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 High-frequency (HF) direction-finding (DF) systems measure the angles of arrival of signals at selected frequencies. With this information, ray tracing can accurately determine the location of the HF transmitters if the three-dimensional (3-D) electron density (Ne) distribution between the DF site and the transmitters is known. The usual approach is to use an ionospheric model like the International Reference Ionosphere (IRI) as a proxy for the density distribution. We describe a more realistic approach developed in cooperation with Codem Systems in Merrimack, New Hampshire. A collocated digisonde at the DF site measures the vertical electron density profile and the local ionospheric tilt, providing, in real time, the inputs for the construction of the 3-D Ne distribution. The vertical profile is automatically obtained from the Automated Real Time Ionogram Scaler with True Height (ARTIST)-scaled ionogram and the local tilt from the sky maps recorded after each ionogram. The characteristics of each layer, for example, critical frequencies and peak heights, are expressed as a function of latitude λ and longitude Ψ. In the neighborhood of the DF site each characteristic, for example, foF2, is given as foF2(λ, Ψ) = foF2m (1 + C7Δλ + C8ΔΨ) (1 + CλΔλ + CΨΔΨ). The coefficients C7 and C8 for any given azimuth direction are determined with the use of the Union Radio Scientifique Internationale/CCIR coefficients (which are also used in IRI), and the calculation of Cλ and CΨ makes use of the measured ionospheric tilt data; foF2m is the local, measured foF2 value. When the measured density profile and tilt data are available, the derived 3-D density distribution represents the instantaneous ionosphere structure near the site. The numerical ray tracing includes the effects of the magnetic field and properly treats the spitze effect, making the ray-tracing program especially useful for small distances. Ray tracing through simulated tilts shows that the differences in ground distances for one-hop high-frequency (HF) propagation vary from about 1 to 100 km depending on the assumed tilts and distances. Operational tests for distances up to approximately 100 km have demonstrated good results in determining the transmitter location in real time and have illustrated the importance of using the actual ionospheric profiles and tilts in the ray tracing.
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 When a radio antenna transmits HF waves, part of the energy enters the ionosphere, where the waves refract, penetrating the ionosphere or returning back to the ground depending on the frequency, the ionospheric condition, and the elevation angle. Only part of the energy in a small solid angle, defined as a ray, can arrive at a specified receiving location. As part of the construction of an approximate solution of Maxwell's equations, ray tracing is a technique to find the traveling path of the radio waves from the transmitter to the receiver [Bennett et al., 2004]. HF direction-finding (DF) systems can detect the signals from HF transmitters and can measure the frequency and the arrival azimuth and elevation angles. With the detected information, ray tracing can accurately determine the location of HF transmitters if the three-dimensional (3-D) electron density distribution between the DF site and the transmitters is known. Usually, ionospheric models like the International Reference Ionosphere (IRI) [Bilitza, 2001; Reinisch and Bilitza, 2004] are used as a proxy for the actual density distribution. Since the actual ionospheric density distribution can differ significantly from the model, the ray tracing may produce false results. A modern ionosonde at or near the DF site can measure the ionospheric density distribution in real time, and we describe below how the 3-D density distribution is constructed from the sounding data. We briefly describe the ray-tracing algorithm and demonstrate that it correctly handles the spitze effect for near-distance ionospheric ray paths.
2. Ionospheric Density Distribution in Real Time
 The vertical electron density distribution in the ionosphere can be measured with ionosondes. In the case of a digisonde [Reinisch, 1996], ionograms are processed with Automated Real Time Ionogram Scaler with True Height (ARTIST) software [Reinisch and Huang, 1983; Reinisch et al., 2005] that deduces the electron density profiles in real time using the program NHPC. The left plot of Figure 1 shows an example from Millstone Hill, Massachusetts, with E, F1, and F2 echo traces; the ARTIST O traces; and the automatically calculated profile plotted as true height versus plasma frequency. The profile for each layer is expressed in terms of the shifted Chebyshev polynomials Ti* [Snyder, 1966],
where h is height, fp is plasma frequency, hm and fm are height and plasma frequency at the layer peak, and hs and fs are height and plasma frequency at the start of the layer. The E–F valley is modeled on the basis of incoherent scatter measurements at Arecibo but is adjusted to best match the lower part of the observed F trace [Huang and Reinisch, 1996]. This measured vertical electron density profile represents the density distribution at the sounding site. When the transmitter and receiver are only a short distance apart, assumption of spherical stratification for the ray tracing generally does not produce much of an error. This is not true, however, in the presence of severe ionospheric tilts, and such tilts must be taken into account in the ray tracing. Since the digisonde measures the F2 layer tilt, we implemented a program that automatically derives the tilt normal vector from the digisonde's sky map data. The sky map in the right plot of Figure 1 shows the locations of the many O echo reflection points in an elevation and azimuth plot. The center of the distribution of the echolocations, marked by an open circle, is in the south-southwest of zenith, indicating that there is a tilt, and the tilt normal to the distribution center is calculated.
 The profile and tilt data represent the density structure over the sounder site. With this information we construct a 3-D density structure that reproduces the measured profile and tilt at the digisonde location (λ0, Ψ0). Assuming a linear variation of the layer parameters with latitude and longitude, the layer parameters at λ = λ0 + Δλ and Ψ = Ψ0 + ΔΨ become
 Here foEm, hmEm, Vwm, VDm, foF2m, and hmF2m are the measured profile parameters at the E layer peak, the valley, and the F2 layer peak, respectively. The coefficients (C1, C2, …, C10) can be determined using the CCIR or Union Radio Scientifique Internationale (URSI) coefficients for the mapping of ionospheric characteristics that are used in the IRI [Bilitza, 2001]. The assumption we make is that the IRI model can describe the variation of the ionospheric characteristics with longitude and latitude in the neighborhood of the sounder location. The coefficients (C11, C12) are determined from
 The vertical profile at any location can then be expressed the same way as in (1), and adjusting the coefficients so that the starting height of a layer is equal to the peak height of the lower layer,
 The linear expansions in (2) assure that the vertical profile calculated from the constructed 3-D density structure equals the measured profile at (λ0, Ψ0). The resulting F2 layer tilt, however, may not equal the measured tilt since the CCIR/URSI coefficients give only average values. A correction factor is therefore introduced in (2) that assures that the resulting tilt at (λ0, Ψ0) equals the measured tilt:
 The constants (Cλ, CΨ) in the correction factor are determined so as to make the zenith and azimuth angles of the normal vector equal to the tilt measured in the middle of the F2 layer.
 To assess the profile changes in the neighborhood of the ionosonde site, we use equation (5) to calculate the profiles at different distances and directions from the site. Figure 2 illustrates the profile variations around Millstone Hill in different directions, northeast, east, southeast, south, southwest, west, northwest, and north. The measured profile at Millstone Hill (0500 LT) is shown in the middle plot and is repeated as a dotted curve in all the plots to show the amount of change. The rest of the plots show three profiles for locations separated by 1°, 2°, and 3° in longitude and/or latitude. The E layer and the E–F valley profiles are modeled for this presunrise time. The measured F layer tilt angles were 2.5° zenith and 90° azimuth; that is, the F2 layer declines exactly toward east. It can be seen from Figure 2 that the F2 layer profiles change very little in the north and south directions, as expected with the indicated tilt. With measurements from a single ionosonde, the constructed 3-D density distribution can be expected to represent the ionosphere in a ∼400 km diameter region.
3. Ray Tracing
 The passage path of a ray in the ionosphere is described by the Haselegrove ray equation system [Haselegrove, 1957]:
 This differential equation system gives the relation of the ray path (r, θ, ϕ) and the wave normal w in spherical coordinates. The refractive index n is computed using the measured 3-D electron density distribution and the International Geomagnetic Reference Field (IGRF) (http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html) for a given radio frequency for the O or X wave propagation modes. When the wave normal at the starting point is given, the ray path can be numerically calculated from this equation system. Analytic or closed-form ray tracing, in general, is impossible, but accurate numerical solutions of the differential equation system can be obtained. The effect of a small ionospheric tilt at Millstone Hill is presented in Figure 3 for propagation in four directions; the measured tilt angles were 1.42° zenith and 119.6° azimuth. The dashed curves show the calculated ray paths for the O mode when the tilt is neglected. The largest ground distance error occurs for westward propagation (bottom left plot), reducing the footprint distance for the 45° elevation ray by 60 km from 580 to 520 km when the tilt is neglected. The simulations for a 5° tilt, the typical value for sunrise/sunset times, produced a footprint change of ∼110 km compared to the no-tilt footprint (not shown here).
 When an O- or X-mode ray enters the ionosphere at sufficiently small elevation angles, it will be reflected from the ionosphere. At any point along the ray path, including the reflection point, the refractive index satisfies n > 0, and the equation system (6) can determine the ray. With increasing elevation angle, the reflection point moves higher, and finally, the ray may penetrate the layer. However, if the electron density is sufficiently large to support the reflection, then the refractive index at the reflection point gets close to zero. When the ray approaches a cutoff point n = 0, the ray will be fully reflected, and the ray direction suddenly changes at the reflection point. The appearance of this reflection is called the “spitze” phenomenon after Poeverlein . The differential equation system (6) is singular at these points, and it cannot give a solution in the neighborhood of the spitze. The general Snell's law must be used for the reflection and refraction of a ray at the boundary of two plasmas. This is illustrated in Figure 4.
 When a ray in a magnetoplasma enters a different plasma, one part of the energy will be reflected, producing a reflected ray; the other part will enter the second plasma, producing the refracted ray. Because of the anisotropy, the ray deviates from the associated wave normal [e.g., Budden, 1985]. For the isotropic case, the general Snell's law reduces to the classical one. In our case, the ray undergoes full reflection when it encounters the cutoff plane n = 0. The constant c1 can be determined using the refractive index surface around the reflection point. The implementation of the general Snell's law into the ray-tracing program allows accurate description of the spitze reflections, as illustrated by the results shown in Figure 5. It should be noted that the spitze phenomenon occurs for the O- and X-mode rays; Figure 5 shows only the O-mode rays.
 As stated earlier, DF systems measure the frequencies and the azimuth and elevation angles of arriving signals. Since the propagation is reciprocal, a ray-tracing technique can trace back from the DF receiving system to the signal sources to find the transmitter location. Codem Systems had installed a digisonde at a DF site to provide the vertical electron density profile and local ionospheric tilt in real time as inputs for the construction of the 3-D Ne distribution. Operational tests were conducted with a multifrequency beacon at a fixed site; several mobile beacons served as interference sources. Using the frequency and arrival angle information from the DF system, the trajectories for the ordinary (O) and extraordinary (X) rays were calculated in real time through the measured tilted ionosphere. Figure 6 shows the results of one such test. In the top plot, the altitude of the O and X rays are plotted versus distance, while the bottom plot shows the projections of the ray paths onto the ground. At the time of these tests, the Codem DF system had no wave polarization discriminator, but the ray tracing shows that for the prevailing ionospheric conditions at this instant, the O wave could not have arrived at the site. The signal frequency of 6.92 MHz was slightly larger than the measured foF2 = 6.5 MHz. The ray tracing showed that the ionosphere peak density was not large enough to support the reflection of the O-mode ray at the detected incidence angle, and it must have been the X ray that had been detected by the DF system. The ray-tracing positions the transmitter at the location marked by the cross in the bottom plot, ∼18 km west from the actual location marked by a solid circle. Notice that the azimuth of the source from the DF site is 245°, while the detected ray arrives with an azimuth of 234°. This large angular deviation is a result of the ionospheric tilt, which was measured as 5° zenith and 158° azimuth. Clearly, without the tilt correction, the ray tracing would have put the transmitter way south of its actual location.
 During a 1-hour fully automated test with several mobile decoy transmitters, the DF/digisonde/ray-tracing system collected the data presented in Figure 7. The triangle shows the location of the DF/digisonde site, and the solid circle shows the location of the transmitter. The ray tracing determined the O- and X-ray footprints that are marked with open circles and crosses, respectively. The footprints cluster near the actual transmitter location; some distant footprints may be the result of bad DF measurements but most likely are the locations of the mobile transmitters.
 Improved accuracy in the location finding of ground-based HF transmitters is obtained by using a DF system in cooperation with a modern ionosonde that can determine the vertical electron density and ionospheric tilt in real time. On the basis of the measured profile and tilt, the 3-D electron density distribution in the neighborhood of the sounder is calculated. A 3-D ray-tracing program then calculates the trajectories for the O and X rays through the 3-D density distribution to determine the location of the HF transmitter from the measured arrival angles of the ray. When the ray elevation angles are very high, a “spitze” occurs at the reflection point, and the Haselegrove  ray equation system is not able to give any solution. General Snell's law is used to handle the spitze reflection. In actual tests, a DF/digisonde system at a distance of ∼100 km from a known HF transmitter location was reliably determining the transmitter location, typically within ∼15 km. When the ray tracing disregards the ionospheric tilt, the error increases to 55 km in some cases. Using measured tilts at the DF site is useful for transmitter distances to ∼400 km. For larger distances it is better to use average tilts derived from models, rather than the measured local tilt, to calculate the 3-D Ne distribution around the measured vertical profile. The presented technique cannot overcome positioning errors in the presence of higher-frequency (short-wavelength) traveling ionospheric disturbances where the instantaneous local tilt does not represent the average tilt over a few hundred kilometers.
 The authors gratefully acknowledge support by AFRL through grant F19628-C-0092 and by Codem Systems, Inc., Merrimac, New Hampshire.