Next: Angle-domain common image gathers Up: Riemannian wavefield extrapolation Previous: 2D point-source ray coordinates

2D finite-difference solution to the equation

The one-way wave fifteen.3d takes in two dimensions the simpler form:
 (82)
where
 (83)
If we substitute the Fourier-domain wavenumbers by their equivalent space-domain partial derivatives, we obtain
 (84)
A finite-difference implementation of fifteen.2d.space involving the Crank-Nicolson method is  ^ _+1-^ _ i2 k_o ^+1_-^-1_+ ^+1_+1-^-1_+14
- i2 k_o 2 k_o^2- ^-1_-2^ _+^+1_+ ^-1_+1-2^ _+1+^+1_+12^2 .

If we make the notations

&=& i2 k_o 4
&=& - i2 k_o 2 k_o^2- 2^2 ,

we can write fifteen.2d.findif as

^ _+1-^ _&& ^+1_-^-1_+ ^+1_+1-^-1_+1
&+&^-1_-2^ _+^+1_+ ^-1_+1-2^ _+1+^+1_+1,

or, if we isolate the terms corresponding to the two extrapolation levels as:

^ _+1&-& ^+1_+1-^-1_+1- ^-1_+1-2^ _+1+^+1_+1=
^ _&+& ^+1_-^-1_+ ^-1_-2^ _+^+1_.

After grouping the terms, we obtain which is a finite-difference representation of the solvable using fast tridiagonal solvers.

Next: Angle-domain common image gathers Up: Riemannian wavefield extrapolation Previous: 2D point-source ray coordinates
Stanford Exploration Project
11/4/2004