, S is the curvelet transform, and b is seismic data. While there are several popular algorithms that solve (1), e.g.…”