In this abstract, we present a nonlinear curvelet-based sparsity-promoting formulation of a seismic processing flow, consisting of the following steps: seismic data regularization and the restoration of migration amplitudes. We show that the curvelet’s wavefront detection capability and invariance under the migration-demigration operator lead to a formulation that is stable under noise and missing data.
In this abstract, recent applications of the discrete curvelet transform (see e.g. Candes et al., 2006; Hennenfent and Herrmann, 2006b) are presented that range from data recovery from acquisitions with large percentages of traces missing to the restoration of migration amplitudes. Our approach derives from two properties of curvelets, namely the detection of wavefronts, without prior information on the positions and local dips (see e.g. Candes et al., 2006; Hennenfent and Herrmann, 2006b) and the relative invariance of curvelets under wave propagation (see e.g. Cand`es and Demanet, 2005). These properties render this transform suitable for a robust formulation of data regularization (see e.g. Hennenfent and Herrmann, 2006a; Herrmann and Hennenfent, 2007); primary-multiple separation (Herrmann et al., 2007); of migration-amplitude recovery (Herrmann et al., 2006) and of wavefield extrapolation (Lin and Herrmann, 2007). All these methods derive from sparsity in the curvelet domain that is a consequence of the above properties. This sparsity corresponds to a rapid decay for the magnitude-sorted curvelet coefficients and facilitates a separation of (coherent) ’noise’ and ’signal’. This separation underlies the successful applications of this transform to exploration seismology (see e.g. Hennenfent and Herrmann, 2006b; Herrmann et al., 2007).
Curvelets are localized ’little plane-waves’ (see e.g. Hennenfent and Herrmann, 2006b) that are oscillatory in one direction and smooth in the other direction(s). They are multiscale and multi-directional. Curvelets have an anisotropic shape – they obey the so-called parabolic scaling relationship, yielding a width µ length2 for the support of curvelets. This anisotropic scaling is optimal for detecting wavefronts and explains their high compression rates on seismic data and images (Candes et al., 2006; Hennenfent and Herrmann, 2006b; Herrmann et al., 2007). Curvelets represent a specific tiling of the 2-D/3-D frequency plane into strictly localized multiscale and multi-angular wedges. Because the directional sampling increases every-other scale doubling, curvelets become more anisotropic at finer scales. Curvelets compose an arbitrary column vector f, with the reordered samples, according to f = CTCf with C and CT, the forward/inverse discrete curvelet transform matrices (defined by the fast discrete curvelet transform, FDCT, with wrapping Candes et al., 2006; Ying et al., 2005). The symbol T represents the transpose, which is equivalent to the pseudoinverse for our choise of discrete curvelet transform, which is a tight frame with a moderate redundancy (a factor of roughly 8 for d = 2 and 24 for d = 3 with d the number of dimensions). Tight frames (see e.g. Daubechies, 1992) are signal representations that preserve energy. Consequently, CTC = I with I the identity matrix. Because of the redundancy, the converse is not the identity, i.e., CCT¹ I.