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.
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 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  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 , 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.
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.
2. Fermat's Principle
 Fermat's principle is the starting point for the development of most ray tracing approaches to radio wave propagation. For isotropic media, the principle is given by
where μ is the refractive index of the medium, A and B are the end points of the ray and s is the distance along the ray. Essentially the principle states that the ray between two points is that for which the phase distance is stationary. Unfortunately, in the case of ionospheric propagation, the medium is non-isotropic due to the effects of Earth's magnetic field. In this case, Fermat's principle is given by
where is a position vector on the raypath and is a vector that is normal to the wavefront with the property that u = || = μ on a raypath. Equation (2) is the starting point for the Haselgrove equations for ray tracing in magneto-ionic media. The refractive index μ is derived as a root of the equation [Haselgrove and Haselgrove, 1960]
 Quantity X = relates the refractive index to the plasma frequency fp and wave frequency f. Vector is parallel to the Earth's magnetic field with magnitude Y = fH/f where fH is the gyro frequency.
 In a direct approach to ray tracing, the ray is found by searching over the possible raypaths that join the end points (A and B) to find the one that makes the phase distance stationary. (Note that Fermat's principle is sometimes stated in terms of the raypath that makes the phase distance minimum, but there are some important cases under which the desired stationary path does not correspond to a minimum.) For an isotropic ionosphere, μ = , and so equation (1) is independent of and we need only deal with variations in ray geometry. For the full non-isotropic case, however, we also have to consider variations of the vector . We would like to eliminate and hence reduce the number of dependent variables to that of the isotropic case. In the manner of complementary variational principles [Arthurs, 1980], it turns out to be possible to eliminate the variations in and hence produce a variational equation in terms of variations in ray geometry alone. The approach is to restrict variations in to those that partially satisfy the Euler-Lagrange equations of (2). In the derivation of her equations, Haselgrove wrote (2) as
where G = u/μ, and t parameterizes the path. (It should be noted that G is homogeneous of degree 1 in and has value 1 on a raypath.) On noting that μ only depends on through Yp = · where = /u (i.e., μ() = μ(, Yp)), the Euler-Lagrange equations for variations in imply
 We can recast (11) and (12) in terms of the unit tangent = /| on the ray and obtain
where Yt = · and pt = · is the cosine of the angle between the ray and the wave normal. Equations (13) and (14) are a pair of simultaneous equations for pt and Yp when and are given (assuming Y and X are known functions of ). We can use (14) to directly eliminate pt from (13) and hence obtain an equation for Yp alone when and are given. This value of Yp can then be substituted back in (14) to produce a value for pt. These values of pt and Yp will be then compatible with the Euler Lagrange equations for variational principle (2). In order to implement the above procedure, however, we still need values for μ2 and ∂ℓn(μ2)/∂Yp. The square of refractive index μ2 is, from (3),
where the + signs correspond to an ordinary ray and the − sign corresponds to the extraordinary ray [Budden, 1985]. Further, by differentiating expression (3) with respect to Yp, we obtain that
 Variational principle (2) can now be written as
where s is the geometric distance along the path of the integration. Given a particular curve joining end points A and B, we can use (13) to (14) to obtain pt and Yp at all points along the curve. By using such values in variations of (17), we restrict ourselves to variations that are compatible with Euler-Lagrange equations expressed by (8) (in the manner of complementary variational principles). In this way, we have removed the dependence from the integrand and obtain a variational principle in terms of raypath geometry alone. In the field free case pt = 1 and μ = and so the variational principle reduces to that of (1). In the case where Earth's magnetic field is present, however, we need to solve (13) and (14) at each point along the path.
3. Solving the Variational Equation
 In order to solve the variational equation (17), we will need to consider the variations of the functional P = μ(, Yp)pt ds over a field of paths that join points A and B. For practical purposes, these paths will need to be approximated by a field that is described by a finite set of parameters = (α1, α2, …, αN). For given parameter values, we can evaluate P using the path geometry, together with equations (13) and (14) to provide values for pt and Yp (providing that X and Y are known functions of position). That is, P is a function of parameters α1 to αN and the path that makes it stationary has parameter values that can be found by solving the system of equations
 In practice, (18) represents a highly nonlinear system of equations and will require an iterative solution that starts from some initial estimate 0. In the current work, we use a Newton-Raphson procedure, which can be summarized as
where superscript I denotes a quantity evaluated at the Ith stage of iteration. This provides a linear system of equations for the increment in from the Ith iteration to the I + 1th iteration. Fortunately, the matrix of this linear system is both sparse and symmetric. Consequently, in the current work, the linear system is solved by a conjugate gradient algorithm that has been adapted to take advantage of these properties. The resulting algorithm is both fast and stable. Typically, the algorithm takes four or five iterations to satisfactorily converge. As with all iterative schemes, convergence requires a good starting point and, in the current work, we have used an analytic solution based on a quasi-parabolic approximation to the ionospheric above the mid point of the path [Croft and Hoogasian, 1968]. This approach works well for the sort of benign ionospheres provided by the IRI90 model [Bilitza, 1990], but could require modification if additional complexity, such as traveling ionospheric disturbances, be present. In such cases there can be multipath and so a possible strategy is to cycle through a set of alternative starting points that are perturbations to the path based on a quasi parabolic approximation to the background ionosphere. A suitably chosen set of perturbations will then enable the iteration to converge to the various extrema. Another option might be to consider the use of gentic optimization techniques [Sadeghi et al., 1999]. Multipath is already an issue as, even with benign ionospheres, there can often be two extrema for the phase path. These two extrema are frequently referred to as the high and low rays [Budden, 1985] and are discussed in greater detail below.
 As mentioned in the previous section, Fermat's principle is often stated in terms of the path that makes the phase distance minimum. Indeed, in the absence of magneto-ionic effects, the functional certainly satisfies the Legendre property [Fox, 2010; Gelfand and Fomin, 2000] of minima for frequencies above the plasma frequency. Consequently, for such situations, one would be tempted to use standard minimization techniques to find the raypath instead of directly solving equations (18). This works well for transionosphere propagation and high rays but, unfortunately, breaks down for low rays. The reason for this can be seen by studying some ray tracing that was produced by the numerical solution of the standard Haselgrove equations [Coleman, 2008]. Figure 1 shows some Haselgrove equation based ray tracing through a spherically stratified ionosphere and from which it will be noted that the penetrating and high rays do not cross over their neighbors. Low rays, however, cross their neighbors before reaching the ground and therefore contain focal points. According to the Jacobi test for a minimum [Fox, 2010; Gelfand and Fomin, 2000], it is necessary for there to be no focal points on the ray for the Legendre condition to predict a minimum and explains why an optimization approach will not work for low rays (the Legendre condition is necessary, but not sufficient, for a minimum). For low rays, we must resort to the solution of equations (18). This turns out to be an advantage as the numerical solution of (18) tends to be lured toward low rays. Consequently, to find low rays we use a numerical solution of (18) and to find high rays (or transionospheric rays) we perform a direct numerical optimization of the phase path. In the current work, the solution of equations (18) was achieved through a Newton-Raphson iterative solver (as described by (19)) and the optimization through a conjugate gradient minimization algorithm [Press et al., 1992].
 The most important choice in the direct approach to Fermat's principle is the field of curves that approximately represents the candidate rays. There are many possible choices for the field, but a prudent choice can lead to accurate ray tracing with a low parameter count. In the current work, we have chosen a parameterization based on sample points along the great circle path between the end points A and B. This parameterization is ideal for applications where the rays have distinctly different ground coordinates at their end points. To discretize, we choose M representative points on the great circle path (these do not need to be uniformly spaced) and then, at each of these points, choose optimization parameters that are the height above the ground and the normal deviation from the great circle. Together, these height and deviation samples constitute 2N continuous parameters that describe a 3D ray field. To turn these discrete parameters into candidate rays, we interpolate over the segments between the sample points. We have investigated two forms of interpolation in the current work, piecewise cubic (PC) and piecewise quintic (PQ) interpolation. PC has continuous first derivatives and PQ has continuous second derivatives. The relative performance of these interpolations will be discussed in the next section, as will the influence of the number of interpolation points. For most ionospheric propagation, the deviation of the ground coordinates from the great circle is relatively small. Consequently, a much lower level of approximation can often be adopted for the deviation parameter. For the simulations in the current paper, the number of sample points for this parameter was taken to be one tenth of the number for the height parameter.
3.2. Some Examples and Comparison With Traditional Ray Tracing
 In the current section we consider some examples of ray tracing using the above procedures. First, we consider a realistic propagation scenario for radio waves propagating from Alice Springs (−23.5°S, 133.7°E) in central Australia to a location that is 10° in latitude to the North. This requires propagation into the equatorial anomaly and so the propagation will experience strong ionospheric gradients. Figure 2 shows a simulation of propagation at a frequency of 16 MHz using the direct method (the ionosphere was defined by Chapman layers using coefficients derived from the IRI90 model [Bilitza, 1990]. The frequency is close to that where high and low rays merge and both types of ray are shown in the figure (ordinary and extraordinary rays represented by white and black curves respectively). In the simulation, 100 segments with constant ground range length (measured on the great circle path) were used, but results with double and half the segments showed the results to be well converged. Figure 3 shows the same rays (together with some others), but calculated using a homing approach [Strangeways, 2000] together with a conventional Haselgrove type ray tracing procedure [Coleman, 2008]. The results are almost identical and so provide an independent check on the direct method. In particular, the direct approach provides a very effective method for high rays. For high rays, a very small change in launch elevation can lead to a massive change in the landing point and hence be problematic for the stability of homing. (In Figure 3, the two penetrating rays were launched at elevations only 1 degree higher than the high rays in order to demonstrate this.)
 In order to investigate the convergence of the method, a series of simulations was performed in which the discretization parameters were varied over a considerable range. These simulations were performed in ionospheres that were based on a uniform Chapman F2 layer (foF2 = 7 MHz, hmF2 = 350 km and ymF2 = 100 km) with strong uniform gradients superimposed. First we consider a strong gradient in the direction of propagation (the peak plasma frequency varied by 0.375 MHZ per degree of latitude). Northward propagating O and X low rays at 10 MHz were simulated using a Haselgrove equation based algorithm [Coleman, 2008]. Both rays were launched with an initial elevation of 25°, having landing points at ground ranges of around 1100 km. A refined approximation was used in the Haselgrove algorithm in order to produce rays with landing points known to a very high degree of accuracy. Such rays could then be simulated using the direct algorithm of the current paper in order to test its accuracy. For simulations with the direct algorithm, the rays were divided into segments with uniform ground range along the great circle path joining the raypath end points. Table 1 shows the phase path and maximum ray height for numerous ray simulations. The first entry is the highly accurate Haselgrove ray tracing result, followed by direct simulations with 24, 50, 100 and 200 segments respectively. It will be noted that the most refined simulations are within meters of accurate Haselgrove results. These first four simulations were performed using a PQ (piecewise quintic) interpolation. In order to test the effect of the order of interpolation, however, two additional simulations (50 and 100 segments) were performed using a PC (piecewise cubic) interpolation. These are the last two entries in the table and show that there is a significant improvement in moving from PC to PQ interpolation.
Table 1. Phase Path (PP) and Maximum Height (hmax) in Kilometers for Northerly Gradients
O Ray PP
O Ray hmax
X Ray PP
X Ray hmax
 In the above simulations, the deviations of the paths from a great circle were no more than a few hundred meters in all cases. This is not surprising since there was negligible lateral ionospheric gradient and mainly longitudinal magnetic field. In order to investigate the effect of lateral ionospheric deviations, we now consider propagation in the above uniform ionosphere, but with a strong transverse gradient superimposed instead of the longitudinal gradients of the previous simulations (the peak plasma frequency varied by 0.375 MHz per degree of longitude). As before, accurate rays with known ends points were simulated using a Haselgrove approach and then the current direct approach used to simulate the rays for the same set of discretizations as above. Table 2 shows the results for ordinary rays of both the high and low variety (low rays launched at an elevation of 30° and high rays at 43°) and from which it will be noted that most refined simulations are within meters of the accurate Haselgrove results. (The high rays have been included as they are most strongly affected by the lateral ionospheric gradients.) It will be noted that, for the high rays, the lateral deviations are less well converged than other parameters and this is due to the relative insensitivity of the phase path to their variation. In general, refinements in the approximation require the phase path to be converged to ever finer levels (10−5 m for the 200 segment approximation) and so rounding error becomes an issue. One possibility is to make better use of a lesser number of segments by means of a non-uniform distribution of sample points along the great circle. In such an approach, longer great circle segments could be used outside the ionosphere where the path is linear and the approximation steadily refined as the path becomes more curved (mainly around the ray apex). Table 3 shows some results for the case where the division of the great circle has been done in a fashion that makes the center segment have half the length that would arise if the same number of equal length segments were used. This approach is obviously at the expense of the outer segments, segment length increasing as the end points of the ray are approached. From Table 3, it can be seen that the simulations maintain good accuracy, even for a very low number of segments.
Table 2. Ray Phase Path (PP), Normal Deviation (dmax) and Maximum Height (hmax) in Kilometers for Westerly Gradients
Table 3. Phase Path (PP) and Maximum Height (hmax) in Kilometers Using Nonuniform Segments
O Ray PP
O Ray hmax
X Ray PP
X Ray hmax
 Simulation speed is another important issue in ray tracing. For the case of uniform segments, and for the same order of accuracy, the direct approach has a similar speed to that of the traditional homing approach. From Table 3, however, it can be seen that a small number of non-uniform elements can provide good accuracy. Further, it turns out that the speed of simulation falls quadratically with the number of elements. Consequently, a small number of non-uniform elements provide accurate results with much greater speed.
4. Some Further Developments
 The above techniques can also be used for multihop propagation, but this is a little bit more involved. For a geographically varying ionosphere, the point of ground reflection will now become an additional variable. This new variable can be introduced into the overall optimization, but tends to slow down convergence and cause divergence unless a good initial estimate is available. In practice, a more stable approach was found to consist of iterating the reflection point separately. First, the great circle range Rr for this point is estimated and the variation equations solved with Rr fixed (lateral variations at the reflection point are allowed). After this, the variation equations are solved with all variables but Rr fixed. This process is then repeated until Rr has converged. Figure 4 shows an example of a 2 hop simulation for propagation northward from Alice Springs. It will be noted that the increase in ionospheric density toward the equator has caused a reduction in the range of the second hop and also in its height.
 Trans-ionospheric propagation is another possible application for the direct approach and one for which the minimum property of the phase path can be used. Point-to-point ray tracing problems arise when investigating propagation between a satellite and its ground station. Figure 5 shows some transionospheric ray tracing from Alice Springs northward into the equatorial anomaly at frequencies of 20 and 40 MHz (both ordinary and extraordinary rays are shown). It will be noted that the raypaths at 20 MHz shows considerable deviation from those at 40 MHz. For transionospheric applications, it will be noted that the great circle discretization will break down when the rays have end points with the same ground coordinates. Consequently, for transionospheric propagation, a better discretization might be that which is based upon a finite number of sample points on the line that connects the end points. Longitude and latitude deviations from this line, at the sample points, will then provide a discrete number of continuous parameters that describe the candidate rays. Such a discretization would also be required for propagation with the same ends point (i.e., the kind of vertical propagation that is used by vertical incidence ionosondes). In this case, the propagation path would need to be broken by the reflection point and the coordinates of this point, since unknown, would also need to be parameters in the discretization. This discretization would have similarities with that of the two hop rays above.
 We have developed a form of variational principle that can be used as the basis of a direct approach to the calculation of ionospheric rays when their end points are known. The method has been demonstrated for numerous ionospheric propagation scenarios and has proven to be an effective alternative to the more traditional method of homing. In particular, the method is effective for the high rays that can put a strain on the homing approach. One improvement to the method is the introduction of a non-uniform discretization in which the discretization points become denser in regions of high curvature in the raypath. This leads to more accurate results with fewer discretization variables. Ultimately, the positioning of mesh points could be chosen to further minimize the phase distance and hence their determination would become part of the solution process for the variational equation.