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 $(\omega, x, z)$ domain is of the following form:

$${\partial^2 P\over\partial x^2}+{\partial^2 P\over\partial z^2} = -{\omega^2 \over v^2}P,$$ (1)

where P is the pressure field, $\omega$ 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

$${\partial P\over\partial z}=\pm i{\omega \over v}\sqrt{1 + {v^2 \over w^2}{\partial^2 \over\partial x^2}}P,$$ (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,

$$R=\sqrt{1-X^2},$$ (3)

where

$$X^2 = -{v^2 \over w^2}{\partial^2 \over \partial x^2}.$$

The expansion is

$$R_n = 1- {X^2 \over {1+R_{n-1}}},$$ (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

$$\sqrt{1-X^2}= 1 - {X^2\over 2} - {X^4 \over 8} - {X^6 \over 16} - \cdots.$$ (5)