L p ‐NORM DECONVOLUTION 1

A bstract The purpose of deconvolution is to retrieve the reflectivity from seismic data. To do this requires an estimate of the seismic wavelet, which in some techniques is estimated simultaneously with the reflectivity, and in others is assumed known. The most popular deconvolution technique is inverse filtering. It has the property that the deconvolved reflectivity is band‐limited. Band‐limitation implies that reflectors are not sharply resolved, which can lead to serious interpretation problems in detailed delineation. To overcome the adverse effects of band‐limitation, various alternatives for inverse filtering have been proposed. One class of alternatives is L p ‐norm deconvolution, L 1 norm deconvolution being the best‐known of this class. We show that for an exact convolutional forward model and statistically independent reflectivity and additive noise, the maximum likelihood estimate of the reflectivity can be obtained by L p ‐norm deconvolution for a range of multivariate probability density functions of the reflectivity and the noise. The L ∞ ‐norm corresponds to a uniform distribution, the L 2 ‐norm to a Gaussian distribution, the L 1 ‐norm to an exponential distribution and the L 0 ‐norm to a variable that is sparsely distributed. For instance, if we assume sparse and spiky reflectivity and Gaussian noise with zero mean, the L p ‐norm deconvolution problem is solved best by minimizing the L 0 ‐norm of the reflectivity and the L 2 ‐norm of the noise. However, the L 0 ‐norm is difficult to implement in an algorithm. From a practical point of view, the frequency‐domain mixed‐norm method that minimizes the L 1 norm of the reflectivity and the L 2 ‐norm of the noise is the best alternative. L p ‐norm deconvolution can be stated in both time and frequency‐domain. We show that both approaches are only equivalent for the case when the noise is minimized with the L 2 ‐norm. Finally, some L p ‐norm deconvolution methods are compared on synthetic and field data. For the practical examples, the wide range of possible L p ‐norm deconvolution methods is narrowed down to three methods with p = 1 and/or 2. Given the assumptions of sparsely distributed reflectivity and Gaussian noise, we conclude that the mixed L 1 norm (reflectivity) L 2 ‐norm (noise) performs best. However, the problems inherent to single‐trace deconvolution techniques, for example the problem of generating spurious events, remain. For practical application, a greater problem is that only the main, well‐separated events are properly resolved.