# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
## 3 Methodology for the Elevation Angle Characterization
### Orbit Coverage  
Figure 9a illustrates (without scale) passes of a satellite in each one of the six altitudes and for all the inclinations mentioned in Table 2. This figure also shows dashed lines indicating the ES's latitudes mentioned in Table 3.  
In Figure 9a as in Table 2 the orbit inclination parameter, \(i\), for each altitude goes from 20\({}^{\circ}\) to 85\({}^{\circ}\), then, this figure can be redrawn as the grid shown in Figure 9b, where the curved lines of Figure 9a are replaced by vertical lines indicating the orbit inclination. Additionally, the ES latitude lines were inverted to start with the lowest value at the top. From Figure 9a it can be observed that satellites with certain orbit inclination \(i\), reach (in its orbit trajectory) at most the latitude that coincide with the inclination value. For example, a satellite with \(i=20^{\circ}\) will not be able to pass over and ES located at \(\phi=80^{\circ}\), since the maximum reached latitude in that orbit will be \(\phi\approx i\approx 20^{\circ}\), then, that satellite will be best suited to provide coverage at ES's located at latitudes \(\phi\leq i\) or at most \(\phi\approx i\). This coincides with basic knowledge of satellite coverage; whereas polar orbits can cover almost all the globe, lower inclination orbits cover smaller portions of the Earth. Then, we can redraw Figure 9b to have a coverage grid by discarding points corresponding to cases of \(\phi_{ES}>i_{SAT}\), this resultant grid is shown in Figure 9c. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c} \hline
**ES\({}_{1}\)** & **20\({}^{\circ}\)** & **25\({}^{\circ}\)** & **30\({}^{\circ}\)** & **35\({}^{\circ}\)** & **40\({}^{\circ}\)** & **45\({}^{\circ}\)** & **50\({}^{\circ}\)** & **55\({}^{\circ}\)** & **60\({}^{\circ}\)** & **65\({}^{\circ}\)** & **70\({}^{\circ}\)** & **75\({}^{\circ}\)** & **80\({}^{\circ}\)** & **85\({}^{\circ}\)** \\ \hline
7378 km & 1.39 & 1.50 & 1.48 & 1.31 & 1.30 & 1.34 & 1.35 & 1.35 & 1.35 & 1.37 & 1.37 & 1.36 & 1.36 & 1.37 \\
7578 km & 1.62 & 1.79 & 1.80 & 1.65 & 1.56 & 1.61 & 1.62 & 1.63 & 1.63 & 1.64 & 1.64 & 1.64 & 1.64 & 1.64 \\
778 km & 1.57 & 1.77 & 1.82 & 1.73 & 1.55 & 1.61 & 1.63 & 1.64 & 1.64 & 1.65 & 1.65 & 1.66 & 1.66 & 1.66 \\
7978 km & 1.63 & 1.74 & 1.83 & 1.79 & 1.62 & 1.60 & 1.63 & 1.65 & 1.65 & 1.66 & 1.66 & 1.66 & 1.66 & 1.67 \\
8178 km & 1.70 & 1.66 & 1.83 & 1.82 & 1.70 & 1.55 & 1.63 & 1.65 & 1.66 & 1.67 & 1.67 & 1.67 & 1.67 & 1.67 \\
8378 km & 1.75 & 1.67 & 1.82 & 1.84 & 1.75 & 1.58 & 1.63 & 1.66 & 1.67 & 1.68 & 1.68 & 1.68 & 1.68 \\ \hline \end{tabular}
\end{table} 
Table 4: Rate parameter, \(a\), of the gamma distribution to characterize the elevation angle distribution at ES\({}_{1}\), for different altitudes, \(h\), and orbit inclinations, \(i\). Figure 8: Kolmogorov-Smirnov statistic for the performed simulations at different altitudes. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline
**ES\({}_{1}\)** & **20\({}^{\circ}\)** & **25\({}^{\circ}\)** & **30\({}^{\circ}\)** & **35\({}^{\circ}\)** & **40\({}^{\circ}\)** & **45\({}^{\circ}\)** & **50\({}^{\circ}\)** & **55\({}^{\circ}\)** & **60\({}^{\circ}\)** & **65\({}^{\circ}\)** & **70\({}^{\circ}\)** & **75\From the simulation results obtained for the grid points of Figure 9c, matrices \(A_{h}\) and \(B_{h}\) containing the shape and scale parameters of \(f_{\Theta}(\theta)\) and \(F_{\Theta}(\theta)\) can be defined as follows  
\[A_{h}= \begin{bmatrix}a_{1,1}&a_{1,2}&a_{1,3}&\dots&a_{1,14}\\ a_{2,1}&a_{2,2}&a_{2,3}&\ddots&a_{2,14}\\ a_{3,1}&a_{3,2}&a_{3,3}&\ddots&\vdots\\ a_{4,1}&a_{4,2}&a_{4,3}&\ddots&\vdots\\ a_{5,1}&a_{5,2}&a_{5,3}&\ddots&\vdots\\ NaN&a_{6,2}&a_{6,3}&\ddots&\vdots\\ NaN&NaN&a_{7,3}&\ddots&\vdots\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ NaN&NaN&NaN&\dots&a_{18,14}\end{bmatrix} ,\quad B_{h}=