# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
## 3 Methodology for the Elevation Angle Characterization
### Elevation Angle Time Series Analysis  
Each elevation angle data set (obtained from the simulations) was individually analyzed to observe its statistical values of interest (mean, median, standard deviation, quantiles). Those time series were fitted to several distributions using the maximum likelihood estimation method. The goodness of fit of the proposed distributions was evaluated for each elevation angle time series using the Kolmogorov-Smirnov test, and it was observed a better performance of the gamma distribution, \(Gamma(a,b)\), for most of the data sets. Other distributions that showed good fit based on the Kolmogorov-Smirnov test were the beta, \(B(a,b)\), and Weibull, \(Weib(a,b)\). Figure 7 shows the values of \(\theta\) as well as the PDF and CDF for one elevation angle time series, \(\theta(t)\) using the proposed distributions. Figure 8 shows the Kolmogorov-Smirnov statistic, often indicated as \(D\), for the time series obtained for different altitudes. The Kolmogorov-Smirnov statistic is an indicator of the maximum distance between the actual time series distribution and the proposed gamma distribution. A low value of the Kolmogorov-Smirnov is often desirable to indicate a better goodness-of-fit.  
The gamma PDF, \(f_{\Theta}(\theta)\), and CDF, \(F_{\Theta}(\theta)\), are defined as follows  
\[f_{\Theta}(\theta)=\frac{1}{b^{a}\Gamma(a)}\theta^{a-1}\exp(-\theta/b), \tag{15}\]  
and  
\[F_{\Theta}(\theta)=\frac{1}{\Gamma(a)}\gamma\left(a,\frac{\theta}{b}\right), \tag{16}\]  
respectively, where \(\Gamma(\cdot)\) is the gamma function, \(\gamma(\cdot)\) is the incomplete gamma function, \(a\) is the shape parameter, and \(b\) is the scale parameter. By definition both \(a\) and \(b\) are greater than zero.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Latitude** & **Longitude** & **Latitude** & **Longitude** \\ \hline ES\({}_{1}\) & 0\({}^{\circ}\) & 276.7121\({}^{\circ}\) & ES\({}_{10}\) & 45\({}^{\circ}\) & 259.7121\({}^{\circ}\) \\ ES\({}_{2}\) & 5\({}^{\circ}\) & 276.7121\({}^{\circ}\) & ES\({}_{11}\) & 50\({}^{\circ}\) & 259.7121\({}^{\circ}\) \\ ES\({}_{3}\) & 10\({}^{\circ}\) & 276.7121\({}^{\circ}\) & ES\({}_{12}\) & 55\({}^{\circ}\) & 259.7121\({}^{\circ}\) \\ ES\({}_{4}\) & 15\({}^{\circ}\) & 267.7121\({}^{\circ}\) & ES\({}_{13}\) & 60\({}^{\circ}\) & 259.7121\({}^{\circ}\) \\ ES\({}_{5}\) & 20\({}^{\circ}\) & 276.7121\({}^{\circ}\) & ES\({}_{14}\) & 65\({}^{\circ}\) & 259.7121\({}^{\circ}\) \\ ES\({}_{6}\) & 25\({}^{\circ}\) & 261.7121\({}^{\circ}\) & ES\({}_{15}\) & 70\({}^{\circ}\) & 265.7121\({}^{\circ}\) \\ ES\({}_{7}\) & 30\({}^{\circ}\) & 259.7121\({}^{\circ}\) & ES\({}_{16}\) & 75\({}^{\circ}\) & 265.7121\({}^{\circ}\) \\ ES\({}_{8}\) & 35\({}^{\circ}\) & 259.7121\({}^{\circ}\) & ES\({}_{17}\) & 80\({}^{\circ}\) & 265.7121\({}^{\circ}\) \\ ES\({}_{9}\) & 40\({}^{\circ}\) & 259.7121\({}^{\circ}\) & ES\({}_{18}\) & -85\({}^{\circ}\) & 259.7121\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: ES locations for the performed simulations specified through their latitude, \(\phi\), and longitude, \(\lambda\).  
Figure 7: (**a**) Histogram and (**b**) theoretical densities for one of the elevation angle data sets obtained from the performed simulations.  
The shape, \(a\), and scale, \(b\), parameters for the gamma distribution were obtained for each of the elevation angle time series of each ES. Those parameters were organized into several matrix arrays as shown in Table 4, which contains the shape parameter for the simulated orbits as seen from the ES\({}_{1}\), and as in Table 5, which contains the scale parameter also for ES\({}_{1}\).