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Refractive Index or Refractivity. The radio properties of the quasi homogeneous layered troposphere are characterized by the refractive index n, related to the dielecp tric permittivity of the air, er , as n er . The refractive index n of the Earth s atmosphere is slightly greater than 1, with a typical value at the Earth s surface of around 1.0003. Since the value is so close to unity, it is common to express the refractive index in N-units, usually called refractivity [1 3,30], which is the difference between the actual value of the refractive index and unit in parts per million: N n 1 106 6:1
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Thus, at the ground surface the refractivity equals N NS % 315 N-units. In a real atmosphere, refractivity N varies with gas pressure and temperature and with water vapor pressure in the atmosphere. The variations of temperature, pressure, and humidity from point to point within the troposphere cause the variations of the refractivity N, which can be calculated according to the semiempiric Debye formula [1 9,30] N 77:6 pa 4810 pw =T T 6:2
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where T is the absolute temperature in Kelvin [K], pa is the atmospheric pressure in millibars [mb], and pw is the water vapor pressure in millibars [mb]. There are seasonal and daily variations of the refractivity measured at the surface of the ground, N0 . More important is the decrease of the refractive index with height. Usually, we can neglect the horizontal variations of N and consider the troposphere as a quasihomogeneous spherically layered medium. If so, the dominant variation of N is vertical with height above the Earth s surface: N reduces towards zero (n becomes close to unity) as the height is increased. The variation is approximately exponential within the rst few tens of kilometers of the Earth s atmosphere, that is, this region is called the troposphere [1,2,30]: & ' h 6:3 N NS exp H where h is the height above sea level, and NS % 315 and H 7:35 km are standard reference values; H is de ned as the height scale of the standard atmosphere. Equation (6.3) is called the standard exponential model of the troposphere. Tropospheric Refraction. The refractive index variations with height cause the phase velocity of radio waves to be slightly slower and closer to the Earth s surface, such that the ray paths are not straight but tend to curve slightly towards the ground. In other words, the elevation angle a1 of the initial ray at any arbitrary point (see Fig. 6.2) is changed after refraction at angle a2 . The same situation will be at the next virtual layer of atmosphere with other refractive index n. Finally, the ray launched
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from the Earth s surface propagates over the curve, whose radius of curvature, r, at any point, is given in terms of the rate of change of n with height [1,2,30]:   cos a1 dn 1 r n dh 6:4
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As a result, a ray passing through the troposphere, instead of the apparent direction, propagates in a direction far from that towards the satellite. The resulting ray curvature is illustrated in Figure 6.1. The gradient of the refractivity is given by g h dN=dh Usually it is assumed [1 9] that near the Earth s surface this gradient varies exponentially as gs h 0:04 exp 0:136 h ; km 1 6:5
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Linear approximation. According to (6.5), the gradient depends nonlinearly with height. However, in the rst approximation we can use the linear model, setting the gradient as a constant equal to its value at h 0: g g 0 . This occurs for small heights, when the standard atmosphere in (6.3) can be approximated as linear, as shown in Figure 6.3, and according to the following equation [1 9,30]: N % NS NS h H 6:6
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