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where E0 is the amplitude of the incident wave on the boundary of the layer; g and F are the logarithm of wave amplitude and the wave phase, respectively. By putting expression (7.61) in (7.60), the following system is obtained
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Let us suppose that k 1 and e1 are small, and present functions g and c 0 as perturbation sums (see perturbation method description in Sections 3.2 and 3.4) g g 0 g1 ; F F0 F1 7:63
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Then for the rst and second order approximation of (7.62) the wave phase can be given as [11,12]
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Parameter F0 de nes the phase of a nonperturbed wave and F1 de nes the disturbance of phase in the ionospheric layer with small-scale inhomogeneities. The function g1 in expansion (7.63) can be expressed as [11,12]
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The Cross-correlation Function of Phase Disturbances. Using expressions (7.63) (7.64) we can nd the cross-correlation function of phase disturbance F1
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of the radio wave re ected from the layer with thickness D
Z0 Z0
GF x; Z k0
0 Z 0 D
0 Z 0 D
dZ2 he1 X1 ; Y1 ; Z1 e1 X2 ; Y2 ; Z2 i e e
Z 0 =2
2 2k0 Ge x; Z;  d 0
7:66a D= Z Z0  2 =4 dZ
Z 0 D =2
where x X1 X2 ; Z Y1 Y2 ;  Z1 Z2 . The second integral in formula (7.66a) equals
Z 0 =2
Z 0 D =2
D= Z Z 0  2 =4 dZ Dfln D =2 D2  2 =4 1=2 ln =2 g % D ln 4D= 7:66b
For the following calculations, as in References, [21 27], we consider that the shape of the inhomogeneities of plasma density dN  N1 < N0, distributed according to the Gaussian law inside inhomogeneous ionospheric layer at the height Z0 , as & 2' & Ge x; Z; & Ge x; Z exp 2 l where l is the characteristic scale of jN1 j2 changing along the Z-axis. If   o4 pe Ge 0; 0; 0 e2 4 e1 o *  + N1 2   N  0 7:68 7:67
does not depend on Z, then the maximum of the cross-correlation function of the phase uctuations is GF 0; 0  F2 1 
2 p1=2 k0 lD
2 where C is the Euler constant [11]. Expression (7.69) takes into account that hN1 i is the same for all altitudes of the ionosphere. If relative uctuations of plasma density
   2 8D C e1 ln e l 2
hjN1 =N0 j2 i depend on the height, then he2 i e1 is the function of Z. In this case hF2 i 1
*  + N1 2 1 e0   e N  0
2 p1=2 k0 lD
From formula (7.71) it is seen that the main contribution to the phase uctuations of the radio wave are determined by the inhomogeneities placed near the re ected level 0 e Z0 , where e0 $ 0. If the following form of spectrum of plasma density uctuations U N K or of dielectric permeability uctuations U e K is used, then U N K $ U e K M e 1 K 2 K 2 K 2 L2 =4p2 p=2 X Y Z 0 7:72
*  +   N1 2   ln 8D C 3 N  l 2 0
where L0 is an external (outer) scale of inhomogeneities and M e is determined by the condition he1 i U e K dKX dKY dKZ , then the correlation function of phase e2 uctuations can be presented as
Z0 2 GF x; Z 2k0 DMe
In Reference [23] it was pointed out that the formula (7.73) could not be analytically integrated. Moreover, for the case L0 ( D calculations of the spectrum of phase uctuations F1 have shown that spectrum UF1 does not reproduce spectrum UN . The Thin Screen (Kirchhoff) Approximation. Now, using the formulas presented above, the question of phase and amplitude uctuations for the case of low-orbit satellite communication can be investigated in more detail, using the thin screen approximation method [33,40 41] presented schematically in Fig. 7.8. Let us suppose that the satellite trajectory and the receiving point on the Earth s surface are in the magnetic meridian plane. To simplify the problem, it is assumed that geometric eld lines are vertical at point O on the Earth s surface (see Fig. 7.9). The z-axis is directed along the magnetic eld lines and the x-axis lies in the meridian plane. The case o0 ) ope , a real case in satellite mobile communication, is now discussed. Due to diffraction and scattering effects at the small-scale inhomogeneities, radio waves from a satellite located at point P have a stochastic modulation of phase after passing through the ionospheric layer having thickness L. For the case of waves from VHF to X-band, that is, for l ( l, where l is the scale of inhomogeneity that is from a few meters up to a few centimeters and for weak