Next: 15-DEGREE DEPTH MIGRATION Up: Ji and Biondi: Explicit Previous: Introduction

# ONE-WAY WAVE EQUATION AND ITS APPROXIMATION

Most extrapolation methods used in migration and modeling are based on the scalar wave equation. For a 2-D acoustic earth model, the two-way scalar wave equation in the domain is of the following form:
 (1)
where P is the pressure field, is the frequency, z is depth, and v is velocity. Solving the scalar wave equation requires the first depth derivative of the wavefield. To obtain a one-way solution to the two-way equation, without having to know the vertical derivatives, we can use the one-way scalar wave equation
 (2)
where the positive sign is used in modeling, the negative sign in migration, and the velocity is half of the true velocity. While the one-way equation eliminates the need to know the vertical derivatives of the wavefields, it contains a square-root of a differential operator that cannot easily be implemented numerically.

There are two methods of rationalizing the square-root equation: by Taylor series and by continued-fraction (Claerbout, 1985). The Taylor-series method leads to an explicit scheme whereas the continued-fraction method leads an implicit scheme. The continued-fraction expansion was first used by Muir (Claerbout, 1985) to approximate the square-root operator,
 (3)
where

The expansion is
 (4)
where n is the order of approximation, and usually R0=1 (Claerbout, 1985). In implicit algorithms, the fraction expansion method for approximation is implemented recursively. In the explicit scheme this method makes implementation difficult because it produces another function of a matrix and as a result, a Taylor-series expansion is generally used. By Taylor-series expansion, the square root operator given in equation (3) can be written as
 (5)

Next: 15-DEGREE DEPTH MIGRATION Up: Ji and Biondi: Explicit Previous: Introduction
Stanford Exploration Project
12/18/1997