Two point raytracing for reflection off a 3D plane |

for some scalar . To determined let and be the velocities of the source and receiver paths respectively and and be the corresponding angles of incidence and reflection. Then Snell's Law says

.

By our definition of , we also have the identities

which, using the identity,

,

gives the relation for

which, combined with

,

produces a fourth order equation for .

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 |

2012-10-29