О ПОГРЕШНОСТЯХ ИЗОТРОПНОГО ПРИБЛИЖЕНИЯ ПРИ ГЕОМЕТРООПТИЧЕСКОМ ОПИСАНИИ РАСПРОСТРАНЕНИЯ РАДИОВОЛН В ИОНОСФЕРЕ

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1. Introduction The problem of electromagnetic-wave propagation in anisotropic plasma cannot usually be described by exact analytical solutions even in simplest model cases. One of main approximate methods is a geometric optics which provides an illustrative description of radio wave propagation in terms of ray paths [1-3]. Applicability boundaries of the method (wavelength is smaller than the typical scales of the medium) generally work quite well in the problem of ionospheric propagation of radio waves. An allowance for magneto-ionic effects can be made by using the Appleton-Hartree formula [4]: (1) where is the ordinary component refractive index; ; ; is the plasma frequency; f is the operating frequency; fH is the gyrofrequency. If f is much greater than fH, the expression for the refractive index of the ordinary wave component can be simplified: (2) Relation (2) represents the isotropic (no-field) approximation. Since the isotropic approximation simplifies the software implementation and reduces the execution time of calculation of the geometric optics method, it is widely used in practical applications. However, the simulation shows that there can be considerable discrepancies between the ray parameters in anisotropic and isotropic cases. In [5-7], oblique incidence ionograms were synthetized from vertical incidence ionograms using the Smith’s method [8]. In [6] the Smith’s method is used for a long path of Khabarovsk-Tory (ground range ), which seems well founded from this point of view. The effect of the magnetic field is also ignored in [9-10] considering paths over 1000 km long. Authors of [11] deal with oblique propagation of radio waves in the isotropic medium both for the long paths of Magadan-Tory and Norilsk-Tory as well as for the short path of Usolye-Tory (120 km). However, for example, the widespread Huang-Reinisch method for calculating electron density profiles [12] considers an anisotropic ionosphere and assumes a magnetic field to be constant in magnitude and direction. Authors of [13] also performed numerical ray tracing considering the Earth’s magnetic field and gave a specification for the ray paths for near-vertical ionospheric sounding. Authors of [5] developed a frequency scaling technique, which is based on the use of equivalent operating frequency , which leads to the same results in the absence of the magnetic field, as the actual frequency f in the presence of magnetic field. The additive correction is in particular a function of magnetic dip angle at the ray mid point. 2. Numerical Simulation For simulations we will use a typical winter mid-latitude plasma frequency profile from the Digisonde DPS-4 sited in Irkutsk (fig. 1), the points were joined using cubic spline interpolation to form a continuous function. The peak of the F2 layer is , the critical frequency of the F2 layer , the critical frequency of the E layer. рис Fig. 1. Plasma frequency profile from DPS-4 ionosonde, Irkutsk, December 02, 2011, 06:30 UT The magnetic field will be considered to be constant in magnitude and direction (as in [12; 13]) with parameters typical for midlatitudes: the gyrofrequency , the magnetic inclination 2.1. Initial value problem (Cauchy problem). Let us state the initial elevation angle (in relation to the vertical plane) and vary the operating frequency. Compare such characteristics of ray paths as group path and ground range (tab. 1). As a comparison, we will give characteristic ray paths for a number of frequencies (fig. 2). The results presented in tab. 1 allow us to plot the relative error versus frequency. Fig. 3 shows that at frequencies close to critical ones of the E () or () layer (an equivalent of the critical frequencies for oblique propagation according to the relation [14]), the error increases sharply, whereas, say, at the relative error for ground range is equal to 4.4 %. Consider now the isotropic approximation error for the initial value problem at and the alternate elevation angle (tab. 2). The characteristic ray paths are as follows (fig. 4). Tab. 2 yields the following dependences (fig. 5). Raw relative errors at low angles of radiation in fig. 5, b are associated with very short paths (near-vertical sounding). However, local maxima in the vicinity of 8º-9º in both the plots of fig. 5, a, fig. 5, b, call for special consideration. Table 1 Frequency dependence of group path and ground range for ordinary wave and for isotropic approximation Frequency, MHz Group path, km (ordinary wave) Group path, km (isotropic approximation) Ground range, km (ordinary wave) Ground range, km (isotropic approximation) 2.5 548.7 484.3 151.2 121.9 3 482.9 477.3 136.8 119.9 3.5 451.0 459.4 125.0 115.3 4 436.4 445.5 118.1 111.8 4.5 437.9 443.2 116.6 111.2 5 443.7 446.2 117.0 111.9 5.5 451.8 452.0 118.3 113.3 6 461.9 460.2 120.5 115.2 6.5 475.8 471.6 123.7 118.0 7 490.7 484.8 127.3 121.2 7.5 507.7 499.9 131.5 124.9 8 528.7 518.3 136.8 129.3 8.5 557.7 542.5 144.1 135.1 9 603.0 578.2 155.6 143.8 9.5 724.8 653.1 185.5 161.8 2a a 2b b 2с c Fig. 2. Examples of ray paths for ordinary component (solid lines) and isotropic approximation (dashed lines) for the profile shown in fig. 1: а - 3 MHz, b - 6 MHz, c - 9 MHz 3a a 3b b Fig. 3. The relative error versus frequency: а - group path; b - ground range Table 2 Group path and ground range versus elevation angle Angle, degrees Group path, km (ordinary wave) Group path, km (isotropic approximation) Ground range, km (ordinary wave) Ground range, km (isotropic approximation) 1 629.6 592.8 11.5 9.9 2 629.5 592.5 21.3 19.8 3 629.0 592.0 31.7 29.8 4 628.8 591.3 42.1 40.0 5 626.2 590.4 53.0 49.4 6 624.1 589.4 64.0 59.1 End table 2 Angle, degrees Group path, km (ordinary wave) Group path, km (isotropic approximation) Ground range, km (ordinary wave) Ground range, km (isotropic approximation) 7 621.8 588.2 75.8 69.3 8 622.1 587.0 86.3 78.5 9 619.7 585.7 97.1 87.9 10 617.0 584.3 107.5 97.4 12 611.4 581.4 127.4 116.1 15 603.0 578.2 155.6 143.8 19 595.3 575.3 191.9 180.1 24 590.5 575.2 236.2 225.2 30 593.0 581.9 290.1 280.4 37 610.3 602.0 358.0 349.6 45 650.7 644.3 447.5 440.4 54 734.5 729.9 577.4 571.8 64 930.0 926.9 812.0 807.9 4a a 4b b 4с c Fig. 4. Examples of ray paths: а - φ0 = 4º; b - φ0 = 24º; c - φ0 = 45º. The solid line is the ordinary wave; the dashed line, the isotropic approximation 5a a 5b b Fig. 5. The relative error versus elevation angle for f = 6 MHz: а - group path; b - ground range It should be noted that at low elevation angles (near-vertical propagation), the ordinary ray path has a typical needle point (see fig. 4, а) [4; 13]; yet the reflection height in near-vertical sounding is equal to the reflection height in vertical incidence and is independent of the elevation angle. The critical angle (relative to the vertical plane), at which the needle point disappears and the reflection heights begin to decrease, is determined by the relation [4; 13]: (3) where fH is the gyrofrequency; f is the operating frequency; χ is the magnetic inclination. In our case hence (3) yields . The local maxima in the plots of fig. 5, а, b approximate to this angle. Thus, we may conclude that the closer the elevation angle to the larger error given by the isotropic approximation. With increasing the elevation angle (i. e. for more oblique paths), the isotropic approximation error becomes insignificant. 2.2. Two-point boundary value problem. Consider now a two-point boundary value problem in which for each operating frequency we carry out “homing-in” to the given ground distance (in our case, 120 km) by varying the initial elevation angle of the ray [15]. We will compare the group path and the elevation angle (tab. 3). In the problem thus stated, characteristic ray paths take the following form (fig. 6). According to the data from tab. 3, the corresponding curves are as follows (fig. 7). Table 3 Frequency dependence of group path and elevation angle for ordinary wave and no-field approximation Frequency, MHz Group path, km (ordinary wave) Group path, km (isotropic approximation) elevation angle, (ordinary wave) elevation angle, (isotropic approximation) 2.5 538.5 481.7 12.14 14.75 3 476.1 477.4 13.27 15.02 3.5 449.2 460.9 14.43 15.57 4 437.0 447.9 15.23 16.04 4.5 438.8 445.6 15.42 16.13 5 444.5 448.1 15.38 16.04 5.5 452.2 453.5 15.21 15.86 6 461.9 461.1 14.94 15.60 6.5 475.2 472.0 14.55 15.24 7 489.6 484.6 14.13 14.85 7.5 506.4 499.3 13.67 14.42 8 527.5 517.3 13.11 13.93 8.5 558.4 542.0 12.37 13.30 9 613.6 580.9 11.24 12.42 6a a 6b b 6c c Fig. 6. Examples of the ray paths: homing-in to the distance of 120 km: а - 3 MHz; b - 6 MHz; c - 9 MHz 7a a 7b b Fig. 7. The frequency dependence of the relative error for the 120-km path: а - group path; b - elevation angle 3. Conclusion The isotropic approximation is widely used for simplifying the ray-optics description of ionospheric propagation of radio waves. However, this description under certain conditions may produce a significant error. So, we have revealed that when an operating frequency approaches the critical frequencies of the E or layer, the no-field approximation error increases sharply for near-vertical sounding. On the other hand, the simulation has shown that when varying the elevation angle the relative error has a local maximum in the vicinity of the critical angle determined by (3). With increasing the elevation angle (i. e. for more oblique paths), the isotropic approximation error decreases monotonically, practically not being able to distinguish between isotropic and anisotropic media for paths over 1000 km long.

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О. А. Ларюнин

Институт солнечно-земной физики СО РАН

Email: laroleg@inbox.ru
Российская Федерация, 664033, г. Иркутск, ул. Лермонтова, 126а

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