INITIAL VALUE RAY TRACING

When the medium is horizontally layered and isotropic, tracing a ray across the different interfaces is easy. From the medium velocities and the direction of the ray at the source (p), successive applications of (4) tells us how a ray changes its direction as it travels. The only angles that matter are those of the incident and refracted rays. If the layers are dipping the problem is slightly more complicated because p has the be calculated every time the ray reaches a new interface and we have to deal with three instead of two angles at each boundary.

When the medium is anisotropic, we have to deal with seven angles at each interface to figure out how a ray changes its direction. All these angles (positive counterclockwise) are shown in Figure . The continuity of the slowness across the interface (equation 3) is expressed as follows:

$$p \ =\ \frac{ \sin (\theta_i + \gamma_1)}{v_1( \theta_i )} \ =\ \frac{ \sin (\theta_t + \gamma_2)}{v_2( \theta_t )} \,$$ (5)

where $v_1( \theta_i )$ and $v_2( \theta_t)$ are the phase velocities in the corresponding media evaluated at the incident and transmitted phase angles respectively. The sums $(\theta_i + \gamma_1)$ and $(\theta_t + \gamma_2)$ have to be done considering the proper signs of the angles. From equation (5) we notice that when p=0 (normal incidence) the angle of the transmitted phase does not depend on the phase velocity and is $\theta_t = - \gamma_2$. Any other relation derived from (5) must reproduce this simple result.

Given the function $v(\theta)$ for each medium, equation (5) tells us how the incident phase ${\bf \tilde{k}}$ changes its direction when crossing the interface between two media. However, it does not say what happens to the direction and velocity of the associated ray. These two quantities can be calculated from the equations that relate phase and ray velocities (Byun, 1984):

$$\begin{eqnarray} v (\theta) & = & V(\phi) \cos(\phi-\theta) \\ \tan (\phi - \t... ...& = & v^2 (\theta) + \left( \frac{dv (\theta)}{d \theta} \right)^2\end{eqnarray}$$ (1) (2) (3)

where $V ( \phi )$ is the ray velocity along the ray angle $\phi$.

 

all-angles
Figure 3 Incident (i) and transmitted (t) group and phase angles of a ray impinging on a plane interface between two transversely isotropic media.

Equations (5) and (6) are the basic relations to trace a ray in a medium where $v(\theta)$ and $\gamma$ are given for each layer. Byun (1984) proposes the following procedure to do that: (1) Assign a value to $\theta = \theta_i$ at the source position. (2) Repeat the next steps for each layer: (a) Evaluate $v(\theta)$ and $\frac{dv(\theta)} {d \theta}$ at the corresponding $\theta$. (b) Find $\phi_t$ and $V (\phi_t)$ form equation (6). (c) Trace a ray along the angle $\phi_t$until it reaches the next interface. (d) Find p of the incident wave using equation (5). (e) Solve equation (5) for $\theta_t$. (3) Go to step 2-a.

The previous ray tracing procedure is valid for any phase velocity function $v(\theta)$. What changes from one choice of the velocity function to another is how to solve step 2-e (to find $\theta_t$ as a function of p using equation (5)). The next section focuses on this step when the medium is transversely isotropic.