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# Introduction

Depth migration algorithms are important for imaging in areas with strong lateral velocity gradients. Wavefield extrapolation algorithms in the 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 methods that they can incorporate lateral velocity variations in a single migration.

One of the aims of this thesis is to facilitate 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 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 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.

Next: Direct solution of Poisson's Up: Helical factorization of the Previous: Helical factorization of the
Stanford Exploration Project
5/27/2001