SH waves

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 $\phi_t$ is simpler, because the phase angle $\theta_t$ 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

$$p \ =\ \frac{v_{\parallel}^{-2} \sin (\gamma_2) + v_{\perp}^... ...\sqrt{v_{\parallel}^{-2} + v_{\perp}^{-2} \tan^2 (\gamma_2)}}.$$ (10)

where $v_{\parallel}$ and $v_{\perp}$ are the SH wave velocities in the direction parallel and perpendicular to the axis of symmetry. After some algebra it is possible to solve for the ray angle $\phi_t$resulting

$$\tan (\phi_t) \ =\ \left( \frac{v_{\perp}}{v_{\parallel}} \r... ...{\parallel} v_{\perp})^2}} {p^2 v_{\perp}^2 - \cos^2 (\gamma)}.$$ (11)

Since equation (11) is independent of the sign of p, it gives the solutions of (10) for both positive and negative p. The correct $\phi_t$ is the one that gives the expected sign of p and correspond to a ray in the desired direction.

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.

 

which-root
Figure 5 Roots of equation (5) (scattered phase angles) for negative p when the slowness surface is elliptical. Only the slowness surface in medium 2 is shown. In each case, the incident p is measured from the center of the ellipse to the vertical line. Continuous arrows show the scattered phase directions that correspond to transmitted rays (dashed arrows with black head). Dashed arrows with white head are ray directions with the same p but pointing towards medium 1. (a) Most common case. (b) Normal incidence (p=0). (c) Phase direction parallel to the interface, ray not. (d) Two phase directions less than 90 degrees. (e) Ray direction parallel to the interface, phase not. (f). Both ray and phase directions parallel to the interface. (g) Two phase directions greater than 90 degrees. (h) One ray goes backwards.

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.