Seismic Wavelet Estimation: A Fast Learning Algorithm Using Cumulant Matching and Natural-Gradient

Abstract In this paper, a simple, fast and local learning algorithm for estimating mixed-phase seismic wavelets is developed. This learning algorithm minimizes the nonlinear cumulant matching criterion using the natural gradient. It is shown that the nonlinear cumulant matching criterion has a Riemannianstructure. Therefore its steepest-descent is given by its natural gradient descent. The Riemannian metric tensor of the pth-order cumulant matching criterion, p?3, is determined. Then thenatural-gradient learning algorithm minimizing the pth-order cumulant matching criterion is derived for recovery of mixedphase seismic wavelets. Computer simulations, for the extraction of a broadband mixed-phase seismic wavelet from synthetic noisy seismic data, illustrate the powerfulness of the developed natural-gradient learning algorithm with respect to the standard-gradient learning algorithm. Introduction Seismic wavelet estimation is required for solving many seismic signal processing problems. Such problems may includetime-lapse seismic inversion, linearized seismic AVO amplitude-varying-with-offset) inversion, mono-/multi-channelseismic deconvolution, phase correction of stacked seismic sections, seismic forward modeling, etc. To estimate mixedphase seismic wavelets, the cumulant matching criterion iscommonly used(2)(3)(5). In this paper, a simple, fast and local learning algorithm for recovery of mixed-phase seismic wavelets is developed. This learning algorithm minimizes the nonlinear cumulant matching criterion using the natural gradient. It is shown that the nonlinear cumulant matching criterion has a Riemannian structure. Therefore the steepest-descent of the nonlinear cumulant matching criterion isnot defined by its standard-gradient descent, as suggested in Lazear (1993) (2), but rather by its natural-gradient descent. The Riemannian metric tensor of the pth-order cumulant matching criterion, p?3, which is a positive-definite matrix, is determined. Then the natural-gradient learning algorithmminimizing the pth-order cumulant matching criterion is derived for the extraction of mixed-phase seismic wavelets. The standard-gradient learning algorithm is derived as a special case of the natural-gradient learning algorithm by substituting the Riemannian metric tensor by the identity matrix. This paper is organized as follows. In section 2, a general deterministic cost function is proposed, and then its natural-gradient descent isdetermined and analysed. In section 3, the nonlinear cumulant matching criterion for the extraction of mixed-phase seismic wavelets is treated as a special case of the deterministic cost function proposed in section 2. Then, the natural-gradient descent of this nonlinear cumulant matching criterion is derived from the natural-gradient descent of the deterministic costfunction proposed in section 2. In section 4, a couple of computer simulations for estimating a broadband mixed-phase seismic wavelet from synthetic noisy seismic data are carried out. In the first computer simulation, the seismic wavelet is recovered by minimizing the fourth-order cumulant matching criterion using both natural- and standard-gradients. In the second computer simulation, the seismic wavelet is extracted by minimizing the joint third- and fourth-order cumulant matching criterion using both natural- and standard-gradients. Both computer simulations illustrate that the natural-gradient learning algorithm requires much less iterations to converge than thestandard-gradient learning algorithm. Section 5 concludes the paper.

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