Progress in radio ray tracing in the ionosphere

The theoretical basis of ray tracing as part of the construction of an approximate solution of Maxwell's equations is discussed. The rays can then be seen as arising as characteristic curves of a partial differential equation for the phase of the wave. If carried through in detail, this approach leads to an additional phase shift that has been discussed previously. Methods of efficiently calculating field strength, Doppler shift, and absorption of the waves are discussed, based on a variational formulation. Some history of the evolution of ionospheric radio ray tracing is given. Numerical and analytic or closed-form ray tracing are contrasted, and guidance is given on the choice of appropriate technique in a particular application. Finally, some suggestions are given for possible useful extensions of the ray-tracing approach.