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FIGURE 6.3. Linear and exponential height dependence of the refractive index.
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The refractivity thus has nearly a constant gradient of about 43 N-units per km. If so, the curvature of the ray trajectory is constant (this follows from (6.3) for dn=dh const:). A common way to take this factor into account is to introduce, instead of the actual Earth s radius, the effective Earth s radius [1 9,30]: Reff kRe 6:7
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where Re 6375 km, and k is the Earth radius factor. As was shown in References [1,2], the large values of the k-factor facilitate the propagation over long paths and small values may cause obstruction fading. In order to predict such fading, the statistics of the low values of the k-factor have to be known. However, since the instantaneous behavior of the k-factor differs at various points along a given path, an effective k-factor for the path, ke , should be considered. In general, ke represents a spatial average and the distribution of ke shows less variability than that derived from point-to-point meteorological measurements. The variability decreases with increasing distance. The effective factor is given by [1,2,30] ke 1 Re Re dn Re 1 1 r dh 6:8
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As the variation of refractive index is mostly vertical, rays launched and received with the relatively high elevation angles usually used in xed satellite communication links (see 14) will be mostly unaffected. But for the near horizontal rays, where r % 106 =g we obtain Reff ke Re where now the effective earth-radius factor is ke 1 10 6 gRe 1 Another form of this relation reads ke 0:157 0:157 g 6:11b 6:11a 6:10 6:9
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For the standard atmosphere and in limits of a linear model (g 3:925 10 2 1=km) one can immediately obtain from (6.11b) ke 4=3, so the effective radius from (6.10) is about 8500 km. Although the linear model leads to an excessive ray bending at high altitudes, this is not that important in our calculations, because the critical part of the trajectory is located near the ground antenna.
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Usually, it is assumed that for the radio path with length d greater than about 20 km, the standard deviation of the effective gradient, ge , tends to the normal distribution with the mean value g0 as in standard atmosphere and rms deviation (see de nitions of statistical parameters and distribution functions in 1): s0 se % p 1 d=d0 6:12
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where d0 % 13:5 km for European climate conditions. Estimations show that for a radio path of length d 150 km we have se % 0:3 s0 , and for d 350 km we have se % 0:2 s0 . The reasonable estimate of s0 is s0 % 0:04. In 1, the Gaussian probability density function (PDF) was introduced, the cumulative distribution function F x of which can be presented by the error function (erf) in the following manner [30,33,34]:  ! 1 x m p F x 1 erf 2 s 2
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where the error function is de ned as
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Then, the characteristic Q-function of the normal distribution is given by Q p 2 erfinv 2t 1 6:14
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where erfinv x is the inverse error function, and t is the time availability expressed in relative units (if t is in percentage there is a need to divide this value by 100%). Thus, for the 95% time availability we get Q 1:64 (see References [30,33,34]) and ge % g0 1:64 se 6:15
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Therefore, ge % 0:020 (ke % 1:14) for d 150 km, and ge % 0:027 (ke % 1:21) for d 350 km. For the 99% time availability we get Q 2:33 (see [30,33,34]) and ge % g0 2:33 se 6:16
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That leads to a ge % 0:012 (ke % 1:08) for d 150 km, and a ge % 0:021 (ke % 1:15) for d 350 km. We can see that for the real model of the spherical layered troposphere, the median value of ke differs from 4=3, which follows from the linear model of the re ectivity pro le.