## 4 Results and Discussions
### Discussions  
The maximum transmit power (\(P_{t}\)), gains of transmit and receive antennas (\(G_{t}\), \(G_{r}\)), and antenna beam widths (\(\omega\), \(\phi\)) are the system design parameters that are mostly fixed for a given tropospheric link. The outer scale of turbulence (\(L_{0}\)) and turbulence wave number (\(\kappa\)) are considered random numbers due to the time-varying nature of atmospheric turbulence. The calculation of received power for any beam elevation angle involves multiple iterations with a randomly selected value of \(L_{0}\) and \(\kappa\) for each iteration and averaging over all of  
Figure 3: Plots of power received in the lowest elevation angle beam set and the highest power received amongst all 11 beam sets for various days of the year 2006 for the Ahmedabad link. the iterations. Therefore, the effects of \(L_{0}\) and \(\kappa\) on the received power are the same for any beam elevation angle. The parameters in the expression for received power given in Equation (9) that actually affect the powers received for different sets of transmit and receive beam elevation angles are the vertical gradient of refractive index (\(\delta n/\delta z\)), sub volume distances (\(R_{t},\ R_{r}\)), and scatter angle (\(\psi\)). \(R_{t}\), \(R_{r}\), and \(\psi\) have the effect of decreasing the received power with an increase in the elevation angle. Hence, for the received power in any elevated beam to be higher than the lowest elevation beam, \(\delta n/\delta z\) in the common volume enclosed by the elevated beam should be large enough to deliver a higher power in spite of increased losses due to elevation. Observations made on the Ahmedabad link for morning time of 5 June 2006 are discussed for a clear understanding of the reason behind the elevated beams delivering maximum received power. On 5 June 2006 at 0530 IST, for the Ahmedabad link, the maximum power was received in the beam set 8 and was 3.78 dB more than the power received in the lowest (first) beam set, as shown in the highlighted portion of Figure 4. In order to understand the reason behind this high power in an elevated beam, the variation of \((dn/dz)^{2}\) with height on 5 June 2006 at 0530 IST is shown in Figure 4. Taking the average values of \(\psi\), \(R_{t}\), and \(R_{r}\) for different beam sets, it was seen that these parameters induce an additional loss of 6 dB for beam set 8 when compared to lowest beam set. However, as seen in Figure 4, \(\left(dn/dz\right)^{2}\) has a distinct peak at around 1935 m along with other smaller peaks that fall in the height range of the common volume formed by the beam set 8 and are above the common volume height range of the lowest beam set. The average value of \(\left(\delta n/\delta z\right)^{2}\) for beam set 8 was found to be approximately 9.5 times higher than for the lowest beam set, which gives a gain of 9.78 dB in the power received for beam set 8. Hence, as a result, the received power for beam 8 was 3.78 dB higher than the lowest beam. The main reason for this peak in \(\left(\delta n/\delta z\right)^{2}\) is a sharp inversion in the vertical gradient of the atmospheric refractive index. It was observed that \(n\) decreases gradually from Earth's surface up to a height of 1935 m and then suddenly increases between 1935 m and 1983 m. This inversion in vertical gradient of \(n\) is associated with a variation in the atmospheric vapor pressure, which was found to increase from 16.5 mb at 1935 m 20.13 mb at 1980 m. Smaller peaks observed after the main peak in Figure 4 are due to a sharp decrease in the refractive index in height ranges of 2115-2252 m and 2539-2855 m, attributable to a sharp fall in the vapor pressure in both these height ranges. Thereafter, \(n\) decreases gradually up to an observed height of 3332 m. Both a sharp increase and a sharp decrease in the vapor pressure result in a peak in \(\left(\delta n/\delta z\right)^{2}\). However, increase in the vapor pressure cause an inversion in the slope of vapor pressure with height.