## First-order ray computations of coupled S waves
in inhomogeneous weakly anisotropic media

**Veronique Farra** **&**
**Ivan Psencik**
### Summary

We propose an approximate procedure for computing coupled S waves in inhomogeneous
weakly anisotropic media. The procedure is based on the first-order ray tracing (FORT) and
dynamic ray tracing (FODRT), which was originally developed for P waves. We use the
so-called common ray tracing concept to derive approximate ray tracing and dynamic ray
tracing equations, and an approximate solution of the transport equation for coupled S waves
propagating in laterally varying, weakly anisotropic media. In our common ray tracing, ray
equations are governed by the first-order Hamiltonian formed by the average of first-order
eigenvalues of the Christoffel matrix, corresponding to the two S-wave modes propagating
in anisotropic media. The solution of the transport equation for the coupled S waves leads
to a system of two coupled frequency-dependent, linear ordinary differential equations for
amplitude coefficients, which is evaluated along the S-wave common ray. For derivation of the
FORT and FODRT equations, we use the perturbation theory in which deviations of anisotropy
from isotropy are considered to be first-order perturbations. To derive the coupled differential
equations for S-wave amplitudes, we assume that the first-order perturbations are of order
O(*omega*^{-1}), where *omega* is the circular frequency. This makes it possible to express the amplitude
coefficients in the coupled differential equations in terms of geometrical spreading and other
quantities related to the common ray. The proposed procedure removes problems of most
currently available ray tracers, which yield distorted results or even collapse when shear waves
propagating in weakly anisotropic media are computed. The first-order approximation leads to
simpler ray tracing and dynamic ray tracing equations than in the exact case. For anisotropic
media of higher-symmetry than monoclinic, all equations involved differ only slightly from the
corresponding equations for isotropic media. If the anisotropy vanishes, the equations reduce
to standard, exact ray tracing and dynamic ray tracing equations for S waves propagating in
isotropic media. The proposed ray tracing and dynamic ray tracing equations, corresponding
traveltimes and geometrical spreading are all given to the first order. The accuracy of the
traveltimes along S-wave first-order common rays can be increased by calculating a secondorder
traveltime correction.

### Keywords

Body waves, seismic anisotropy, wave propagation.

*Geophys. J. int.*, **173** (2008), 979-989.

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