INTRODUCTION

A solved problem is the factorization of a positive real autospectrum into a minimum-phase wavelet and its adjoint. The most practical method is that of Kolmogoroff. Here I extend the Kolmogoroff method to cross-spectra.

This problem arises in the extrapolation of 3-D wavefields where we need to factor an operator like .We get a band matrix to solve. In principle, we factor it into lower and upper triangular band matrices which we then backsolve. Except at the ends (ends of the helix which are the two side boundaries of a 2-D space), this is equivalent to a filter problem where the two backsubstitutions are polynomial divisions, one causal, the other anticausal. Although $-\nabla^2$ is an autocorrelation, is not, so we need two different minimum-phase filters whereas the Kolmogoroff method gives us the same one for both the causal and anticausal operations.