4. LREI Model

[19]  In the LREI model (11) the spatial coordinates are h and the distance to the epicenter axis l (see Figure 1). To obtain the explicit solution of equations (3) and (4) by the method of variable separation [Semenov, 1974] in the analytical model (11), one is able only in the case of remote from the epicenter vertical sounding of LREI, or in the cases ( J_Mb, J_MN D H²/ 4b₀ ), when one can neglect the horizontal gradients. The latter situation takes place during SNE of relatively small equivalents occurring at considerable distances from the Earth when the LREI influence on the radio wave propagation is insignificant. We here are interested in cases of the strongest impact of LREI on the operation of radio systems when one cannot separate the variables in equations (3) and (4) for model (11). However, one can easily see (see, e.g., Table 1 and equations (2) and (11)) that under t 1 s and w 10⁹ s ^-1 a_M = max |a(h,l,t)| 10^-2; so to obtain approximate solutions of this equations, one should use the small perturbation method [Kravtsov et al., 1983]. Actually, even the first approximation of this method makes it possible to provide the needed accuracy of the calculations of the main radiophysical effects occurring at the propagation of the radio waves with w 10⁹ s ^-1 through LREI.

[20]  To develop methods of calculations of radiophysical effects on the basis of the analytical models of N(h,l,t), one has to obtain the dependence of its spatial coordinates h and l on the angular coordinates of the object observation ( M and F ) measured by a radio engineering device and on the current distance r (0 r L ) under the given values of the geographical coordinates of the explosion epicenter ( j₁, l₁ ) and the radio engineering device in question ( j₂, l₂ ). In Figure 1, L designates the distance to the observed object, the elevation angle M and azimuth angle F are calculated from the horizontal plane in the point of the radio engineering device location and from the direction to the explosion in this plane, respectively. We have the following expressions for the above indicated dependencies

(12)

where

(13)

and R_E is the Earth radius.

[21]  Solving equation (3) by the perturbation method the following formulae were obtained for the refraction errors of the measurements of the observed object angular coordinates ( D M = M- M_i, D F= F- F_i ) [Gdalevich et al., 1963]:

(14)

(15)

For the errors in the group delay D L_gr the phase path length D L_ph and the component of the Doppler frequency shift caused by the medium nonstationarity D w_n (they determine the error in the determination of the distance and velocity of the observed object) the following formulae are obtained in the first approximation:

(16)

(17)

The medium parameter distributions a (h,l,t) and their partial derivatives with respect to the spatial coordinates and time are determined by

(18)

(19)

(20)

(21)

One can neglect the terms proportional to h'(t) and c' (t) in equation (21) not exceeding the error of the order of a few percent. In equations (18)-(21),

where

It should be noted that the Doppler shift of the frequency caused by the influence of the medium is due to both the motion of the object itself and nonstationarity of the medium along the radio wave propagation path. The first component of this shift is of the same order of magnitude as for the natural ionosphere [see, e.g., Kravtsov et al., 1983], so here we consider only the second component (17). Calculating the integrals included to equations (14)-(17), each of them is split to two parts. The first is determined by the LREI parameters and integration is performed from r₁ to r_c. The calculation of the second part determined by the parameters of the natural ionosphere is performed from r_c to L. If the condition L < r_c is fulfilled, integration in equations (14)-(17) is limited only by the boundaries of the lower region and is performed from r₁ to L. The values of r₁ and r_c are given by

(22)

into which either h = 30 km or the height of the conjugation of the lower region to the ionosphere h = 90 km is substituted, respectively.

Figure 4

[22]  Solution of equation (4) for the field amplitude is written as [Semenov, 1974]

Expanding the Jacobian D(t ) in terms of the small parameter a_m, one can show [Kravtsov et al., 1983] that for a regular inhomogeneous ionosphere the correction of the first order D₁(t ) is small as compared to the zero approximation D₀(t ). Therefore calculating the field amplitude F in our case it is enough to take into account only the spherical divergence of the rays and calculation of the attenuation d in the first approximation ( dt dr )

(23)

Here expression (18) is used for a (h,l,t). Writing equation (23), we did not neglect S² in comparison to 1 in the imaginary part of e (see equation (1)) because at heights of 30-40 km the inequality n _eff,en/w 1, generally speaking is not fulfilled for w 10⁹ s^-1. According to the experimental data [Al'pert, 1972; Gringauz, 1966] obtained by different methods for h 100 km the effective collision frequency n _eff fairly well is described by the exponential dependence (see Figure 4)

(24)

where the values n₀ =1.1 10¹¹ s ^-1 and H₀ = 7.1 km are fitted for the n₀ and H₀ parameters. Calculating the attenuation the usage of the LREI analytical model (11) does not provide the relative error required because its value is determined not only by the accuracy of the N(h,l,t) approximation, but depends on the quality of the approximation of the product N(h,l,t) n(h). To provide the methodical error of the order of a few percent while calculating the attenuation one should model the N values by the methods described by Kozlov [1967, 1971], Kozlov and Kudimov [1969], and Kozlov and Raizer [1966] along the ray connecting the radioengineering device and the observed object. In other words, the initial electron concentrations are determined along the ray in question and then the decrease of N in these points is calculated up to the time moment needed. Doing this one has to obtain the distance R_i from the explosion point to the points with the coordinates M, F, and r_i at the ray having a height of h_i,

where l_i is determined by equations (22) and (12). Further calculation of the attenuation (23) is performed in a similar way to the calculation of the integrals in equations (14)-(17). Such approach provides the required methodical error in simulation of the wave attenuation in LREI, however increases the time needed for the calculations.

Figure 5

Figure 6

Figure 7

Figure 8

[23]  Using the method described above, the calculations of the attenuation, group delay, Doppler frequency shift and azimuthal refraction as functions of the radiotechnical coordinates of the object in the wide range of H and q were performed. The results show that the main radiophysical effect influencing propagation of the radio waves with w 10⁹ s^-1 through LREI is the attenuation. For the Doppler systems operating at frequencies w 10⁹ s^-1, the Doppler frequency shift reaching at w=10⁹ s^-1 at t 1 s the value of D w_L 1 kHz may present an exception. As an example, Figures 5, 6, 7, and 8 show the values of these effects at the working frequency w = 10⁹ s^-1 for the explosion with q = 5000 kT at H = 150 km, distanced from the radio engineering device by b = 335 km, at different elevation angles of the observed object M and F = 10^o. The value of b was found by