Corresponding author: V. G. Galushko, Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, Ukraine. (email@example.com)
 This paper analyzes the effect of ionospheric refraction on the scattering of high frequency (HF) signals by random field-aligned irregularities in the upper ionosphere. Ray optics calculations are made using the perturbation method for a plane-stratified (on average) ionosphere, i.e., the incident and scattered waves are both supposed to propagate along the undisturbed trajectories with neglect of the geomagnetic field effect. The equation for the so-called cone of aspect-sensitive scattering is derived to relate the trajectory characteristics of the incident and aspect-sensitive scattered signals. The Born approximation is applied to calculate the scattering cross-section for the anisotropic power law model spectrum of random irregularities of the upper ionosphere. The possibility of excitation of the ionospheric interlayer waveguide by the aspect-sensitive scattered HF signals is analyzed in detail for the specific conditions of the HF heating experiment at European Incoherent Scatter (EICSAT).
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 Aspect-sensitive scattering of high-frequency (HF) (3–30 MHz) and very high frequency (VHF) (30–300 MHz) waves by field-aligned irregularities of the ionospheric plasma are widely used for diagnostics of turbulent processes in the near-Earth plasma of both natural [Lyon, 1965; Bourdillon et al., 1995; Bezrodny et al., 1997] and artificial [Djuth et al., 2006; Hysell, 2008; Yampolski et al., 1997; Koloskov et al., 2002] origins. Theoretical premises for this method of diagnosing the inhomogeneous structure of the ionosphere were developed within the single-scattering approximation [Rytov et al., 1987]. As a rule, in the relevant studies the condition is assumed, where ω is the cyclic frequency of the sounding signal, ωcr is the critical frequency of the ionospheric layer (i.e., the maximum plasma frequency ωp max), and ωH the electron gyrofrequency. This assumption makes it possible to ignore the effects of the regular ionospheric refraction and background geomagnetic field. Consequently, the wave trajectories are approximated by straight lines, and ionospheric plasma is assumed isotropic.
 However, to be precise, in order to ignore the effects of refraction stronger conditions have to be satisfied, which can be derived as follows. The equation for the incident ray trajectory in a plane-stratified ionosphere can be written as [Kravtsov and Orlov, 1990]:
where θ0 is the angle of incidence at the lower boundary of the ionospheric layer, z0, counted from the vertical. Obviously, in order for a wave to penetrate through the ionosphere (with no reflection point at which the ray curvature would be the greatest), the following condition must be satisfied: ω > ωcr/cos θ0; while in order to approximate the ray trajectories by straight lines, an even stronger condition should hold: ω > > ωcr/cos θ0. Here is an example from practice. During most of the experiments on HF signal scattering from artificial ionospheric turbulence (AIT) produced by the Sura HF heater (Nizhny Novgorod, Russia) [Yampolski et al., 1997; Myasnikov et al., 2001; Koloskov et al., 2002], the geometry was such that θ0 ≈ 70° (for the probe signal) and fcr = ωcr/(2π) ≥ 4.5 MHz. Therefore, in order to be able to ignore the effects of refraction, the probe signal frequency should be much higher than ~13 MHz, which was rarely the case. Therefore, for correct analysis and interpretation of experiments of the kind, the refraction effects need be taken into account.
 It is especially important to account for the ionospheric refraction in analyzing the scattering of powerful HF emissions on the ionospheric irregularities produced by the same radiation. The fact that the HF pump wave scatters on ionospheric irregularities is rather trivial and has been observed a long time ago [Belikovich et al., 1975; Erukhimov et al., 1980). It was not clear, however, whether the scatter signals can travel over large distances away from the heater, for example, through the ionospheric waveguide. Such effect was apparently first experimentally discovered by Zalizovsky et al.  using the European Incoherent Scatter (EISCAT) HF heater (Tromso, Norway) and was given the name “self-scattering effect.” In their experiment, the EISCAT transmission was monitored at three greatly dispersed receiving sites: at the Ukrainian Antarctic station “Akademik Vernadsky” (UAS); at the Radio Astronomical Observatory of the IRA NASU (RAO) near Kharkov, Ukraine; and near Sankt-Petersburg, Russia (SPB). Typically, the received signals contained two characteristic spectral components. One component was a relatively narrow line with insignificant Doppler shift variations characteristic of the skywave HF propagation in middle latitudes. The spectrum of the other component was much broader, reminiscent of that of a signal scattered at frequencies above the maximum usable frequency (MUF). The broadband (scattered) spectral component exhibited strong frequency fluctuations with amplitude rate occasionally exceeding 10 Hz. However, the most notable observational fact was that the Doppler frequency shifts and intensities of this component varied practically synchronously at all three receive sites with nearly the same oscillation rate. The authors concluded that the observed Doppler shifts of the broadband spectral component were likely induced within a wave trajectory segment that was common to all the propagation paths, i.e., where the HF pump wave travels from the heater to the scattering region with the AIT. Simultaneous observations by the EISCAT incoherent scatter radar showing strong variations in the electron density above the HF heater are supporting the suggested mechanism. However, the exact mechanism of the super long range propagation of the HF heater signals (the EISCAT-UAS propagation distance was ~16,300 km) still has not been determined, and it is quite possible that ionospheric refraction may play an important role since the experiments at the EISCAT location were conducted at frequencies below the critical frequency of the ionosphere (ω ≤ ωcr). Galushko et al.  had also observed the self-scattering effect by monitoring the High Frequency Active Auroral Research Program (HAARP) transmission at several remote sites in the USA, Europe, and Arctic using digital Doppler receivers of the Institute of Radio Astronomy (Kharkov, Ukraine) and Lowell Digisondes.
 The aim of the current paper is to develop a general theory for the aspect-sensitive scattering of electromagnetic waves taking into account ionospheric refraction effects on the incident and scattered wave trajectories. We will specifically analyze a possibility of signal channeling in a given direction as a result of such scattering. The work on HF signal trapping in ionospheric waveguides due to scattering was pioneered in the 1970s [e.g., Erukhimov et al., 1975; Gurevich et al., 1975]. Our approach, however, is quite different from those studies as we will analyze the range of incidence angles at the lower boundary of the ionosphere responsible for channeling signals in a given direction rather than evaluate a “trapping coefficient” for HF waves. The gyrotropic effects of the ionospheric plasma on the radio wave propagation will be assumed negligibly small. Note that this is a quite legitimate first-step approach used in earlier works as well [e.g., Gurevich et al., 1975].
2 Statement of the Problem
 Let a plane monochromatic electromagnetic wave be propagating in a horizontally stratified ionosphere from the lower half-space through the ionospheric layer (see Figure 1). The wave is characterized by a frequency ω and wave vector .Here , and , where k0 = ω/c, and angles θ0 and ϕ0 determine the direction of the wave vector at the lower boundary of the ionosphere z = 0. The wave frequency ω is assumed to be much higher than the electron gyrofrequency ωH, i.e., ω ≫ ωH. This makes it possible to treat the ionosphere as an isotropic medium and to assume the propagating waves to be transversal. In order to avoid any confusion, the following nomenclature will be used in our analysis. The HF waves will be regarded as “incident” and “scattered” with respect to the scattering on the ionospheric irregularities. With respect to the propagation in the ionosphere, the waves will be considered as “direct” and “reflected.”
 The ionosphere will be specified as a plane-stratified collisionless dielectric medium containing random irregularities:
 Here are horizontal coordinates; ε0(z) is a regular (i.e., without irregularities) component of the dielectric permittivity of the ionosphere,
where N0(z) is a regular electron density profile; and e and m are the electron charge and mass, respectively; and is a random addition due to electron density fluctuations , characterized by zero mean and variance (the angular brackets < … > stand for statistical averaging). Note that the fluctuations are related to the relative electron density variations as
 Electron density irregularities (or irregularities in the dielectric permittivity ) of the magnetized plasma in the upper ionosphere are highly anisotropic, stretched along the geomagnetic field direction . The position of the unit vector in the plane of the geomagnetic meridian y = 0 (see Figure 1) is specified by the inclination angle I (−90° ≤ I ≤ 90°), counted from the horizontal plane [Akasofu and Chapman, 1972]. Positive values of I correspond to pointing downward (Northern Hemisphere), while the negative ones correspond to the upward direction (Southern Hemisphere). Then, for the selected system of coordinates (the x axis in Figure 1 points to geomagnetic north) we have
 In order to calculate characteristics of scattering of the incident electromagnetic wave by such ionospheric irregularities, let us apply the single scattering approximation method [e.g., Rytov et al., 1987]. In this case it is assumed that , and therefore, it is possible to ignore the effects of the ionospheric irregularities on the wave trajectories and calculate them within the geometrical optics approximation.
3 Derivation of the Scattering Characteristics
Trajectory Parameters of the Aspect-Sensitive Scattered Signals
 It is known [Gershman et al., 1984] that HF waves incident on the field-aligned irregularities of the upper ionosphere are predominantly scattered in the direction determined by the so-called aspect condition:
 Here is the scattering vector, which is equal to the difference of the wave vectors of the scattered and incident fields at the scattering point zs:
 As is evident from Figure 2, equation (4) is satisfied when the angles made by the magnetic field vector with the incident and scattered wave vectors are equal. As a result, the set of vectors forms a characteristic angular cone around the magnetic field with the apex angle ν.
 In the coordinate system shown in Figure 1, and with account of ((5)), equation (4) can be written as
where θ(s), ϕ(s) and θ(i), ϕ(i) are the angles specifying the wave vector positions of the scattered (superscript s) and incident (superscript i) fields at the scattering point zs (hereafter, symbol zs is omitted to simplify the equations). Solving equation (6) for sin θ(s) and cos θ(s), it is possible to derive the direction of the wave vector of the scattered field. Using the relation sin 2θ(s) + cos 2θ(s) = 1, two possible solutions are obtained:
 Here the top sign stands for , while the lower sign stands for . It is also taken into account that ϕ(i) = ϕ0 according to the equations of geometrical optics for a plane-stratified medium [e.g., Kravtsov and Orlov, 1990]. Note that the incident wave can be scattered either upward (), or downward (), i.e., . Therefore, the solution space of equations ((7а)) and ((7b)) with respect to ϕ(s) is limited by the condition:
 Another necessary condition for the existence of a real solution of equations ((7а)) and ((7b)) is a non-negative argument of the square root in equations ((7а)) and ((7b)), i.e.,
 The value of sin θ(i) at zs is related to sin θ0 at z = 0 through Snell's law [Kravtsov and Orlov, 1990]
 Whence it follows that if at a certain altitude z, then the value of sin θ(i) is equal to 1 at that altitude (i.e., θ(i)(z) = π/2). Using equation (1) this condition can also be written as
 Thus, equation (11) determines the critical reflection height zcr for the ray with the incidence angle θ0 at the lower boundary of the ionospheric layer. This is why when we determine cos θ(i)(zs) two situations are possible, depending on the ratio between ω cos θ0 and ωcr (see Figure 3). If ω cos θ0 > ωcr, then the wave penetrates through the ionospheric layer (no reflection). In this case, within the entire volume of the ionospheric layer with irregularities, the scattering occurs only at the ascending part of the wave trajectory (the direct wave is scattered alone), and the value of cos θ(i)(zs) according to Kravtsov and Orlov  is
 If ω cos θ0 ≤ ωcr, then the wave is reflected at z = zcr, determined by equation (11), and the range of zs is limited as 0 ≤ zs ≤ zcr. As a result, at the scattering point two waves are scattered: one on the ascending part of the trajectory (direct wave) and the other on the descending path (reflected wave). Then for cos θ(i)(zs), one gets
where “+” and “−” stand for the direct and reflected waves, respectively.
 Now, consider the scattered wave. Let the background electron density profile N0(z) be a smooth function with a single maximum Nm(zm) at zm (single layer model). If ωcr ≥ ω, then the waves scattered downward (cos θ(s) ≤ 0) and upward (cos θ(s) > 0) both reach the lower boundary of the ionospheric layer at z = 0 with an angle , which can be found from an equation similar to equation (10), viz.,
 If ωcr < ω, then only the waves scattered at angles can reach the lower boundary of the ionosphere (z = 0). Here is a certain critical angle whose value, with account of equation (1) and Snell's law ((10)), can be determined as
 If , the scattered wave goes into the upper half-space and then penetrates through the ionosphere.
 Note that in the case of a multi-layer ionosphere (e.g., E and F layers are present), the waves scattered in certain directions can be “trapped” by the interlayer ionospheric waveguide. Such a possibility will be investigated in the next section, but first let us analyze the amplitude characteristics of the aspect-sensitive scattered signals.
 The effective differential cross-section of a random medium is used to characterize the energy scattered by a unit volume into a unit solid angle in a given direction with a unit flux density of the incident radiation [e.g., Rytov et al., 1987]. In the approximation of a single scattering of electromagnetic waves in random isotropic medium with a regular dielectric permittivity ε0 = const, the scattering cross-section is [Rytov et al., 1987; Gershman et al., 1984]
 Here is a three-dimensional spatial spectrum of fluctuations δε; is the scattering vector given by equation (5); and is a polarization factor, where is the unit vector of the electric field polarization of the incident wave. Using the same geometrical optics assumptions as in Rytov et al. , it can be shown that equation (16) is applicable to the analysis of the scattering of a single quasi-plane wave in the medium with smooth spatial variations in ε0(z).
 The factor P(zs) is dependent on the incident wave polarization. For instance, according to Rytov et al.  in the case of a linear polarization
and for a circular polarization
where χ(zs) and γ(zs) denote the angles made by the scattered field wave vector with the polarization vector and wave vector of the incident field, respectively.
 The anisotropic power law model [e.g., Gershman et al., 1984] is a conventionally used approximation for the spatial spectrum of the fluctuations in the magnetized plasma of the upper ionosphere for the inertial interval of the wave numbers. In this model, can be written as
where is a normalization factor, with representing the variance of the fluctuations δε; K|| and K⊥ are longitudinal and transversal (with respect to ) components of the scattering vector ; L|| and L⊥ are characteristic longitudinal and transversal external scale sizes of the ionospheric turbulence with L|| ≫ L⊥; and 3 < p < 4.
 If condition (4) is satisfied, then equation (16) yields
 Equation (21) is inapplicable in the specific case of the horizontal magnetic field, h0z = 0, (at the magnetic equator). It can be shown that in this case becomes
 Therefore, from equation (20) it is possible to determine the scattering cross-section in the direction defined by the aspect condition ((4)) for a plane electromagnetic wave propagating in an ionospheric layer with anisotropic electron density irregularities. Note that since ϕ(i)(z) = ϕ0 and , all the terms in equation (20) can be expressed through the position angles of the incident and scattered field wave vectors at the lower boundary z = 0 of the ionospheric layer.
 As an example, let us first consider scattering of a plane circularly polarized electromagnetic wave propagating vertically in an ionospheric layer with anisotropic irregularities. The wave frequency is assumed to be lower than the critical frequency, ω ≤ ωcr. Note that these are typical conditions for the HF heating of the ionosphere.
 In a single layer model ionosphere, two components will be scattered in the height range 0 ≤ zs ≤ zcr (zcr is determined from ((11)) with cos θ0 = 1), which are the direct and reflected waves. Then equations ((7а)) and ((7b)) with account of equation (13) yield
where the “+” and “−” signs stand for the scattering components of the direct (subscript d) and reflected (subscript r) waves, respectively.
 As can be seen from equation (23b), in the Northern Hemisphere (h0xh0z < 0), the direct wave is scattered southward (− 90° ≤ ϕ(s) ≤ 90°), while the reflected is scattered northward (90° ≤ ϕ(s) ≤ 270°). In the Southern Hemisphere the situation is reversed. Near the magnetic equator (hz ≈ 0), as follows from equation (7b), both components are scattered predominantly within the plane perpendicular to the geomagnetic field lines at all angles θ(s) ∈ [0,π].
 To analyze the orientation of the scattered field wave vector at the exit from the ionospheric layer, it is convenient to introduce an angle . Then, according to Snell's law and using equations ((23b)) and ((1)), one gets
 As an example, Figure 4 shows isolines for ϕ(s) = 75°(255°), 45°(225°), and 0°(180°), which are plotted in the coordinates and . The inclination angle I of the geomagnetic field is set equal to 77.5° (recall that h0x = cos I and h0z = − sin I), which corresponds to the location of the EISCAT HF heater (69°35 ′ N, 19°14 ′ E). In this case, the wave vector of the scattered component at the lower boundary of the ionosphere (z = 0) can be significantly off the vertical despite strongly vertical propagation of the primary wave. The value of this offset increases as α decreases (i.e., with decrease of the scattering altitude zs), and/or as the scattering direction approaches the plane of the magnetic meridian. In this example, the maximum offset is equal to π − 2I = 25°, for zs = 0 and ϕ(s) = 0, π. Analysis of equation (24) shows that such a dependence is valid for the geomagnetic field inclinations |I| ≥ π/4. Otherwise, if |I| ≤ π/4, the maximum offset of the aspect-sensitive scattered wave from the vertical, (i.e., the scattering within the horizontal plane) will occur for zs = 0 and cos ϕ(s) = ± h0z/h0x.
 To calculate the scattering cross section in the direction defined by equations ((23а)) and ((23b)), we combine equations ((18)), ((21)), ((23а)), and ((23b)) and Snell's law (equation (10)) obtaining for the vertically propagating wave
 Substitution of these expressions together with equation (22) into ((20)) yields
where . For quiet ionospheric conditions, it is usually assumed that the plasma density fluctuations are proportional to the background electron density, i.e., . Thus one can write .
 Figure 5 shows the angular distribution of the value 10 log[Q(zs)/Q0(zs)], calculated with equation (25) with p = 11/3 and L⊥/λ0 = 1 (Figure 5a), and L⊥/λ0 = 5 (Figure 5b) for the vertical incidence case for the latitude corresponding to the EISCAT location (I = 77.5°). Here λ0 = f/c = 2π/k0 is the free space wavelength. The results are shown for the scattering of the reflected wave (scattering in the northward direction, ϕ(s) ∈ ± 90°). For the direct wave (scattering in the southward direction, ϕ(s) ∈ 180° ± 90°), the angular distributions are completely symmetric. In order to be able to at least roughly relate these data to the scattering heights zs (or the corresponding plasma frequency ωp(zs)), Figure 5 shows isolines for (solid line), α = 0.75 (dashed line), and α = 0.99 (dotted line). Note that , ϕ(s), and ωp(zs) are related through equation (24). As is evident from Figure 5, the value of the scattering cross-section increases with α (i.e., with the scattering height zs), reaching its maximum at α = 1, which corresponds to . This is because increases with height (since it is assumed that ), and also because the second term in the square brackets of equation (25) gets smaller (since ε0(zs) → 0 as zs → zcr). The latter effect can be treated as an increase of the signal wavelength in the ionosphere as it approaches the reflection point since . Apparently, this is also the reason for the widening of the azimuthal distribution of the scattering cross-section with the decrease of . For example, already for the scattering cross-section is practically isotropic in the azimuthal plane, and smaller L⊥/λ0 values lead to greater values of the scattering cross-section. This would suggest that in the presence of several ionospheric layers (e.g., E and F), a certain fraction of the scattered wave energy can be captured by the interlayer ionospheric waveguide. Since such waveguide propagation is characterized by a very small attenuation, it is quite plausible that this mechanism supported the super long distance propagation of the EISCAT signals observed in the “self-scattering” experiment [Zalizovski et al., 2009]. The potential of exciting the ionospheric interlayer waveguide by the aspect-sensitive scattered HF signals requires greater investigation. It should be noted that the role of scattering in HF radio wave trapping into ionosphere waveguides was rather intensively examined as early as the 1970s [e.g., Gurevich et al., 1975; Erukhimov et al., 1975]. However, in contrast to the previous studies, we will be primarily interested in determining the range of θ0 and ϕ0 responsible for channeling scattered signals in a given direction rather than in estimating the trapping coefficient for an incident wave.
 Let the ionosphere be represented by two layers, e.g., E and F, each characterized by their respective critical frequencies ωcrE and ωcrF and critical heights zmE and zmF. We denote the minimum plasma frequency inside the E-F valley at the height zv as ωv. Further assume that ωcrF ≥ ω, i.e., the wave with frequency ω is reflected from the F layer (the upper “wall” of the waveguide) at any incidence angle. Obviously, the interlayer waveguide can be excited by the waves scattered in the height range zmE < zs < zmF only. Therefore, it is necessary for the direct wave to penetrate the E layer, i.e.,
 As follows from Snell's law, in order for the scattering component to be confined inside the waveguide, i.e., to be reflected from the E layer (the lower “wall” of the waveguide), the following condition must be met:
which, with account of equation (1), can be recast in a form similar to equation (15):
where sin θ(s)(zs) is determined by equation (7b). Since sin θ(s)(zs) ≤ 1, it follows from equation (27b) that the scattering should occur within the height range zmE < zs < zF, where zF ∈ [zv,zmF], and can be determined from ωp(zF) = ωcrE.
 Further analysis will be carried out for the specific conditions of the “self-scattering” experiment conducted by Zalizovski et al. . We are primarily interested in the possibility of channeling the EISCAT signals inside the interlayer waveguide in the directions of the receive sites located at UAS, SPB, and RAO. In the chosen coordinate system, ϕ(s) ≈ 135° for the UAS station and ϕ(s) ≈ 225° for the other two stations. In most heating sessions, the EISCAT heater transmitted toward magnetic zenith at one or two close frequencies around 4.04 MHz. The measurements from the collocated ionosonde showed the presence of strong E layers, with the critical frequencies close to and even above the heating frequencies. For the analysis fcrE = 3.9 MHz and fH ≈ 1.4 MHz are assumed and the calculations are not limited to the vertical incidence case, making it possible to investigate the effect of the angles of direct wave incidence. As known [Gurevich, 2007], artificial ionospheric turbulence is effectively produced by the HF heating in the height range with the lower boundary zUH, determined from the upper hybrid resonance condition , where fH is the gyrofrequency, and the upper boundary zcr determined by the wave critical reflection condition equation (11). For this reason, the plasma frequency fp(zs) in the calculations is assumed to vary within the scattering region from MHz to fp(zcr) = fcrE = 3.9 MHz. Figure 6 shows results of the calculations for such conditions in the system of coordinates with θ0 (counted radially) and ϕ0 (counted counterclockwise). The plots show the distributions of sin θ(s) values which satisfy the conditions of excitation of the interlayer ionospheric waveguide (see equation (27b)). The calculations were made for f = 4.04 MHz, ϕ(s) = 135° and fp = 3.79 MHz (Figure 6а) and fp = 3.87 MHz (Figure 6c). Figures 6b and 6d show the respective distributions of the value 10 log[Q(zs)/Q0(zs)] for p = 11/3 and L⊥/λ0 = 1. The values are given in grayscale (see the legends on the right). As follows from equations ((7а)), ((7b)), and ((21)), the distributions of sin θ(s) for ϕ(s) = 225° will be the same as in Figures 6a and 6c, while the distributions of 10 log[Q(zs)/Q0(zs)] will look like mirror images of Figures 6b and 6d with the reflection line at ϕ0 = 180°. Note that limits of the region from which the scattered signals are captured by the interlayer waveguide are determined at the level (see equation (27b)), which is approximately equal to 0.754 for fp = 3.79 MHz and 0.909 for fp = 3.87 MHz. The dashed line in Figure 6 shows the EISCAT antenna beam at the half-power level. As can be seen, the signal propagation through the interlayer duct channel in the azimuthal directions ϕ(s) = 135° and ϕ(s) = 225° is possible for the given orientation of the heater antenna only for a very narrow range of angles near θ0 ≈ 15° and ϕ0 = 180°, with characteristic sizes (for fp = 3.87 MHz) of ± 15° in azimuth and less than 1° in θ0 (see Figure 6а). Accordingly, the linear size of the scattering region at 200 km is approximately 45 to 50 km across and 3 to 4 km along the magnetic meridian. Note that, as follows from equations ((7а)) and ((7b)), this case corresponds to the scattering of the ionospherically reflected signal. The sharp cutoff at θ0 ≈ 15° is determined by condition ((26)) for the direct wave to penetrate the E layer. The value of 10 log[Q(zs)/Q0(zs)] in this angular range varies insignificantly, from about − 4 to − 12 (see Figure 6b). As fp increases, i.e., as the scattering height zs approaches the critical reflection height zcr, the angular and linear sizes of the scattering region decrease by approximately a factor of 2 (see Figure 6c), while the cross-section changes insignificantly (see Figure 6d).
 The above analysis supports the mechanism of the “self-scattering effect” that was suggested by Zalizovsky et al.  for the explanation of their experimental results of monitoring the EISCAT transmission at three remote locations (SPB, RAO, and UAS). Recall that during the experiment the variations of the Doppler frequency shifts recorded at all the sites were almost identical. During the experiment, significant variations of the electron density were also observed by the EISCAT incoherent scatter radar in the height range from 100 to 200 km. Such variations can significantly affect the Doppler frequency shift and amplitude of the HF heating signal as was demonstrated by the authors using computer modeling. With the above analysis, we have demonstrated that the angular sizes of the scattering region responsible for the excitation and subsequent channeling of the HF signals through the interlayer waveguide to the receive sites are rather small (see Figure 6). Therefore, it is reasonable to assume that all remotely recorded signals first travel through the same irregularities in the lower ionosphere and are equally affected by them. Then, the signals reflected from the ionospheric F layer are scattered by field-aligned irregularities produced by the same EISCAT heating transmission and propagate toward the receive sites through the ionospheric waveguide. Since the scattering region guiding the signals into the interlayer duct channel for propagation into different directions is the same, it is plausible to expect that the variations of the Doppler frequency shifts and signal amplitudes recorded at greatly dispersed receive sites will be very similar, as was observed during the “self-scattering” experiment reported by Zalizovski et al. .
5 Summary and Conclusions
 In this paper, the ray optics approximation and small perturbation method were used to analyze the effect of refraction on the scattering characteristics of HF signals scattered by random field-aligned irregularities of the ionospheric plasma. The calculations are carried out for an isotropic plane-stratified (on average) ionosphere, i.e., the incident and scattered waves were both assumed to propagate along unperturbed trajectories. The equation of the so-called aspect-sensitive scattering is derived which relates the trajectory characteristics of the incident and aspect-sensitive scattered waves. The scattering cross-section is calculated within the Born approximation for a plane electromagnetic wave propagating in the ionospheric layer with anisotropic electron density irregularities. For the vertical incidence scenario, the intensities of the aspect-sensitive scattered wave (cross-section) and its exit angle are calculated as functions of the scattering direction and height. A possibility of excitation of the interlayer ionospheric waveguide by the aspect-sensitive scattered signals with a selected orientation of the horizontal projection of the wave vector is investigated in dependence on the angles of the direct wave incidence. It is demonstrated that such possibility can be implemented using powerful HF heaters like EISCAT or HAARP. Under certain conditions (e.g., specific orientation of antenna beam), the scattered HF signals can be trapped inside the waveguide and subsequently channeled to very long distances because of the very small attenuation characteristic of the interlayer waveguide.
 Thus, the aspect-sensitive scatter mechanism can explain the “self-scattering effect” which was observed using the EISCAT transmission by Zalizovsky et al. . Therefore, there exists the potential for the practical applications in HF communication, although further studies are required for a more reliable assessment. In particular, it is necessary to consider the suggested mechanism with taking into account the magnetic field effects which may influence propagation of the incident and scattered waves. Addressing this issue properly would require application of full scale 3-D numerical ray tracing. The authors plan to carry out the respective analysis in the future and present the results in a separate publication. As a first approximation, however, we think that neglecting the geomagnetic field effect on radiowave trajectories is a legitimate approach, which was also taken by other authors [e.g., Gurevich et al., 1975].
 The results of this work can be useful for the analysis and interpretation of the experiments on HF signals scattering from artificial and natural ionospheric irregularities in the ionosphere, for the development of new methods of diagnostics of the near-Earth plasma turbulence and also for the investigation of the mechanisms of long and super long range propagation of HF electromagnetic emission.
 This work was conducted in the frame of the Projects “Yatagan-2” (registration 0011U000063) and “Spitsbergen-12” (registration 0112U004096) and with a partial financial support by the STCU Partner Projects Р-330 and P-524.