## Numerical comparison of the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory

**Petr Bulant** ** & **
**Ludek Klimes**
### Summary

In the common-ray approximation of the coupling ray theory, the
S-wave traveltimes in the coupling equations are approximated by the
first-order perturbation expansion from the common reference ray.
The common-ray approximation eliminates problems with ray tracing
through S-wave singularities and also considerably simplifies the
numerical algorithm of the coupling ray theory for S waves, but may
introduce errors in traveltimes due to the perturbation from the
common reference ray. The errors of S-wave traveltimes due to the
common-ray approximation may be approximated by the second-order
terms in the perturbation expansion. The first-order and
second-order terms in the perturbation expansion can be calculated by
simple numerical quadratures along the common reference ray.

The common reference ray may be represented by the isotropic common
ray or by the anisotropic common ray. The equations for the
first-order and second-order perturbation expansions of traveltime
from both the isotropic and anisotropic common ray to the
anisotropic-ray-theory rays have already been derived. We
numerically test the equations in three smooth 1-D velocity models of
differing degrees of anisotropy.

We first compare the first-order and second-order perturbation
expansions of traveltime from the isotropic common rays and from the
anisotropic common rays to the anisotropic-ray-theory rays with the
correct anisotropic-ray-theory traveltimes in order to illustrate
the accuracy of the perturbation expansions and reliability of the
error estimates.

Then, the synthetic seismograms obtained by the isotropic-common-ray
and anisotropic-common-ray approximations of the coupling ray theory
are compared with the more accurate coupling-ray-theory synthetic
seismograms simulated by the second-order perturbation expansion of
traveltime from the anisotropic common rays. In the numerical
examples, the errors of the anisotropic-common-ray approximation of
the coupling ray theory are considerably smaller than the errors of
the isotropic-common-ray approximation.
### Whole paper

The expanded abstract is available in
PostScript (4 334 kB !) and
GZIPped PostScript (990 kB).

*Expanded Abstracts of 10th Int. Congress of SBGf (Rio de Janeiro)
(CD-ROM)*, Brazilian Geophysical Society, 2007, SBGF0329_07 (4 pp.).

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