# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
## 3 Methodology for the Elevation Angle Characterization
### PDF and CDF of the Elevation Angle  
In addition to \(h_{s}\), we can define two additional sets containing the orbit inclinations (as in Table 2) and ES's latitudes (as in Table 3) at which simulations were performed, as  
\[i_{s}=\{20^{\circ},25^{\circ},\ldots,85^{\circ}\} \tag{20}\]  
and  
\[\phi_{s}=\{0^{\circ},5^{\circ},\ldots,85^{\circ}\}, \tag{21}\]  
respectively.  
Depending on the input values of \(\phi_{k}\), \(h_{k}\), and \(i_{k}\), if those are part of the previously defined sets (\(i_{s}\), \(h_{s}\), \(\phi_{s}\)) eight cases can arise as follows:  
* Case 1. \(  
The shape and scale parameters for the case study are shown in Table 9. First, parameters for the four-points (as in Figure 11b) are listed for each altitude (1400 km and 1600 km). Then, the resultant parameters obtained from interpolation were computed. Finally, the shape and scale parameters for the query altitude can be obtained using a weighted arithmetic mean were the weights can be obtained from the distance of the query altitude to the closest characterized orbit altitudes. In this case, the distances from the query altitude to the closest characterized altitudes are the same, then we obtained the shape and scale for the query altitude using the arithmetic mean of the shape and scale values at 1400 km and 1600 km.  
Both the elevation angle CDF obtained with the proposed methodology and the empirical CDF are shown in Figure 11b. The distance magnitude between those two curves is also shown in Figure 11b and labeled as absolute error (\(|\)Error\(|\)). The distance (or error) between the gamma CDF and the empirical CDF is very small, and its maximum value is around 0.025 (2.5%).  
Figure 11: Example case results for \(\phi_{ES}=22^{\circ}\), \(i_{SAT}=43^{\circ}\) and \(h_{k}=1500\) km: (**a**) four-point mesh to calculate the shape, \(a\), and scale, \(b\) parameters of \(f_{\theta}(\theta)\) and (**b**) empirical CDF vs. CDF obtained with the proposed methodology.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \multicolumn{6}{c}{**Four-Point Mesh Setup**} \\ \hline  & 1400 & 1600 & & 1400 & 1600 \\ \hline \(a(\phi_{1},i_{1})\) & 1.9515 & 1.9552 & \(b(\phi_{1},i_{1})\) & 9.3194 \(\times 10^{-2}\) & 8.8794 \(\times 10^{-2}\) \\ \(a(\phi_{1},i_{2})\) & 1.9515 & 1.9552 & \(b(\phi_{1},i_{2})\) & 9.3194 \(\times 10^{-2}\) & 8.8794 \(\times 10^{-2}\) \\ \(a(\phi_{2},i_{1})\) & 1.9515 & 1.9552 & \(b(\phi_{2},i_{1})\) & 9.3194 \(\times 10^{-2}\) & 8.8794 \(\times 10^{-2}\) \\ \(a(\phi_{2},i_{2})\) & 1.7462 & 1.7822 & \(b(\phi_{2},i_{2})\) & 9.4112 \(\times 10^{-2}\) & 9.2215 \(\times 10^{-2}\) \\ \hline \multicolumn{6}{c}{**Obtained parameters from interpolation**} \\ \hline \(a(\phi,i)\) & 1.8776 & 1.8929 & \(b(\phi,i)\) & 9.3529 \(\times 10^{-2}\) & 9.0025 \(\times 10^{-2}\) \\ \(a(\phi,i)\) & & 1.8853 & \(b(\phi,i)\) & 9.1777 \(\times 10^{-2}\) \\ \hline \multicolumn{6}{c}{**Empirical parameters**} \\ \hline \(a(\phi,i)\) & & 1.9804 & \(b(\phi,i)\) & 9.5987 \(\times 10^{-2}\) \\ \hline \end{tabular}
\end{table}
Table 9: Shape and scale parameters for the elevation angle PDF and CDF for \(\phi_{ES}=22^{\circ}\) and \(i_{SAT}=43^{\circ}\) at \(h_{k}=1500\) km, corresponding to the four-point mesh shown in Figure 11a.