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FIGURE 6.24. The total eld pattern consisting of the coherent part (Ico ) and incoherent part (Iinc ).
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propagating wave are due to random interference of a large number of diffraction scattering of the small eddy cells. Zero-inner-scale Model for Plane Wave. For the case of l0 ! 0, L0 ! 1, and 2 2 2 k Cn const, the plane-wave coherence radius is r0 1:46Cn px 3=5 , in both weak and strong uctuation regimes, whereas the Fresnel zone x=k de nes the correlation length in only weak irradiance uctuations. Cell sizes smaller than the Fresnel zone cause diffractive distortions of the wave, whereas those larger than the Fresnel zone cause refractive distortions such as focus and tilt. At the onset of strong uctuations, the coherence radius approaches the size of the Fresnel zone, and all three cell sizes are roughly equal (i.e., l1 $ l2 $ l3 ). This happens in the vicinity of the focusing regime. For conditions of stronger uctuations, the correlation length is de ned by the spatial coherence radius r0 , which is now smaller than the Fresnel zone, and the scattering disk L=kr0 is larger (i.e. l1 < l2 < l3 ). Let us consider the scintillation index of a plane radio wave that has propagated a distance x through unbounded turbulent atmosphere. (A) Weak uctuations. Under the weak- uctuation theory and the Rytov method, the scintillation index can be expressed in the form: s2 exp s2 I 1 s2 I ; I ln ln s2 ( 1 1 6:99
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where s2 I is the log-irradiance variance de ned under the Rytov approximation by ln
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Z 11=6 1 cosZx dZ dx 6:100
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In the last step, we have assumed a conventional Kolmogorov spectrum and 2 z introduced the nondimensional quantities: Z xk and x x. Performing the k integration above, we obtain the result: s2 s2 0:847 x 5=6 ; I 1 s2 ( 1 1 6:101
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(B) Moderate uctuations. At the other extreme, the asymptotic behavior of the scintillation index in the saturation regime is described by s2 1 I 0:86
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The resulting log-irradiance scintillation is s2 I ln
2 2 L 1 0 0
8p k
1:06 s2 1
1  2  7=6 1 L k x2 k4=3 exp 2 dZ dx 0:15s2 Z7=6 1 x kx k 0 0
 z k n k Gx k 1 cos k2 dk dz k
where Zx xk2 =k. x (C) Strong uctuations. In the case of strong turbulence regime, the scintillation index for a plane wave in the absence of inner scale is given by s2 exp I " 0:54s2 1
12=5 7=6
6:104 An example of signal intensity scintillation index computation according to (6.104) from 1 GHz to 50 GHz versus the refractive-index structure parameter varied from 10 13 to 10 11, for the distance x 10 km and the inner scale l0 0 mm, is shown 2 in Figure 6.25. It is clearly seen that the scintillation index for any Cn const: 2 12 by the vertical line) becomes twice as strong as the (denoted, e.g., for Cn 10 frequency increases from 20 GHz to 50 GHz. This result is very important for predicting the fast fading of the signal within land aircraft and land satellite radio communication links passing through the turbulent troposphere and operating at frequencies in the L/X-band (i.e., more than 1 GHz). Nonzero-inner-scale Model for Plane Wave. When inner-scale effects become important ( l0 6 0), the atmospheric power spectrum is more strictly described by a modi ed spectrum with high wave-number rise, that is, the traditional Tatarskii spectrum [44].
1 1:22s1
0:509s2 1 1 0:69s1
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s2 < 1 1
0.3 0.3 0.2 0.2 0.1 0.1 0.05 0 10 13
f = 1 GHz f = 10 GHz f = 20 GHz f = 30 GHz f = 40 GHz f = 50 GHz
50 30 40 GH GH GH z z z
1 GHz
10 12
2 Cn
10 11
FIGURE 6.25. Index of signal intensity scintillations versus the intensity of refractive index scintillations for different frequencies from 1 GHz to 50 GHz.
Under weak irradiance uctuations in the case of an unbounded plane wave, the scintillation index based on the modi ed spectrum is described for s2 < 1 by 1 &     11 1 1:507 3 tan Ql sin tan 1 Ql s2 x 3:86s2 1 Q 2 11=12 sin I 1 l 6 4 1 Q2 1=4 l # )   0:273 5 5=6 3:5Ql sin tan 1 Ql 6:105 2 7=24 4 1 Ql Here, Ql 10:89x=kl2 is a nondimensional inner-scale parameter. Asymptotic 0 expressions for the scintillation index in the saturation regime, which are based on the modi ed atmospheric spectrum, are s2 1 I 2:39