Abstract-A new algorithm for simultaneous wavelet estimation and deconvolution of seismic reflection signals is given. To remove the inherent ambiguity in this blind deconvolution problem, we introduce relevant a priori information. Our major assumption is sparseness of the reflectivity, which corresponds to a layered earth model. This allows non-minimum phase wavelets to be recovered reliably and closely spaced reflectors to be resolved. To combine a priori knowledge and data, we use a Bayesian framework and derive a maximum a posteriori estimate. Computing this estimate is a difficult optimization problem which is solved by a sub-optimal iterative procedure. The procedure alternates steps of wavelet estimation and reflectivity estimation. The first step only requires a simple least squares fit, while the second step is solved by the iterated window maximization algorithm recently proposed by Kaaresen. This enables better efficiency and optimality than established alternatives. The resulting optimization method can easily handle multichannel models with only a moderate increase of the computational load. Lateral continuity of the reflectors is achieved by modeling local dependencies between neighboring traces. Major improvements in both wavelet and reflectivity estimates are obtained by taking the wavelet to be invariant across several traces. The practicality of the algorithm is demonstrated on synthetic and real seismic data. An application to multivariate well log segmentation is also given.

This research report is organized as two separate papers. The first paper describes a new deconvolution algorithm for sparse spike trains. The second paper compares the new algorithm to a number of existing alternatives.

The ability of pulse-echo measurements to resolve closely spaced reflectors is limited by the duration of the ultrasonic pulse. Resolution can be improved by deconvolution, but this often fails because frequency selective attenuation introduces unknown changes in the pulse shape. In this paper we propose a maximum a posteriori algorithm for simultaneous estimation of a time varying pulse and high-resolution deconvolution. A priori information is introduced to encourage estimates where the pulse varies only slowly and the reflectivity sequence is sparse. This adds sufficient regularization to the problem, and no further assumptions on the pulse such as minimum phase or a particular parametric form are needed. The joint pulse and reflectivity estimate are computed iteratively by alternating steps of pulse estimation and reflectivity estimation. The first step amounts to only a linear least squares fit. The second step is a difficult combinatorial optimization problem that we solve by a suboptimal but efficient search procedure. Due to the sparseness assumption, our approach is particularly suited for layered media containing a limited number of abrupt impedance changes. This is a situation of interest in many applications of nondestructive evaluation. Synthetic and real data results show that the algorithm works well.

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