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Introduction

Spectral factorization is the task of estimating a minimum-phase signal from a given power spectrum. The advent of the helical coordinate system (Mersereau and Dudgeon, 1974; Claerbout, 1998) has led to renewed interest in spectral factorization algorithms, since they now apply to multi-dimensional problems. Specifically, spectral factorization algorithms provide the key to rapid multi-dimensional recursive filtering with arbitrary functions, which in turn has geophysical applications in preconditioning inverse problems (Clapp et al., 1998; Fomel and Claerbout, 2003), wavefield extrapolation (Zhang and Shan, 2001; Zhang et al., 2000; Rickett, 2000; Rickett et al., 1998), and 3-D noise attenuation (Rickett et al., 2001; Ozdemir et al., 1999a,b).

The Kolmogoroff (cepstral or Hilbert transform) method of spectral factorization (Oppenheim and Shafer, 1989; Kolmogoroff, 1939; Claerbout, 1976) is often used by the geophysical community because of its computational efficiency. However, as a frequency-domain method, it has certain limitations. For example, the assumption of periodic boundary conditions often requires extreme amounts of zero-padding for a stable factorization. This is one of the limitations which make this method inconvenient for multi-dimensional applications.

The Wilson-Burg method, introduced in this paper, is an iterative algorithm for spectral factorization based on Newton's iterations. The algorithm exhibits quadratic convergence. It provides a time-domain approach that is potentially more efficient than the Kolmogoroff method. We include a detailed comparison of the two methods.

Recent surveys (Sayed and Kailath, 2001; Goodman et al., 1997) discuss some other methods for spectral factorization, such as the Schur method (Schur, 1917), the Bauer method (Bauer, 1955) and Wilson's original method (Wilson, 1969). The latter is noted for its superb numerical properties. We introduce Burg's modification to this algorithm, which puts the computational attractiveness of this method to a new level. The Wilson-Burg method avoids the need for matrix inversion, essential for the original Wilson's algorithm, reduces the computational effort from $ O(N^3)$ operations to $ O(N^2)$ operations per iteration. A different way to accelerate Wilson's iteration was suggested by Laurie (1980). We have found the Wilson-Burg algorithm to be especially suitable for applications of multidimensional helical filtering, where the number of filter coefficients can be small, and the cost effectively reduces to $ O(N)$ operations.

The second part of the paper contains a practical example of the introduced spectral factorization method. The method is applied to the problem of two-dimensional smooth data regularization. This problem often occurs in mapping potential fields data and in other geophysical problems. Applying the Wilson-Burg spectral factorization method, we construct a family of two-dimensional recursive filters, which correspond to different values of tension in the tension-spline approach to data regularization (Smith and Wessel, 1990). We then use the constructed filters for an efficient preconditioning of the data regularization problem. The combination of an efficient spectral factorization and an efficient preconditioning technique provides an attractive practical method for multidimensional data interpolation. The technique is illustrated with bathymetry data from the Sea of Galilee (Lake Kinneret) in Israel.


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2014-02-15