Two point raytracing for reflection off a 3D plane

Converted wave reflection

The same approach applies to $P$ -to-$S$ or $S$ -to-$P$ reflection as well with one important difference--the angle of reflection differs from the angle of incidence. Now

$$\begin{array}{ccc} ( R - P ) & = & \alpha ( {\bf n} + {\bf w... ...\bf w} & = & 0 \\ {\bf n} \cdot (P - P_0 ) & = & 0 \end{array}$$

for some scalar $\zeta$ . To determined $\zeta$ let $v_s$ and $v_r$ be the velocities of the source and receiver paths respectively and $\theta_s$ and $\theta_r$ be the corresponding angles of incidence and reflection. Then Snell's Law says

$$\frac{\sin \theta_s}{v_s} = \frac{\sin \theta_r}{v_r}$$    .

By our definition of ${\bf w}$ , we also have the identities

$$\begin{array}{ccc} \vert{\bf w}\vert & = & \tan \theta_r \\ \vert\zeta {\bf w}\vert & = & \tan \theta_s \end{array}$$

which, using the identity,

$$\sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}$$ ,

gives the relation for $\zeta$

$$\frac{1}{\zeta^2} = \left(\frac{v_r}{v_s}\right)^2 + \left({\,\left(\frac{v_r}{v_s}\right)^2 - 1}\right) \vert{\bf w}\vert^2$$

which, combined with

$$( R - S ) = ( \alpha - \beta ) {\bf n} + ( \alpha + \zeta \beta ) {\bf w}$$    ,

produces a fourth order equation for $\zeta$ .

The fourth order equation can be solved directly using algebraic formulas. Lanczos (1956) provides a clean, efficient numerical approximation, reproduced in Appendix A, that is about 10 times faster than using a general purpose numerical root finder. (Appendix B shows how to make it free of floating point divisions.)

An interesting alternative to direct solution is to apply Newton iterations to the shooting method wherein source ray parameters are repeatedly adjusted to return very near to the target receiver. This approach applies to multiple layers and multiple reflections, not just a single interface. In Appendix C, I demonstrate global convergence of that method when applied to forward ray tracing through a stack of horizontal layers.

Two point raytracing for reflection off a 3D plane