Seismic imaging in the curvelet domain and its implications for the curvelet design
Résumé
This paper is a first attempt towards the migration of seismic data in the curvelet domain for heterogeneous background velocity models. We first explain how to build a simple curvelet decomposition/reconstruction code, based on the use of Fast Fourier Transforms (FFTs). It is directly related to ideas from Candés (Candés et al., 2005), but appears to be slightly more general. We then show how to derive simple operations (image shift, rotation and stretch) directly in the curvelet domain. The basic transforms are needed for subsequent seismic migration. We demonstrate how to interpolate in the curvelet domain, using the Shannon interpolation formula. It appears that the interpolation scheme imposes some specific conditions on the shapes of the filters used for the curvelet decomposition. In particular, the transform has to be redundant. Finally, we combine the basic operations to migrate seismic data in the curvelet domain. The 2‐D common offset section is first decomposed into curvelet coefficients. All coefficients are map‐migrated independently using the notion of direction inherent to curvelets. The final depth section is obtained by performing an inverse curvelet transform. The curvelet migration is applied on a very simple background velocity model but for a complex reflectivity structure. The results are as good as for the Kirchhoff migration, but are obtained by map‐migrating the curvelet coefficients instead of smearing energy along isochrones. In previous papers (Candés and Demanet, 2005; Douma and de Hoop, 2005), the authors had the hope that it would be possible to only consider after map‐migration the nearest coefficient in the curvelet domain. We show that this approach unfortunately does not provide in practice the expected quality for the result. An interpolation in the curvelet domain as developed here is indeed necessary.