## 1. Introduction

[2] Geometric optics is the appropriate limit of electromagnetism for the study of many radio wave propagation problems at high frequencies (HF) and above. In this limit, the propagation is described in terms of rays and the major task is that of ray tracing. In the case of ionospheric propagation, the raypaths are often heavily distorted and quite sophisticated ray tracing techniques can be required (usually of a numerical variety). Since their introduction more than 50 years ago, the equations of Haselgrove [*Haselgrove*, 1955, 1963; *Haselgrove and Haselgrove*, 1960] have provided the main tool for ray tracing. These equations are ideally suited to the situation where the initial point and launch direction of the ray are specified, but the final point is unknown. When the initial and final points of the ray are specified, the standard approach is to send out rays in a variety of directions and use their landing points to estimate the direction for which the ray would land at the desired end point. These predictions are steadily refined until the desired landing point is achieved. This ‘homing-in’ approach has been well developed by several authors [*Reilly*, 1991; *Strangeways*, 2000; *Vasterberg*, 1997] and is the major approach to point-to-point ionospheric ray tracing at the current time. It is to be noted, however, that the Haselgrove equations can themselves be derived from a variational principle [*Haselgrove and Haselgrove*, 1960] and such principles are well suited to problems where the end points are known. Through direct methods such as Rayleigh-Ritz [*Fox*, 2010], variational principles can provide effective approximate solutions to such problems. Such an approach is well known in seismology where it is often called the bending method [e.g., *Cerveny*, 2001]. There have been several important developments of this approach including the pseudo-bending method [*Um and Thurber*, 1987] and methods that overcome convergence problems using genetic algorithms [*Sadeghi et al*., 1999]. In the case of ionospheric propagation, magneto-ionic effects introduce additional complications that make variational techniques difficult to implement. *Smilauer* [1970] has considered an approach in which he derives some ordinary differential equations from the variational principle and then solves these equations by a Galerkin technique. In the present work, however, we take an alternative approach in which we discretize the functional before variation. We consider a set of candidate raypaths that are labeled by a finite set of parameters and use these to directly turn the functional, used by *Haselgrove* [1955], into a function of these parameters. The approximate raypath is then found by using the parameters at a stationary point of this function. For paths with only slight deviation from linear (propagation well above HF) this is a fairly easy proposition. For the heavily distorted rays of HF propagation, however, this posses a myriad of technical difficulties. In this paper we show that the technical difficulties can be overcome and that this direct method can indeed provide effective point-to-point ray tracing.

[3] Section 2 of this paper discusses Fermat's principle, the basic variational principle of ray tracing. It is shown that a partial solution of the Haselgrove equations can be used to reduce this variational principle to a form that is more convenient for studying ray tracing. Section 3 discusses the techniques whereby the variational principle can be discretized and the resulting discretized equations solved. Some examples of direct ray tracing are given in section 4 and there is also some discussion of the convergence and accuracy of the direct approach, including comparisons with ray tracing via the Haselgrove equation method. Section 5 discusses some extensions of the method to trans-ionospheric ray tracing and multihop ray tracing. Section 6 concludes the paper and discusses some possible future directions.