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This appendix details the computations associated with the
finitedifference solution
to the equation in a 2D orthogonal Riemannian space.
The 3D wave equation (22)
takes in two dimensions the simpler form:
 
(38) 
If we substitute the Fourierdomain wavenumbers by their equivalent
spacedomain partial derivatives, we obtain
 
(39) 
A finitedifference implementation of equation (39)
involving the CrankNicolson method is
 

 (40) 
If we make the notations
 

 (41) 
we can write equation (40) as
 

 (42) 
or, if we isolate the terms corresponding to the two extrapolation levels as:
 

 (43) 
After grouping the terms, we obtain
which is a finitedifference representation of the solvable using
fast tridiagonal solvers.
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Up: Sava and Fomel: Riemannian
Previous: 2D pointsource ray coordinates
Stanford Exploration Project
10/14/2003