SH waves in a transversely isotropic medium have elliptical slowness and wave surfaces. An elliptical slowness surface may not be a good global (ie., for all angles) approximation for a P or SV slowness surface in a transversely isotropic medium, but it is a good paraxial approximation (Dellinger and Muir, 1991). Hence the importance of elliptical anisotropy in data fitting, specially when the measurements have limited aperture, as explain by Dellinger Muir (1991) and Michelena (1992).
Byun (1982) shows that when the slowness surfaces are elliptical the calculation at each interface of the refracted ray angle is simpler, because the phase angle does not need to be explicitly computed. Byun's expression for the ray parameter as a function of the ray is angle at each interface is
If we take the extra step of solving explicitly for the phase angle at each interface, we have to look among the four different points in the slowness surface whose angles satisfy equation (5) and keep the one that produces a ray pointing in the desired direction. The ``desired direction'' depends on the type of problem where the ray tracer is used. Figure shows how to pick the roots that correspond to transmitted rays, the ones needed in transmission traveltime tomography.
In this figure, the horizontal distance from the center of the each ellipse to the vertical line is equal to the incident p. As we see, to obtain transmitted rays (dashed arrows with black head), it is enough to pick the root that produces a ray pointing towards medium 2. In some cases the transmitted ray can travel parallel to the interface with the phase direction (continuous arrows) parallel to it or not (Figures -f and -e respectively). The emergent phase angle may be greater than 90 degrees for the transmitted ray, as shown in Figure -g. The ray pointing towards medium 2 may also go backwards, as Figure -h shows. When the vertical line doesn't intersects the ellipse, there are not transmitted but only reflected rays (angle of incidence greater than the critical angle). However, in order to keep track more easily of the rays that hit the interface at post-critical angles, I will assume that those rays continue traveling along the interface. The inversion procedure I describe in a separate paper (Michelena, 1992) doesn't use these rays.