Two point raytracing for reflection off a 3D plane

Globally convergent Newton's method for ray shooting

Quite some time ago, Bob Keyes at Mobil mentioned that Newton's method applied to shooting rays to solve the two-point problem in horizontally layered media is globally convergent, assuming, of course, that there is a solution. Specifically, there must be a solution if an initial guess at the ray parameter overshoots the target.

Formally, let the ray parameter $p$ be in the open interval $(0,1/v_{max})$ . Starting from the origin, Snell's law $pv = \sin \theta$ says that

$$x = \int_0^z \tan \theta \, dz = \int_0^z \frac{pv}{{(1-p^2 v^2)}^{1/2}} \, dz$$

gives the horizontal displacement of the ray from the origin when it reaches depth $z$ . Taking two derivatives of this formula with respect to $p$ , we have

$$\frac{d x}{dp} = \int_0^z \frac{v}{{(1-p^2 v^2)}^{3/2}} \, dz$$    , $$\frac{d^2 x}{dp^2} = \int_0^z \frac{3pv^3}{{(1-p^2 v^2)}^{5/2}} \, dz$$    .

At a glance one sees that the second derivative is a quantity guaranteed to be positive in $(0,1/v_{max})$ . By Thorlund-Petersen (2004), Newton's method applied to finding the $p$ for a ray that reaches a given $x$ at given depth $z$ is therefore globally convergent. (Technically, we do need to ensure that the Newton update doesn't overshoot the range $(0,1/v_{max})$ .)

Two point raytracing for reflection off a 3D plane