A new technique for diagnostics of traveling ionospheric disturbances (TIDs) uses measurements of arrival angles and Doppler frequency shift variations of HF radio signals reflected from the ionosphere. This technique is a further improvement of a frequency-and-angular sounding method previously suggested. Here we solve the respective inverse problem of reconstructing TID parameters by using a physically realistic three-dimensional model of electron density waves propagating through an ionospheric layer. The developed diagnostic algorithms were tested through computer simulations demonstrating the high efficiency of the proposed method. We also reprocessed the data collected in an earlier experiment carried out at the Millstone Hill Observatory in March 2001 using the new algorithms. In that campaign, HF signal trajectory parameters were recorded using the Digisonde instrument as a receiver monitoring continuous transmission from the CHU station. Comparison of the TID parameters determined with the frequency-and-angular sounding method and simultaneous incoherent scatter radar observations showed good agreement between the two diagnostic techniques with the differences of no greater than 10%.
 The ionosphere exhibits a wide spectrum of waves with wavelengths ranging from hundreds of meters to thousands of kilometers and with time periods from minutes to several hours and even days. Among these wavelike disturbances atmospheric gravity waves (AGWs) and their ionospheric signatures, traveling ionospheric disturbances (TIDs), prominent over the periods between several minutes and 2–3 hours, remain among the most interesting phenomena observed in Earth's upper atmosphere [e.g., Williams, 1996]. These large-scale features are believed to be responsible for the energy and momentum exchange between different regions of the ionosphere, especially important during ionospheric storms [Prölss, 1993]. In addition, TIDs carry along information about their sources of excitations which may be either natural (energy input in the auroral region, earthquakes, hurricanes, solar terminator, and others) or artificial (industrial accidents, ionospheric modification experiments, powerful blasts, chemical releases, etc.). Thus creating a global system of TID diagnostics becomes an important task for further development of ionospheric science and establishing the concept of “space weather.” Such a system shall make it possible to continuously monitor the dynamics of the ionosphere over a large scale. The focus is on creating relatively inexpensive, but robust and effective systems for remote diagnostics of this type of ionospheric disturbances. One such technique, frequency-and-angular sounding (FAS), was developed by the authors in earlier works [Beley et al., 1995, 2000]. It is on the basis of measuring trajectory parameters (angles of arrival and Doppler frequency shift) of the probe HF signals propagating obliquely. The FAS technique offers a possibility of using transmissions from broadcasting stations as probing signals leading to reduced overall system costs and expenditures by using a single receiving site to monitor several transmitters making measurements over a large area. In the previous work, a model of corrugated reflecting surface was used for representing the TID disturbance, this approach made it possible to solve the inverse problem analytically and determine the main TID parameters. Such a model is simple and intuitive, yet it permits an adequate interpretation of the measured data. Results of computer simulations [Beley et al., 1995] demonstrated that TID parameters can be recovered with reasonable accuracy even for volumetric (three dimensional) TIDs, still using the algorithms developed within the “effective reflecting surface” model. In early 2001 the FAS technique has been implemented in the Digisonde Portable Sounder (DPS) system [Reinisch, 1996] and was used during the trial measurements together with the Millstone Hill Incoherent Scatter Radar [Galushko et al., 2002, 2003]. The major TID parameters (direction of motion, wavelength, and propagation speed) measured simultaneously by the incoherent scatter radar (ISR) and DPS instruments were in reasonable agreement thus demonstrating the effectiveness of the FAS technique. Certain difficulties, however, were faced in the reconstruction of the disturbance amplitudes that resulted from the use of a simplified “effective reflecting surface” model. Namely, the variations in the reflection height are not readily related to the real fluctuations of electron density. In order to remove this shortcoming we have developed a new version of the FAS oblique diagnostic techniques that uses a more physical model of the TIDs by representing them as three-dimensional plasma density waves traveling through a real ionospheric layer.
2. Problem Formulation and Basic Relations
 The suggested TID diagnostics technique is on the basis of monitoring the trajectory parameters of a radio signal propagating through the ionosphere in the presence of TIDs. In this and the next sections we explain how a mathematically accurate set of equations relating measured signal parameters to the parameters of TIDs is derived. We start from discussing the direct problem of HF radio wave propagation through the three-dimensional non-uniform ionosphere. In the following analysis the CGS system of units is used.
 Let the transmitter, located at the origin of a Cartesian frame (as shown in Figure 1), radiate a quasimonochromatic signal at a carrier frequency f0. The receive site is at a distance D0 along the x-axis and serves to record trajectory parameter variations (specifically the arrival angles of the wave and Doppler frequency shifts) of the ionospherically reflected sounding signals. The ionosphere is characterized by its permittivity function ɛ(, t), which can be represented as a sum of two terms
of which ɛ0(z) is the dielectric constant height profile in the non-disturbed ionosphere, and ɛ1(, t) is an addendum characterizing the effect of large-scale ionospheric disturbances. Further, we assume that
where α = fcr/f0 is the ratio of the ionospheric critical frequency, fcr (plasma frequency corresponding to the peak electron density of the ionosphere, N0max), to the diagnostic frequency, f0, and F(z) = N0(z)/N0max.is the normalized non-disturbed height distribution of the ionospheric electron density. Finally, for ɛ1 (, t) we write
where Φ(z) is the normalized height profile of the disturbance amplitude and ν(,t) is the disturbance waveform.
 To derive the set of equations relating the measured signal parameters (i.e., angles of arrival and Doppler frequency shifts) to parameters of the disturbance, we will make use of the perturbation theory for the wave eikonal. It is assumed that the characteristic size of the disturbances is much larger than the first Fresnel zone, and the disturbance amplitude is small, ∣maxɛ1∣ ≪ ɛ0, at any point along the signal trajectory. This allows using the first-order approximation and representing the eikonal L as a sum of the regular (non-disturbed) component, L0, and the addendum L1 due to the disturbances, viz.
In what follows, the analysis will be restricted to the case of a single-skip propagation for which the eikonal components can be written [Kravtsov and Orlov, 1980] as
(The integration is along the non-disturbed ray (group path) ρ.) Accordingly, the Doppler frequency shift is then
where λ denotes the signal wavelength.
 To estimate the fluctuations in the angles of arrival measured at the point D0, it is helpful to write the wave vector components in polar coordinates as
These relations and the fact that the gradient of the wave eikonal ∇L represents the wavefront normal are used to establish a relationship between the eikonal and the signal arrival angles:
where θ and ϕ stand for zenith and azimuthal angles of the received signal measured at the receiver site D0 (see Figure 1). Hereafter all the derivatives are taken at the point D0. The sine and tangent values of the angles of wave arrival will then read
Making use of the composite eikonal L = L0 + L1 one obtains
Using , , and (θ0 stands for the launch angle of the unperturbed trajectory and D ≡ D(θ0) is the skip distance as a function of the launch angle for the unperturbed ionosphere), and also, and , we finally have
Further, let us assume that and , which implies ∇x,yL1 ≪ ∇xL0; that is, the angle of arrival fluctuations are small and the observation point is sufficiently far away from the caustic surface of the wavefield. Then we can write θ = θ0 + Δθ and linearize the above equations by neglecting terms of the second and higher orders in Δθ ≪ 1. The resulting estimate for fluctuations in the vertical-plane angle of arrival is
In the same way, the azimuthal-plane fluctuations can be written as
Here again all the derivatives are calculated at the location of the receive site, and ϕ0 denotes the initial azimuth of the signal trajectory (in our problem it is assumed that ϕ0 = 0).
 Expressions (6), (16), and (17) constitute the equation set needed for deriving parameters of three-dimensional TIDs from the signal trajectory parameters variations.
3. The Inverse Problem
 Our previous experience suggests that it is more convenient to treat the diagnostic inverse problems in the spectral domain. Therefore we will need to transform the derived equations (6), (16), and (17). Using equation (3), the eikonal equation (5) can be written as
The disturbance waveform ν ((ρ), t) is represented in terms of its Fourier image ν(Ω, ) as
where Ω denotes the circular frequency of electron density fluctuations; is the wave vector of the disturbances (the components are κx, κy, and κz), and is the position vector. The function ν (, t) is assumed to be stationary and statistically uniform, with 〈ν (, t)〉 = 0 (the angular brackets stand for statistical averaging). Then,
The integral over ρ will be denoted (, θ0, ϕ0), viz.
In general, the accuracy of the TID parameter reconstruction depends on the TID vertical wavelength (or TID vertical velocity component). In the first approach, (assuming TID moving almost horizontally) one can expect that the reconstructed TID wavelength and velocity will be overestimated proportionally to the angle between the TID velocity vector and a horizontal plane. However, in the frame of this work we do not specifically study this question in details. At the same time, from our earlier experimental works [Galushko et al., 1998, 2003] it follows that most of the time the TID vertical wavelength can be assumed to be nearly infinite (i.e., κz → 0) and TID disturbances to be traveling horizontally. Thus we write
permitting to obtain explicit formulas for the frequency spectra of the measured signal parameter fluctuations. We substitute equation (25) into equations (6), (16), and (17) which makes is possible to derive the expressions for the spectral representations Sθ(Ω), Sϕ(Ω), and SF(Ω) of the signal trajectory parameters (zenith, azimuth, and Doppler frequency shift, respectively). The result is
where κ and γ denote TID wave number and direction of motion respectively, Aθ = (2 cos θ0)−1, Aϕ = , and AF = (here θ0 is again the launch angle of the unperturbed trajectory and ϕ0 = 0; that is, the radio propagation path is along the x-axis). We further assume that for TID disturbances each fluctuation frequency Ω can be associated with a single spatial harmonic plane wave with the wave number κ(Ω), traveling in the direction γ(Ω); that is, the disturbance Fourier image can be written as
Whence it follows that TID wave number κ (Ω) and the direction of motion γ (Ω) are expressed through the measured signal spectra as
Note that κ(Ω) and γ(Ω) can be found without the knowledge of the unperturbed electron density profile. It is sufficient to have an estimate of θ0 which, owing to the stated stationarity and statistical uniformity of the disturbances, is θ0 = 〈θ〉. From equations (28e) and (28f), and with the use of equation (22), the complex spectrum of the electron density fluctuation can be written as
where Ψ = 1/2D0 κ cos γ according to equation (24) and assuming ϕ0 = 0. To determine the fluctuation spectrum Sν (Ω), we now face a difficulty of evaluating the integral equation (23). This can be done either numerically, or analytically using certain model profiles of electron concentration in the unperturbed layer.
 Now we can outline the scheme for solving the inverse problem of TID diagnostics. First, the spectra of measured variations of the radio signal parameters are substituted in equation (30) to restore the wave numbers κ and directions of motion γ of the wave disturbances. Then the recovered values and known parameters of the non-disturbed layer component are used to estimate the integral G which is to be used in equation (31) for evaluating the real and imaginary parts of the time-and-space TID spectrum.
4. Computer Simulation and Experimental Testing
 The algorithm developed was tested through computer simulation of the problem of interest. Using a specially prepared computer code for numerical integration of the ray trajectories in an inhomogeneous medium we calculated angles of arrival and Doppler frequency shifts for specific propagation paths in the presence of modeled TID disturbances. A parabolic profile was used to represent the undisturbed electron density distribution N0 (z), and the disturbance amplitude was specified as a relative fluctuation with respect to the background electron density:
Here N0(z) is the undisturbed (parabolic) vertical electron density profile; ai is the relative fluctuation of the electron density of the ith spectral component; Ωi and ψi are circular frequency and initial phase; and γi is the direction of motion of the disturbance with the wave number κi with respect to the radio path. Since for the isotropic ionosphere ɛ(, t) = 1 − Ne(, t), equation (32) implies the following permittivity function
where e and m are electron charge and mass, respectively, and ω0 = 2πf0. The calculated signal parameters were then used as an input for solving the inverse problem in accordance with the above-described procedure. The results of the TID parameter reconstruction were compared with the initial disturbance parameters simulated with the use of equation (32). Altogether, 30 simulation runs were performed for a variety of parameters of the ionosphere and TID disturbances. Figure 2 presents parameters of the TID disturbances used in our computer simulation. TID wavelength is on the x-axis, amplitude is on the y-axis, velocity of the motion is shown by the length of the arrow, and TID propagation direction is indicated by a rotation angle of the arrow, with the vertical orientation corresponding to the northward propagation. We mainly concentrated on varying TID wavelength and amplitudes which were of major interest in this work. The simulation results confirmed the proposed diagnostic method to be effective, as the errors in reconstructing the TID speed of motion, wavelength and amplitude did not exceed 3%, while the errors in determining TID propagation directions were less than 1–2°. These results are summarized in Figure 3 which presents TID parameter reconstruction errors as functions of assumed TID parameters. It should be noted, however, that the error distributions presented in these plots do not allow drawing a statistically significant conclusion about the functional dependence of the accuracy of the proposed technique on the TID parameters. This is due to a rather small total number of simulation runs and the fact that several TID parameters were varied simultaneously. However, typical and maximal errors of the TID parameter reconstructions observed in the computer simulations are easily seen in the plots. By way of example, Table 1 shows the modeled TID parameters and those restored using the proposed method in one of the numerical experiments. In this case, the TID disturbance was modeled with a single spatial spectral component and its parameters were chosen such that only a single solution existed for the two-point radio wave propagation problem (single-mode propagation). Results given in the table illustrate high accuracy of the restoration of the modeled TID parameters provided by the developed algorithm. Figures 4a and 4b present both the modeled and reconstructed distributions of the disturbed electron density component in a horizontal plane near the height of the maximum of the background ionospheric layer. Clearly, both presented distributions are practically identical. Similarly satisfactory results were obtained for the vertical ionospheric profile (along the x-axis) as illustrated in Figures 4c and 4d. The developed algorithms also allow calculating the time-evolution of the disturbance electron density distribution which is visualized using a designed software package. Summarizing, one concludes that computer simulation results confirm the effectiveness and robustness of the suggested technique of TID diagnostics based on the model of three-dimensional density waves propagating in a realistic ionospheric layer.
Table 1. Reconstruction of the TID Parameters in the Model of Three-Dimensional Electron Density Waves
 To further test the validity of the suggested technique, we made use of the data collected in the frame of a dedicated campaign that was carried out at the Millstone Hill Observatory in March 2001 near equinox conditions. For the detailed description of the experiment the reader is referred to [Galushko et al., 2003], and here we only outline the experimental layout. In the 2001 experiment aimed at recovering large-scale wavelike processes in the upper atmosphere a Digisonde Portable Sounder made simultaneous observations with the incoherent scatter radar. The DPS system, modified to be used as a narrowband receiver for ionospheric HF signals from broadcast stations, recorded signal trajectory parameters (Doppler frequency shift and angles of arrival) making one observation every 4 min. The Canadian time-service broadcast station CHU, located near Ottawa (45.3°N, 75.8°W) was used as a source of the probing signals, operating at the transmission frequencies of 3330 kHz, 7335 kHz and 14679 kHz.
 The ISR steerable antenna was cycled through three tightly-spaced positions with the horizontal beam separation at 250 km altitude of about 70 km and with the complete data-taking cycle of 5 min. Such three-position experiment made it possible to use the triangulation technique to determine the direction and speed of the observed wavelike ionospheric disturbances from the ISR data. Further details of the ISR instrument settings and the recorded electron density variations can be found in our earlier study [Galushko et al., 2003].
 Unfortunately, during this relatively short campaign no very strong, clearly defined TID events were observed, but the best TID candidate was recorded on March 15, 2001 (magnetically quiet day, KP < 2.3) between 12–16 UT (07–11 LT). Figure 5 presents the CHU signal parameters (Doppler frequency shift and angles of arrival) measured by the DPS during this time interval. The strongest wavelike disturbance was observed soon after sunrise and had a period of about 30 min which is characteristic of TIDs associated with the solar terminator at the middle latitude F-region [Galushko et al., 1998]. Our previous analysis made it possible to determine the amplitude of the TID disturbance, its wavelength, speed and direction of motion using the FAS technique with the perfectly reflecting surface model as well as through the ISR triangulation method. The results of the comparison are given in Table 2, (two upper rows) showing a good agreement between TID speed and direction, but only fair agreement between the amplitudes of the disturbance determined with the two techniques. This observed disagreement in the measured amplitudes was one of our motivations to develop a more sophisticated model for TID analysis with FAS technique. In this work we applied the newly developed FAS algorithms based on the model of electron density waves to the 15 March 2001 data. Results of these calculations are also given in Table 2 (bottom row) demonstrating a noticeable improvement of the accuracy of the TID amplitude determination. The disturbance velocity and wavelength parameters measured with the new technique are also somewhat closer to the ISR results compare to those obtained with the FAS technique using the surface model.
Table 2. Comparison of the 30-min TID Parameters Obtained With FAS and ISR Techniques for 15 March 2001 Data
Propagation direction, deg
 This work is a further development of the frequency-and-angular sounding technique for TID diagnostics that was originally proposed in [Beley et al., 1995, 2000]. The FAS technique is on the basis of monitoring the variations of the angles of arrival and Doppler frequency shifts of the HF radio-signals propagating obliquely. In this paper, the FAS algorithms are extended to include the TID model of three-dimensional electron density waves propagating in a real ionospheric layer that provides more accurate reconstruction of the disturbance parameters. Such model is also more physically realistic in comparison to the earlier model of the corrugated reflecting surface used to model the TID disturbance. The inverse problem of determining the TID parameters from the observed fluctuations of the signal trajectory parameters is solved using the eikonal formalism. The solution obtained allows reconstructing the time variations of the TID parameters and therefore, to visualize the ionospheric disturbances. In the frame of this work we developed a software package that includes a numerical ray tracing code for modeling the trajectory parameters of the signal propagating in a real ionospheric layer (i.e., solving the direct problem); reconstructing TID parameters (i.e., the inverse problem) and visualizing the disturbances. Results of the numerical simulations and trial measurements at Millstone Hill demonstrated the effectiveness and reliability of the developed technique and suggest a possibility of establishing a large-scale diagnostic system for TID monitoring. The existing worldwide network of Digisonde sounders [Galkin et al., 2006] with its dense coverage in certain sectors (e.g., European, South-African sectors) is an excellent choice for laying out a base for such an ambitious project.
 The authors wish to thank Dr. A. V. Koloskov (IRA NASU) for his help in computer simulation. We are also thankful to Dr. J. Foster and the entire MIT Haystack Observatory team for their participation in the joint experiment in 2001. B.W.R. and V.V.P. were in part supported by AF Research Laboratory grant FA8718-06-C-0072.