In order to simplify the computations, I introduce the following notation:

&=& 1 Å^2 ,

&=& GJ^2 ,

&=& EJ^2 ,

&=& FJ^2 ,

&=& 1ÅJ JÅ ,

&=& 1ÅJ
GÅJ -
FÅJ ,

&=& 1ÅJ
EÅJ -
FÅJ .
All quantities in coefs.3d
are only function of the chosen coordinate system, and
do not depend on the extrapolated wavefield.
They can be computed by finite-differences for any choice of
Riemannian coordinates
which fulfill the orthogonality condition indicated earlier.
In particular, we can use ray coordinates to compute those
coefficients. With these notations, the acoustic wave-equation
can be written as:

(7) |

(8) |

From weqrc.3d.coefs, we can directly deduce the modified form of the dispersion relation for the wave-equation in a semi-orthogonal 3D Riemannian space:

(9) |

(10) |

For the particular case of Cartesian coordinates (), the one-way wavefield extrapolation equation takes the familiar form

(11) |

We can use the wavenumber for recursive wavefield extrapolation of the data recorded on the acquisition surface using the relation

(12) |

We can simplify the Riemannian wavefield extrapolation method by dropping the first-order terms in oneway.3d. According to the theory of characteristics for second-order hyperbolic equations (27), these terms affect only the amplitude of the propagating waves. To preserve the kinematics, it is sufficient to keep only the second order terms of oneway.3d:

(13) |

The coordinate system coefficients for
Riemannian wavefield extrapolation given by coefs.3d
have singularities at caustics,
e.g., when the geometrical spreading term *J*,
defining a cross-sectional area of a ray tube, goes to zero.
In the following examples, I use simple numerical
regularization to avoid this problem,
by adding a small non-zero quantity to the
denominators to avoid division by zero.

11/4/2004