(1) |

The Laplacian operator acting on a scalar function in an arbitrary Riemannian space with coordinates takes the form

(2) |

The expression simplifies if one of the coordinates, e.g. the coordinate of one-way wave extrapolation ,is orthogonal to the other coordinates . The metric tensor reduces to

(3) |

E& = & _k _k _k ,

F& = & _k _k _k ,

G& = & _k _k _k ,

Å^2 & = & _k _k _k .

The associated metric tensor has the matrix

(4) |

(5) |

Substituting ametric and det into laplac, and making the notations , , and , with orthogonal to both and ,we obtain the Helmholtz wave helm for propagating waves in a 3D semi-orthogonal Riemannian space: 1ÅJ

JÅ + GÅJ - FÅJ + EÅJ - FÅJ

= - ^2v^2 .

In weqrc.3d, is the wave propagation velocity mapped to Riemannian coordinates.

For the special case of two dimensional spaces
(*F*=0 and *G*=1),
the Helmholtz wave equation reduces to the simpler form:

(6) |

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 (19). The coefficients*E*,*F*,*G*, and*J*are easily computed by finite-difference approximations with the Huygens wavefront tracing technique (92). If the propagation parameter is taken to be time along the ray, then equals the propagation velocity*v*.

11/4/2004