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Notes: 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 Lp-norm deconvolution, L1norm 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 Lp-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 L2-norm to a Gaussian distribution, the L1-norm to an exponential distribution and the L0-norm to a variable that is sparsely distributed. For instance, if we assume sparse and spiky reflectivity and Gaussian noise with zero mean, the Lp-norm deconvolution problem is solved best by minimizing the L0-norm of the reflectivity and the L2-norm of the noise. However, the L0-norm is difficult to implement in an algorithm. From a practical point of view, the frequency-domain mixed-norm method that minimizes the L1norm of the reflectivity and the L2-norm of the noise is the best alternative.Lp-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 L2-norm.Finally, some Lp-norm deconvolution methods are compared on synthetic and field data. For the practical examples, the wide range of possible Lp-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 L1norm (reflectivity) L2-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.

Notes: An efficient full 3D wavefield extrapolation technique is presented. The method can be used for any type of subsurface structure and the degree of accuracy and dip-angle performance are user-defined. The extrapolation is performed in the space-frequency domain as a space-dependent spatial convolution with recursive Kirchhoff extrapolation operators.To get a high level of efficiency the operators are optimized such that they have the smallest possible size for a specified accuracy and dip-angle performance. As both accuracy and maximum dip-angle are input parameters for the operator calculation, the method offers the possibility of a trade-off between these quantities and efficiency. The operators are calculated in advance and stored in a table for a range of wavenumbers. Once they have been calculated they can be used many times.At the basis of the operator design is the well-known phase-shift operator. Although this operator is exact for homogeneous media only, it is assumed that it may be applied locally in case of inhomogeneities. Lateral velocity variations can then be handled by choosing the extrapolation operator according to the local value of the velocity. Optionally the operators can be designed such that they act as spatially variant high-cut filters. This means that the evanescent field can be suppressed in one pass with the extrapolation. The extrapolation method can be used both in prestack and post-stack applications. In this paper we use it in zero-offset migration. Tests on 2D and 3D synthetic and 2D real data show the excellent quality of the method. The full 3D result is much better then the result of two-pass migration, which has been applied to the same data.The implementation yields a code that is fully vectorizable, which makes the method very suitable for vector computers.