Introduction

Geophysical spectral factorization is the task of estimating a minimum-phase signal with a given power spectrum. The advent of the helical coordinate system Claerbout (1997) 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 applications in preconditioning inverse problems, and wavefield extrapolation.

The Kolmogoroff algorithm Claerbout (1976); Kolmogoroff (1939) is widely used by the geophysical community since it is the most computationally efficient. However, it is not without its problems: as with all frequency domain methods, it assumes circular boundary conditions. Time-domain functions, especially those with zeros close to the unit circle, often require extreme amounts of zero-padding before they can be safely factored.

The Wilson-Burg method, introduced below, provides an algorithm for spectral factorization that is based on Newton's iteration for square-roots. Despite its iterative nature, we show that convergence is quadratic, providing a time-domain algorithm that is potentially cheaper than Kolmogoroff. In addition we describe how the algorithm can be extended to factor cross-spectra.