Application of the optimization method to the point-to-point radio wave ray-tracing problem

Point-to-point ray tracing is an important problem in many fields of science. In direct variational methods, some trajectory is transformed to an optimal trajectory. While these methods are routinely used in calculations of pathways of seismic waves, chemical reactions, diffusion processes, etc., these approaches are not widely known in ionospheric point-to-point ray tracing. A two-dimensional representation of the optical path functional is developed, and used to gain insight into the fundamental difference between high and low ionospheric rays. We conclude that high and low rays are minima and saddle points of the optical path functional, respectively. An optimization method for point-to-point ionospheric ray tracing, based on the direct variational principle for the optical path, is proposed. This is applied to calculations of high, trans-ionospheric, and multi-hop rays.