Methods of inverting high-frequency radar land and sea backscatter to obtain ionospheric vertical profiles offer an important means of remote sensing the ionosphere up to thousands of kilometers from the transmitter/receiver location. This new inversion technique requires the received elevation angle as well as the time delay or group path of the backscattered signal. The technique can be used with both fixed-frequency and swept-frequency radars. The advantage of using swept-frequency radar is that the downrange gradients in electron density can more readily be determined.
 The case we consider is ground backscatter signals received from distant locations on the Earth's surface via reflection by the ionosphere. Both the transmitted and received signals traverse the ionosphere and these signals contain useful information regarding the state of the intervening ionosphere at the time, and over the range of the returned signal which can be up to a few thousand kilometers from the transmitter/receiver location. In this paper a new HF radar backscatter inversion method is presented. It is similar in concept to that of Rao , Bertel et al. , and Norman , where inversion techniques were derived for backscatter ionogram inversion. This new inversion technique requires HF radar data in the form of time delay or group path, P′, as a function of elevation angle, β, from each of the ionospheric layer echo traces. This data is then inverted to form a multiple quasi-parabolic segment ionospheric profile, using the quasi-parabolic segment (QPS) model derived by Dyson and Bennett . This QPS model uses five QPSs to represent the E, F1 and F2 ionospheric layers.
 The characteristics of the backscattered echoes depend on the radar system characteristics and ionospheric characteristics as well as the backscattering properties of the ground. Important radar characteristics are the wavelength, pulse length, antenna beam width, antenna gain, and transmitted power. HF radars that utilize ionospheric propagation at oblique incidence are usually capable of illuminating the ionosphere over a relatively wide range of elevation angles, giving rise to the possibility of ground backscatter being detected over a wide span of ranges. The radar characteristics largely affect the signal strength of echoes and the ionospheric structure determines the P′ elevation angle characteristics at each frequency so throughout this investigation we consider only these ionospheric effects. Displays of echoes detected by ionospheric HF radars are commonly called ionograms and here we are primarily concerned with fixed-frequency backscatter ionograms displaying echo traces formed by the variation of P′ with elevation angle.
 The inversion technique was tested using real backscatter data from the Tasman International Geospace Environment Radar (TIGER) [Dyson and Devlin, 2000]. TIGER is an oblique incidence, ionospheric backscatter radar, designed to study the physical processes occurring in space weather systems such as radio aurora. It is situated on Bruny Island (43.4°S, 147.2°E; 55°Λ) and transmits southward and is part of the Super Dual Radar Network [Greenwald et al., 1985, 1995] consisting of six radars in the Southern Hemisphere and nine in the Northern Hemisphere. Throughout this investigation we concentrate on the ionospheric effects on the propagation; that is, it is assumed that the radar characteristics are known and can be accounted for separately.
 This new inversion technique derives quasi-parabolic ionospheric layer parameters from the layer echo traces in the form of group path and elevation angle. At least three group path elevation angle data points are required from each layer echo trace. The maximum elevation of each of the ionospheric layer echo traces is also a required input. The inversion technique is based on using a fixed operating frequency. However, by varying the operating frequency of the HF radar it is possible to determine the downrange ionospheric gradients and thus produce more realistic ionospheric mappings.
electron density at a radial distance r from the Earth's center;
maximum electron density at the radial height rm;
radial base height of the ionospheric layer;
layer semilayer thickness.
 The QPS model, presented in the work of Dyson and Bennett , consists of smoothly joined QP segments, where QPSs are used to describe each of the ionospheric layers at and immediately below their peaks, and additional QPSs are used as joining segments that smoothly join together the QP descriptions of the layer peaks. The equations describing the QPS ionospheric model may be written as follows.
 The E layer is
the joining layer is
and the F layer is
where a = Nm and b = Nm (rb/ym)2 and
where the joining point, rc, at which the joining layer connect to the F layer may be written as
 If an F1 layer is present then QPS joining layers are used between the F1 and F2 QPS layers and between the F1 and E QPS layers.
 The equation for calculating the group path may be written as
where rt is the radial height, from the Earth's center, at which the ray is reflected ro is the radius of the Earth and
where μ is the refractive index and β is the elevation angle.
Dyson and Bennett  presented explicit equations for the ray parameters in their QPS model for a spherically stratified ionosphere, where if the ray propagates through n segments the group path may be written as
where Un and Ln represent the values of the integral in equation (1) at the upper and lower bounds.
3. New Radar Backscatter Inversion Technique
 The method of inversion of the backscattered echo traces presented here requires the determination of the three layer parameters for each of the quasi-parabolic layers, namely the critical frequency fc, base height rb, and height of maximum electron density rm.
 The inversion technique begins with inversion of the backscattered signal refracted from the E layer. Once the parameters for the E layer have been calculated the inversion technique inverts the F1 layer echo trace using the E layer parameter results as well as the QPS equations for the joining layer, which smoothly joins the E layer and the F1 layer. The process is then repeated for the F2 layer where the layer parameters already determined for the E and F1 layers, as well as the equations for the joining layer that joins the F1 and F2 layers are required for the inversion process.
 The inversion technique requires n (where n is at least 3) data points from each of the layer echo traces, where the n group paths chosen P′1, P′2,…,P′n correspond to the n elevations β1, β2 …βn, respectively. The method then sets out to find a set of values for the layer parameters, which yield to within a specified accuracy, to the data points chosen. To accomplish this an initial guess of the ray parameters (rc, rb, rm) is made.
 The maximum elevation angle of each of the layer echo traces, βmax is a required input and is used to determine the initial guess ionospheric parameters and aids in stabilizing the technique. A raypath having an elevation angle of βmax reaches an apogee height equal to the peak height, or maximum height, rm of the layer. Raypaths with elevation angles greater than βmax penetrate the ionospheric layer.
 Then, using the maximum elevation angle of the layer echo trace and an educated guess of the peak height rm and the base height rb of the ionospheric layer, the maximum plasma frequency of the layer can be determined using the equation
The initial guess values of rm and rb can be incremented so that the process homes in to the best possible solution.
 These new layer parameters are then substituted into the analytic expressions for the group path for the quasi-parabolic layer. Let the computed minimum group paths, which most likely will differ from the real values P′1, P′2, …, P′n, be P′c1, P′c2, …, P′cn, respectively, and the corresponding differences between the real and computed values be ΔP′1, ΔP′2, …, ΔP′n, respectively. Then, to a first approximation the amount δxi ≡ (Δfc, Δrb, Δrm) by which the assumed layer parameters should be incremented, so that ΔP′1, ΔP′2, …, ΔP′n are a minimum is given by (refer also to Bertel et al.  and Norman )
and may be rewritten in matrix form as
or in expanded form as
or more simply as Z = SK, where
Z = [ΔP′]
K = [dxi]
 The inversion of this equation is K = [STS]−1STZ, where ST is the transpose matrix of S and the matrix STS is a square matrix capable of inversion.
 The assumed layer parameter values are then incremented by Δfc, Δrb and Δrm. The entire procedure begins again with the new layer parameter values until the differences in group path ΔP′1, ΔP′2, …ΔP′n converge to a small specified minimum, thereby obtaining the final solution of the layer parameters for that ionospheric layer.
 Once the layer parameters have been calculated the next ionospheric layer parameters are then solved for using the technique above and the layer parameters already determined. For example let us assume, that the E layer parameters rbE, rE, foE have already been evaluated using the method shown above. The peak elevation angle of the F1 layer echo trace, βmaxF1, may be determined directly from the backscattered trace and at least three data points are required from the F1 layer echo trace. The unknowns to be solved for are foF1, rbF1 and rF1.
 Two additional QP segments are now involved where one represents the joining layer, which smoothly joins the E and F1 layers and the other QPS representing the F1 layer from the peak of the F1 layer down to the point where these two layers are smoothly attached. The joining layer parameters depend on the E and F1 layer parameters [see Dyson and Bennett, 1988]. Thus three QP segments are required to determine the F1 layer parameters using the inversion technique described here. The equation for the group path in determining the F1 layer parameters contains the parts of the raypath in (1) the free space (region between transmitter and base of the ionosphere), (2) the QPS representing the E layer, (3) the joining QPS layer joining the E and F1 layers, and (4) the QPS representing the F1 layer.
 The equation for the group path of the propagated raypaths being reflected from the F1 layer may be written as
 Once the F1 layer parameters are known the F2 layer parameters may be calculated using the same procedure. In all, up to five QP segments are required to determine the F2 layer parameters.
4. Inversion of a Synthesized BSI
 To test the inversion technique and its stability and inherent accuracy, a synthesized backscatter echo trace with known QP layer parameters was used. The synthesized group path versus elevation angle echo traces, shown in Figure 1, were determined using an analytic ray-tracing program and the ionospheric layer parameters: foE = 3.32 MHz, ymE = 14 km, and rE = 101 + ro km (E layer); foF1 = 4.75 MHz, ymF1 = 43 km, and rF1 = 164 + ro km (F1 layer); and foF2 = 6.45 MHz, ymF2 = 68 km, rF2 = 214 + ro km (F2 layer); where ro = 6370 km (the radius of the Earth).
 An operating frequency of 12 MHz was used in the determination of the backscattered trace in Figure 1. The layer echo traces are clearly visible where the E layer echo trace is the trace between 0° and 12° elevation, the F1 layer echo traces is the trace between 12° and 19.7° and the F2 layer echo trace is between 19.7° and 29.3° elevation. Raypaths for this frequency that have elevation angles greater than 29.3° will penetrate the ionosphere specified by the above layer parameters.
 The inversion technique was then applied to this synthesized data. Table 1 shows the three data points chosen from each of the layer echo traces in Figure 1 which is the minimum number of points required for the inversion technique. The maximum elevation angle from each of the layers is also a required input. In this initial test, high-accuracy input data were used to validate the technique.
Table 1. Three Data Points Chosen From Each of the Layer Echo Traces
Elevation Angle β°E, deg
Group Path P′E, km
 The results in Table 2 clearly demonstrate that the inversion technique converges correctly for the first stage of the inversion, that is, the determination of the E layer. After only five iterations the technique has homed into the desired layer parameters. The error in group path between the computed and true values in group path is negligible after only five iterations.
Table 2. Convergence of the E Layer Parameters to the Correct Result After Each Iteration
 The results in Table 3 clearly demonstrate that the inversion technique converges correctly for the second stage of the processes determining the F1 layer parameters. After only six iterations the technique has homed into the desired F1 layer parameters. The difference in group path after only six iterations is negligible.
Table 3. Convergence of the F1 Layer Parameters to the Correct Result After Each Iteration
 The results in Table 4 also show that the inversion technique converges correctly in the final stage of the inversion as after only five iterations the technique has homed into the desired F2 layer parameters. The difference in group path after only five iterations is again negligible.
Table 4. Convergence of the F2 Layer Parameters to the Correct Result After Each Iteration
Tables 2–4 clearly validate the inversion method by showing that the new computed layer parameters converge closer to the true values after each iteration. Figure 2 shows the three-layer ionospheric profile having the resultant layer parameters.
5. Inversion of Synthesized Three-Dimensional Ray-Tracing Results Using the IRI Model
 The backscatter results in Figure 3 show synthesized one-hop sea scatter using a frequency of 8 MHz where each point corresponds to a ray tube determined from the three-dimensional (3-D) numerical ray-tracing program HIRT (Homing-In Ray Tracing) [Norman et al., 1994]. The International Reference Ionosphere (IRI) model [Bilitza, 2001] was used to provide a 3-D representation of the Earth's ionosphere. The IRI settings were: year 2000; day 50 (local summer); time 1100 LT; geographic transmitter coordinates (43.4°S, 147.2°E; 55°Λ) corresponding to the TIGER SuperDARN radar at Bruny Island, Tasmania, Australia. The synthesized raypaths are determined in the direction of the south geomagnetic pole. The squares in Figure 3 represent the points chosen for the inversion technique.
Figure 4 compares the resultant three-layer ionospheric profile obtained from the inversion technique (thicker line) with the thin curve representing the IRI ionospheric profile at the transmitter location. These results clearly show that this new inversion technique produces accurate results. The results below highlight the three sets of layer parameters representing the E, F1 and F2 ionospheric layers determined from the inversion technique using the data points represented by the squares from Figure 3 and the small number of iterations required as well as the high level of numerical accuracy obtained (Tables 5, 6, and 7)
 The results in Figure 5 compare the analytic ray-tracing results obtained using the three layer parameters determined from the inversion process (light diamond points) with the 3-D numerical ray-tracing results obtained using the IRI model (dark dashed points). The data represented by the squares show the values used for the inversion. It is only in the low-angle F1 and F2 layer echo traces where there is a significant difference in results. Ray tracing shows that there is considerable spreading of the ray tubes reflected at these levels and this causing the signals to be weak and unlikely to be important for propagation experiments or communications.
6. Example Using Real Other Data
 The inversion technique was tested using real backscatter data from the Tasman International Geospace Environment Radar TIGER [Dyson and Devlin, 2000]. The TIGER data used is from 29 February 2000 at 1200 LT and an operating frequency of 13.095 MHz. The square data points in Figure 6 represent the backscattered results received from TIGER and these are the data points used in the inversion program to produce the following ionospheric layer parameters: foE = 3.41 MHz, ymE = 17.8 km, and rE = 107.0 + ro km (E layer); foF1 = 6.60 MHz, ymF1 = 84.4 km, and rF1 = 197.5 + ro km (F1 layer); and foF2 = 6.90 MHz, ymF2 = 55.9 km, and rF2 = 279.5 + ro km (F2 layer); where ro = 6370 km (the radius of the Earth).
Figure 7 shows the ionospheric profile obtained by inversion and represented by the above layer parameters. The diamonds in Figure 6 represent analytic ray-tracing results obtained using the inverted profile shown in Figure 7. The data points match up well with the synthesized results. It is only in the E layer where there is a difference between the synthesized layer echo trace and the real data.
7. Mapping/Modeling the Ionosphere
 The previous examples show that a spherically stratified three-layer model representation of the ionosphere in the region of the radar can be determined using the inversion technique. It will now be shown how this new inversion technique can be used to determine a 3-D grid point model of the ionosphere. The inversion technique described above assumes that the ionosphere is horizontally stratified over the region traversed by the signals used in the analysis. However, if horizontal gradients exist, different raypaths will be affected differently and this will limit the accuracy of the profile obtained by the inversion technique described here.
 On the other hand, it is possible to determine the horizontal ionospheric gradients in the coverage area of the radar by applying the inversion technique to results obtained at a number of different radar operating frequencies. Raypaths with different frequencies will be reflected from different regions of the ionosphere. For example, raypaths with frequencies of, say, 8 MHz will in general be reflected from the E layer at ranges closer to the transmitter and with higher elevation angles than raypaths having a frequency of, say, 12 MHz. By stepping through the radar operating frequencies and determining the E layer parameters at each frequency, it is possible using the inversion technique, to determine the E layer parameters at specified range bins, in the direction of the propagated signal, from the transmitter location.
 Once having obtained a grid point map/model of the E layer in this way, the F1 layer can then be similarly mapped/modeled. The raypaths reflected from the F1 layer will have passed through the E region at two different ranges, as shown in Figure 8. The F1 layer parameters can then be determined using the inversion technique and the appropriate two sets of E region layer parameters determined in the analysis of the E region data. By stepping through in frequency the F1 layer parameters can be determined at each specified range bin. Similarly the same process can be applied for the raypaths reflected from the F2 layer where these raypaths will have passed through each of the F1 and E regions at two different ranges. The F2 layer parameters can then be determined using the appropriate two sets of E region parameters and two sets of F1 region parameters.
 Note that, as shown in Figures 8 and 9, the range grids define specific regions in height and range such that many raypaths may travel through a single grid component. The Segment Method Analytical Ray Tracing (SMART) technique [Norman and Cannon, 1997, 1999; Cannon and Norman, 1997] which performs analytic ray tracing in a grid point ionosphere is ideal in this situation and can be combined with the inversion technique to produce a better grid point ionospheric map/model and also to perform 2-D analytic ray tracing using this new ionospheric map/model.
 Many HF radars consist of multiple beams and this approach can be used for each beam. This means that not only the ionospheric gradients in the direction of the beam can be determined but also the gradients at right angles to the beam. The coverage area of the radar can be divided into a grid consisting of latitude, longitude and altitude, where each grid location will be assigned a set of the layer parameters. Thus obtaining a complete map of the ionosphere in the coverage area is possible. The Segment Method Analytical Ray Tracing (SMART) technique [Norman and Cannon, 1997, 1999; Cannon and Norman, 1997] which has now been upgraded to perform 3-D ray tracing can be used in conjunction with the grid point ionosphere to obtain realistic 3-D ray-tracing results and to compare these results with real radar data.
 It should be noted that several other inversion methods have been proposed in the past; for example, Landeau et al.  developed a technique known as elevation scan backscatter sounding. Their approach was to apply Bayesian methodology described by Tarantola  to the inverse problem, where they applied it to only a single QP layer representing the F layer of the ionosphere. Caratori and Goutelard  developed a relatively simple technique to determine the spatial ionospheric gradients however it requires the addition of vertical soundings and the inversion is limited to the F region of the ionosphere.
 This new inversion technique is robust in that it converges quickly to an accurate result. It has the ability to use many data points from each layer echo trace and in general the more data points used the better the inversion results will be. The technique adds a further dimension to the analysis of HF radars, such as the SuperDARN radar network, where the elevation angle of the received signal is measured and recorded.
 This research was supported by the USAF AOARD (Asian Office of Aerospace R&D) and VPAC (Victorian Partnership for Advanced Computing) Expertise Program Grant Scheme.