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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.

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An alternative to the conventional time series approach to single‐trace modeling and inversion by convolution and inverse filtering is a parametric approach. To obtain insight into the potential of the parametric approach, the solution of the single‐trace forward problem is formulated in matrix terms. For the nonlinear reflector lag time parameters this is achieved by linearization, which is shown to be a valid approximation over a sufficiently large region. The matrix forward operators are analyzed by means of the singular value decomposition (SVD). The SVD can be considered a generalization of the Fourier transform of convolution operators. On the basis of the SVD analysis, inverse operators are designed which combine stability with high resolving power. A method to determine the resolving power of the parametric inverse operators is presented. Several examples show how wavelet bandwidth, data noise level, and model complexity influence the resolving power of the data for the reflection coefficient and the lag‐time parameters. The most important result is that the resolution obtained in parametric inversion is, in most cases, superior to and, at worst, equal to the resolution obtained with wavelet inverse filtering. The explanation is that in parametric inversion a different representation of the reflectivity function is used which, in practical situations, involves fewer unknowns. In wavelet inverse filtering the reflectivity function is represented as a regularly sampled function where every sample point represents an unknown. In practical applications of parametric inversion the reflectivity function is represented as a model with a limited number of reflectors as unknowns. To formulate parametric models, a priori information is required. The effort of collecting sufficient a priori information is the cost of increasing resolution beyond the resolution offered by wavelet inverse filtering.