There are two different high-frequency asymptotic ray theories, the "isotropic ray theory" assuming equal velocities of both S-wave polarizations and the "anisotropic ray theory" assuming both S-wave polarizations strictly decoupled. In the isotropic ray theory, the S-wave polarization vectors do not rotate about the ray, whereas in the anisotropic ray theory they coincide with the eigenvectors of the Christoffel matrix which may rotate rapidly about the ray.
In weakly anisotropic models, at moderate frequencies, the S-wave polarization tends to remain unrotated round the ray but is partly attracted by the rotation of the eigenvectors of the Christoffel matrix. The intensity of the attraction increases with frequency. The isotropic and anisotropic ray theories are thus limiting cases and the gap between them has to be filled. A ray theory providing continuous transition between the isotropic and anisotropic ray theories is called the "coupling ray theory".
There are many possible modifications and approximations of the coupling ray theory. For example, the reference ray may be calculated in different ways, the Christoffel matrix may be approximated by its quasi-isotropic projections onto the plane perpendicular to the reference ray and onto the line tangent to the reference ray, travel times corresponding to the anisotropic ray theory may be approximated in several ways, e.g., by linear quasi-isotropic perturbation with respect to the density-normalized elastic moduli, etc.
Commonly used quasi-isotropic approximations of the coupling ray theory are discussed. The coupling ray theory and its quasi-isotropic approximations are then numerically compared with the exact analytical solution for the plane S wave, propagating along the axis of spirality in the 1-D anisotropic "oblique twisted crystal" model.
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