Ionospheric HF channel reciprocity is investigated at middle latitudes on the basis of ionospheric oblique incidence sounding experiments. Two identical Wuhan Ionospheric Oblique Incidence Sounding Systems (WIOISS), located at Wuhan (30°32′N, 114°21′E) and Wanning (18°58′N, 110°31′E), are used to carry out the campaign. Comparisons of group distance and Doppler shift between Wuhan-Wanning and Wanning-Wuhan HF ionospheric propagation paths indicate that the reciprocity of the ionospheric HF channel is satisfied at midlatitude region. A 3-D ray tracing simulation is also implemented to evaluate the group distances of the two paths. Midlatitude ionospheric HF channel reciprocity, as verified both experimentally and theoretically in the present study, can be useful for HF communication systems and sky wave over-the-horizon radars.
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 In principle, the reciprocity theorem that can be derived from the Maxwell equations is applicable to electrical systems, accounting for the reciprocal relationship between stimulus and response in an electromagnetic field. On the basis of the theorem, the relationship between an oscillating current and the resulting electric field remains unchanged if the locations of the current and the field measurement exchange with each other. In the ionosphere, which is an anisotropic medium because of the ambient magnetic field, the condition for reciprocity frequently breaks down for HF systems and sky wave over-the-horizon (OTH) radars that rely on high frequency (3–30 MHz) radio wave propagation in the ionosphere. However, under certain circumstances the reciprocity of the ionospheric channel still needs to be valid to play a role in the application of short-wave communication channel simulation, HF direction finding, and sky wave OTH radar.
 Several studies have been performed to explore both theoretical and experiment aspects of the reciprocity of HF ionospheric paths. Balser et al. , Jull and Pettersen , and Jull  investigated the HF ionospheric radio circuits through the fading features of signal pulses and the characteristics of antenna polarization. Later, Ginsburg  and Budden  developed the ionospheric reciprocity theorem that the dipole aerials were parallel or perpendicular to the local magnetic plane in the theoretic approximation of ray theory. Heading  provided a complete generalization of the reciprocity problem in a stratified, inhomogeneous, anisotropic, ionized medium. Numerical analysis of the reciprocal problem was also conducted by Tateiba [1982, 1991] from the point of wave propagation in the inhomogeneous random media, although realistic circumstances of radio wave propagation for sky wave radar and HF systems were not taken into account in his studies. Coleman  successful extended reciprocity ideas to perturbed nonisotropic media and derived several important reciprocity-related results.
 Sky wave OTH radar and HF communication systems both take advantage of ionospheric reflection to explore remote targets and communicate with distant terminals. Characteristically, the background ionosphere is turbulent, permeated by various random irregularities. As a result, the scattering function, which describes the group delay spread and Doppler spread of pulse signals propagation in a turbulent ionosphere, is of great importance to characterize the HF ionospheric channel and acquire a better understanding of HF channel reciprocity. By performing scattering function modeling of a reflective and fluctuated ionospheric channel, Watterson et al.  first established the statistical stationary narrowband HF channel model, followed by the development of the wideband model of the HF channel by Vogler and Hoffmeyer . In terms of theoretical analysis the phase-screen-diffraction-layer method [Kiang and Liu, 1985a, 1985b; Wagen and Yeh, 1986, 1989a, 1989b] was used to compute the transfer functions and coherence functions along the oblique HF ray path. Gherm and Zernov [1995, 1998] and Gherm et al. [1997a, 1997b] adopted the Rytov approximation to pursue systematic studies of the two-frequency, two-position, time coherence function and the ionospheric scattering function that describe the HF ionospheric fluctuating radio channel. They also derived the analytical expressions for the power spectra of the amplitudes and phase fluctuations of the received signals. However, all these studies assumed that the background medium is isotropic, which is always not the actual case. Therefore, the anisotropy of the ionosphere should be included to better understand the reciprocity problem in the ionosphere.
 Ionospheric oblique incidence sounding provides a traditional but effective means to recognize the background conditions for radio propagation [Davies, 1989; Reinisch, 1986]. A bistatic system configuration can be used to acquire the ionospheric electron density profile and estimate the quality of the ionospheric HF channel by taking advantage of radio wave oblique transmission and reflection in the ionosphere. The Wuhan Ionospheric Oblique Incidence Sounding System (WIOISS) is just such an ionospheric sounding system. It was recently developed by the Ionospheric Lab of Wuhan University on the basis of the monostatic Wuhan Ionospheric Oblique Backscattering Sounding System [Chen et al., 2007, 2009a, 2009b; Shi et al., 2009; Zhang et al., 2010]. The WIOISS has the following characteristics: (1) the power of transmitter can be as low as only a few hectowatts; (2) the ionospheric parameter can be acquired in real time, including the classical oblique ionogram and Dopplionogram; (3) the global positioning system (GPS) technique is used in WIOISS for time and frequency synchronization between the transmitter and the receiver, which considerably improves the performance of the entire system. The WIOISS not only can be used for real-time ionosphere research and HF channel modeling but also provides essential data for the frequency management systems of sky wave OTH radar.
 The main objective of the present study is to use the WIOISS to investigate the applicability of the reciprocity on the HF ionospheric channel in the midlatitude region. A two-way oblique sounding campaign is carried out by utilizing two identical WIOISSs located separately in the midlatitude region, that is, at Wuhan (30°32′N, 114°21′E) and Wanning (18°58′N, 110°31′E), China. Obtained scattering functions are then analyzed to examine the ionospheric HF channel reciprocity. A general description regarding reciprocity of the HF channel scattering function in an anisotropic media is given in section 2. Section 3 presents the details of the WIOBSS experiment arrangements, followed by the 3-D ray tracing simulation of HF signals using the IRI2007 and IGRF model in section 4. The experimental results are shown and discussed in section 5. In section 6 we make our main conclusions.
2. Analysis of Scattering Function
 The scattering function and correlation function are pairs of Fourier transformation, which are usually used for interpreting the time and frequency spread in wave propagation and scattering in random media. The scattering function, which is also known as the delay-Doppler power spectrum of the received signal, is the most fundamental quantity describing the two-dimensional characteristics and the comprehensive effects of the ionospheric HF channel. As a time- and space-varying anisotropic inhomogeneous medium, the ionosphere can affect HF signals both deterministically and stochastically. Under quiescent conditions, the deterministic model is used to interpret the ionospheric reflective channel. In contrast, the effects of the random fluctuations in the ionospheric density can be principally represented by a stochastic model.
 In the circumstance of ionospheric oblique incidence sounding, the HF channel is exposed to both deterministic and stochastic influences of the ionosphere. The HF channel scattering function contains not only information on propagation time delay and Doppler frequency but also the characteristics of the delay and Doppler spread. Therefore, the reciprocity of an ionospheric two-path channel can be explored from the aspect of the scattering function.
 The scattering function in the HF band depends both on the spectrum of ionospheric fluctuation and the propagation path through the medium. According to Budden , the propagation path does not change in reciprocal propagation in a magnetic, stratified medium. Gherm and Zernov  derived the expression of the scattering function of HF wave propagation in the fluctuating ionosphere based on the Rytov approximation for the case of an isotropic medium. They concluded that the scattering function must be purely symmetric in shape in the geometrical optics approximation. However, it can also be applied to HF wave propagation under anisotropic conditions if the spectrum of ionospheric fluctuations is reciprocal.
 The effect of ionospheric dielectric fluctuations on the radio wave is finally determined by the correlation function Bɛ(d) which represents the random fluctuation of Δ(), where Bɛ(d) = 〈Δɛ(r1′)Δɛ(r2′)〉 [Yeh and Liu, 1972; Ishimaru, 1978]. According to the Wiener-Khinchin theorem, the spectrum Φ(κ) is the Fourier transformation of the correlation function Bɛ(d). Based on a number of observations, the spectrum Φɛ(κ) tends to hold a power law form Φɛ(κ) ∝ κ−p. A generalized turbulence space-correlation and wave number spectrum function that can be used in HF ionospheric propagation was proposed by Shkarofsky  and also adopted by Gherm et al. [1997a, 1997b] and Gherm and Zernov :
where r0 is the inner scale, l0 = 2π/κ0 is the outer scale, and K is a Hankel function of imaginary argument.
 Anisotropy spectrum-correlation pairs can be reformed by replacing r2 by x2/lx2 + y2/ly2 + z2/lz2 and replacing κ2 by lx2κx2 + ly2κy2 + lz2κz2, where lx, ly, and lz denote characteristic spatial scales of the ionospheric random fluctuation in Cartesian coordinates, as shown in Figure 1. Thus,
 Upon the assumption that the two geometry paths of the oblique propagation are identical, which is the usual situation, the incident direction and reflective direction can be considered to be reversed when the locations of transmitter and receiver are mutually replaced; that is,
 It is obvious that the spatial characteristic scales lx, ly, and lz do not change with the direction of the wave vector. This indicated that the inversion → − did not change the shape of the spectrum of the medium fluctuation. As a consequence, the spectrum of dielectric fluctuations is independent to mutual replacement of the transmitter and receiver.
 On the basis of the above analysis, the delay-Doppler spectrum related to the propagation path and the spectrum of the dielectric fluctuations do not change shape with respect to the inversion → −. However, further theoretical studies and numerical simulation are needed in future work.
3. Experimental Deployment
 The principle of ionospheric oblique incidence sounding radar is very similar to the ionosonde, but the bistatic ionospheric oblique incidence sounding radar is capable of long-distance measurements. The WIOISS has two typical operation modes, a fixed-frequency mode and a swept-frequency mode with Doppler resolution of 0.0367 Hz and group distance resolution of 6.25 km. In the fixed-frequency mode, the channel scatter function is obtained by the Fourier transformation and the correlation of the received signals. The relationship between group delay of signals and sweeping frequencies is calculated in the swept-frequency mode. Thus, the WIOISS provides an outstanding opportunity to study the reciprocity of two ionospheric HF paths between remote locations. The detailed specification of WIOISS is demonstrated in Table 1.
Table 1. WIOISS Basic Parameters
Horizontal polarization log-periodic antenna
GPS clock stability
41.66 μs × N (N is a integer)
Variable, 20% typically
16 bits, 20 MHz
Binary phase coding up to 1024 elements max
 To validate the reciprocity of a midlatitude ionospheric HF channel, two identical WIOISSs were operated individually at the Wuhan and Wanning sites for three consecutive days, 27–29 March 2009. Since the horizontal polarization log-periodic antenna was shared by the transmitter and the receiver at each location, the radio signals were first transmitted from Wuhan and the antenna at Wanning was used for receiving, and then the antenna at Wanning was used for transmitting while the antenna at Wuhan received the signals. The ionosphere was considered to be stationary for the time interval of about two minutes. Swept-frequency experiments were also designed to test the reciprocity of two time delays, and the scatter functions were calculated by later fixed-frequency experiments. The arrangement of Wuhan-Wanning ionospheric HF reciprocity experiments is depicted schematically in Figure 2.
4. Results of the Experiments
 As with the operation modes, the experimental results are divided into two categories: the swept-frequency experiment and the fixed-frequency experiment. The detailed parameters of the experiments are summarized in Table 2.
Table 2. Experiment Specification
27–29 March 2009
Wanning: 18°58′N, 110°31′E
Wuhan: 30°32′N, 114°21′E
magnetic inclination: 25.00°, magnetic declination: −1.40°
magnetic inclination: 45.98°, magnetic declination: −3.73°
Figures 3a–3d show the swept-frequency oblique ionograms recorded on 28 and 29 March 2009. In Figures 3a–3d the left graph presents the oblique ionogram obtained at the Wanning site, and the right graph presents the ionogram obtained at Wuhan site. The horizontal axis represents the radio frequency, and the vertical axis represents the group distance. The frequency ranges of the swept-frequency experiments are 6.2–18.2 MHz and 9.0–19.0 MHz on 28 and 29 March 2009, respectively, with the step frequency as 200 Hz for both days. From Figures 3a–3d, the group distance of the two paths varies with frequency. Two traces are present in each plot as a result of magnetic splitting, which is more unambiguous in the high elevation. Additionally, there are straight lines at a fixed group distance in each plot, which is supposed to be a result of sporadic E layer reflection. The different signal strength discrepancies are mainly due to the different background noise around each receiver. However, it is also evident that the difference of group distance between the two ionospheric paths, Wuhan-Wanning and Wanning-Wuhan, is quite small, suggesting the applicability of ionospheric HF channel reciprocity during the experiments. Such a feature is also verified by the ionospheric ray tracing technique, the results of which are shown in section 5.
4.2. Fixed-Frequency Operations
 One hundred and twenty four pairs of fixed-frequency two-path experiments were carried out on 27–29 March 2009. The distribution of the Doppler shifts difference between the two paths is calculated and shown in Figure 4. The distribution is calculated by subtracting the Doppler shifts of the Wuhan-Wanning path from those of the Wanning-Wuhan path. About 82% of the Doppler shift difference is between ±0.1 Hz and about 92% of the Doppler shift difference is between ±0.15 Hz. Apparently, the Doppler shift difference between the two paths is statistically minor. This results indicate that the Doppler shift difference is negligible under most circumstances of our experiments. However, the discrepancies really exist; we speculate that this is due to the inaccuracy of measurement.
Figures 5a–10a present the examples of the scattering function derived from the fixed-frequency experiments. In Figures 5a–10a the left graph shows the scattering function obtained at Wanning, and the right graph shows the scattering function obtained at Wuhan. The Doppler spectra of the two paths are presented in Figures 5b–10b, which correspond respectively to the scattering functions in Figures 5a–10a. From Figures 5a and 5b through Figures 10a and 10b, the Doppler spread and shift obtained at the Wanning receiver are consistently highly similar to those obtained at the Wuhan receiver. Since the scattering function of extraordinary waves is relatively indistinct, only the scattering functions of ordinary waves are plotted for the fixed-frequency operations of the WIOISS. Although the analysis in section 2 indicates that the delay-Doppler power spectrum is indeed reciprocal, it should be noted that signal strength discrepancies exist between two paths. The receivers located at each site (Wanning and Wuhan) are totally identical. However, the different electromagnetic environments of Wanning and Wuhan, which influence the SNR, might be the primary reason.
5. Ray Tracing Simulation
 Ray tracing techniques are widely used for HF radio wave propagation in the ionosphere under the approximation of geometrical optics. The ionospheric ray path trajectories in the spherical coordinate system are described by the group of Haselegrove equations [Haselegrove, 1957].
 The refractive index n in equation (5) is usually calculated using the International Reference Ionosphere 2007 (IRI2007) model and the International Geomagnetic Reference Field (IGRF) [Huang and Reinisch, 2006]. The group delay of the Wuhan-Wanning and Wanning-Wuhan ionospheric transmission links is calculated separately by using the IRI2007 model and the IGRF with the Jones-Stephenson ray tracing code based on the Haselegrove equations [Jones, 1966] for both O-wave and X-wave modes operating with sweeping frequencies. The one-hop oblique ionograms from the Wuhan and Wanning terminals are then simulated for the reciprocity validation with the model results shown in Figures 6a and 6b. For the sake of comparison with the experiment results, we have chosen the simulation time as 1000 LT on 28 March 2009 for IRI2007.
 To fulfill a more realistic simulation more adapted to practical two-way oblique propagation, the ray tracing simulation is carried out by the following procedure. First, the two locations, Wanning and Wuhan, are fixed in the geographical map. Then the Wanning and Wuhan geographical coordinates are used separately as the transmitter location in our simulation. The elevation angle of simulated rays began at 25° and ended at 55° with a step of 0.5°, since the elevation of the experimental transmitted beam is about 40° with a half-power width of 30°. In a similar way, the azimuth angle of simulated rays began at 3° (this angle is measured to the south direction, the same below) and ended at 33° with a step of 0.5°. Finally, the rays which land at the closest position (the threshold of difference we chose is 1 km) to the corresponding receiver are selected for simulating the oblique ionograms (Figure 11).
Figures 12a and 12b demonstrate, respectively, the difference of group distances in O-mode and X-mode. In principle, the difference of group distances of two paths is very small. There occur rather large differences in group distance of the two paths, which may potentially be attributed to different propagation modes for the corresponding frequencies. After removing these few irregular points, the main results for the different wave propagation modes can be stated as follows: (1) for the low-elevation O-wave propagation mode, the average value of the two-path difference is 1.5779 km, with a standard deviation of 2.4837 km; (2) for the low-elevation X-wave propagation mode, the average value of the two-path difference is 1.8657 km, with a standard deviation of 4.0584 km; (3) for the high-elevation O-wave and X-wave propagation mode, the average values of the two-path difference are larger than those in the low-elevation mode; (4) for the high-elevation O-wave mode the average value of the difference is 9.2628 km, with a standard deviation of 3.7101 km; (5) for the high-elevation X-wave propagation mode, the average value of the two-path difference is 11.0145 km, with a standard deviation of 10.2481 km.
 It is clear that for low-elevation propagation the group delay differences for both O-wave mode and X-wave mode of the two-path can be negligible for HF systems and sky wave radar, suggesting that the time delay of the ionospheric HF channel can be reciprocal for low-elevation propagation. For high-elevation propagation, although the differences in group delay of the two paths are relatively elevated, they are not large enough to affect the capability of certain types of HF communication systems and sky wave OTH radars. It is worthwhile to note that the power of low-elevation waves is much higher than that of high-elevation waves for most HF communication systems and sky wave OTH radars; for instance, high-elevation echoes are at least 3 dB lower in power than low-elevation echoes because of the defocus effect [Tornatore, 1972], which may offset the outcome of the relatively large difference in group delay of two paths for high-elevation waves.
 In the present study the reciprocity of a midlatitude ionospheric HF channel is validated by theoretical analysis and experimental verification. The reciprocity campaign has been conducted by using two identical WIOISSs. Two terminals are located at Wuhan and Wanning, both in the midlatitude region of China. Comparisons of group distance and Doppler shift between the Wuhan-Wanning and Wanning-Wuhan HF ionospheric propagation paths indicate that the reciprocity of the ionospheric HF channel is satisfied at the midlatitude region. Oblique ionograms of two radio links are also simulated by a 3-D ray tracing technique to compare the simulated group distances of two paths. Midlatitude ionospheric HF channel reciprocity verified both experimentally and theoretically in the present study can be useful for HF communication systems and sky wave over-the-horizon radars.
 This research is supported by the Chinese National Natural Science Foundation (40804042). The authors would like to thank the two reviewers for quite helpful comments.