Physics of auroral phenomena : proceedings of the 36th Annual seminar, Apatity, 26 February – 01 March, 2013 / [ed. board: A. G. Yahnin, A. A. Mochalov]. - Апатиты : Издательство Кольского научного центра РАН, 2013. - 215 с. : ил., табл.

G.F. Remenets and A.M. Astafiev Solution of the VLF inverse problem of first kind The results of the solution of the VLF inverse problem of first kind for the event on May 13, 1987 are presented at Fig.2. It was gotten by the self-consistent VLF method [1-5], in which were used a diffraction ray (Watson-Fock wave) and two rays reflecting from the ionized middle atmosphere [8]. UT time --- 1----1----1— 17:10 17:30 UT time 18 30 £ 0.6 Fig. 2 Variations of the effective height h(t) (left) and the reflection coefficient modular R(t) (right) o f a wave-guide, as functions of universal time (UT) for the auroral radio-pass Aldra- Apatity. The first part of the disturbance (before 17:00) was gotten by an analysis at positive direction o f time and with the determination of the initial values h(tu) and R(t0), where t0= 16:30 UT. The second part of the disturbance (after 17:00) was gotten by an analysis at negative direction of time and with the determination of the initial values for analysis h ( t j and R (tJ , where tm= 18:30 UT. The values of breaks of the continuous curves in 17.00 UT on Fig. 2 indicate on the error order of the method of analysis used. For the effective height it is about 3 km, maximum variation being 27 km. For the reflection coefficient the estimation is about 0.04, maximum variation being 0.18. These results are quite satisfactory, if to consider that at the maximum of disturbance (UT = 17:00) the signal/noise ratio was about several units for the amplitudes. The error of the phases measurements was ± 0.5 mks). Solution of the VLF inverse problem of second kind Let us imagine a model of a wave-guide with length S2, which consists of two parts. The southern part of the wave­ guide with length S2 - Daw was modeled with the help of the middle latitude ionosphere and the northern part with length Davr which did not depend on time. The latitude of the pointed boundary was chosen equal to 62° N. To this undisturbed model a third section with variable length D (D = 0 + S2) was introduced in the vicinity of the receiver. This North terminal part of the wave-guide homogeneous at every time moment was characterized by the changes in time of the electric properties of the sporadic D-layer of conductivity, which the VLF inverse problem of first kind had given, Fig 2. The calculations of the relative amplitude and phase variation (-<pcc;f ) were fulfilled according to the following approximate formulas for: £'cai<r(tn.D ) = £ '« e x p h n (v )(S 2 - Dair) Im (vavr)(Davr - D) * exp R R R <Pcalc(tn-D ) = + G ll(D) = y 1f o f t n ) - Е«»с(*л)]г L , n = 0 ^ ( h ) 2 R v W u n=0 R <Pcalc (in )]2 ^caic ап^ (• <Pcaic ) are ^ e amplitude and the phase calculated of a signal with frequency f 4 (long radio- pass) for a moment of disturbance t„ and a fixed length D; v -is an eigenvalue with zero number for a model of undisturbed middle latitude wave-guide; -is analogous one for an undisturbed part of auroral wave-guide; i’djjr ( tn) -is an eigenvalue for a disturbed part of the wave-guide at a moment t n of the disturbance; R -is the Earth radius. The last eigenvalues were calculated according to the generalized Schumann’s method [9]. G ll(D ) is a discrepancy function of parameter D, which was minimized relative to the value of D. The minimum was achieved at the relative distance D / S2 = (45 ± 5) % and for the latitudes (62 ± 1)° N correspondingly, Fig. 3, right. A discrepancy- function G ll(D), which contained only phase data, was minimized too, and the corresponding result was practically the same: D / S2= (46 ± 5)%; (61.7 ± 0.8)° N. Continuation of the calculations for a function-discrepancy GII(D), which contained only the amplitude data (Fig. 3, left), gave the following result: D / S2 = (42 ± 6)%; (62 ± 1)° N. It concludes, that 164