First-order S-wave ray computations in inhomogeneous weakly anisotropic media

Veronique Farra and Ivan Psencik


We propose an extension for S waves of the first-order ray tracing (FORT) and dynamic ray tracing (FODRT), which was originally developed for P waves. We present approximate ray-tracing and dynamic ray-tracing equations and approximate solution of the transport equation for coupled S waves propagating in laterally varying, weakly anisotropic media. For their derivation we use the so-called common S-wave ray-tracing concept. In it ray equations are governed by the Hamiltonian formed by averaging eigenvalues of the Christoffel matrix, corresponding to the two S-wave modes. Solution of the transport equation assuming the common S wave leads to the system of two coupled frequency-dependent, ordinary differential equations for S-wave amplitude coefficients, which should be solved along a common ray.

In contrast to the standard common S-wave ray tracing, we work with the first-order approximations of the exact S-wave eigenvalues. We use the perturbation theory, in which deviations of anisotropy from isotropy are considered to be first-order perturbations, of order O-1), where ω is the circular frequency. This makes possible to interprete the coefficients of the coupled differential equations for S-wave amplitudes in terms of the geometrical spreading and other quantities related to the common ray. First-order approximation also leads to considerably simpler ray tracing and dynamic ray tracing equations than the exact ones. Although it is not shown explicitly here, for anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to standard, exact ray-tracing and dynamic ray-tracing equations for isotropic media.

The proposed ray-tracing and dynamic ray-tracing equations, corresponding traveltimes and the geometrical spreading are all valid to the first order. Accuracy of the traveltimes can be simply increased by calculating a second-order correction along first-order common S-wave rays. This extends considerably the applicability of the proposed procedure.


Inhomogeneous media, common S wave, coupling, perturbation methods, weak anisotropy.

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In: Seismic Waves in Complex 3-D Structures, Report 17, pp. 135-148, Dep. Geophys., Charles Univ., Prague, 2007.
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