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This paper outlines an exciting new approach for carrying out blind seismic deconvolution. In this algorithm, overlapping source wavelets are modeled as amplitude-modulated sinusoids, and blind deconvolution is carried out by initially determining the seismogram's principle phase components. Once the principle phases are determined, a Rao-Blackwellized particle filter (RBPF) is utilized to separate the corresponding overlapping source wavelets. This deconvolution technique is referred to as principle phase decomposition (PPD). The PPD technique makes use of the fact that in reflection seismology the discrete convolution operation can be represented as the summation of several source wavelets of differing arrival times. In this algorithm, a jump Markov linear Gaussian system (JMLGS) is defined where changes (jumps) in the state-space system and measurement equations are due to the occurrences and losses of overlapping source wavelet events. The RBPF obtains optimal estimates of the possible overlapping source wavelets by individually weighting and subsequently summing a bank of Kalman filters (KFs). These KFs are specified and updated by samples drawn from a Markov chain distribution that defines the probability of the overlapping source wavelets that compose the JMLGS. In addition, hidden Markov model filters are utilized for refining the principle phase estimates. Index Terms--Blind deconvolution, hidden Markov models (HMMs), jump processes, Rao-Blackwellized particle filter.

An iterative nonparametric approach to spectral estimation, useful for estimation of line spectra, is presented. It reduces a cost function from the Bayes' theorem, and allows phase information to be recovered and used to expand original window data. Hyperparameters, controlling the degree of fit and spectral resolution, formulate the objective function, and the estimated Fourier transform yields the periodogram.

Levinson-type algorithms facilitate the derivation of the Q decomposition and Cholesky factorization of Vandermonde and Hankel matrices, respectively, and the fitting of data to polynomials. Multiply and divide operation (MDO) involving different quadratic equations are essential for the derivation of the least-squares solutions of the Hankel equations and Cholesky factorization. Fortran subroutine versions and polynomial fittings for different orders help test and validate the algorithms.