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Equation (6) can be used to describe
twoway propagation of acoustic waves in a semiorthogonal
Riemannian space.
For oneway wavefield extrapolation, we need to modify
the acoustic wave equation (6) by selecting a single
direction of propagation.
In order to simplify the computations,
we introduce the following notation:
 

 
 
 
 
 
 (8) 
All quantities in equations (8)
can be computed by finitedifferences for any choice of
a Riemannian coordinate system
which fulfills the orthogonality condition indicated earlier.
In particular, we can use ray coordinates to compute those
coefficients. With these notations, the acoustic waveequation
can be written as:
 
(9) 
For the particular case of Cartesian coordinates
(),
the Helmholtz equation (9) takes the familiar form
 
(10) 
From equation (9), we can directly
deduce the modified form
of the dispersion relation for the waveequation in a
semiorthogonal 3D Riemannian space:
 
(11) 
For oneway wavefield extrapolation, we need to solve
the second order equation (11)
for the wavenumber of the extrapolation direction ,and select the solution with the appropriate sign to
extrapolate waves in the desired direction:
 
(12) 
The solution with the positive sign in equation (12) corresponds to
propagation in the positive direction of the extrapolation axis .
For the particular case of Cartesian coordinates
(),
the oneway wavefield extrapolation equation takes the
familiar form
 
(13) 
Next: Mixeddomain solutions to the
Up: Sava and Fomel: Riemannian
Previous: Acoustic waveequation in 3D
Stanford Exploration Project
10/14/2003