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The Laplacian operator of a scalar function in an arbitrary
Riemannian space with coordinates has the form
 
(1) 
where g^{ij} is a component of the associated metric tensor,
and is its determinant Synge and Schild (1978).
The expression simplifies if one of the coordinates
(e.g. the coordinate of oneway wave extrapolation)
is orthogonal to the other coordinates. Let ,, and ,
with orthogonal to both and .
Then the metric tensor has the matrix
 
(2) 
where E, F, G, and
are differential forms that can be found
from mapping Cartesian coordinates
to the general coordinates
, as follows:
 

 
 
 (3) 
The associated metric tensor has the matrix
 
(4) 
where .
The metric determinant takes the form
 
(5) 
Substituting equations (4) and (5) into
(1), we can modify the Helmholtz wave equation
for propagating waves in a 3D Riemannian space:
 
(6) 
In equation (6),
is temporal frequency,
is the wave propagation velocity, and
represents a propagating wave.
For the special case of two dimensional spaces
(F=0 and G=1),
the Helmholtz wave equation reduces to the simpler form:
 
(7) 
which corresponds to a curvilinear orthogonal coordinate system.
Particular examples of coordinate systems for oneway wave propagation are:
 Cartesian (propagation in depth):
 , , ,
 Cylindrical (propagation in radius):
 , , ,
 Spherical (propagation in radius):
 , , ,
 Ray family (propagation along rays):
 and represent
parameters defining a particular ray in the family (i.e. the ray takeoff
angles), J is the geometrical spreading factor, related to the
crosssectional area of the ray tube Cervený (2001). The coefficients
E, F, G, and J are easily computed by
finitedifference approximations
with the Huygens wavefront tracing technique Sava and Fomel (2001). If
the propagation parameter is taken to be time along the ray, then
equals the propagation velocity v.
Next: Oneway waveequation in 3D
Up: Sava and Fomel: Riemannian
Previous: Introduction
Stanford Exploration Project
10/14/2003