[1] An improved method of estimating ground-scattered power using high-frequency (HF) ray tracing techniques that overcomes the limitations of the derivation presented by Bristow and Greenwald (1995) is presented. The improved method is applied toward identifying the effects of an artificial ionospheric density layer on measured ground scatter power. The presence of artificial density layers induced at the High Frequency Active Auroral Research Program (HAARP) station in Gakona, Alaska, are observed through the ground-scattered power received by the Kodiak Super Dual Auroral Radar Network (SuperDARN) HF radar. The location and physical dimensions of the artificial layers are estimated by simulating radar returns using HF ray tracing through a model ionosphere that includes a model artificial density layer. Simulation results of ground-scattered power as a function of range are compared to the measured ground-scattered power as a function of range during a time period when artificial layers were evident in ionogram data. It is shown that a model artificial density layer based on research by Pedersen et al. (2009) produces simulation results that approximate the mean of the measured results.

[2] An improved method of calculating ground-scattered power as a function of time delay range using ray tracing techniques is presented in this paper. The method was developed in response to limitations in the derivation presented byBristow and Greenwald [1995]and is applied to the identification of the effects of artificial density layers on the observed ground-scattered power by high-frequency (HF) radar.

[3] High-power HF heating experiments recently have been shown to produce artificial density layers in the ionosphere [Pedersen et al., 2010]. These artificial layers have been observed using both optical instruments and ionosondes [Pedersen et al., 2009, 2010]. The ability to generate artificial layers may have important radio applications, particularly for HF communications.

[4] Super Dual Auroral Radar Network (SuperDARN) radars have been used extensively for observations of HF heating experiments. Coherent backscatter from artificial field-aligned irregularities (FAIs) generated by the SPEAR and EISCAT high-power radar systems has been observed by both the Þykkvibær and Hankasalmi SuperDARN radars [Robinson et al., 2006; Wright et al., 2009, 2006]. The observations made by these radars have been used to infer FAI excitation thresholds and the presence of hysteresis effects during ionospheric modification experiments [Wright et al., 2009, 2006]. Similarly, the Kodiak SuperDARN radar has been used to observe effects of heating experiments conducted by the Frequency Active Auroral Research Program (HAARP) radar system [Kendall et al., 2010].

[5] In this paper, the measured return power from the Kodiak SuperDARN radar [Greenwald et al., 1985, 1995] whose propagation path intersects an artificial layer region has been investigated. The measured return power as a function of range was compared to a numerical estimate of the return power profile generated using ray tracing techniques and a model of the artificial layer. The model of the artificial layer is based on ray tracing conducted by Pedersen et al. [2009]that demonstrated optical observations of artificial layers are consistent with a density perturbation Gaussian distributed in the horizontal and vertical directions. Results are provided that illustrate the numerical estimates of ground-scattered power approximate the mean of the observed ground-scattered power profiles.

2. Calculation of Radar Return From Ray Tracing Results

[6] A method for calculating ground-scattered power as a function of time delay range from ray tracing results was derived byBristow and Greenwald [1995]. However, during the course of our research it was found that this method could produce spurious results for ground-scattered power.Section 2.1 briefly discusses the Bristow and Greenwald [1995] limitations before presenting an alternative derivation that overcomes these shortcomings.

2.1. Bristow and Greenwald Method

[7] Consider the radar geometry illustrated in Figure 1 where a pulse of length τ is transmitted from the origin at the elevation angle α, is refracted by the ionosphere, and scatters from the ground at a distance x. The radar transmits over a beam broad in elevation but restricted in azimuth to a beam width of φ_{b}. The area subtended by the pulse on the ground is a function of the incidence angle θ, total raypath length R, beam width φ_{b}, and pulse length τ. Given the radar geometry illustrated in Figure 1, Bristow and Greenwald [1995]illustrated that the ground-scattered power could be calculated as

where σ^{0} is the surface backscattering coefficient, P_{t} is the total transmitted power, and G(α, φ) is the antenna gain pattern as a function of take off angle α and azimuth angle φ. The term f(α/θ), labeled the focusing factor, in (1) accounts for the power focusing that is a function of the reflecting layer structure. The focus factor term was derived by Bristow and Greenwald [1995] as

f(Î±)=Râˆ‚xâˆ‚Î±sin2(Î¸)âˆ’1

and contains discontinuities at points where âˆ‚xâˆ‚Î±sin2(Î¸) approaches zero. In practice, it is assumed that the incidence angle θ equals the take off angle α which is restricted to the angular extent 5–45 degrees by the antenna gain pattern so that discontinuities in the focus factor are only introduced by sign changes in the term ∂x/∂α. Sign changes in ∂x/∂α are the result of a nonmonotonic relationship between ground range and take off angle and physically correspond to the intersection of rays on the ground. Numerically, these points of discontinuity result in arbitrarily large calculated power returns when using the Bristow and Greenwald [1995] method.

[8] There are many common situations where the ionospheric conditions result in a nonmonotonic relationship between ground range and take off angle. For example, a strong E layer in conjunction with an F layer can produce two distinct regions of ground scatter where there exists a nonmonotonic relationship between ground range and take off angle. Figures 2a–2c illustrate a raypath plot, x(α) plot, and the calculated ground-scattered power plot in such a situation. Note that the ground-scattered power plot inFigure 2c contains curves for two sets of take off angles and demonstrates that discontinuities in f(α) result in an arbitrarily large received power, which in the limiting case approaches infinity.

2.2. A New Approach

[9] The derivation presented by Bristow and Greenwald [1995] is simply obtained from theory and can be readily applied to the results of a ray tracing analysis, but it fails to gracefully handle the common case where there is a nonmonotonic relationship between ground range and take off angle. A new approach based on simply the conservation of energy is presented here.

[10] The distribution of power per unit solid angle of a transmitted pulse of power P_{t} is S_{t}(α) = P_{t}G(α){4π}^{−1} (W/Ω), where G(α) is the normalized antenna gain. In an unconstrained medium, power spreads in azimuth and elevation such that the power per unit area at a distance of R is S_{t}R^{−2} (W/m^{2}). Here we assume power spreads normally in the azimuth direction so that the power per unit length-angle in elevation isS_{t}R^{−1} (W/(rad m)). However, the spreading in elevation is a function of ionospheric structure. Assuming that the target fills the beam in azimuth, the total power scattered from the illuminated area per unit elevation angle is (S_{t}R^{−1})(Rφ_{b}) = S_{t}φ_{b}, which has units of (W/rad) and illustrates that scattered power is conserved in the azimuth dimension. The total scattered power from the illuminated area is

Ps(t)=Ï†bâˆ«Î±2Î±1Ïƒ0(u)St(u)du,

where σ^{0} is the surface backscattering coefficient to be discussed in sections 2.3 and 2.4 and the limits of integration correspond to the illuminated angular extent. Given a transmitted pulse of length τ and the time delay to the scattering point as a function of elevation angle, t_{delay}(α) or equivalently α(t_{delay}), the angular extent in elevation that is illuminated at time t is α(t_{delay})|_{t−τ} ≤ α ≤ α(t_{delay})|_{t}. Figure 3 graphically illustrates the relationship between total scattered power and the plots of S_{t}(α)R^{−1}(α) and α(t_{delay}). Note that ionospheric focusing occurs through the spacing of the limits of integration in equation (3). In situations where t_{delay}(α) is not monotonic, t_{delay}(α) can first be divided into monotonic segments so that α(t_{delay}) is a single-value function andP_{s}(t) is then the sum of the results of equation (3) for each segment.

[11] Given the expression in (3) for the scattered power, the power received is found in a similar fashion. Assuming isotropic reradiation in the upper half plane, the distribution of scattered power per unit solid angle is a constant S_{s} = P_{s}/2π(W/Ω). Again we assume power spreads normally in azimuth but is a function of ionospheric structure in elevation so that the power per unit length-angle in elevation isS_{s}R^{−1} (W/(rad m)). The radar presents a fixed effective area A_{A} (m^{2}) assumed to be a function of elevation angle but uniform with respect to azimuth within the beam width. Let L_{φ} (m) be the effective length of the antenna in the azimuth dimension so that the power per unit elevation angle is (S_{s}R^{−1})(L_{φ}) = L_{φ}S_{s}R^{−1} (W/rad). The total power received may then be found by integrating over the elevation angular extent from the scattering area that subtends the illuminated length of the antenna in elevation which may be formulated as

In (4) we include the antenna gain pattern G(α) in the integrand to account for the dependence of the antenna illuminated area on elevation angle. The effective length of the antenna in elevation is proportional to the wavelength and on the order of tens of meters in this study. An effective length on this scale size corresponds to a minute angular extent from the scattering area so that (4) may be approximated as

PR(t)=CLÏ†SsRâˆ’1(Î±(t))G(Î±(t)),

where we assume that the integrand in (4) is constant over a minute angular extent C (rad). The expressions in (3) and (5) can be combined to give

Note that the all of the constants that appear in (6) are inconsequential as results in this study are expressed in decibels above an arbitrary noise level rather than absolute power.

[12] A comparison of the calculated received power from a smooth ionosphere given identical parameters using the alternative derivations presented in equations (1) and (6) is illustrated in Figure 4. Figure 4 illustrates that the new method produces a power profile that decays more rapidly than the Bristow and Greenwald [1995] derivation and differs near the skip distance in amplitude and trend. The Bristow and Greenwald [1995] derivation results in a peak value at the skip distance range of ∼1050 km with an amplitude of 55 dB that exceeds the dimensions of the graph in Figure 4 before monotonically decreasing. The large amplitude at the skip distance in the Bristow and Greenwald [1995] derivation is the result of the focus factor approaching zero at this range as discussed in section 2.1. The alternative derivation in equation (6) results in a power profile that increases to a local maximum at ∼1100 km of 30 dB before monotonically decreasing at a slightly greater rate than given by the Bristow and Greenwald [1995] derivation. The smearing of the skip distance peak in the power profile given by (6) is the result of a convolution effect of the spatial dimensions of the transmitted pulse and the illuminated scattering area. Near the skip distance, the illuminated area is smaller than the projection of the transmitted pulse on the ground. Equivalently, at t_{0}, where (t_{0} − t_{min}) < τ, the limits of integration in (3) are [α_{1}, α_{2}], where and The effect is that the scattered power increases up until the point that the entire transmitted pulse is projected onto the ground and the limits of integration in (3) are governed by the pulse dimensions. A similar effect is not witnessed in the Bristow and Greenwald [1995] derivation because the assumption is made that the illuminated area is always determined by the dimensions of the transmitted pulse. The presence of a skip distance maximum preceded by increasing power samples is consistent with measured results. The discrepancy between the trend of the power profiles given by the Bristow and Greenwald [1995] derivation and the method presented here are not investigated further as it will be shown in section 3 that power profiles generated using the new method produces curves that accurately approximately the trend of measured results.

2.3. Terrain Cross Section

[13] An influential factor in the received ground scatter power is the ground cross section σ. Given that the ground is a distributed target, the ground cross section is a function of the effective scattering area and can be written as

Ïƒ=Ïƒ0AC,

where σ^{0} is the surface backscattering coefficient and A_{C} (m^{2}) is the effective scattering area [Peebles, 1998]. In general, σ^{0} is a function of angle of incidence, polarization, frequency, roughness, and permittivity at the scattering point [Moore et al., 2008]. Although various theoretical approximations of σ^{0} have been derived, the complicated nature of the scattering coefficient lends itself to an experimental approach where distributions of σ^{0} are measured and used to derive empirical models. Empirical models of σ^{0} for various terrain classes, polarization, and frequency are provided by Moore et al. [1980], Nathanson et al. [1991], and Ulaby [1980]. Figure 5 illustrates a comparison of empirical models of σ^{0} as a function of incidence angle for similar parameters given by the aforementioned authors.

2.4. Application of Terrain Cross-Section Models

[14] Several methods of estimating and incorporating a terrain cross section into the simulation of the ground scatter return profile were investigated with the goal of isolating power enhancements due to ionospheric phenomena from those due to variations in σ^{0}.

[15] Ground cross-section estimates based on empirically fitted curves for ÏƒÂ¯0(Î¸,f) given by Ulaby [1980] and Nathanson et al. [1991] were both implemented. The empirically fitted curve for ÏƒÂ¯0(Î¸,f) provided by Ulaby [1980] is valid for θ ∈ [0 , 80] degrees and f ∈ [1 , 18] GHz and the empirically fitted curve provided by Nathanson et al. [1991] is valid for θ ∈ [30 , 90] degrees and f ∈ [1 , 2] GHz. However, lacking an empirical model of ÏƒÂ¯0(Î¸,f) for HF band, the curve of ÏƒÂ¯0(Î¸,1GHz) from both models was used for calculations. The value of incidence angle θ with respect to the surface normal was calculated from ray tracing parameters for both a smooth, spherical earth and also from a terrain profile generated with digital elevation model (DEM) data at a 3 arc second (∼100 m) resolution. Given a terrain profile, the surface normal can be calculated using the geometry depicted in Figure 6a and equations (8)–(12).

a=REarth+h1,

b=REarth+h2,

c2=a2+b2âˆ’2abcosÏ†,

Î³=sinâˆ’1(acsinÏ†),

Î¸=Ï€âˆ’Î³+Ï€2.

Note that the surface backscattering coefficient as defined in (7)) is given per unit area. However, in this application the total scattered power is found by integrating over an angular extent as depicted in (3). Conversion of σ^{0} to units of per radian is achieved by multiplying the backscattering coefficient by the area per unit elevation angle. Considering the geometry of Figure 6b, the illuminated area can be expressed in terms of elevation angle as

A=RÏ†bX=RÏ†bRÎ”Î±sin(Î±)âˆ’1

so that the surface backscattering coefficient per unit radian is

3. Application: Observations of Artificial Layers in Ground-Scattered Power Profiles

[16] HF ionospheric heating experiments at HAARP appear to produce bottomside plasma density enhancements. The signature associated with the enhancement is the formation of an optical ring that results from the excitation of electrons at energy levels consistent with ionization production [Pedersen et al., 2009]. The results of ray tracing illustrate that modeling the artificial bottomside density layer as a density perturbation Gaussian distributed in the horizontal and vertical directions results in the deflection of rays into a ring structure consistent with optical observations during heating experiments [Pedersen et al., 2009].

[17] Independent observations of the production of artificial density layers at the HAARP station located in Gakona, Alaska, were sought from the ground scatter return measured by the Kodiak, Alaska, SuperDARN. The Kodiak SuperDARN is uniquely located to observe density perturbations due to its field of view with respect to the Gakona HAARP station and has been used in previous experiments to analyze the time scales of irregularities produced by heating experiments [Kendall et al., 2010]. The potential effects of an artificial density layer on received ground scatter at the Kodiak SuperDARN were investigated through ray tracing by inserting a model artificial layer into the density profile and using the resultant raypath lengths and time delays to calculate from (6)the expected ground scatter return. The expected ground scatter return was then compared to the observed ground scatter return during a period of time when high-power HF heating experiments were performed.

[18] One period that high-power HF heating experiments were conducted with the potential to produce artificial layers was 18 November 2009 between 01:17 and 01:34 UT. A RTI plot of the return power observed by the Kodiak SuperDARN during this interval is illustrated inFigure 7.

[19] During the interval 01:17–01:34 UT, HAARP was operated at 2.92 MHz with O-mode polarization at a power of ∼440 MW ERP, which is consistent with test conditions under which artificial layers had been generated previously [Pedersen et al., 2010]. In Figure 7, the high-power backscatter observed in the range [500, 800] km is an indicator of high-power HF heating and results from specular reflection from plasma irregularities in regions of instability. Note that the backscatter power present from [1100, 1800] km can be divided into two regions, the first with a skip distance in the range [1100, 1200] km and the second with a skip distance between [1500, 1600] km in range. Ionograms measured during the period 1:15–1:35 UT demonstrate that the background ionosphere is composed of distinctiveE and F layers. The results of a ray tracing simulation based on measured ionogram characteristics are illustrated in Figure 8. Note that the raypath plot in Figure 8 clearly depicts two distinct regions of ground scatter return from E region deflected rays at low elevation angles and from F region deflected rays at higher elevation angles which supports the structure of the RTI plot in Figure 7. Researchers have postulated that artificial density layers form when the transmit frequency, f_{T}, matches the upper hybrid frequency, f_{uh}, and/or multiples of the electron gyro frequency, f_{ce} [Pedersen et al., 2010]. These characteristic plasma frequencies correspond to electron densities obtained in the F region during the time period in question and consequently the simulation results will be limited to calculating the expected ground scatter return due to rays deflected from the F region.

[20] A comparison of the F region simulated ground scatter return with the measured return power during the interval UT 01:32:34–01:32:35 on 18 November 2009 is illustrated in Figure 9. Note from Figure 9 that the measured profile features very large fluctuations in power between range bins. Furthermore, a comparison of consecutive integration intervals also demonstrates that there is a large variation in power at the same range bin over time. Despite the large natural fluctuations in the measured profile in Figure 9, note that the simulated ground-scattered power profiles accurately approximate the skip distance location and the trend of the mean of the measured profile regardless of terrain cross-section model implemented. Because of the high natural fluctuations between range gates of the measured return power and the comparatively small difference between the simulated results using either theNathanson et al. [1991] or Ulaby [1980]ground cross-section models instead of a uniform ground cross section, a uniform ground cross section will be assumed for all results presented here.

[21] Given a brief comparison of the simulated and measured ground-scattered power as a function of range due to the background density profile, the effects of a model artificial layer on the simulated ground-scattered power can be investigated.Figure 10 illustrates an ionogram from 18 November 2009 at UT 01:26:40 that illustrates an artificial density layer in addition to the background density profile. The artificial density layer present in Figure 10was simulated in ray tracing by inserting a perturbation in the density profile grid. The perturbation was Gaussian distributed in horizontal and vertical directions. The peak density, peak density altitude, and half-widths in the horizontal and vertical directions were all free parameters. In order to maintain consistency with previously published results, the horizontal and vertical half-widths were chosen to be 30 km and 8 km, respectively [Pedersen et al., 2009]. The peak density was estimated from ionograms to be in the range 2.5–3.0 MHz. The effects of an artificial density perturbation at various altitudes with these parameters on the simulated ground-scattered power is illustrated inFigure 11a. Note from Figure 11athat a model artificial density layer produces a power reduction region with a width of 60–100 km followed by a smaller amplitude power enhancement region with a width of 80–120 km. The width of the power reduction and enhancement regions increases as the artificial layer descends in altitude because lower-elevation angle rays subtend a larger time delay region than the same angular extent at higher elevation angles.

[22] Before comparing simulation results to the measured data, the format of the measured data must be considered. In this study, the measured data from the Kodiak SuperDARN was analyzed in fittedform. Fitted data consists of the measured return signal from a number of transmit sequences averaged over an integration interval. The fact that the return signal is being averaged over a period of time has practical implications. Optical observations during high-power HF heating experiments demonstrate that after artificial density layers form, the layers rapidly descend at rates up to 0.26 km/s until reaching an altitude of ∼150 km where recombination processes exceed ionization production and the density perturbation dissipates [Pedersen et al., 2010]. The effect of integration can be approximated in the simulated ground-scattered power results by averaging the ground-scattered power profiles calculated with the density perturbation vertically displaced by the product of the rate that the density layer descends and the length of the integration interval. The effect of the vertical displacement of an artificial layer over an integration interval on the simulated ground-scattered power profile is illustrated inFigure 11b. Figure 11billustrates that the simulated profiles generated with the artificial layer displaced by 0.25 km in altitude differ by fractions of a decibel. As the effect of averaging produces a negligible change in the ground-scattered power profile compared to natural fluctuations, it will be ignored in all results presented in this paper.

[23] Given a description of the effects of a model artificial density layer on the simulated ground-scattered power, the measured data was examined for power reductions that may be due to focusing by an artificial layer. A challenge faced in determining possible power reductions due to focusing by an artificial layer was the natural temporal distribution of received ground-scattered power. For example,Figure 12 illustrates the mean and standard deviation of backscattered power as a function of range on 18 November 2009 during the interval UT 01:20:00–01:25:00, which is during the period HAARP was heating but prior to any evidence of artificial layers in ionograms. In Figure 12, the mean and standard deviation in the range [1600, 1800] km where focusing from an artificial layer might be expected to create power enhancements is approximately 7 dB and 4 dB, respectively. Application of Chebyshev's inequality with these parameters then implies that one could expect received power levels anywhere between 1 and 13 dB 56% of the time at these ranges, which demonstrates the ambiguity in discerning a power enhancement rather than a natural fluctuation.

[24] Despite the difficulties mentioned, several measured profiles were identified that contained possible power reductions due to focusing by an artificial layer. Simulated profiles were generated to recreate the power reductions observed in these measured profiles. A model artificial layer was included in the density profile and the parameters of the artificial layer were varied in an attempt to fit the resultant simulated profile to the mean of the measured profile. A comparison of the simulated profiles with and without a model artificial layer and the measured profile during the integration interval at UT 01:26:58 is depicted in Figure 13. Note in Figure 13 that both simulated profiles accurately approximate the location of the skip distance maximum and the trend of the mean of the measured profile outside of the regions annotated as power reduction or enhancement regions. At ranges between [1620, 1700] km corresponding to the artificial layer induced power reduction region, the simulated profile generated without a model artificial layer overestimates the mean of the measured profile whereas the simulated profile generated with a model artificial layer approximates the mean of the measured profile. Similarly, at ranges between [1700, 1780] km corresponding to the power enhancement region, the simulated profile without a model artificial layer underestimates the mean of the measured profile whereas the simulated profile generated with a model of the artificial layer approximates the mean of the measured profile. Given the similarity between the simulated profile including a model artificial layer and the measured profile, an estimate of the actual dimensions and location of the artificial layer is provided by the model parameters listed in Figure 13.

[25] Some discrepancy between the location, width, and amplitude of the simulated and observed power reductions is due to the oversimplification of the artificial density layer model. While the evolution of the artificial density layer in altitude has been discussed and found to be negligible, the spatial evolution of the density perturbation has been neglected. Optical observations during HF heating experiments illustrate that the interior of an artificial density layer is highly structured [Pedersen et al., 2010]. Specifically, the interior of an artificial density layer contains field-aligned filaments that evolve over the lifetime of the artificial density layer [Pedersen et al., 2010].

4. Conclusions

[26] In this paper, an improved method of calculating ground-scattered power using ray tracing techniques has been presented along with an application of the method in identifying and characterizing artificial density layers. The estimated ground-scattered power profile generated using a simple model of an artificial layer was compared to the observed return power profile during a time interval where an artificial layer was present in ionograms. It was found that a simple model of the artificial layer could be used to generate a simulated ground-scattered power profile that contained power reductions and enhancements of the appropriate width and location as those observed in the measured data.

[27] However, a number of challenges presented ambiguity in the results. Foremost, it was found that the observed return power naturally fluctuates over a wide range of values, which limits the ability to discern enhancements. Future work might include quantifying the natural variation in ground-scattered power through a Monte Carlo approach of varying the ionospheric density in a random fashion and observing the resulting spread of simulated ground-scattered power. In addition, the model of the artificial density layer used in simulations is an oversimplification that does not account for the evolution of the artificial layer density and structure. Although some encouraging simulation results have been illustrated, the combination of these factors make the identification of power enhancements in ground-scattered power return due to artificial layers a challenging task.

Acknowledgments

[28] This work was supported under award N000140711081 from the Office of Naval Research. Operation of the Kodiak radar system is supported primarily through a grant from the National Science Foundation under award 0520546.