Helical factorization of the Helmholtz equation

James Rickett and Jon Claerbout

james@sep.stanford.edu, jon@sep.stanford.edu

ABSTRACT

The accuracy of conventional explicit wavefield extrapolation algorithms at high dips is directly related to the length of the convolution filters: increasing the dip range leads to increased cost. Recursive filters have the advantage over convolutional filters in that short filters can move energy long distances. We discard both Crank-Nicolson and McClellan transforms, and extrapolate waves by factoring the 3-D Helmholtz equation in a helical coordinate system. We show that one of the minimum-phase factors provides a 90$^\circ$extrapolator, that can be applied recursively in the ($\omega-{\bf x})$domain. By developing a purely recursive wavefield extrapolator, we hope to achieve accuracy at high dips with shorter filters than is possible with explicit methods.