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The Fourier transform of Equation (7.83) gives the spectrum of amplitude oscillations Ug f $ Ue Kx0 ; Ky0 ; Kz0 dKy0 7:84
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Now, at point O, the angle i1 is the angle between the magnetic eld and ray OP (see Fig. 7.9). In the coordinate system fx0 ; g, with the -axis along the geomagnetic eld and with the base at point O, there are two cases (1) For the case i1 0 when the ray is parallel to the magnetic eld B0 lines, the disturbance of plasma density is averaged and the eld correlation function GE is determined [36,37] by the scales of the inhomogeneities perpendicular to B0 .
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(2) In the other critical case when i1 90 , the density oscillations are averaged in the direction transverse to the magnetic eld and the spectrum GE is expressed from the spectrum of the scales of the inhomogeneities elongated along B0 . One should note that expressions (7.74) and (7.77) were obtained for the smallscale inhomogeneities (l ( dF ). In real cases of satellite experiments a wide spectrum of scales are observed from centimeters up to kilometers. For a more general two-dimensional (2D) case, the spectrum of plasma density UN K of inhomogeneities was calculated by Shkarofsky [38]. At the same time the spectrum of amplitude scintillations, measured in satellite experiments, shows that for the 1 spatial frequencies L 1 ( K ( l0 the spectrum of plasma density UN K can be 0 presented in a simpler manner in one-dimensional (1D) case as [38] p 2 2 hN1 iG p 1 K0 1 p p 22 % K UN K 2 K 2 p 2 =2 pG 2 K0
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where l0 , L0 2p=K0 are the inner and outer scale of inhomogeneities, respectively; K 2p=l is the scale of inhomogeneity perpendicular to the geomagnetic eld B0 , and G w is the gamma-function. It was also obtained from satellite experiments that in the direction parallel to B0 for the longitudinal scales ljj > dF , the spectrum of inhomogeneities UN Kjj is Gaussian: n o 2 UN Kjj $ exp Kjj l2 jj 7:86
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The transformation from spectrum (7.85) to spectrum (7.86) was observed in satellite experiments at the angles i01 between the radio ray and the geomagnetic eld B0 : i01 $ tan 1   l ljj 7:87
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In many experiments it was shown that ionospheric inhomogeneities of the F-region, which give radio scintillations, are stretched along the geomagnetic eld ( l =ljj ( 1 and the angle i1 has a value of only several degrees. Thus, the spectrum of eld amplitude uctuations is determined by the spectrum of transversal scales of inhomogeneities. It should also be noted that for the metric and decimetric wave band the case l0 ( dF ( l occurs and more general formulas than (7.85) (7.86), such as (7.72), must be used. But analytical calculation for the more general case of anisotropic inhomogeneities and of oblique incidence of radio waves on the ionospheric layer is a very complicated mathematical problem. The analytical result was obtained only
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EFFECTS OF THE INHOMOGENEOUS IONOSPHERE ON RADIO PROPAGATION
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for the case when the wave was incident on the ionospheric layer perpendicular to its surface and when the magnetic eld lines were also perpendicular to the ionosphere surface. In this case, one can present the spectra of signal logarithmic amplitude g, Ug K , and signal phase S1 , US1 K , uctuations as  2 2 K dF US1 K 2  2 2 K d F 2 Ug K k0 LUe K sin2 2
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Expressions (7.88) show that for a wide spectrum of inhomogeneities the amplitude and phase spectra are determined by the spectra Ue K of inhomogeneities and by 2 2 2 2 lter functions cos2 K dF =2 and sin2 K dF =2 . The Scintillation Index. In an observation of the fading of radio signals passing the ionosphere, usually used a scintillation index [29 41], which is denoted by s2, I in the same manner as in 6, because it also determines the strength or power of inhomogeneities (how they are strong) inside the ionospheric layer, and therefore can be de ned by formula (6.97) from Section 6.3.2 as the dispersion of the radio wave intensity I uctuations. We will rewrite this formula for our future discussions: s2 I  2 I hIi2 hIi2   2 I hIi2 1 7:89
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For weak scintillations this index was obtained in References [29,39 41] s2 I  2 4 g 4 Ug K dKx dKy 7:90
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Here again, g is the logarithm for the amplitude of the radio signal. The above mentioned allows us to conclude that it the small-scale inhomogeneities exist, then two conditions are followed (see also Section 6.3): The inhomogeneity scale, l, is smaller than the rst Fresnel zone, dF . l is larger than the wavelength l. The in uence of small-scale inhomogeneities was mainly manifested in phenomena like diffraction and scattering [29 41]. However, the effect of small-scale inhomogeneities on wave propagation in the ionosphere is not well recognized. Although a lot of literature exists about large-scale inhomogeneities, the role of small-scale inhomogeneities seems to be less well understood. As the spectrum of small-scale inhomogeneities gives more complicated effects, in the next section we will focus mainly on the effects of these phenomena, will analyze them according to
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