A graphical method for the construction of rays and travel times in spherically layered media
Authors: Helbig Klaus 
Published: July 2021

abstract
The graphical method developed in the first part of this series of articles is generalized to shells of ‘tangential isotropy’, the spherical analog to transverse isotropy. In order to obtain as complete a list as possible of deviations from waves in isotropic shells, all mathematically possible types of tangential isotropy are investigated, irrespective of their likeliness to occur in some part of the earth. The quantitative background for the classification of transversely anisotropic media implied here is presented in an appendix.
In tangentially isotropic spheres, rays are curved even if the material has the same elastic properties everywhere (with the exception of a degenerate case, where SV-rays are straight lines).
The geometric shape of SH-rays is controlled by a single parameter. Their curvature has the same sense throughout, and the corresponding t(Δ) curves measured on the sphere are, as on isotropic homogeneous spheres, quarter sine waves, which only differ in range.
P-rays generally change their sense of curvature twice, and the wave front charts are considerably more complicated than those of SH-waves. The travel-time curves differ in details from those in isotropic spheres or those of SH-waves, but the general appearance is the same.
The most striking differences occur for SV-rays, particularly if the slowness surface of the medium has points of inflection. If the concave part lies about π/4 from the axis of symmetry, cusps in the wave front are formed in the vicinity of the focus. For some distance rays cross over and back, but finally the cusps disappear. There are no major indications of these cusps present at the surface or in the t(Δ)-curve. If the concave part lies in the vicinity of the axis of symmetry, the t(Δ)-curve ends in a receding branch. Correspondingly, the surface of the sphere consists of a singly-covered zone, a doubly-covered zone, and a shadow zone. A concave section near the plane of symmetry of the slowness surface is tantamount to the coexistence of two different SV-rays with different direction of wave normal and polarization; these two rays travel along different paths, but cover the same distance in the same time and thus correspond to a single valued branch of the time versus distance curve.
If one or more spherical shells are tangentially isotropic, refraction and reflection at the interfaces cause further deviations from wave propagation in isotropic shells since Snell's law refers to wave normals, which in anisotropic media do not generally coincide with the direction of the ray. A concave section close to the plane of symmetry of the slowness surface of SV-waves can lead to a ray traveling along the inner boundary of the shell. Unlike channel rays this ray loses energy to normal rays and thus should lead to “ghosts” of diminishing strength in the corresponding parts of the time versus distance curves. For shells with sufficiently narrow slowness graphs there exists in addition a double refraction for SV-rays traversing the shell which should lead to multi-valued branches of the time versus distance curve.