## 2 Methodology  
The aim of this study was to investigate the impacts of rain height variability on XPD at some selected frequencies in Pretoria. Seasonal rain heights reported by [PERSON] et al., [13] in Table 1 were used to compute rain-induced attenuations. The procedure for calculating rain-induced attenuation, which is the major component of CPA, was followed as provided by the most recent ITU-R P618-14 recommendations [23]. The seasonal rain-induced attenuations \(A_{p}\) and CPA (CPA) for various percentages of time \(p\) at 11.7 GHz, 15 GHz, 20 GHz, 25 GHz, 30 GHz, and 35 GHz were computed using Pretoria seasonal rain heights and other parameters presented in Table 1[13]. The recommended seasonal rain heights used for this study were reported in the recent study conducted by [PERSON] et al. [13] in Pretoria. Frequencies above 10 GHz were selected for the study because the dual polarization method often applied in the frequency reuse technique is usually threatened by cross-polarization effects at these higher frequencies. The 5 GHz interval was adopted to reveal the effects of XPD due to gradual increase in frequency.  
The procedure for the computation of XPD from rain-induced attenuation statistics for earth-space link operating with look-up angle less than 60\({}^{\circ}\) is as follows [23].  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Input Values** \\ \hline Ground station latitude & 25.542\({}^{\circ}\) S \\ Antenna elevation & 31.5\({}^{\circ}\) \\ Antenna height above sea level (m) & 1346 \\
1-min rain rate (mm/hr) at 0.01\% exceedance & 84.5 \\ Autumn rain height (m) & 4732 \\ Winter rain height (m) & 4308 \\ Spring rain height (m) & 4853 \\ Summer rain height (m) & 5217 \\ ITU annual mean rain height & 4540 \\ Satellite name & IS-20 \\ Polarization & Circular \\ Satellite position & 68.5\({}^{\circ}\) E (GEO) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The Earth station and satellite parameters used for computation of rain-induced attenuation [13; 30].  
Step 1: Calculate the frequency-dependent term \(C_{f}\) using Equation (1):  
\[C_{f}=\begin{cases}60\text{log}f-28.3&6\leq f<\text{ 9 GHz}\\ 26\text{log}f+4.1&9\leq f<\text{ 36 GHz}\end{cases} \tag{1}\]  
where \(f\) is the frequency in GHz.  
Step 2: Calculate the rain attenuation-dependent term \(C_{A}\):  
\[C_{A}=V(f)logA_{p} \tag{2}\]  
where  
\[V(f)=\begin{cases}30.8f^{-0.21}&6\leq f<\text{ 9 GHz}\\ 12.8f^{0.19}&9\leq f<\text{ 20 GHz}\\ 22.6&20\leq f<\text{ 40 GHz}\end{cases} \tag{3}\]  
\(A_{p}\) is the rain-induced attenuation statistics for \(p\) percentage of time.  
Step 3: Calculate the polarization improvement factor (PIF):  
\[C_{\tau}=-10\text{log}[1-0.484(1+\text{cos}4\tau)] \tag{4}\]  
The improvement factor \(C_{\tau}=0\) for \(\tau=45^{\circ}\) for circular polarization and reaches a maximum value of 15 dB for \(\tau=0^{\circ}\) or \(90^{\circ}\).  
Step 4: Calculate the elevation angle-dependent term \(C_{\theta}\):  
\[C_{\theta}=-40\text{log}(cos\theta)\qquad for\ \theta<60^{\circ} \tag{5}\]  
where station is the earth station look-up angle also known as path elevation angle.  
Step 5: Calculate the canting angle dependent term \(C_{\sigma}\):  
\[C_{\sigma}=0.0053\sigma^{2} \tag{6}\]  
where \(\sigma\) is the effective standard deviation of the raindrop canting angle distribution, expressed in degrees and it takes the values \(0^{\circ}\), \(5^{\circ}\), \(10^{\circ}\), \(15^{\circ}\), and \(20^{\circ}\) for 1%, 0.1%, 0.01%, 0.001%, and 0.0001% of the time, respectively.  
Step 6: Calculate \(XPD_{min}\) due to rain not exceeded for p% of the time:  
\[XPD_{rain}=C_{f}-C_{A}+C_{\tau}+C_{\theta}+C_{\sigma} \tag{7}\]  
Analysis of the results obtained are presented and discussed in the subsequent section.