Introduction

Depth migration algorithms are important for imaging in areas with strong lateral velocity gradients. Wavefield extrapolation algorithms in the $(\omega,x)$ domain have the advantage over Kirchhoff depth migration methods that they are based on finite bandwidth solutions to the wave-equation, rather than asymptotic approximations. Additionally, they have the advantage over $(\omega,k)$ methods that they can incorporate lateral velocity variations in a single migration.

One of the aims of this thesis is to facilitate $(\omega,x)$implicit finite-difference depth migrations, by providing a fast and efficient solution to wave-equation PDE's. To illustrate the underlying concepts, I begin with a review of the spectral factorization of Poisson's equation discussed by Claerbout (1998b). I then show how similar techniques can be used to solve the full Helmholtz wave-equation.

Traditional $(\omega,x)$ wavefield extrapolation algorithms are based on solutions to one-way wave equations, which can be derived by making paraxial approximations to the Helmholtz equation. For example, Claerbout (1985) describes implicit 2-D wavefield extrapolation based on the Crank-Nicolson formulation. Later, Hale popularized explicit extrapolation in 3-D by improving its stability and efficiency.

Rather than making a paraxial approximation, in this chapter, I construct a finite-difference stencil that approximates the two-way Helmholtz operator in the $(\omega,x)$ domain. The helical coordinate system then allows me to remap the multi-dimensional operator into one-dimensional space, where I can find two minimum-phase factors using a conventional spectral factorization algorithm.

The factorization provides a pair of filters: one causal minimum-phase and one anti-causal maximum-phase. I show that recursive application of the former propagates waves vertically downwards, and the latter propagates waves vertically upwards.