4. Inverse Problem

[20]  The quantitative analysis in our case is the solving of the inverse LF problem in the set of the unmonotonous electron concentration profiles [Remenets, 1994, 1997]:

(1)

where b=-0.14 is the increment of the electron collision frequency profile n_eff(z)

(2)

The profiles (1) with b< 0 are the rough approximations to two layers of the ionized atmosphere. The top layer ( z > z₀ ) is a normal low ionosphere and the bottom layer ( z < z₀ ) is a sporadic layer, which exists because of the anomalous ionization. When z₀ is equal to z₁ and b> 0, the set of profiles (1) turns to the set of monotonous profiles which are suitable to the solving of LF inverse problems for all kind of LF disturbances at the polar zone except HEREP and UrEP disturbances [Remenets, 1994, 1997]. The attractiveness of such set of monotonous N_e(z) is that for them the point junction z₀ is approximately equal (with the accuracy of 1-1.5 km) to the effective height h' [Remenets, 1997] of the radio waveguide.

[21]  The quantitative analysis relative to the PwD is connected with the following evident apparatus difficulty: a conversion into "zero" of one or more amplitude values makes the input data for the inverse problem indefinite, and the corresponding quantitative analysis becomes an estimation one [Beloglazov et al., 1998]. However, this "zero" fact is a very strong fact in the physical sense, as we have mentioned above. The fact may have only one meaning that the ground wave is compensated by the ionospheric wave. According to the calculations of the optical paths of the waves this compensation is an interference minimum of zero order in the curved ground-ionized layer with an effective height near 25 km. So this extremely low altitude of the atmospheric ionized layer should be considered as an absolutely reliable experimental fact just as the oscillation character of the electromagnetic field as a function of distance from a ground radio source is an experimental proof of the existence of the ionosphere.

[22]  Some of the results of the LF inverse problem solution are presented in Table 4 for three PwDs at the time moments of their maximum amplitude and phase variation for all three radio signals ( i=1, 2, and 3 for f=10.2, 12.1, and 13.6 kHz, respectively). One event was analyzed for two "undisturbed" initial monotonous profiles N_e(z) [Remenets, 1997] with z₀=55, 62 km (profiles 1 and 2) for daytime and two events with z₀=70, 75 km (profiles 3, 4, 5, and 6) for the nights.

[23]  In Table 4, z₁ and b are the parameters of the effective unmonotonous electron concentration profile N_e(z) (1). Their values have been found by the minimization of a functional G [Remenets, 1997] containing the differences between the experimental and theoretical values of relative amplitude changes and phase changes of the radio signals for three frequencies. The parameter z₁ according to (1) is a bottom level of a layer with homogeneous electric conductivity at the range z₀-z₁ km. The parameter b is a height increment of the bottom part of the effective profile (1). If it is negative, the corresponding profile N_e(z) is unmonotonous.

[24]  The parameter h' is an effective height of the waveguide formed by the Earth and the ionized middle atmosphere. The calculation of the parameter h' is not necessary for solving the inverse problem, but it is methodologically useful because as it is shown in the scientific literature, this effective height is located inside the so-called essential region of the electric conductivity layer relative to the radio wave reflection and insignificantly differs from another effective height h [Beloglazov and Remenets, 1982; Remenets, 1997; Remenets and Beloglazov, 1992].

[25]  The experimental amplitude A_i and phase j_i data in comparison with the calculated ones (the values in the parentheses) are given in the Table 4. The numerical results of the Table 4 are part of the Nemirov's thesis [Beloglazov et al., 2000]. The calculated data are the inverse problem solution results. Indexes "c" and "d" indicate the "calm" (before a disturbance) and disturbed conditions on a radio trace.

[26]  In Table 4 the profile parameters obtained have estimation character. In the minimization of the functional G the "experimental" amplitudes A_i were set equal to the noise level. For the 27 March 1988 case it has been deduced from the experimental values of the amplitudes before a UrEP that the ionosphere was not quiet, that it was disturbed by auroral particles. So it was necessary to realize the calculations with a model of an initial ("calm") ionosphere ( b=0.05 km^-1 ) for which the effective height ( z₀=55 km) is lower than for the initial undisturbed ionosphere ( z₀=62 km). The result turned to be positive in the sense that the value of the functional G( z₀ ) has diminished by 3.5.

[27]  In Table 4, profiles 3-6 correspond to the night radiotrace conditions. According to our estimation [Remenets and Beloglazov, 1985] the auroral night effective height h on quiet and moderately disturbed conditions is equal to 75 and 70 km, respectively. The ratio of the functional values G(z₀=70 km)/ G(z₀=75 km) was equal to 2.0 and 0.45 for 21 January 1992 and 22 January 1992 data, respectively. So we would accept that the effective height h before the disturbances was near 75 and 70 km, respectively. The calculations for profiles 3-6 were performed with b=0.39 km for the bottom part of the initial ("calm") monotonous night profile N_e(z) as a representation of (1) with z₁=z₀.

[28]  We finish this section devoted to the inverse problem with a preliminary result in Table 5. In Table 5 the ranges of values of the parameters h' and b are indicated for four types of anomalous LF disturbances due to the inverse problem solution (20-30 cases).