First-order ray computations of coupledSwaves in inhomogeneous weakly anisotropic media

Abstract

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(ω −1 ), where ω 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.