Next: One-way wave-equation in 3-D Up: Sava and Fomel: Riemannian Previous: Introduction

# Acoustic wave-equation in 3-D Riemannian spaces

The Laplacian operator of a scalar function in an arbitrary Riemannian space with coordinates has the form
 (1)
where gij 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 one-way 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 3-D 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 one-way 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 take-off angles), J is the geometrical spreading factor, related to the cross-sectional area of the ray tube Cervený (2001). The coefficients E, F, G, and J are easily computed by finite-difference 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: One-way wave-equation in 3-D Up: Sava and Fomel: Riemannian Previous: Introduction
Stanford Exploration Project
10/14/2003