Quasi‐shear wave coupling in weakly anisotropic 3‐D media

SUMMARY Zeroth‐order ray theory is frequently used to calculate synthetic seismograms in media which are both anisotropic and inhomogeneous. One of the principal features of such media is that the polarization vectors of the two quasi‐shear ( q S) waves are determined by the nature of the anisotropy. Thus, a shear wave entering a region of anisotropy will generally be split into two separate polarizations. Ray theory predicts that these two waves will propagate independently, at different velocities, throughout the anisotropic region. Ray theory solutions also show that in inhomogeneous media, the polarization vectors will rotate along the ray. The rotations of these polarization vectors are strongly influenced by the symmetry and orientation of the anisotropy system, but only weakly depend upon the strength of the anisotropy. In contrast, in isotropic media the polarization of S ‐waves is determined from the initial conditions and only varies slowly due to the ray curvature. The polarization only changes in the ray direction and at any point does not rotate about the ray. In this paper we show that in the limit of infinitely weak anisotropy, solutions calculated using ray theory in anisotropic media conflict with the known results calculated for a similar isotropic medium. We show this fundamental breakdown in ray theory occurs because coupling between the qS waves is ignored in the zeroth approximation. Thus, the isotropic limit is not equivalent to the high‐frequency limit of anisotropic ray theory. The coupling is particularly important in weakly anisotropic media, where the qS velocities are similar, but the same effect is still present in media exhibiting stronger anisotropy. This coupling must be taken into account when calculating waveforms. We show that this coupling may be modelled by treating the ‘error’ terms, produced by substituting a zeroth‐order ray theory Green's function into the wave equation, as source terms distributed throughout the medium. For weakly anisotropic media where the qS ray paths are similar, this volume integral may be simplified using perturbation and asymptotic methods and evaluated as a simple integral along the ray path. In the isotropic limit this expression correctly describes the polarization of shear waves along the ray. This integral is easy to compute, requiring only quantities already used in ray tracing and traveltime calculations. A prior knowledge of the location, or even the existence of kiss, intersection, point or other singularities along the ray path, is not required for the method to give accurate results. We present some numerical examples for some simple cases previously investigated by less general or more expensive techniques.