A ray-tracing model to account for off–great circle HF propagation over northerly paths

Authors


Abstract

[1] Off–great circle HF propagation effects are a common feature of the northerly ionosphere (i.e., the subauroral trough region, the auroral zone, and the polar cap). In addition to their importance in radiolocation applications where deviations from the great circle path may result in significant triangulation errors, they are also important in two other respects: (1) In systems employing directional antennas pointed along the great circle path, the signal quality may be degraded at times when propagation is via off–great circle propagation modes; and (2) the off–great circle propagation mechanisms may result in propagation at times when the signal frequency exceeds the maximum usable frequency along the great circle path. A ray-tracing model covering the northerly ionosphere is described in this paper. The results obtained using the model are very reminiscent of the directional characteristics observed in various experimental measurement programs, and consequently, it is believed that the model may be employed to enable the nature of off–great circle propagation effects to be estimated for paths which were not subject to experimental investigation. Although it is not possible to predict individual off–great circle propagation events, it is possible to predict the periods during which large deviations are likely to occur and their magnitudes and directions.

1. Introduction

[2] Various reports have been made by the authors, their colleagues and others over a number of years concerning off–great circle propagation over northerly paths (i.e., those paths impinging upon the polar cap and auroral ionospheres and that within the subauroral trough region) [see, e.g., Bates et al., 1966; Whale, 1969; Warrington et al., 1997; Rogers et al., 1997; Stocker et al., 2003; Hunsucker and Hargreaves, 2003; Siddle et al., 2004a]. Recent papers by the authors' research group have included the results of modeling these effects (in particular, see Zaalov et al. [2003] and Siddle et al. [2004b]), however space constraints did not enable the mathematical detail of the ray-tracing models to be included. It is the purpose of this paper to report this in sufficient detail for the work to be replicated by others and for the effects to be considered in other circumstances. In particular, the work should be useful in developing future generations of HF prediction codes where the signals impact on the northerly ionosphere.

2. Numerical Ray Tracing

[3] Modeling the influence of ionospheric structures on HF propagation requires the construction of a model of the electron density distribution within the ionosphere traversed by the radio waves together with a versatile method of solving the wave equation. Many ionospheric propagation problems have been investigated using the method of geometrical optics [see, e.g., Kravtzov and Orlov, 1980], which has been found to be acceptable within the following limitations: (1) The parameters of the propagation medium must only vary slightly on the scale of a wavelength, and (2) the scale size of any irregularities must be large compared with the main Fresnel zone.

[4] A numerical code (commonly referred to as the Jones 3D program) developed by Jones and Stephenson [1975] has been employed in this investigation. This is a versatile FORTRAN program for tracing rays through an anisotropic medium whose refractive index varies continuously in three dimensions. For each path, the program can calculate the group path length, the phase path length, absorption, Doppler shifts due to a time-varying medium, and the geometrical path length.

[5] The program calculates each ray path by numerically integrating the Hamilton-Jacobi equations for a user-specified model of the ionosphere given the transmitter location (latitude, longitude and height above the ground), the frequency of the wave, the direction of the transmission (azimuth and elevation angles), the receiver height, and the maximum number of hops. The refractive index equations used in the Jones 3D ray-tracing program are based on the Appleton-Hartree formula [Budden, 1961], or the Sen-Wyller formula [Sen and Wyller, 1960]. For the purposes described here, it was necessary to make a few alterations to the Jones 3D program as outlined below.

2.1. Time-Varying Parameters

[6] An additional loop in time was implemented in the program in order to provide a flexible way of investigating time-dependent processes in the ionosphere. Since many geophysical parameters are time-dependent, values at specific times are input to the program (e.g., Kp and By with 6 hour steps) and interpolated values calculated for each time step.

2.2. Alternative to a Homing Procedure

[7] Since the northerly ionosphere is a disturbed region containing irregularities on various scales and amplitudes, it difficult to construct an acceptable homing algorithm because of the very complicated structure of the rays. As an alternative, the calculation of the number of rays falling in the area close to the receiver produces reasonable results. This area has the shape of an ellipse, oriented along the azimuthal angle of arrival, with the major axis (b) depending on the elevation angle of arrival (ɛ) as

equation image

where a is the minor axis of the area in which rays are counted, and is an input parameter to the program.

2.3. Variation of the Azimuth and Elevation Step Sizes

[8] The program run time increases rapidly whenever a ray travels through regions with large gradients of electron density, a situation which commonly arises in the high-latitude ionosphere. To improve execution speed, an adaptive procedure to adjust the step sizes in both azimuth and elevation was incorporated into the code. In the ray-tracing program, the outer loop is in elevation angle, and the loop in azimuth angle executes immediately before. In each time step Ti+1 (excluding the first) the nominal steps in both elevation angle En(Ti+1) and azimuth, An(Ti+1) depend on number of the rays reaching the receiver in previous time step Ti as

equation image
equation image

where n(Ti) is the number of rays reaching the receiver in previous time step, A0 and E0 are user-defined steps in azimuth and elevation, respectively, and

equation image

where D is the distance between transmitter and receiver.

[9] The actual angle steps employed depend on additional conditions. When the loop in elevation angle has been executed, the step changes depending on how close the specific ray goes to the receiver. The shaded area in Figure 1 corresponds to the takeoff angles of rays that fall into the elliptical area surrounding the receiver described above. If the ray passes through the shaded area, then the elevation step is reduced relative to the nominal step by a factor of 2,

equation image

otherwise, if it passes through the unshaded region (which has a collecting area twice as large as the shaded region) the step is

equation image

and if the ray passes outside of the dashed region the step is increased by a factor of 2,

equation image

The subsequent azimuthal step also changes according to this algorithm. The bold stepped line at the top of Figure 1 separates two ranges in the elevation angles at the transmitter. If all of the rays in a specific range of azimuths (this is 20° in Figure 1) penetrate the ionosphere in the previous time step (Ti), the elevation angle step above the bold line in the same azimuthal range at Ti+1 is

equation image
Figure 1.

Schematic to illustrate the adaptive algorithm for angles step.

3. Computational Model of Electron Density

[10] The ray-tracing program requires a model of ionosphere in which the electron density (Ne) profile and the gradient of the refractive index are continuous. It is convenient from an implementation point of view to approximate the electron density distribution with analytical expressions that are amenable to mathematical manipulation. Furthermore, it is important that the model can be implemented within the ray-tracing code since calling an external program at each step of the ray calculation results in unacceptable calculation times.

[11] For the model described here, the background electron density profile is first defined and then perturbed by imposing changes representing electron density variations associated with the midlatitude trough, the presence of polar patches and arcs, and precipitation within the auroral zone. In brief:

[12] 1. The background model is determined by measurements of E and F layer critical frequency (see sections 3.1 and 3.2).

[13] 2. The position of the midlatitude trough is based on the observations of Halcrow and Nisbet [1977] (section 3.3), while the magnitude of the electron density perturbation varies with sunspot number and Kp (section 3.4). Density irregularities are also added to the wall of the trough (section 3.3).

[14] 3. The model used to simulate the presence of density irregularities in the auroral oval is described in section 3.5. The maximum density enhancement associated with the irregularities in the model depends upon sunspot number and Kp (section 3.6).

[15] 4. The patches and arcs found in the polar cap F region are simulated by a randomized distribution of electron density enhancements (section 3.7). This distribution then varies with time according to the convection flow patterns given by Lockwood [1993]. The number and size of the patches and arcs can be varied for different runs of the simulation program (section 3.8).

[16] In order to make credible use of the model, it is necessary to find appropriate links between the geophysical parameters that control the HF propagation, and the computational parameters that define the details of the ionospheric model. The process of formation and evolution of the high-latitude ionosphere involves many different physical mechanisms and is a subject of extensive investigation by many researchers. Progress in auroral global imagery, photometry and spectrometry, in rocket experiments in the auroral zone, and in theory provide the opportunity to define the main parameters of the model with acceptable reliability. Most of these parameters are specified as inputs to the ray-tracing code to facilitate flexible use of the model.

3.1. Background Ionospheric Model

[17] A number of ionospheric models exist [e.g., Bilitza, 1990; Chasovitin et al., 1987] that describe the ionosphere under undisturbed conditions. However, none of them reproduces the ionospheric electron density profile in the high-latitude region with good accuracy [Egorova et al., 1995]. Furthermore, they do not reflect the day-to-day and hour-to-hour (or even more rapid) variations of the real ionosphere.

[18] An adjustable model of electron density profile was constructed on the basis of data from vertical soundings from a number of stations. The input parameters of the model are the maximum (fmax) and minimum (fmin) values of the critical frequencies occurring during the day, the heights and height scales (if available) of the F2 and E layers for three reference points and the times of the day when fmax and fmin occur. One reference point is located at the geographical North Pole, while the two others are chosen depending on the region of interest and the availability of the vertical sounding data. Sounding data are not available for the North Pole and therefore a compromise solution was adopted whereby data for South Pole for the period between 1 July 1957 and 30 June 1965 were employed. Extrapolation of these data, on the assumption that both poles will exhibit similar ionospheric variations, was used in the modeling.

[19] The longitudinal dependence of electron density (Ne) was derived from an approximation of the time dependence of the layer's critical frequency at the chosen reference points. Three slightly different approximating functions reflecting the diurnal variability of the electron density profile have been used:

equation image
equation image
equation image

where ϕ is the current geographical longitude, ϕdel is the longitudinal delay relative to local midnight for which the critical frequency reaches the minimum value, and

equation image

where t is the universal time in hours. In the calculations, the function, f1, f2, or f3 that best resembles the experimental profile of Ne was used (this was determined “by eye,” but it would be possible to produce a correlation algorithm to undertake this function).

[20] The critical frequency of the layer as a function of longitude and time at the latitude of the ith reference point is

equation image

where fmax and fmin are maximum and minimum values of critical frequency at the each reference point. An example of the time dependence of fcr for each of the three approximating functions (for 0° longitude and 1.5 hours delay) is given in Figure 2.

Figure 2.

Value of fcr for the three approximating functions f1 (solid curve), f2 (dash-dotted curve), and f3 (dashed curve). Parameters for the curves are 0° longitude, 1.5 hours delay.

[21] Analysis of vertical sounding data coupled with typical values from various ionospheric models indicates that the latitudinal dependence of the critical frequency of each layer can be approximated as

equation image

where θ and ϕ are the current geographical colatitude and longitude, respectively, and fpl is the critical frequency at the North Pole. The parameters c1 and c2 are determined by substituting into equation (14) the values of the critical frequencies at the reference points,

equation image

The parameters fpl, ko, and θ0 are input parameters that allow the user to adjust the model.

[22] A double Chapman profile [see, e.g., Davies, 1990] with the height and half thickness of the layers dependent on latitude was used as an approximation of the vertical Ne profile:

equation image

where h is current height, z = (hHm)/H0, Hm is the height above the Earth of the peak electron density, H0 is the half thickness of the layer, and N0 is the peak electron density.

[23] The height of the layer, replacing Hm in equation (16), is a function of latitude,

equation image

where HmN and HmS are the height of the layers at the northern and southern reference points, respectively, θS, and θsc,S are the latitude and latitudinal scale of the southern border of the midlatitude trough, respectively, and fNS and fEW are the functions of ϕ defining the structure of the trough for which exact expressions are given in a later section.

[24] The half thickness of the layer Hsc, replacing H0 in equation (16), depends on θ and ϕ as

equation image

θS and θN depend on ϕ. Hsc,T, Hsc,N and Hsc,S are half thicknesses of the layers inside the trough and at the northern and the southern reference points, respectively, and are user input parameters.

3.2. Parameter Selection for the Background Ionosphere Model

[25] The input parameters defining the background ionosphere in the model are the maximum (fmax) and minimum (fmin) values of the critical frequencies of the F2 and E layers for three reference points, and the times of day (tmax and tmin) when fmax and fmin occur during a specific day. As an example, Figures 3 and 4 show the variations of these parameters for two reference points during September 2002. In Figures 3 and 4, the vertical sounding data are indicated by dots, and the data smoothed with a two-day window are shown by the curves. The variations of fmax, fmin, tmax and tmin during one month are significant even at a midlatitude station (Chilton in this case). This contrasts with the IRI model, which only has a monthly resolution. The resulting latitudinal dependence of the critical frequency of the F layer is given in Figure 5 (for a geographic longitude of 0° and parameters fitted to Tromsø and Chilton).

Figure 3.

(top) Variations in MHz of fmax (solid curve) and fmin (dashed curve) and (bottom) variations in UT of tmax (solid curve) and tmin (dashed curve) for a midlatitude station (Chilton, UK, 51.5°N, 358.7°E).

Figure 4.

(top) Variations in MHz of fmax (solid curve) and fmin (dashed curve) and (bottom) variations in UT of tmax (solid curve) and tmin (dashed curve) for a high-latitude station (Tromsø, Norway, 69.7°N, 19.0°E).

Figure 5.

Latitudinal dependence of the critical frequency of the F layer based on equation (14). Asterisks mark the values of the critical frequency at two reference points (Tromsø and Chilton).

3.3. Midlatitude Trough

[26] A parameterized model of the location of the walls of the midlatitude trough based upon topside sounder measurements was published by Halcrow and Nisbet [1977]. The model assumes a constant electron density depletion along the bottom of the trough with a linear rise in electron density in each of the boundary walls from the fully depleted value at the bottom to the undepleted value at the top. The invariant latitudes (in degrees) of the top (subscript t) and bottom (subscript b) of the north (subscript N) and south (subscript S) walls are given therein as

equation image
equation image
equation image
equation image

where T = LT + 12 if LT < 12 and T = LT − 12 if LT ≥ 12 (LT is local time in hours).

[27] The trough closes at around sunrise at local times when the solar zenith angle lies between 95° and 87°.

[28] Opening of the trough begins at a local time 1.5 hours after the solar zenith angle reaches

equation image

and the trough is fully formed at a local time 1.5 hours after solar zenith angle reaches

equation image

While the Halcrow and Nisbet model has an advantage of being based on observations, it is a statistical model with a physically unrealistic profile that does not reproduce the diurnal variations of the propagation characteristics along the trough, and is unsuitable for ray-tracing applications. In order to achieve the required flexibility in the simulation, an analytical approximation defining the location of the trough has been used. The input parameters to the model which define the location of the trough are: the geomagnetic coordinates of the central point of the southern (ΘcS and Φc) and northern (ΘcN and Φc) borders; the eastward and westward extent of the trough (ΔE and ΔW); and their scales (scE and scW). The south and north borders of the trough are formed by the composition of the Epstein functions with parameters δΘS, δΘN, δΦES, δΦEN, δΦWN, δΦWS, SES, SEN, SWN, SWS. The meaning of these parameters is given in equations (32)–(35) and Figure 6. The geomagnetic coordinates of the four points defining the position of the trough depend on the level of geomagnetic activity Kp as

equation image
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equation image
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The longitude of the center of the trough depends on time as

equation image

where ΦGM is the geographic longitude of the geomagnetic pole, and t is the universal time in hours.

Figure 6.

Schematic to illustrate definition of the trough parameters in equations (32)–(35).

[29] The southern border of the trough is given in these terms by

equation image

where

equation image

The function fEW determines the dynamics of the opening and closing of the trough in time,

equation image

where ϕ is current geomagnetic longitude and SW and SE are the longitudinal scale of the western and eastern ends of the trough, respectively.

[30] Similarly, for the northern border

equation image

where

equation image

The latitudinal dependence of the electron density depletion inside the trough is derived from that function as

equation image

where SN and SS are the latitudinal scales of the north and south walls of the trough, corresponding to (Λt − Λb) in the Halcrow and Nisbet model.

[31] The electron density with the presence of the subauroral trough according to equations (33) and (34) is given by

equation image

where Ne0 is the background electron density and ΔNe is the maximum value of the electron density depletion inside the trough.

[32] The Halcrow and Nisbet model is trapezoidal in form whereas our model has a smooth variation in electron density perturbation that is more physically realistic and is in a form suitable for ray tracing. In the real ionosphere, the structure of the trough is significantly more complicated because of vertical and horizontal transport processes that lead to the development of irregularities in the electron density on a smaller scale. In order to reflect this, a spline approximated longitudinal random function has been superimposed on the smooth borders of the trough:

equation image
equation image

where ΔSb and ΔNb are the amplitudes of the latitudinal variations of the southern and northern border of the trough, frand is a random function in the range ±1 with a longitudinal scale that is a user input parameter. The scale, in this context, is the longitudinal distance between the nodes of the random function.

[33] The resulting expressions fNStr and fEWtr replacing fNS and fEW, respectively, are given by

equation image
equation image

where ΔNwall and ΔSwall are random functions of latitude with scale SNS, and range ±ΔN0 and ±ΔS0, respectively. ΔEW represents the random functions of longitude with scale SEW and range ±ΔoEW. ΔN0, ΔS0, ΔoEW and σ are all user-defined parameters.

[34] The resulting expression for the electron density, Ne, in the ionosphere in the presence of the midlatitude trough with irregularities is

equation image

The factor fRtr is included to correct the vertical profile of the depletion in Ne. In the simulations, fRtr is set to 1 since there is little information about the vertical profile of the electron density inside the trough.

3.4. Parameter Selection for the Midlatitude Trough

[35] The parameters needed to define the statistical properties of the trough depend on geophysical parameters. In periods of high levels of geomagnetic activity, the structure of the trough becomes more irregular and the walls of the trough become filled with medium-scale irregularities. On the basis of observations, it is reasonable to assume that the electron density depletion inside the trough depends on sunspot number (see, for example, the ray-tracing studies for two paths at different times in the solar cycle reported by Siddle et al. [2004b]). Furthermore, ion convection, which depends on the level of geomagnetic activity, plays a crucial role in the trough development. The depletion of electron density (ΔNe) increases when Kp increases, and decreases when sunspot number increases because of transport of plasma from the south:

equation image

where R is the current sunspot number, equation image is the diurnal mean value of Kp index, d is the day number, and R0, R00 and Kp0 are adjusting parameters.

[36] The expression for the intensity of the irregularities inside the walls of the trough was adopted in the model as

equation image

with typical latitudinal (SNS) and longitudinal (SEW) scales of 0.2° and 2°, respectively. These produce a landscape of patches along each wall that are elongated in the direction of the trough.

[37] The latitude of both walls of the trough was perturbed by three smooth random functions of longitude with a zero mean. The longitudinal scales of these fluctuations of the border were set to 18°, 6° and 2° with typical latitudinal deviations of 1.5°, 0.5° and 0.2°, respectively.

3.5. Model of Electron Density Irregularities in the Auroral Oval

[38] Auroral emissions due to precipitation occur in oval-shaped bands lying approximately between 65° and 75° magnetic latitude and centered on the magnetic pole. This region, which consists of a more or less continuous band of faint diffuse emissions within which brighter discrete arcs are embedded on both the dayside and the nightside, is known as the auroral oval. These diffuse emissions result from a relatively steady flow of electrons and protons that are precipitated out of the central plasma sheet by interactions with plasma waves [Elphinstone et al., 1994; Bates and Hunsucker, 1974]. A sharp increase in electron concentration over the height range 100–180 km provides a clear signature of auroral precipitation (see the results from the EISCAT UHF radar (69.6°N, 19.2°E) reported by Jones et al. [1997]).

[39] In the computational model, the southern border of the oval is detached from the northern wall of the midlatitude trough by the scale value of the north wall of the trough (SN). The latitudinal width of the region of the precipitations (δθp) is a user specified parameter.

[40] In latitude-longitude coordinates the expression for electron density irregularities is

equation image

where

equation image
equation image

where θNp and θBp are the latitudinal positions of the auroral oval and

equation image

In the case of strong events when auroras can be observed at latitudes south of the midlatitude trough

equation image

or

equation image

in the situation when additional auroras could be observed moving in northward direction.

[41] The widths of the auroral oval zones are δθNp, and δθSp, respectively, and Sp is their latitudinal scale, δNlat, and δBlat are the random function with range ±1 and user-defined latitudinal scales, specifying the distribution of the electron density irregularities in the latitudinal direction, while δlon fulfils this role for longitude, and Δpr is the intensity of the irregularities.

[42] In the vertical plane, a Gaussian distribution along the magnetic field line with a user specified height scale has been adopted.

3.6. Parameter Selection for the Electron Density Irregularities in the Auroral Oval Region

[43] The basic model of the enhancement of electron density inside the auroral oval is a trapezoidal function of geomagnetic latitude. As a function of distance in the vertical plane along the magnetic field lines, the density enhancements were modeled as starting at heights of around 100–120 km, having several peaks at around 100–150 km, and then decaying slowly toward 200 km. These background enhancements were perturbed by the product of smoothed random functions of longitude and latitude. Typical scales of the variations in the horizontal projection were 0.2° and 2° for latitude and longitude, respectively. The peak value of the electron density enhancement depends on sunspot number and Kp as

equation image

where R is sunspot number, equation image is daily mean value of Kp, and R1 and Kp1 are adjusting parameters.

3.7. Model of F Layer Patches and Arcs Inside the Polar Cap

[44] At present, there exists no reliable self-consistent model of F layer patches and Sun-aligned arcs of enhanced electron density in the polar ionosphere. Nevertheless, ray-tracing simulations require a continuous, three dimensional distribution (with derivatives) of Ne in the area of calculation. The computational model of the electron density irregularities responsible for off–great circle HF propagation must adhere to the general understanding of the structure of the high-latitude ionosphere, but on the other hand be simple enough to be incorporated into the ray-tracing code.

[45] The process of formation and evolution of F layer patches and Sun-aligned arcs has an irregular, stochastic character. The F layer patches are formed on dayside of the Earth and travel across the polar cap in accordance with the convection flow patterns. However, the number of patches, their specific trajectory, intensity and spatial scale of the electron density disturbance are not well defined.

[46] A quasi-statistical approach has been adopted in modeling F layer patches and arcs, in which their distributions inside the polar cap are determined by one of a number of different scenarios. The size of the region in which the patches and arcs are distributed is a function of Kp. The Sun-aligned arcs move slowly across the polar cap in the direction of the By component of the IMF, while the trajectories of the patches are deformed ellipses resembling the convection flow patterns given by Lockwood [1993] (reproduced here in Figure 7). The speeds of the patches have a Gaussian distribution, with the parameters of the distribution specified as a user input.

Figure 7.

Model of the cross–polar cap convection flow [Lockwood, 1993].

[47] Each patch in the convection cell is composed of an arbitrary number of three-dimensional Gaussian distributions with approximately equal scale in each of the horizontal directions. The number of the cells is a user-specified parameter, which is dependent on the scenario and defined by the IMF. The position of each component distribution in the patches is defined as a random function with a specific spatial scale around the regularly distributed ‘nodes’ in the area surrounding the geomagnetic pole. The temporal evolution of the patches depends on the movement of the component distributions forming the patch coupled with the rotation of the Earth beneath the convection flow patterns. The speed inside each cell was assumed to form a Gaussian distribution. The maximum value of the speed (Vo) and the position where this value is attained (ρVm) are both user input parameters. Two versions of the speed distribution have been incorporated into the code. In the first case, all component distributions move with the linear speed depending on their distance from ρVm. In the second case, the distributions belonging to the specific patch travel with equal angular velocity around the center of the cell. The intensity of the patches depends exponentially on the time from the moment of their formation with a user-defined lifetime. The numbers of the nodes in the Sun-Earth and dawn-dusk directions are user input parameters.

[48] Each arc consists of a number of three-dimensional Gaussian distributions with different scales in the dawn to dusk and noon-midnight directions. The position of each distribution in the arcs is defined as a random function with specified spatial scale around the quasi-regularly distributed nodes in the polar cap area.

[49] The size of the convection flow area (polar cap area) Δ depends on Kp and corresponds roughly to the distance between the geomagnetic pole and the northern border of the midlatitude trough,

equation image

The positions of the nodes for each cell in quasi-Cartesian coordinates:

equation image
equation image

where k is the number of nodes (patches or arcs) in dawn-dusk direction, and i runs from 1 to k, l is the number of nodes in noon to midnight direction, and j runs from 1 to l. The position of the nth component forming the ijth patch are

equation image
equation image

where frand is a random function generated in the computer code with a range of ±1 (this is implemented using the FORTRAN rand function with the argument acting as a seed), δij is the maximum distance of the components from each node, and n runs from 1 to m, where m is the number of the components forming the patch or arcs. In terms of polar coordinates the expressions for the position of the components are

equation image
equation image

Then the expression for angular speed can be given in the first case of the velocity field distribution as

equation image

and in the second case as

equation image

The coordinates of the components as function of time t are

equation image

while ρijn is independent of time.

[50] These formulae give a random distribution of components moving on a circular trajectory around the geomagnetic pole. To make the trajectories more realistic, it is necessary to shift the center of the cell by Δc, depending on the By component of IMF, and change the shape of the convection flow in accordance with Figure 7. This is illustrated diagrammatically in Figure 8.

Figure 8.

Schematic to illustrate the coordinates of the patches.

[51] In Cartesian coordinates the position of the components will then be

equation image
equation image

where the factor γ compresses the circle in the dawn-dusk direction to form an ellipse, and the term Δcequation image shifts the cell center and adjusts the cell shape.

[52] In the case of Sun-aligned arcs, the numbers of nodes in noon-midnight and dawn to dusk directions are 1 and k, respectively. Each arc is composed from m components distributed in noon-midnight direction around k nodes. The expressions for the positions of the arcs in Cartesian coordinates as function of time t follow from equations (54) and (55) and are

equation image
equation image

where V00 is the speed of the arcs in dawn to dusk direction.

3.8. Parameter Selection for F Layer Patches and Sun-Aligned Arcs

[53] Between three and five individual components in the patches are usually used in the model, with typical horizontal scales of the components in both directions of about 50 km. These produce patches of diverse shape and a size of about 300 km. The vertical scale of the patches was about 30 km.

[54] Each arc consists of a number of three-dimensional Gaussian distributions (typically around 8) with a horizontal scale of 50 km in the dawn to dusk direction and 400 km in the noon-midnight direction. The vertical scale of the arcs was about 30 km. Each fragment of the arc was separated in the noon-midnight direction by a distance of about 300 km, thereby producing structures elongated in the Sun-Earth direction. The number of the arcs inside the polar cap is a user input parameter, and the distance between them depends on the scenario.

[55] An example map of the electron density distribution at a fixed height of 220 km produced by the model is shown in Figure 9. This particular example includes the midlatitude trough and F layer patches, but excludes Sun-aligned arcs.

Figure 9.

Map of the electron density distribution produced by the model inside the northerly ionosphere at 0100 UT for a height of 220 km and a Kp of 1.6. The midlatitude trough and F layer patches are shown.

4. Concluding Remarks

[56] Off–great circle propagation effects often need to be accounted for when planning and operating HF radio links. Usually, these effects are considered as only being of importance in radiolocation (HF-DF) where deviations from the great circle path may result in significant (sometimes intercontinental) triangulation errors. It is also noteworthy that in systems employing directional antennas pointed along the great circle path the signal quality may be degraded at times when propagation is via off–great circle propagation modes. Furthermore, the off–great circle propagation mechanisms may result in propagation at times not anticipated (i.e., when the signal frequency exceeds the maximum usable frequency along the great circle path).

[57] In order for these propagation effects to be properly taken into account in system design and operation, it is necessary for the propagation mechanisms to be fully understood and incorporated into prediction tools. To this end, the model described here has been developed. The results obtained using this model are very reminiscent of the characteristics observed in the experimental measurement programs [Warrington et al., 1997; Rogers et al., 1997; Stocker et al., 2003; Siddle et al., 2004a], and enable the nature of off–great circle propagation effects to be estimated for paths which were not subject to experimental investigation. Although it is not possible to predict individual events, due, for example, to the unpredictable nature of the precise positions of polar patches and arcs, it is possible to predict the periods during which the large deviations are likely to occur, their magnitudes and directions.

[58] Ray tracing through model ionospheres is computationally intensive. Consequently, it is not envisaged that most propagation prediction tools developed as a follow-up to this research will contain ray-tracing elements. An alternative approach is being considered by the authors whereby a large number of ray-tracing results will be included in a database which will form part of a rule base for predicting the effects of off–great circle propagation on any path impinging on the northerly ionosphere.

Acknowledgments

[59] The authors are grateful for the support of the EPSRC under grants GR/N16877 and GR/M35025 and the EPSRC and the United Kingdom Ministry of Defence under the Joint Grants Scheme grant GR/N66056. We are also grateful for the support of the Canadian DND/DRDC. The authors would also like to thank the various organizations that have hosted the transmitting and receiving systems employed in various investigations into the propagation characteristics: the Auroral Station in Adventalen, Svalbard; the Swedish Institute of Space Physics, Kiruna; the Swedish Meteorological Institute, Uppsala; and the Norwegian Defence Research Establishment.

Ancillary