Time of flight and direction of arrival of HF radio signals received over a path along the midlatitude trough: Theoretical considerations



[1] Observations from an HF radio experiment on a subauroral path between Sweden and the UK near sunspot maximum in 2001 are compared with the position of the midlatitude trough according to a statistical model. Periods of off-great circle propagation, occurring predominantly in winter and equinoctial nights at frequencies 7–11 MHz, show characteristics consistent with scattering from field-aligned irregularities in the northern trough wall and/or auroral oval. Very little reflection and/or scattering was apparent from directions to the south of the great circle path. These results are in marked contrast with those from a similar experiment conducted near sunspot minimum in 1994 in Canada, during which both southerly and northerly deviations were observed in the 5–15 MHz range. The contrasting results were simulated using ray tracing through a model ionosphere incorporating a model of the trough and, optionally, precipitation. The observed off-great circle propagation features on the European path could only be reproduced when precipitation within the northern trough wall/auroral zone was included, whereas features of the northerly and southerly deviations observed in the Canadian experiment could be simulated by the presence of the trough walls and without the need for precipitation.

1. Introduction

[2] A number of measurements of the direction of arrival of HF signals received over paths oriented along the midlatitude trough have been reported by Rogers et al. [1997] and Siddle et al. [2004]. The first paper describes measurements made between Halifax and Leitrim in Canada near solar minimum in 1994, while the latter describes an experiment undertaken between Uppsala, Sweden and Leicester, UK in 2001 close to sunspot maximum. In both cases, large deviations from the great circle path were observed, both to the south and to the north in the Canadian measurements and mostly to the north for the European path.

[3] This paper investigates the different mechanisms that can cause deviations from great circle propagation for each of these experiments. In the European experiment, time of flight data enable a simple estimation of the location of the point of reflection for one-hop signals. This reflection point correlates well with the position of the trough's northern wall as estimated by a statistical model [Halcrow and Nisbet, 1977]. Further analysis shows that the plane of reflection assumed in this simple reflection model is oriented along the bearing of geomagnetic declination. These findings and the Doppler spread characteristics of the signals point to scattering by irregularities as the mechanism for azimuthal deviation in this case. Time of flight measurements were not available for the Canadian path, so such considerations were not possible for this path.

[4] Ray-tracing studies were also performed using a model ionosphere based on ionospheric sounding data. The model also contained an electron density depletion associated with the trough based on the Halcrow and Nisbet [1977] model and optionally, the effect of precipitation. Random density irregularities at appropriate scales were also added to the model. The off-great circle propagation features on the Uppsala to Leicester path could only be simulated when irregularities in the north wall of the trough or the auroral zone caused by precipitation were introduced into the model ionosphere. In contrast, many of the features apparent in the Canadian measurements were reproduced through reflection by trough-scale electron density gradients only, without the need for precipitation in the model.

2. Reflection Point Estimation for the Uppsala-Leicester Path

[5] As reported in the work of Siddle et al. [2004], during winter and equinoctial nights signals in the range 7.0 to 11.1 MHz were often not able to propagate along the great circle path (GCP) and arrived instead from directions well to the north. Large Doppler spreads and shifts were evident at these times. Below this range, at 4.6 MHz, propagation along the GCP was supported throughout most nights, whereas at 14.4 and 18.4 MHz, nighttime propagation rarely occurred. Two possible mechanisms for the deviation to the north for the 7.0–11.1 MHz signals have been considered, namely reflection from the tilted electron density gradients forming the north wall of the trough, and scattering from irregularities embedded in the north wall or within the auroral oval. To investigate which of the mechanisms was responsible for the deviations from the GCP, the measurements of time-of-flight (TOF), azimuth and elevation for signals identified as single-hop reflections were used to estimate the location of the ionospheric reflection point. The calculations assume a single specular reflection, linear propagation and take into account the curvature of Earth. Examples of the time-dependence of the reflection points estimated by these means are shown in Figure 1. As well as estimating the reflection point, the calculation also gave the orientation of the tilt of the ionosphere at the point of reflection. Whilst the implicit assumption of specular reflection is clearly an oversimplification, this procedure does yield interesting results. A more sophisticated simulation, which takes into account the refraction of the ray through the gradients in electron density, is reported later in this paper.

Figure 1.

The latitude of the calculated point of virtual reflection (dots) and of the model trough walls (bold lines) between noon on 26 December 2001 (day 360) and the end of 31 December 2001 (day 365). From top to bottom: frequency 7.0, 10.4, and 11.1 MHz. The horizontal line at 56.5°N represents the latitude of the midpoint of the GCP.

3. Comparison With the Halcrow and Nisbet Trough Model

[6] An empirical model of the trough, based on satellite measurements is given by Halcrow and Nisbet [1977]. This model, which is parameterized by Kp, may be used to predict the latitude of the top and bottom of the north and south walls of the trough as a function of local time. The walls are modeled as regions where the perturbation in electron density increases linearly from the constant (reduced) level within the trough to zero outside. Kp, local time and the solar zenith angle are used to estimate the location of the sunrise and sunset walls of the trough. Siddle et al. [2004, Figure 1] show an example of the position of the trough at 0000 UT in March 2001.

[7] Shown as bold lines in Figure 1 are the latitudes of the top (i.e., unperturbed) and bottom (i.e., fully perturbed) edges of the north and south trough wall, as derived from the Halcrow and Nisbet model. The local time employed in the calculations was that for the midpoint of the GCP (approx. 7.3°E, 56.5°N) and since the reflection points (shown as dots) inferred from the observations lay between longitudes of 1°W and 18°E, there may be a local time discrepancy of up to half an hour between the model and data. Note that in Figure 1, the southern wall disappears during the daytime whereas the northern wall continues from one day to the next. This arises since because the model specifies the eastern and western terminations of the trough in terms of solar zenith angle which is satisfied in the south, but not in the north which may be in the polar night. In reality, the cessation of the southern wall probably signifies the recession of the trough to higher latitudes or an absence of the trough.

[8] Good agreement between observations and model is evident in Figure 1. The best example of agreement is on day 363–364, where the signal follows the model north wall for about ten hours, as it decreases in latitude during the night and then rises just before dawn. However, there are clearly factors outside the model, such as the trough's responsiveness to Kp changes and the electron density in the auroral oval, which control the position of the reflection point. Some contrast is seen between the higher frequencies and 7.0 MHz in that the latter shows a more continuous change between northerly and GCP propagation (e.g., in the early hours of day 361), a later change from GCP to northerly propagation (e.g., at the end of day 363 where both GCP and off-GCP propagation exist simultaneously) and is less likely to be reflected from more distant points (e.g., no feature near midnight on day 362–363 and shorter tails on the features on the last two nights). There is also good agreement between sunrise (as evidenced by the poleward recession of the trough) and the onset of GCP propagation.

4. Analysis of the Orientation of the Reflecting Medium

[9] Two mechanisms by which deflections from GCP can occur are considered here. The first is through reflection from the smooth electron density gradient in the trough walls, while the second is by scattering from field-aligned irregularities (FAIs). FAIs are regions of enhanced electron density which are found mainly in the north wall of the trough and within the auroral oval (which are often coincident). They are strongly localized on the scale of tens of kilometers and aligned along the geomagnetic field lines [Jones et al., 1997]. Observations [see Siddle et al., 2004, Figure 4] show that FAIs can last several hours, are quite strong and subject reflected/scattered signals to Doppler spreading due to their dynamic origin. The scattering is also fairly independent of signal frequency. These characteristics suggest that the scattering surface is less smooth than the trough wall, but not sufficiently varying on the decameter-scale to cause Bragg scattering. Thus FAIs can be represented in the ray-tracing model (see next section).

[10] The geometry of reflection from FAIs is depicted in Figure 2. As the ray travels upward, it enters a (reduced) F2 layer, and is refracted toward the horizontal. Since electrons are confined to move along the geomagnetic field lines, rather than across them, the reflection coefficient from irregularities increases rapidly if the incident ray is perpendicular to the field. Also shown in this figure is the virtual plane, P, of reflection, assuming no refraction. The tilt, t′ of P can be derived from the elevation angle of the received rays, and will differ from the geomagnetic dip angle, t, due to refraction. The bearing b′ of P can be derived from the positions of the transmitter, receiver and virtual reflection point using Snell's law. Apart from some refraction, b′ will be the same as the magnetic declination, b. The refraction is, however, more likely to alter the ray's elevation than its azimuth.

Figure 2.

A ray (solid line) travels upward from the transmitter and is refracted toward the horizontal by residual electron density in the trough F layer. It reflects from electron density structures localized along a field line, B with dip t and bearing b, and proceeds to the receiver. A simple model represents the ionosphere as a specular plane P and disregards refraction (dotted line). P has bearing, b′ (the direction of fastest descent) and tilt t′.

[11] At a position (62°N, 5°E) typical of the point of virtual reflection for the northerly signals, the declination of Earth's field is about 25° west of north and its dip is about 74°. In Figure 3, the value of b′ has been calculated for each off-GCP signal at 10.4 MHz in 2001 and the occurrence frequency plotted. The most notable feature here is the narrow peak around 23° west of north, showing that the geomagnetic field is relevant to the azimuth of arrival. This may be due to scattering from FAIs or to reflection from the trough wall, which, when averaged over Kp, will run along lines of constant geomagnetic latitude.

Figure 3.

Plot of bearing of fastest decrease in height of assumed planar ionosphere for all off-GCP signals at 10.4 MHz in 2001 (1 degree bins).

[12] The tilt of the specular plane away from the horizontal (t′ in Figure 2) can also be calculated from the direction of arrival. For the peak described in Figure 3, tilts between 25° and 65° are seen, with a peak in frequency of observations around 55°, disagreeing with the calculated dip of 74°. If, as the Doppler spread suggests, the signals were scattered from FAIs rather than from the trough walls, then the difference is presumed to be due to vertical deviation caused by refraction and to the lower accuracy of the elevation data. No systematic variation in the peak in elevation angle is seen amongst the different frequencies.

5. Ray-Tracing Studies

[13] A ray-tracing study has been undertaken to investigate the contrasting propagation effects observed for the Uppsala (18°E, 60°N) to Leicester (1°W, 52°N) experiment [Siddle et al., 2004], which was conducted close to sunspot maximum, and the Halifax (64°W, 45°N) to Leitrim (76°W, 45°N) experiment [Rogers et al., 1997], which was conducted close to sunspot minimum. The ray tracing was based on the code by Jones and Stephenson [1975].

5.1. Ionospheric Model

[14] A realistic ionospheric model was required and therefore a combined model of the background ionospheric electron densities, the trough and auroral oval was developed. Initially, a bottom-side electron density profile consisting of a single Chapman layer was adjusted to match the ionospheric parameters (foF2, estimated Hmax, etc) measured at ionosonde stations located at latitudes south of, north of, and close to the trough. The longitudinal variation of the background ionosphere was derived from the time variation of these profiles in the relevant period.

[15] The Halcrow and Nisbet [1977] model was used as a basis for the position of the trough walls. Initial studies using an unmodified trough model were unable to reproduce the observations. This was attributed to the unrealistic nature of the smooth walls produced by the model, and therefore smaller-scale structures were added to the modeled electron density profile as follows:

[16] 1. The latitude of the walls was perturbed by two smoothed random functions of longitude; one of typical scale 10°, the other of typical scale 2°. The functions were of zero mean and typical latitudinal size 2°.

[17] 2. The depletion in the walls, as a percentage of that in the center of the trough, was also perturbed. To the linear variation of the model, was added the product of lateral and longitudinal smoothed zero-mean random functions. The typical scales of variation were 0.2° and 2° respectively, and the typical amplitude varied between zero at the top and bottom of the walls and 25% at the wall center.

[18] The latter of these perturbations produced a landscape of patches along each wall, which were elongated in the direction longitudinal to the trough. This created small areas of higher density gradient than existed in the original model which enhance the ability of the wall to reflect rays. No perturbations were added to the floor of the trough. The maximum depletion of the trough was set according to the phase of the solar cycle. Typically, a reduction in electron density of 20–30% was used for times of maximum sunspot number, and a reduction of 60% for low sunspot number. These values represent averages derived from ionograms from ionosonde stations under the trough (Chilton, UK) and close to the southern edge (Pruhonice, Czech Republic), and they differ markedly from the constant reduction of 75% assumed in the Halcrow and Nisbet model.

[19] The auroral oval is an enhancement of electron density caused by particle precipitation in the E region and above. It was modeled as having its equatorward edge coincident with the trough's poleward edge, although it should be noted that the oval is sometimes to the north of the trough wall. The basic model of the oval was, like the trough, trapezoidal in cross section, having a flat top and linearly sloping sides when plotted as electron density against latitude. As a function of distance along (near-vertical) field lines, the density enhancement was modeled as starting 80–100 km from the ground, having one or more peaks of about 1013 electrons/m3 around 110 km, and then decaying slowly toward 200 km [Bates and Hunsucker, 1974]. This enhancement was then also perturbed by the product of lateral and longitudinal smoothed zero-mean random fluctuations of typical size 25%. The typical scales of variation were 0.2° and 4° respectively, again resulting in elongated patches representing the FAIs inferred from the measurements. Although precipitation is known to vary over the solar cycle, this variation was not included in the model since there are few direct observations of this parameter, and it is extremely difficult to predict the intensity and spectrum of this complex phenomenon.

5.2. Ray-Tracing Results for the Uppsala-Leicester Path

[20] Figure 4 illustrates the importance of taking into account precipitation in modeling the off-GCP propagation effects. These plots show azimuthal deviation (top) and TOF (bottom) for the 7.0 MHz signal. For the panels on the left, the model includes the trough but no precipitation, while those on the right were produced using the same parameters but including precipitation. No off-GCP signals are evident when there is no precipitation, while the panels on the right show a deviation of 40°–60° to the north and a TOF which decreases to 6 ms in the evening and then increases again around sunrise, indicating an approach and retreat of the reflection points. These plots show all possible ray paths irrespective of signal strength, and consequently the off-GCP trace overlaps in time with the GCP signal. However, the relative ray densities of the on- and off-GCP propagation in the model output indicate that the off-GCP signal would be much weaker than the on-GCP signal. For comparison with observations, the two panels in Figure 5 show an example of the 7.0 MHz observations for the night of 9th December 2001, which exhibit features in good agreement with the model.

Figure 4.

(top) Simulated azimuth deviation (degrees clockwise from GCP azimuth) and (bottom) time-of-flight (ms) for the 7.0 MHz signal with (left) trough only and (right) with trough and scattering from irregularities within the auroral oval.

Figure 5.

Typical measured nighttime traces for(left) time-of-flight (ms) and (right) azimuth deviation (degrees clockwise from GCP azimuth) for 7.0 MHz.

5.3. Ray-Tracing Results for the Halifax-Leitrim Path

[21] A goniometric direction finder was employed for these measurements in a system capable of providing a single bearing estimate which, in general, is in the direction of the strongest signal component. Measurements were taken at 5.1 MHz, 10.9 MHz and 15.9 MHz. As summarized in Table 1, this path was shorter than the Uppsala-Leicester path, and the data were taken at a different phase of the solar cycle, but the geomagnetic latitude and the orientation of the GCP relative to local magnetic north were quite similar.

Table 1. Comparison of Path Characteristics Between Uppsala-Leicester and Halifax-Leitrim
Length, km1400900
Geomagnetic azimuth of GCP, deg east of magnetic N8092
Geomagnetic latitude of midpoint, deg5756
Geographic latitude of midpoint, deg5645
Geographic longitude of midpoint, deg east7−70
Mean sunspot number11130

[22] Figure 6 shows example observations at 5.1 MHz for the Halifax-Leitrim path. For this path, azimuthal deviations to the south occur more frequently than to the north, in contrast with the Uppsala-Leicester observations where southerly deviations were rarely observed. Variation is seen throughout 1994 in the time of commencement of deviation from GCP (between 2100 and 0800 UT), and in whether the off-GCP signals deviate smoothly from the GCP or change abruptly. The variability of the onset of deviation and its direction and rate of change are exemplified in Figure 6. At 10.9 MHz and 15.9 MHz, southerly deviations are also more common than northerly, and occur mostly between 0000 and 1200 UT. However, at these frequencies, off-GCP features are not so clearly defined.

Figure 6.

Examples of azimuthal deviation (degrees clockwise from GCP) along the Halifax-Leitrim path) at 5.1 MHz for (top left) 8 March 1994, (top right) 26 May 1994, (bottom left) 18 April 1994, and (bottom right) 27 March 1994. All times are UT.

[23] Figure 7 shows the modeled azimuthal deviation of a 5.1 MHz signal for Kp values of 2, 4 and 6. A southerly deviation is seen around 2100 UT for all these values of Kp, and an increasingly strong northerly deviation is seen around 0200 UT for higher values of Kp. This Kp dependence is due to the north wall being further to the south with increased Kp values in the Halcrow and Nisbet model. The time of the southerly deviation appears to coincide with the cessation of one of the GCP propagation modes.

Figure 7.

Simulation of azimuthal deviation (degrees clockwise from GCP) of a 5.1 MHz signal for (top left) Kp = 2, (top right) Kp = 4, and (bottom left) Kp = 6 for actual path length (911 km) and (bottom right) extended path length (1500 km). All times are UT.

[24] To further investigate the reasons for the difference in behavior on the Canadian and European paths, the Canadian model path length was increased from 911 km to 1500 km, similar to that for the Uppsala-Leicester path. Increasing the path length decreases the deviation and makes the propagation last longer into the night (see Figure 7). The principal features apparent in each plot, namely the early southerly deviation and the later northerly one, remain the same. These results were obtained without the effects of precipitation in the auroral oval being included in the model, showing that the experimentally observed features can be reproduced to some extent using only the mechanism of reflection from the trough walls. In reality, precipitation will occur and this will add to the complexity of the simulation results.

6. Conclusions

[25] For the Uppsala-Leicester measurements made close to sunspot maximum, estimates were made of the reflection points and the orientation of the apparent reflecting plane for all of the measurements at frequencies between 7.0 and 11.1 MHz. The latitude of the estimated reflection points was compared to a statistical model [Halcrow and Nisbet, 1977] of the latitude of the trough walls. In many cases, a good agreement was apparent between the estimated reflection point and the location of the north wall. The orientation of the apparent reflection plane was predominantly in the direction of Earth's field, but the calculated tilt differed from the geomagnetic dip angle. It is likely that the strong reflections from the north wall were caused by a combination of refraction through the ionosphere within the trough, and scattering in a plane perpendicular to the geomagnetic field. The high Doppler spread of the off-GCP signals is further evidence for scattering via irregularities. This is consistent with ray-tracing studies, which indicated that the electron density gradients within the trough walls at this phase of the solar cycle are insufficient to allow reflection at 7.0 MHz or above.

[26] Comparisons were made between the nature of the off-great circle propagation for the European path at a time close to solar maximum and for the Canadian path at a time close to solar minimum. Although the geomagnetic latitudes of the paths are similar, far more southerly deviations were observed in the Canadian measurements. Ray tracing produced southerly and northerly deviations similar to those observed, showing that for the Canadian path at the time of the measurements, i.e., close to solar minimum, there is a sufficiently large electron density gradient in the trough walls to reflect the rays. Lengthening the path used in the model led, as expected, to lower azimuthal deviations and a lengthening of the duration of off-GCP signals into the early morning, but did not remove the southerly deviations.

[27] The differences in character of the off-GCP signals between the two sets of data seems therefore to be mainly due to the different levels of depletion of the trough at different phases of the solar cycle. Some of the difference in propagation may also be due to changes in the density and vertical and meridional profiles of the background ionosphere caused by the solar cycle. Although daytime electron density varies dramatically over the solar cycle, differences in the nighttime ionosphere are much less pronounced. These effects are implicit in the model via the electron density background profiles, from which the background ionosphere is constructed. The other differences, noted in Table 1, are the one-degree difference in magnetic latitude and the 12° variation in GCP orientation between the two experiments. Although the trough is only a few degrees wide, its position varies by a few degrees latitude and its orientation by tens of degrees according to geomagnetic conditions, so these are unlikely to fully account for the observed differences. The difference in orientation, though small, may be more important if it alters the angle of incidence of a ray on the southern wall near the critical frequency.

[28] By comparing the propagation characteristics of the midlatitude trough at these two different times and places, a measure of the variability and sensitivity to path geometry, frequency, time of day, season and position in the solar cycle has been gained. Eventually, the authors hope to be able to predict the effect of the trough and auroral oval on any path given its location, bearing, length and the phase of the solar cycle. Such characterization of the effects of the trough will allow future planners and operators of HF systems to take account and mitigate its impact on communications systems.


[29] The authors would like to thank the Swedish Meteorological Institute for hosting the transmitter at their Marsta site. This investigation was supported by a grant from the EPSRC.