Point-to-point ionospheric ray tracing by a direct variational method

While the Haselgrove ray tracing equations are well suited to situations where the ray launch direction is known, they are less effective for situations where only the end points of the ray are known. In such cases, many rays must be traced from the launch point in order to home in on the landing point. An alternative approach is to directly solve the variational principle from which the Haselgrove equations are derived. Such an approach is well suited to the point-to-point ray tracing, but poses several technical difficulties. In this paper we overcome these difficulties and show that a direct approach can indeed provide an effective means of point-to-point ray tracing.