I derive an alternative approach to this problem and and analyze its results in synthetic and real data. This approach is an iterative deconvolution process, that is based on the picking of absolute maxima of the absolute values of a scaled correlation between the residuals and the wavelet, within a pre-specified range (related to the wavelet length). An important property of the proposed method is that the spectrum of the retrieved reflectivity series shows the same frequency distribution as the original reflectivity sequence.
The method is defined first for the simple case, of a known wavelet. In this case, if the reflectivity sequence is sufficiently sparse, the method finds the exact solution in a few iterations. As the reflectivity series becomes less sparse, the number of required iterations increases until it reaches the point where it is unable to find the exact solution within a finite number of iterations. Because of its slow rate of convergence, I combine the method with the usual least-squares inversion after some iterations, to speed up the process.
For the more interesting case of an unknown wavelet, the method requires an initial estimation of the wavelet to start the iterative scheme; the final solution depends on this initial choice. The results obtained when the method is applied to a deep-water CMP gather correspond to a slightly sharper version of the results obtained with predictive deconvolution.