On Common-offset Prestack Time Migration With Curvelets

Recently, curvelets have been introduced in the field of applied harmonic analysis and shown to optimally sparsify smooth (C2, i.e., twice continuously differentiable) functions away from singularities along smooth curves. In addition, it was shown that the curvelet representation of wave propagators is sparse. Since the wavefronts in seismic data lie mainly along smooth surfaces (or curves in two dimensions), and since the imaging operator belongs to the class of operators that is sparsified by curvelets, curvelets are plausible candidates for simultaneous sparse representation of both the seismic data and the imaging operator. In this paper, we study the use of curvelets in pre-stack time migration, and show that simply translating, rotating and dilating curvelets according to the pre-stack map time-migration equations we developed earlier, combined with amplitude scaling, provides a reasonably accurate approximation to timemigration. We demonstrate the principle in two dimensions but emphasize that extension to three dimensions is possible using 3D equivalents of curvelets. We treat time-migration in an attempt to learn the basic characteristics of seismic imaging with curvelets, as a preparation for future imaging in heterogeneous media with curvelets.

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