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[1] An experiment on transionospheric HF propagation was carried out in 1978 using a dedicated ground transmitter at Ottawa and the sounder receivers of the ISIS 1 and ISIS 2 spacecraft. This paper deals with resulting data from ISIS 2 at 1400 km altitude. Fade frequencies of one to a few hertz on 9.303-MHz fixed-frequency carrier signals were measured and were found to be confined to satellite locations between about 3° and 20° equatorward of the transmitter. The majority of the fades were of the Faraday type, involving ordinary (O) and extraordinary (X) wave components. There was also a smaller number of single-mode fades interpreted as a beat interference between two O-mode or two X-mode rays that take different routes to the spacecraft. Rays traced through model ionospheric density distributions based on tomographic data show that rays launched toward the equator are more susceptible to focusing by latitudinal periodicity than rays launched toward the pole. Such refractive effects can produce two same-mode rays with equal intensities and different propagation directions. Swept-frequency ionograms interleaved with fixed-frequency measurements have also been used to model density distributions in altitude and latitude. These distributions were used iteratively with three-dimensional ray tracing to find rays that connected the transmitter with the position of the satellite at times of interest along its path. Faraday fade rates thus predicted agree with those observed equatorward of the transmitter. Geometric optics cannot account for observed X-mode to O-mode signal ratios of at least 10 dB in other parts of the passes, ruling out deep Faraday fades. This research supports planning for coordinated ground-space radio experiments in the upcoming Enhanced Polar Outflow Probe satellite mission.

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[2] One of the proposed goals of the Enhanced Polar Outflow Probe (e-POP) mission [Yau et al., 2006] is to improve understanding of plasma processes in the high-latitude ionosphere through the study of the effects of 1–100-km plasma structure on radio propagation. The e-POP radio receiver instrument (RRI) will measure the HF electric fields of waves from coordinated ground transmitters such as ionosondes, HF radars, and ionospheric heaters. Imaging of ionospheric density structures with this transionospheric propagation tool will be investigated.

[3] Considering the complexity of F region structure that can affect the space-time dependence of signals from the ground detected on an orbital platform, the deduction of ionospheric structure from satellite receiver data is expected to be challenging. The present analysis has been undertaken as a first step toward learning how and where imaging might succeed. The research reported has as its objective to understand the fading of signals. Fading is expected to serve as an indicator of the intersection of separate rays resulting from density structure below the spacecraft. This paper is concerned with a particular subset of the fading data, those with periodic or quasiperiodic time dependence. Even though there are a limited number of examples of this particular case in the data from the 1978 “Northridge” experiment [James, 2006a], this relatively simple situation is a useful first test of the concepts needed for the analysis of periodic and aperiodic fading.

2. Periodic Fade Observations

2.1. Characteristics

[4] The Northridge experiment was carried out using a dedicated HF transmitter at Ottawa (45.35°N, 75.88°W) and the sounder receivers of the ISIS 1 and ISIS 2 satellites [Franklin and Maclean, 1969; Daniels, 1971]. The present investigation is confined to ISIS 2 data. The data contain examples of periodic fading of the Northridge signal in the southern, or equatorward, part of the passes. Fades are smooth variations in the total signal detected at the spacecraft, with fluctuation periods between 0.1 and 1.0 s. These fades are observed on both ascending node (LMT ≈ 2100) and descending node (LMT ≈ 0900) passes, all recorded in June and July 1978. Generally, the fluctuations are maintained over intervals of several seconds and are clearest when the wave frequency is no more than 3 MHz greater than the critical frequency f_{o}F_{2} of the underlying ionospheric F region.

[5]Figure 1 is a portion of a film image of a fixed-frequency frame showing examples of periodic signals found in ISIS sounder receiver data. Other examples can be found in Figures 2 and 6 of James [2006a], where the fade patterns and their dynamic range (20–30 dB), respectively, are discussed. The strength of the desired signal from the dedicated ground transmitter is proportional to the darkness of the ionogram film. The two horizontal dark bands at delays of 9–13 ms and 15–19 ms are the signals for two cycles of the 180-Hz rectangular amplitude modulation of the carrier fixed at 9.303 MHz. Magnetoionic dispersion partially resolves the O-mode and the X-mode pulses [Budden, 1985] through difference in group delay at the spacecraft. At the beginning of each square pulse there is 0.5-ms period of O-mode-only signal; 0.5 ms of X-mode-only signal similarly appear at the end of the square pulse. In between, there is a period of about 1.8 ms of strong O + X signal with periodic fringes at a rate of about 4 Hz. These fringes are attributed to Faraday fading of the O and X modes. Single-mode fades are less obvious. They are present in the successive fades in the X-only signal spaced by about 2 s and in faster fluctuations in both the O- and X-only parts. The purpose of this communication is to explain the single-mode and Faraday fades.

[6] Of the two types of periodic fades just introduced, Faraday fading is expected to be widely observable in a homogeneous ionosphere where waves from a common source in both O and X cold plasma modes coexist and interfere. The simplest Faraday effect results from propagation parallel to the terrestrial magnetic induction B of equal-amplitude O- and X-mode waves. Since both wave modes have circularly polarized electric fields E, the total resulting E is linearly polarized. Generally, the interfering waves do not propagate in the same direction, are not parallel to B, and are not of equal amplitude. The total E resulting is then elliptically polarized. Because the orientation of the linear E varies with distance along the phase paths of the waves, a moving dipole detects a periodic signal [James and Calvert, 1998]. In addition to the Faraday fade periodicity, the satellite spin rate of about 3 min^{−1} imposes an additional modulation through rotation.

[7] In an ionosphere structured with both horizontal and vertical density gradients, focusing and defocusing of rays of both modes are expected. Focusing may lead to crossing of same-mode rays, which beat to produce a spatially periodic or quasiperiodic structure of constructive and destructive interferences. Again, a receiving dipole moving through the structure detects a quasiperiodic signal.

2.2. Conditions for Fades

[8] Approximately 50 passes of the ISIS 2 satellite were recorded with its sounder in the optimal mode for this research: alternate swept- and fixed-frequency frames, with a square wave amplitude modulation frequency of 180 Hz. Ionospheric f_{o}F_{2} values varied between about 5 and 10 MHz during these passes. Most of the time, the working fixed-frequency f = 9.303 MHz was at least 2 MHz above f_{o}F_{2}. Consequently, although strong transmission was obtained, the O-X separation shown in Figure 1 was very small, and single-mode fades often were not clearly present. This study of periodic fades is based on short-lived periods of a few to several seconds from passes when f − f_{o}F_{2} ≤ 2 MHz.

[9] A survey of the ISIS data identified 20 instances, on 16 different passes, of clear periodic fades, mostly of the Faraday type. Six of the passes were in ascending node at a local time near 2100. The locations of the spacecraft in geocentric coordinates are shown in Figure 2. These circles and crosses correspond to the center time of observations. The phenomenon is confined to a 30°-wide longitude corridor and to a range of latitudes between 42° and 25°. The fade frequencies lie between 1 Hz and a few Hertz.

[10] The Northridge data show O- and X-mode signals present with similar strengths at ISIS 2 throughout a latitude range of 40° or more. However, deep self and Faraday fades in Figure 2 cluster around 35° latitude. It is inferred that the structure of the ionosphere is important for fades, likely requiring certain B directions and/or particular density distributions. A preliminary iterative ray tracing was carried out to find rays connecting the ground transmitter to ISIS 2 for the locations in Figure 2, assuming a horizontally stratified ionosphere with a typical f_{o}F_{2} = 6 MHz. When attempting to reach satellite locations most distant from the transmitter, rays were reflected to the ground by the ionosphere so that no solutions were available; these are the “R” locations in Figure 2. Solutions were found for the closer locations; in these cases, the computed value of the angle (k, B) in degrees between the wave vector and B at the spacecraft is displayed above the symbol in Figure 2 for the O-mode solution.

[11] From the (k, B) values, k is found to be within about 15° of the B axis much of the time. It was also determined that the angle between k and the axis of the satellite spin s lay between 15° and 40°. With k, B, and s more or less aligned, and with the dipole antenna extended perpendicular to s, this geometry ensured that the receiving dipole lay in a rotation plane sensitive to the orientation of the elliptically polarized total O + X field of the incident waves. Therefore the geometry is more favorable to the detection of Faraday fades at the spacecraft locations in Figure 2 than in the northern parts of satellite passes. The next two sections investigate some first-order hypotheses for the generation of periodic fades.

3. Single-Mode Fades in Two Dimensions

3.1. Effects of North-South Gradients

[12] A first-order consideration of refraction of waves in the ionospheric F region helps to explain why single-mode fading appears equatorward of the Northridge transmitter. Figure 3 is the result of a two-dimensional O-mode ray tracing through an ionospheric density model featuring an array of density irregularities that are periodic with latitude. The structure has the shapes and orientations of mesoscale ionospheric irregularities, elongated along the axis of B. The pattern in Figure 3 also approximates a traveling ionospheric disturbance (TID) density structure, which is observed to tilt over in its equatorward direction of propagation [Pryse et al., 1995] and to have a horizontal wavelength of ∼250 km. A range of starting wave normal elevations at the transmitter, starting at 11° from the north direction on the right side of the diagram and incrementing by 2°, produces the network of rays fanning out from the ground transmitter. Because the density gradients of the irregularities are approximately perpendicular to B, rays and wave vectors make angles of about 0° with the density gradients on the poleward (right) side of Figure 3. Consequently, the waves suffer minimal refraction there. The ray-gradient angles are near 90° on the equatorward (left) side, so the structure is most effective in focusing or defocusing rays in that sector.

[13] Of the several individual density maxima in this model of Figure 3, only two produce appreciable focusing or defocusing at the ISIS 2 altitude of 1400 km. The strongly focused or defocused rays intercept the ISIS 2 orbital height at latitudes between about 5° and 15° equatorward of the transmitter. Focusing means that two or more wave paths can arrive at the spacecraft at a given instant. Ray tracing in the X mode produces similar ray patterns in this model with focusing in the same latitude range.

[14] We next examine more closely the computed characteristics of crossed ray paths in the north-south distance range 1500–1900 km at the ISIS 2 altitude of 1400 km in Figure 3, i.e., where the rays terminate. Two pertinent parameters are plotted in Figure 4: the angle between the wave normal and the vertical direction (Figure 4a), and the Poynting flux magnitude S in (Figure 4b). The numbers beside each point are the starting elevation angle minus 90°; for example, “8” labels the ray with a starting elevation of 98°, i.e., 8° south of the zenith. There is a small range of positions around 1940 km where wave normals make angles of about 22° with the local vertical. More important, pairs of wave vectors cross at angles between them of a few degrees, and the associated S magnitudes are close to each other in the 1940-km neighborhood.

[15] Ray tracing in both X and O modes through F region density minima can lead to more complicated ray focusing and caustics. Nevertheless, the computations show that over some limited range equatorward of Ottawa, there can be crossing rays of comparable field strengths with direction separations of a few degrees. The conditions for same-mode fades thus appear to be met in these circumstances.

[16] A relatively simple two-dimensional geometry can be applied to the data in Figure 4 to model constructive interference of two waves of the same mode. This geometry is illustrated in Figure 5, there being no curvature of wave fronts over the space traversed by satellite orbital motion during the observing period of a few seconds. Absence of curvature means that the wave planes have unchanging and infinite extent. Two waves of the same wavelength λ (neglecting anisotropy) have wave vectors k_{1} and k_{2} separated by angle α. The intersection of the planar wave fronts produces a series of planes where the wave amplitudes interfere constructively. In our diagram, k_{1} and k_{2} lie in the plane of the paper. Also, we use the particular case where the two waves are in phase at the origin. Consequently, the constructive interference planes are perpendicular to the diagram and intersect along lines, of which dashed lines I_{a} and I_{b} are two examples. N is the normal direction to the interference planes. Trigonometry can be used to show that the I_{a} − I_{b} separation along the x axis is w = λ/sinα and that their perpendicular separation is d = λcos(α/2)/sinα. If we had placed the position where the waves are in phase elsewhere, the broken lines would be displaced to the right or to the left but would have the same orientation as in Figure 5, and d would be given by the same expression.

[17] The receiving satellite has a velocity v at an angle γ with N. Speed v = ∣v∣. The time Δt for the spacecraft to move between adjacent interference planes is

[18] As an illustration, take the Figure 1 case of wave fronts rising to ISIS 2, in which a variety of single-mode-fade beat periods between a fraction of 1 and 2 s are observed. Consider a fade period of Δt = 1.0 s. With f = 9.3 MHz, λ = 32 m, v = 7200 m s^{−1}, and γ = 45°, the inversion of (1) gives α = 0.36°. Since this is of the order of wave vector separations shown for conditions near the north-south distance of 1940 km in Figure 4a, this evaluation supports the same-mode interference hypothesis sketched in Figure 5, with reference to the ray optics results in Figures 3 and 4.

4. Faraday Fades in Three Dimensions

4.1. Calculation of Fades

[19] The accuracy of the same-mode analysis associated with Figure 5 is seen to be limited when it is appreciated that wave fronts at the spacecraft are not strictly planar. Three-dimensional ray tracing described above has been carried out to find the parameters of wave fields at the ISIS 2 spacecraft for a number of the Northridge passes. We concentrate here on Faraday fades because they constitute a large fraction of the periodic fades observed.

[20] Rays traced from the Ottawa transmitter to an ISIS satellite location yield orientations of wave vectors that require three-dimensional geometry (see Appendix A). As discussed there, standard concepts of wave dispersion relations and ray tracing allow the E field to be completely defined at the spacecraft. Resulting values of the open-circuit voltage V_{oc} induced on the dipole can subsequently be determined using expression (B2). By computing the variation of the sum of voltages V_{oc} induced by two or more waves at the spacecraft, it is possible to test these ideas against observed periodic fades.

[21] Regarding time dependence, two rays contribute V_{oc} signals that have phase delays equal to the ray phase paths, say, ϕ_{O} and ϕ_{X}. This delay is added to ωt in their exponential time factors in (B2) to account for instantaneous phase at the spacecraft. The phase paths ϕ of solution O- and X-mode rays from the ground to a series of spacecraft positions in the observation region change with time t. To explain fade rates, it is required to know the relative rate of phase change ϕ_{O}′(t) − ϕ_{X}′(t) ≡ F(t).

[22] A basic method for determining relative phase is to use the difference of the integrated phase paths of the two modes. Iterative ray tracing [James, 2006a] was developed in support of the ISIS analysis for cases of smoothly varying ionospheric density distributions that do not produce crossed rays like those in Figure 3. The James [2006a] iterative ray-tracing technique showed that 2π < F(t) < 20π s^{−1}.

[23] The infinite plane wave assumption used in section 3.2 can be used to estimate F(t). In this approximation, the rate of change of phase path for a given mode is given by ϕ′(t) = k•v = 2πfvncosψ/c, where n is the phase refractive index, ψ is the angle between k and v, and c is the free-space velocity of light. Then F = (2πfv/c)(n_{O}cosψ_{O} − n_{X}cosψ_{X}), which is the difference of the O- and X-mode Doppler frequency shifts observed by the receiver. This method produces values of the same order of magnitude as the basic method but different enough to show the importance of curved wave fronts and total phase path.

4.2. Rate of Phase Change F(t) With Horizontal Gradients

[24] The ray-based approach to calculating Faraday fade rates is found to be successful in predicting the magnitude of fade rate change where Faraday fading in observed. This is illustrated in the case of an ISIS 2 pass starting at 1307 UT on 8 July 1978 [see James, 2006a, Figure 6b]. The analysis was based on a two-dimensional distribution of the ionospheric density in latitude and radial distance, summarized in Figure 6. Variables f_{o}F_{2} and f_{os} are the plasma frequencies at the peak of the F_{2} layer and at the spacecraft, respectively. The solid line shows the latitudinal dependence of f_{o}F_{2} and f_{os} determined from the sounder ionograms through the whole period of Northridge signal reception at ISIS 2. For speed of computation with the iterative ray solution finder, it was desirable to have an analytical model of the complete ionosphere. The broken lines show analytic fits to f_{o}F_{2} and f_{os}. The third pair of curves is for the Chapman scale height H given by fitting a Chapman distribution for plasma frequency f_{p} between the F layer peak and the satellite height for zero solar zenith angle [Budden, 1985]. The vertical distribution with height z above the F peak, assumed to be at 300 km, is f_{p} = f_{o}F_{2} exp[(−1/4)(1 − z/H −exp(−z/H))] and a cosine function with a total thickness of 200 km below the peak. The values of H in the broken line are determined from the other two fitted curves, whereas the solid-line values of H are found directly from the scaled f_{o}F_{2} and f_{os} values. No density gradients are assumed in the longitudinal direction.

[25] Solution O- and X-mode rays connecting the transmitter and the satellite were explored for the period when transionospheric transmission was observed to be successful, 1307:10 to 1321:30 UT. This compares with the time interval predicted by ray tracing, 1309:00 to 1320:20 UT. The predicted interval corresponds to a geocentric latitude range of 63.6° to 27.7° during this equatorward pass centered at about 0840 LMT. The satellite suborbital path is in a longitude corridor centered at about 68.2°W.

[26]Figure 7 shows a series of solution rays at 10-s intervals over the latitude range where transionospheric transmission was predicted for the 8 July pass starting at 1307 UT. Outside this predicted range, or iris, rays failed to penetrate the ionosphere and were reflected earthward. In fact, for the last satellite position on the left side of Figure 7, only the O-mode solution was found; hence there is only one line connecting to that point. The rays in Figure 7 are projected onto the vertical meridian plane through the Ottawa transmitter. Owing to the lower electron densities, the O and X rays are not spatially resolved on the poleward side of the diagram, whereas they are seen separately on the left side.

[27] Black dots 1 through 6 in Figure 7 show where O-X Faraday fades have been computed to indicate their evolution across the pass. The open circle close to dot 6 shows the satellite position at the time of recording of the ionogram in Figure 6b of James [2006a].

[28] The telemetry system of ISIS 2 recorded measurements of the input voltage to the sounder receiver V_{in}. This voltage is computed starting with the equivalent circuit of the antenna comprising V_{oc} in series with its antenna impedance, which is evaluated using the vacuum theory of Schelkunoff and Friis [1952]. This antenna circuit is in series with the impedance seen by the antenna at its terminals, as measured before launch. This permits the power P_{in} injected at those terminals to be found. Then V_{in} = (50P_{in})^{1/2} because V_{in} is referred to the 50-ohm input impedance of the receiver.

[29]Figure 8 contains the six plots of total V_{in} corresponding to the six solid dots in Figure 7. V_{in} is plotted as a function of time, for a full satellite spin period of about 20 s. A number of predicted features can be noted. Absolute signal magnitudes maximize at about 3 mV around the point of closest approach of the spacecraft to the transmitter (Figures 8c and 8d). Signal levels decrease by 15–20 dB between the closest approach and the edges of the transmission zone. Signal strength variations from maximum to minimum across the iris of up to about 30 dB are seen and computed in other passes [James, 2006a].

[30] At the beginning of the pass, the predicted Faraday fade rate F is about 0.4 Hz, the number right after “F = ” at the top of each plot in Figure 8. At the end, it has increased by an order of magnitude to about 5 Hz. Increases in F with decreasing latitude arise because of the change of propagation from close to perpendicular to nearly parallel to B, as noted in the discussion of Figure 3. In addition, the proximity of f_{o}F_{2} and f_{x}F_{2} to the wave frequency on the equatorward side of the pass results in more dispersion of both O and X modes. That is, F(t) also increases through the pass as the rise in ambient density moves O and X cutoff frequencies closer to the working frequency 9.3 MHz. This has been demonstrated by rerunning the F calculation for a horizontally stratified density distribution with f_{o}F_{2} = 4.6 MHz at all latitudes. Rather smaller F values are obtained at equatorward latitudes. However, the F values are still greater equatorward where the propagation is essentially parallel to B, as opposed to poleward propagation which is essentially perpendicular to B. The values of F computed without north-south gradients can be read inside the parentheses at the top of each plot in Figure 8, right after the F value computed with gradients.

4.3. Magnitudes of Fading Signals

[31] The computed power into the ISIS receiver, Pinc, is displayed at the top of each plot of Figure 8 for comparison with the observed power Pino, right beside it. Absolute levels Pino are found to be 10–20 dB weaker than Pinc. Best Pinc-Pino agreement is obtained in the south part of the pass. The calculation is based on a nominal transmitter output power of 1 kW. The discrepancies are attributed to peculiarities of the transmitter radiation pattern, which was not measured and is therefore unknown.

[32] There is a important difference between Figures 2 and 8: The observational evidence in Figure 2 confines deep fades to a latitude range of about 17° equatorward of Ottawa, whereas the ray-based computations of Figure 8 predict deep fades through a latitude range from 29 to 55°. Examination of the digital ionogram amplitude scans (A scans) confirms that often the X signal was at least 10 dB stronger than the O signal. We give an illustration of this in Figure 9. Strong transmission was maintained through this fixed-frequency frame (Figure 9b), which was recorded at latitudes around 39° during the same pass as in Figures 7 and 8. At a specific time indicated with a white circle in the ionogram, the A scan is as shown in Figure 9a. The individual pulses all show a small ledge on their rising edges corresponding to the O-only signal. The rise in total signal level from the O-only signal to the maximum pulse level is about 12 dB. Other ionograms show the X-to-O signal strength ratio rising to 20 dB and more.

[33] Given the X-to-O ratios of about 12 dB in Figure 9a, the fade depth through a complete Faraday fade cycle is expected to be shallow compared with deep fades shown in Figure 2. There the X-to-O ratios are less than a few decibels. In Figure 9c we have plotted a “B scan,” showing signal power into the receiver as a function of time, at a fixed apparent range of 1005 km. Aperiodic B scan variations of 10–20 dB, or more, in total signal are found. The comparatively strong signals in Figure 9b, in green-yellow, confirm the deep random variations in signal. Periodic variations like those seen in Figure 8d, the closest one to the time of Figure 9, are difficult to identify in Figures 9b and 9c. Fourier analyses of the B scan in Figure 9c shows a number of peaks, none very close to the 1.3-Hz computed fade frequency. The random noisy nature of the B scan indicates that the Faraday effect is completely masked by some other processes in the determination of the intensity of the O and X signals at the spacecraft.

[34]X-to-O ratios of 10 dB or more, as illustrated in Figure 7a of James [2006a], are found in many locations. This rules out deep Faraday fades which require X and O levels within ∼1 dB of each other. Collisional absorption is estimated by evaluating the square of the complex refractive index n^{2} = (μ − iχ)^{2} [Budden, 1985, equation (4.47)] to determine the imaginary part χ. This is obtained by assigning values to the magnetoionic variable Z = ν_{en}/2πf in the n formula using a model height profile for the electron-neutral collision frequency ν_{en} appropriate for the D and E ionospheric regions [Prikryl et al., 2000]. Spatial attenuation of wave electric field E with incremental distance z is given by E(z) = E(0)exp(−βz), where β = 2πfχcosξ/c. The angle ξ between k and the wave energy flow direction can be set to 0 for f = 9.3 MHz. The corresponding spatial absorption factor A then is

where k_{0} = 2πf/c. Figure 10 is a plot of the O- and X-mode A factors as a function of the angle (k, B) for two heights. Although the daytime height-integrated collisional absorption could be of the order of 10 dB, it is seen that preferential absorption of O over X mode is not predicted; hence X-mode dominance is not explainable by differential D-E region absorption.

[35] As a matter of interest, James [2006a, Figure 7b] illustrates another phenomenon seen in a few ionograms: the O + X main part of the pulse is trailed by ragged, dispersed X-mode delayed signals. This effect was found only on poleward passes (LMT near 2100) at latitudes between 36° and 58°. The recorded data together indicate that other mechanisms than those considered in the present analysis can intervene to affect the X-to-O ratio. Explanation of these effects is left for later research.

[36] Spin modulation is deepest poleward of the transmitter. The angle (k, s) between the O-mode solution wave vector and the satellite spin axis appears at the top of each plot in Figure 8. This angle is closer to 90° on the poleward side of the pass than on the equatorward side. Because the dipole antenna rotates in the spacecraft equatorial plane, deeper spin modulation is to be expected on the north part of the pass. The depth of spin modulation is about 20 log_{10}(1.7/0.4) = 13 dB in Figure 8b. This is of the order of the observed modulation in this pass, and also in Figure 5b of James [2006a] from another ISIS 2 pass.

5. Conclusions

[37] Deep, periodic Faraday fades and self-mode fades of either the O or X mode are found to last a few to several seconds. Their duration and their restriction to locations 10°–15° latitude south of the Ottawa Northridge transmitter imply that B field–aligned density structures focus rays into caustic surfaces near the B direction. Ray tracing through two-dimensional models of the ionospheric density distribution yields predictions of the rate of O-X relative phase change along the ISIS 2 spacecraft orbit. These predict Faraday fade rates close to those observed south of Ottawa.

[38] Deep Faraday fading theoretically expected elsewhere along the orbit is not observed; disparately strong X signals are detected at ISIS. The sporadic nature of low-level Faraday fades in these cases points to local conditions which cause transionospheric ray intensities to vary in a random fashion. For instance, scattering by ionospheric irregularities of transionospheric ray paths, overhead or poleward of the Ottawa transmitter, could be responsible for the unexplained high X-to-O ratios observed. Certainly, X-to-O ratios are not in the right sense to be attributed to D-E region differential absorption of waves.

[39] In the context of the ePOP radio science agenda for the radio receiver instrument (RRI), the equatorward propagation phenomena reported in Figure 2 with k relatively close to B at ISIS 2 are expected to reappear when the source has the upward pointing radiation pattern of the high-latitude Canadian Advanced Digital Ionosonde (CADI) ionosondes. With judiciously chosen fixed frequencies, the CADI signals detected by the RRI should be a sensitive indicator of the horizontal structure in the F region.

[40] The above results are also relevant to Super Dual Auroral Radar Network (SuperDARN) in that its zenith and back lobe radiation are expected to be observable by ePOP/RRI. The present analysis of ISIS data did not uncover many new insights about oblique poleward propagation, which is of considerable interest in SuperDARN-ePOP coordinated investigations. The increased capability of ePOP/RRI for measuring different HF wave parameters should help to redress our lack of understanding about this kind of transionospheric propagation.

Appendix A: Vector Relations for Dipole Antennas and Wave Field Vectors

[41] The details of Faraday fades depend on the orientation of the elliptically polarized E fields of the combined waves relative to the receiving dipole direction. The spatial relation of the receiving satellite to the ground transmitter generally requires a three-dimensional geometry. Significant vectors to be considered are shown in Figure A1. Here the z axis is parallel to B, and the satellite spin direction s is placed in the x-z plane. The three other significant vectors, two wave vectors k_{1} and k_{2} and the satellite velocity vector v, each has a polar angle θ and an azimuthal angle ϕ. Associated with each wave is an electric field E. In the cold plasma theory, in spherical coordinates,

[42] The R ratios on the right side of (A1) are [Quémada, 1968, equation (3.4.40)]

where S, iD, and P are elements of the relative dielectric permittivity tensor for a cold magnetoplasma [Stix, 1992, equations (1)–(28)], n is the refractive index, and θ is the polar angle between k and the B axis. The E vector describes an ellipse having semiaxial lengths of (E_{θ}^{2} + E_{k}^{2})^{1/2} and E_{ϕ}. Evaluations of the ratios for the O or X mode in the topside ionospheric conditions of the ISIS 2 orbit show that the angle between the normal to the plane of polarization and k is less than 1° for 0° < θ < 89° under most experimental circumstances. E_{k} is negligible in most instances of O and X propagation discussed here. The Poynting flux vector associated with (A1) is

where Z_{0} is the vacuum wave impedance. For the present purposes, a wave is fully specified with θ, ϕ, mode (O or X), and E_{θ} or ∣S∣, as defined in (A1) or (A3), respectively.

Appendix B: Voltage Induced on Dipole Antenna

[43] We computed the open-circuit voltage V_{oc} = E•L_{eff} induced on the ISIS 18.8-m receiving dipole for comparison with observations. L_{eff} is the equivalent-length vector of the receiving dipole, and some assumptions were made in calculating it. The expression derived by Kuehl [1962, equation (14)] for E radiated by a short dipole in a cold magnetoplasma has been manipulated under the principle of reciprocity to provide expressions for L_{eff} [James, 2006b]. In the calculation of V_{oc} it is necessary to keep track of orientation and shape of the two elliptically polarized incident wave fields. The present method has the advantage of reproducing the electric field polarization as represented in (A2).

[44] Under Jordan's [1950] concept of the L_{eff} vector and in the vector notation of (A1) and the coordinate system of Figure A1, James [2006b] proposed that V_{oc} be expressed as

where the vector is square brackets is L_{eff}. The coefficients A′, B′, C′, and D′ are defined by Kuehl [1962, equations (15)–(19)]. Ratios of these Kuehl coefficients can be expressed in terms of the polarization ratios in (A2) so that V_{oc} can be written

Coefficient C′ is a function of the dipole moment p = (p_{x}, p_{y}, p_{z}).

[45] Kuehl developed his cold plasma theory in terms of the dipole moment of the antenna. The ISIS 18.8-m dipole is a little longer than half the vacuum wavelength 3 × 10^{8}/9.303 × 10^{6} = 32 m. In a vacuum, the effective length of a finite dipole of half-length h for the θ component of E is

in which θ′ is the angle between the theta component of E and the axis of the dipole and k_{0} is the vacuum wave number. In the present data, propagation is not far off perpendicular to the antenna. For θ′ = 90° in (B3), L_{eff} = 13.4 m at 9.3 MHz, as opposed to what it would be at much lower frequencies where the short dipole approximation holds, L_{eff} = h = 9.4 m.

[46] To take account of the finite-length effects in the dipole theory, the dipole moment required by the Kuehl expression for radiated E field is derived here assuming that the current distribution on the dipole as a function of distance z from the drive point has the sinusoidal form

in which I(0) is the current at the terminals. Following Jackson [1962, p. 271], the linear charge density is

where ω = 2πf. The dipole moment p is

and is used to compute the Kuehl coefficients B′, C′, and D′. In the short-dipole case of k_{0}h ≪ 1, p in (B6) reverts to [Jackson, 1962]

For f = 9.303 MHz, p/p_{s} = 1.41. This ratio has exactly the same value as the ratio of the above mentioned corresponding L_{eff} values at θ′ = 90°.