We use the recently introduced multiscale and multidirectional curvelet transform to exploit the continuity along reflectors for cases in which the assumption of spiky reflectivity may not hold. We show that such type of seismic reflectivity is sparse in the curvelet-domain. This curvelet-domain compression of reflectivity opens new perspectives towards solving classical problems in seismic processing including the deconvolution problem. In this paper, we present a formulation that seeks curvelet-domain sparsity for non-spiky reflectivity and we compare our results with those of spiky deconvolution.
The forward problem of seismic imaging can be written as:
y = AKm+n, (1)
where y is the known data vector, K is the linearized Born scattering operator, A is the convolution operator representing source and receiver frequency characteristics, m is the unknown model vector (reflectivity) and n is zero-centered white Gaussian noise. For now, we assume that the source and receiver are omnidirectional. During the traditional processing, we first deconvolve, followed by migration in which case the estimated reflectivity is given by m= K?A?y, where ? denotes the pseudo inverse (some sort of approximate inverse). On the other hand, simultaneous migration and deconvolution produces the solution m= (AK)?y. Although, it is claimed that both approaches would generate the same result, they are not the same in theory (De Roeck, 2002). Indeed in the situation where both K and A are invertible, we have (AK) = K-1 A-1 justifying the approach of deconvolution first followed by migration. However, we all know that deconvolution is an ill-posed problem whereas least-squares migration entails the inversion of an over-determined system. This means that the above identity may no longer hold as right fully pointed out by De Roeck. This paper is a first attempt to address this issue. To keep our argument simple, we assume for now that the Born scattering operator is given by the Identity operator. This assumption corresponds to assuming a constant velocity model and zero-offset time to depth converted data. The assumption also transforms Eq. 1 to a deconvolution problem. The forward problem becomes:
y = Am + n, (2)
Given A and y, we need to find m. Since the early 80''s, researchers have cast this problem as a l1-norm minimization (Taylor et al., 1979; Oldenburg et al., 1981), where the reflectivity is assumed to be made up of spikes. In recent work by Felix Herrmann, it was shown that the assumption of spiky reflectivity is too limited to describe seismic reflectivity (Herrmann, 2005). This means that in cases where the reflectivity is not spiky, spiky deconvolution may fail (Herrmann, 2005). In our approach, we show that the non-spiky reflectivity is still sparse in the curvelet-domain and that this sparsity can be exploited while solving the deconvolution problem. We start with a brief introduction to curvelets, followed by a presentation of our algorithm and application to synthetic data. Curvelets are amongst one of the latest members of the family of multiscale and multidirectional transforms (Candés et al., 2006).