**
COMMON RAY APPROXIMATIONS OF THE COUPLING RAY THEORY
**

*
Petr Bulant & Ludek Klimes
*

*
Department of Geophysics,
Faculty of Mathematics and Physics,
Charles University
Ke Karlovu 3,
121 16 Praha 2,
Czech Republic
*

The coupling ray theory (Coates & Chapman 1990) provides a continuous transition between the isotropic and anisotropic ray theories, and is particularly important at degrees of anisotropy and frequencies typical in seismic exploration and structural seismology on all scales. This contribution reviews the recent developments in the "common ray approximations" of the coupling ray theory.

The coupling ray theory is applicable to all degrees of anisotropy
if the initial vectorial S-wave amplitude is decomposed
into the eigenvectors of the Christoffel matrix,
and if the coupling equations are solved along the
*anisotropic-ray-theory reference rays*.

In the *common ray approximation*,
only one reference ray is traced
for both anisotropic-ray-theory S waves,
and both S-wave anisotropic-ray-theory travel times are
approximated by the first-order perturbation expansion
from the common reference ray (Bulant & Klimes, 2002).
The common ray approximation thus considerably simplifies
the coding of the coupling ray theory and numerical calculations,
but may introduce errors in travel times due to the first-order
perturbation.
The accuracy of the common ray approximation depends
considerably on the selection of the reference rays.
The accuracy is much worse for the *isotropic common rays*
calculated in the isotropic reference model,
than for the *anisotropic common rays*,
proposed by Bakker (2002).
The anisotropic common rays are traced in the anisotropic model,
using the averaged Hamiltonian of both anisotropic-ray-theory S waves.
The algorithm of the dynamic ray tracing corresponding to
the anisotropic common rays has been proposed by Klimes (2003).
The coupling ray theory along anisotropic common rays
has been numerically tested and compared with the
isotropic common ray approximation by Bulant & Klimes (2004).

Since the anisotropic-ray-theory travel times in the coupling equations are approximated along the reference ray by the first-order perturbation expansion, the error of the common ray approximation may be estimated using the second-order term in the perturbation expansion of the anisotropic-ray-theory travel times. This second-order term can be calculated by simple numerical quadratures along the reference ray (Klimes, 2002). Klimes & Bulant (2004) have estimated the accuracy of both isotropic common ray and anisotropic common ray approximations by quadratures along isotropic common rays. Analogously, the accuracy of both common ray approximations can be estimated by quadratures along anisotropic common rays.

**
References
**

Bakker, P.M.
(2002):
Coupled anisotropic shear wave raytracing in situations where associated slowness sheets are almost tangent.
*Pure appl. Geophys.*, **159**, 1403-1417.

Bulant, P. & Klimes, L.
(2002):
Numerical algorithm of the coupling ray theory in weakly anisotropic media.
*Pure appl. Geophys.*, **159**, 1419-1435.

Bulant, P. & Klimes, L.
(2004):
Anisotropic common ray approximation of the coupling ray theory.
In: *Seismic Waves in Complex 3-D Structures, Report 14*, pp. 107-122,
Dep. Geophys., Charles Univ., Prague.

Coates, R.T. & Chapman, C.H.
(1990):
Quasi-shear wave coupling in weakly anisotropic 3-D media.
*Geophys. J. int.*, **103**, 301-320.

Klimes, L.
(2002):
Second-order and higher-order perturbations of travel time in isotropic and anisotropic media.
*Stud. geophys. geod.*, **46**, 213-248.

Klimes, L.
(2003):
Common ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium.
In: *Seismic Waves in Complex 3-D Structures, Report 13*, pp. 119-141,
Dep. Geophys., Charles Univ., Prague.

Klimes, L. & Bulant, P.
(2004):
Errors due to the common ray approximations of the coupling ray theory.
*Stud. geophys. geod.*, **48**, 117-142.