The three-dimensional mathematical field model of HF sky waves reflected by a spatially nonuniform nonstationary magnetoactive ionosphere is described. The model is based on the structural physical approach, which leads to complete understanding of the field structure in space and time associated with specific geomagnetic conditions.
 In the design of HF communication and direction-finding systems, a sufficiently complete knowledge of the properties of sky wave fields is required in order to develop effective algorithms for space-time processing of signals coming from antenna arrays. Similar problems arise in modem technology design. The traditional approach of natural tests is hardly feasible for ionospheric channels. Here computer simulation can be a very valuable help.
 The paper presents the three-dimensional mathematical model of the field of HF sky waves reflected by a spatially nonuniform nonstationary magnetoactive ionosphere. The model represents the further extension of the two-dimensional one [Barabashov and Vertogradov, 1996, 2000], and like the latter differs in principle from existing phenomenological ones [Goodman, 1992] in that it makes use of actual mechanisms of HF propagation in the time-dependent irregular ionosphere. The present model is based on a structural physical approach, which leads to complete understanding of field structure in space and time for specific geomagnetic conditions. Thus is accomplished the main requirement of the simulation: adequacy of the model.
 Construction of the model is determined by two major moments which represent researchers' latest opinions and are also confirmed by lots of experimental data. The first moment is that the problem of complete description of a sky wave field with all its peculiarities is totally solved by numeric simulation of ray paths in spatially nonuniform magnetoactive plasma. The second moment is that all main characteristics of sky waves can be interpreted in the first order of approximation within the model of medium- and large-scale traveling ionospheric disturbances.
2.1. Basic Points of the Model
 1. The total field at the reception site results from the interference of a few individual rays with different space- and time-dependent amplitudes and phases.
 2. Time variation of individual ray and total field parameters is attributed only to terminator travel and traveling ionospheric disturbances (TID).
 3. The ionosphere is taken as a three-dimensional nonuniform medium along a ray path, thus involving arbitrarily directed gradients.
 4. The multipath field is sought under geometrical optics approximation in the form of a solution of an eikonal equation.
 Experimental validation of the basic points is as follows.
 Discrete structure of the field is confirmed by numerous oblique ionograms.
 Spectra of a multipath field obtained for 30–60 s appear as discrete lines. Line spread for one-hop rays at oblique incidence, as a rule, does not exceed 0.1…0.2 Hz.
 For a fixed length path, a Doppler shift grows with the increase of operating frequency (that is, for a deeper penetration of wave into layer). Doppler shift drops with the increase of a path length.
 Time dependency of Doppler shift and of angles of arrival (elevation and azimuth) has a form of an oscillating process with a quasiperiod from a few minutes up to a few hours.
 Scattering mechanisms do not significantly affect field strength for one-hop paths. The values predicted for no scattering differ from the experimental ones by 2–3 dB on average.
 An ample bibliography can be found in the work by Goodman .
2.2. Coordinate Systems
 The model uses three coordinate systems. The first one is Cartesian with origin in the center of the Earth, the z axis pointing at the North Pole and the x axis lying in the plane of the Greenwich meridian. In this system a ray is described by spherical coordinates (ϕ, θ, r), where θ counts from the z axis to radius vector r of the ray, ϕ describes the direction of r with respect to the x axis in the oxy plane, and r is the length of r. Similar coordinates are used for a wave vector k = (kϕ, kθ, kr). The second system is Cartesian local centered at the transmission point with the z axis pointing upward, the y axis pointing to geographical north, and the x axis pointing eastward. Initial values of the wave vector are given by azimuth α0 and elevation Δ0. Azimuth α counts from the y axis, and elevation Δ starts at the oxy plane. Finally, the third coordinate system is Cartesian physical. It has the z axis in the direction of à wave vector, and the x and y axes are located such that major and minor semiaxes of polarization ellipse are oriented along these axes, respectively.
2.3. Model of Nonstationary Ionosphere Irregular in Three Dimensions
 Spatial distribution of electronic density in the present model is set in nodes of a three-dimensional (3-D) grid for r, θ, and ϕ coordinates incremented by Δr, Δθ, and Δϕ, respectively, with time increment Δt. The number of nodes is not large and is subject to external variation. In the nodes, the nondisturbed electronic density profile is described by the International Reference Ionosphere (IRI-2001) ionospheric model. The values thus obtained are then approximated by 3-D cubic spline, which provides continuity of the function and its derivatives.
 In trial usage of the model the optimum space-time grid increments were found as follows: Δr = 4 km, Δθ = 0.2°, Δϕ = 0.2°, and Δt ≈ 5 min. Such values provide the desired accuracy of modeling at acceptable computational expense.
 Wave disturbances (TID) are modeled by a packet of traveling monochromatic waves:
where N0(r, θ, ϕ, t) is space-time distribution of ionization for nondisturbed ionosphere in altitude r, angle coordinates ϕ and θ, and time t; δi is a relative magnitude of TID having period of Ti; pri, pϕi, and pθi are radial, horizontal, and normal components of the TID wave vector; Φ0i is the initial phase of the ith harmonic, and R0 is the radius of the Earth. Components pri, pϕi, and pθi of the ith harmonic can be expressed in the form
where Λi is the wavelength, βi is elevation from the horizontal plane, and γi is azimuth, starting at the oy axis. Normally, n of 2 or 3 is enough.
 Ionospheric absorption is determined using the collision frequency profile which does not vary along a path and is close to the gas kinetic profile (in MHz)
where h is altitude in km. The possible variation of the v(h) profile due to geocyclic and heliocyclic conditions is simulated by a correction factor K, which is calculated using global empiric International Telecommunication Union recommendation (ITU-R) maps for 2.2 MHz absorption.
2.4. Ray Tracing
 As is known, ray path deflections from the great circle plane (azimuth deflection) due to ionospheric anisotropy become significant (over 0.5°) for traces shorter than 200…250 km. If these deflections are neglected, then there is the possibility of performing ray tracing neglecting the effect of geomagnetic field. Thus the problem of multipath field simulation simplifies substantially, and the influence of the geomagnetic field on other characteristics of a ray such as field strength, elevation angles, group and phase delays, and Doppler frequency shifts can be represented approximately, which is sufficient for simulation accuracy.
 For isotropic ionosphere with three-dimensional irregularity, the set of characteristic equations derived from an eikonal equation takes the following form:
 Here Δ and α are ray coordinates (elevation and azimuth angles, respectively), and group path P′ is chosen to parameterize ray trajectory. The other variables denote the following: P is phase path; (kr, kθ, kϕ) are physical polar coordinates of wave vector k; δf is Doppler frequency shift, μ2 = 1 − (fn/f)2; λ is the wavelength; and f = c/λ and fn are the operating and plasma frequencies, respectively.
 For set (1), initial conditions at ground point with coordinates (ϕ0, θ0, R0) are given as follows:
 Set (1) is solved using the fifth-order Runge-Kutta-Falberg method with variable step, which allows automatic selection and adjustment of the integration step in the ray-tracing process.
2.5. Solution of Boundary Value Problem
 For a given trace, to solve the point-to-point problem in the 3-D ionosphere is to find initial azimuth α0 and elevation Δ0 such that a ray reflected from the ionosphere arrives exactly at the reception site (θk, ϕk). Thus we have a set of two nonlinear equations in independent variables α and Δ:
 Here θ(Δ, α) and ϕ(Δ, α) are current coordinates of ground arrival point of a wave launched at arbitrary azimuth α and elevation Δ. In addition, θ(Δ, α) and ϕ(Δ, α) result from solution of (1) with initial conditions (2), where θ0, ϕ0 are coordinates of the transmission site.
 1. For known coordinates of transmission and reception sites, the trace's geographical azimuth α0 and range S0 are found.
 2. For α0 using the bisection method, elevation Δ0 is found for which the solution of set (1) gives S(Δ0, α0) − S0 ≤ λ. Here S(Δ, α) is the range between the transmission site and ground point (Δ, α). Solution of (1) starts at Δ = 1.0° with further incrementing by 0.5°.
 3. For the set
one step of coordinatewise descent method is performed with initial conditions (Δ0, α0 + δα), where ψ(Δ, α) is the azimuth of the current ground arrival point. The initial value of δα is −15°.
 4. The values (Δ1, α1) are further used as the initial condition for the modified Newton method to find a root of (3) with given error. The accuracy is defined on the condition that the distance between points of arrival being sought and the receiving antenna location does not exceed λ/30.
 5. The other possibly existing rays are searched for in a similar way by setting δα = α1 + 1°.
 6. Steps 3–5 are iterated while δα < 15°.
 7. The procedure of steps 2 through 6 iterates several times as well, with angles Δ = Δ0 until either a ray goes through and out the ionosphere or Δ0 ≥ 85°.
 The above technique allows us to find practically all roots of set (3), that is, all rays, at acceptable computational expense.
 Prior to the ray-tracing and boundary value solution, the corrections δfo,x are found to the operating frequency [Barabashov and Vertogradov, 2000]. As a result, the operating frequency is replaced by two equivalent ones: fo = f + δfo and fx = f − δfx. The subsequent calculation is then performed for isotropic media. This approach allows finding both magnetoionic components having frequency f. The correction values are given by approximating polynomial P(S, γ, I):
Here fh is gyrofrequency, S is path range in 1000 km for ranges below 1000 km and is a unity for longer paths, and geomagnetic azimuth γ and magnetic inclination I are given in radians and are brought to the interval [0, π/2]. Coefficients of the polynomial were obtained using results of calculation of the above corrections for paths ranging from 50 to 2000 km for I and γ varying within 0° … 90°. Magnetic parameters fh, I, and γ are found in nodes of the space-time grid using components of magnetic field set by the global model. In intermediate points, cubic spline approximation is used.
2.6. Field Strength Calculation
 Ionospheric absorption Lf(f, t) accumulated along the trajectory resulting from solution of the boundary value problem is calculated using a generalized theorem of equivalence [Barabashov and Vertogradov, 1989]. The equivalence relationship between integral absorption values for oblique and vertical propagation in spherical horizontally irregular magnetoactive ionosphere is given by
 Here L(f, ν, fh) gives the value of oblique absorption at frequency f with collision frequency profile ν(h) and gyrofrequency fh, and Lv1(fv1, ν′1, f′h1, η1) and Lv2(fv2, ν′2, f′h2, η2) are values of absorption for vertical incidence into the ionosphere which have been calculated for electronic density distribution profiles at the points of an oblique ray entering and exiting the ionosphere, respectively. Equivalent frequencies fv1,2, collision frequency profiles ν′1,2(h), and gyro frequencies f′h1,2 are given by
Here elevation angles Δ1,2 are taken at the points of a ray crossing the lower bound of the ionosphere, as angles between the geomagnetic field and wave normal.
 The value of absorption for each normal wave at vertical incidence is calculated by numeric integration of the imaginary part of Appleton's refractive index with respect to altitude using Gaussian 20-point quadrature formulas. Error of calculating collision loss for both magnetoionic components lies within 1 dB.
 Space loss (ray divergence) LS = 10 lgSe2 is found by numeric differentiating using a simplified expression of Se2:
Here Δ1 and Δ2 are launch and arrival angles, RE is the radius of the Earth, and α1 and α2 are azimuths at points of a ray entering and exiting the ionosphere.
 Total multipath field components En(n = 1, 2, 3) in the local coordinate system at the (x,y) point are
where j(t, f) is the number of rays (roots of boundary value problem (4)); Ej(t, f, x, y) is the amplitude of the jth ray as a function of time t, frequency f, and coordinates x, y; Ajn(t, f) are coefficients of conversion of the physical to local coordinate system; and Pj(t, f, x, y) is the phase delay of the jth ray.
 Amplitude of the jth ray is given (with respect to 1 μV/m), by
where P is transmit power in kW, L is ionospheric absorption in dB, Ls is space loss in dB, Lr is Earth reflection loss in dB, and Lp is loss in dB due to mismatch between polarization ellipses of a wave coming to the ionosphere and a magnetoionic component excited in the ionosphere. The latter loss is sought within the theory of the limiting polarization [Davies, 1990]. The values of Ej(t, f, x, y) are converted to μV/m for determination of En.
 The variation of total field E(t) with time is obtained using series of Ej(t) and Pj(t, f, x, y) calculated with time increment of 0.1 s. This value of time sampling is selected because the maximum Doppler offset of operating frequency does not exceed 1 Hz. Then the E(t) series is obtained using (5). Validity of basic points of the model and properties of wave field predicted using the model are confirmed by a variety of experimental data available at present.
3. Test Results and Conclusions
 A few figures illustrate the results of simulating space-time distribution of wave field on the Earth's surface for the model with two TIDs, typical for middle latitudes: δ = 10%, Λ = 200 km, T = 20 min, β = −45°, γ = 45°, and Φ = 0°; and δ = 5%, Λ = 100 km, T = 30 min, β = −30°, γ = 0°, and Φ = 90°.
 The modeling was performed for the following conditions: path range is 1000 km; arc of the great circle connecting the transmission and reception sites is parallel to the x axis; the operating frequency is 9.0 MHz; for the case of 1F2 and 2F2, rays are simultaneously present with close amplitudes; and elevation angles are ≈23° and 53°, respectively.
Figures 1a and 1b depict simulated equal phase curves of total multipath field in 2π spacing on a 50 × 50 m area. Curves 1, 2, and 3 correspond to successive time increments of 0.1 s. Figure 2 shows simulated time variation of field strength in a local reception point for a period of 300 s, with radiated power of 1 kW. Figure 3 shows the autocorrelation ratio of field strength at the same location.
 The present model can suggest various areas of application. It can be used in the following situations: to obtain complete information on mode, ray, polarization, and space-time structure of the field for specific heliophysical and geophysical conditions; to compute the time series of complex voltage on every element of an antenna array of any size and configuration, which enables us to study spectral, statistical, and correlative (spatial and time) properties of ionosphere-reflected signals; to study both narrowband and wideband signal propagation via the ionospheric channel; to set up requirements for antenna arrays in communication and direction finding; to make expert estimates of algorithms for space-time processing of signals in antenna arrays; to calculate amplitude frequency and phase frequency channel responses and their variation with time; and to “run” signals through an ionospheric channel.
 The authors thank J. M. Goodman (Radio Propagation Services, Inc.) for helpful remarks.