Raytracing
Introduction
Raytracing is one of the first theories of light propagation to ever be developed. A ray, conceptually, represents the flow of power or the path taken by "packet" of electromagnetic energy through a system. Raytracing, more formally, is a reduction of Maxwell's equations to a much simpler and more computationally tractable form, under fairly modest assumptions of smooth variation.
Raytracing has been used to study wave propagation and phenomena in the ionosphere and magnetosphere for over 50 years [Kimura(1966), Haselgrove(1955)]. Initially, the primary motivator was computational tractability. The cost in computing a ray path in the magnetosphere is relatively small compared to the cost of a global solution like that obtained using finite difference or finite element methods. Initial investigations were even carried out graphically [Maeda and Kimura(1956)].
In a modern context, however, raytracing is still a very important computational tool for investigating magnetospheric phenomena. Due to the enormous length scales (many megameters) and timescales (10s of seconds), full simulations of wave propagation over the entire magnetosphere are still difficult if not intractable on modern computers.
Modeling efforts
The Stanford VLF group's raytracing modeling efforts include: Flexible raytracing simulation: The Stanford 3D ray tracer can accommodate any arbitrary magnetic field or plasma density function, opening up the possibility to study the role of largescale inhomogeneities, like plumes or notches, in the propagation of VLF waves.
 Modeling of loss and growth processes: Accurate modeling of loss processes, like Landau and cyclotron damping, is essential to accurate interpretation of raytracing results. The Stanford VLF ray tracer can calculate the losses for any arbitrary distribution function distributed over space, including observationbased models.
 Physicallybased and observationbased models: We have integrated the IGRF, Tsyganenko, GCPM, and our own radial diffusion model in our raytracer.
VLF Raytracing demonstrations
The VLF group at Stanford uses a 3D magnetospheric raytracer to study many different wave propagation effects in the inner magnetosphere. Ray paths in HF (as shown in Figure 1) show characteristic total internal reflection off the ionosphere, an effect that is used for longdistance HF communication.

At VLF frequencies, however, the phenomena encountered are much richer  rays can reflect off specular boundaries, magnetospherically reflect, or even significantly drift in longitude, as shown in Figures 2 and 3.


A short movie here illustrates how VLF waves can propagate and reflect in the magnetosphere, in real time:
The panel on the left shows the refractive index surface, which is a plot of the refractive index as a function of direction. The arrow in blue on this plot shows the group velocity, or the direction of the ray. On the right, we show the actual ray path superimposed on a modelbased electron density distribution with a very sharp plasmasphere "knee". The red arrow shows the direction of the wavenormal, which in a plasma is not, in general, coincident with the group velocity. A few interesting features can be seen clearly in this visualization. Four of the reflections are termed MR (magnetospheric reflections) and occur because the refractive index surface's topology changes from opened to closed, permitting an abrupt change in the group velocity for a relatively small change in the wavenormal at nearperpendicular propagation. The remaining two are pseudospecular reflections at the sharp plasmasphere boundary.
Bibliography
 Haselgrove(1955)
 Haselgrove, J. (1955), Ray Theory and a New Method for Ray Tracing, in Physics of the Ionosphere, pp. 355+.
 Kimura(1966)
 Kimura, I. (1966), Effects of Ions on WhistlerMode Ray Tracing, Radio Science, pp. 269283.
 Maeda and Kimura(1956)
 Maeda, K., and I. Kimura (1956), A Theoretical Investigation on the Propagation Path of the Whistling Atmospherics, Rept. Ionosphere Res., pp. 105123.