# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
## 2 Materials and Methods
### Relevance of the Elevation Angle for the Satellite Channel  
LEO satellites differ to those at GEO in its relative position as seen from a fixed ES. Whereas GEO satellites appear as fixed, those at LEO appear as always-moving points in the sky or are absent causing link unavailability. Link availability can be seen as an ON-OFF process, where the link is ON if the satellite is visible in the sky, and it is OFF if the satellite is absent. Figure 3b shows the ON-OFF process for both a LEO and GEO satellite. In addition to the short visibility that LEO satellites have, conditions in this lapse are always varying as a consequence of the changing position of the satellite. When the satellite is first visible from an ES, it is at the largest distance supported by the link, then, the satellite starts approaching the ES until reaching the shortest distance in that particular contact. Finally, the satellite moves away after being at its minimum contact distance. It is important to note that the minimum distance from an ES to a LEO satellite, \(\min(r_{E,S})\), varies from one contact to another, as well as \(\max(r_{E,S})\) does. Differences between the Earth's rotation rate and LEO orbiting velocities cause an always-varying link length (as seen from a fixed ES), and then, an always-varying elevation angle. An important relation that needs to be accounted for the elevation angle, \(\theta\), and the link length, \(r_{S,E}\) is given by the implication that when the angle is minimum, the \(r_{S,E}\) distance is maximum, i.e.,  
\[\theta\rightarrow\min(\theta)\Longleftrightarrow r_{S,E}\rightarrow\max(r_ {S,E}), \tag{2}\]  
similarly  
\[\theta\rightarrow\max(\theta)\Longleftrightarrow r_{S,E}\rightarrow\min(r_ {S,E}); \tag{3}\]  
where the maximum link length, \(\max(r_{S,E})\), occurs at some value of \(\theta\) such that \(\min(\theta)<\theta\leq 90^{\circ}\); and \(\min(r_{S,E})\) occurs at some value of \(\theta\geq 0^{\circ}\) lower than \(\theta_{max}\). The implications in (2) indicate that when values of \(\theta\) diminish, \(r_{S,E}\) increases; similarly, in (3), when values of \(\theta\) increase then \(r_{S,E}\) decreases. As a satellite moves away from an ES, \(r_{S,E}\) and the free-space path loss, \(L_{FS}\), increases. Longer paths occur for lower values of \(\theta\), then, the atmospheric attenuation, \(A_{Atm}\), increases for larger paths (assuming similar atmospheric conditions for different link paths). On the other hand, when \(\theta\) takes values close to \(90^{\circ}\), \(r_{S,E}\) will be at its minimum, and so does \(A_{Atm}\). Then, low elevation angles cause a greater \(L_{FS}\), and more atmospheric attenuation, \(A_{atm}\) (assuming that the atmospheric conditions are approximately the same for two distinct paths). For LEO it is important to consider link interference, \(I\), usually addressed as noise in link-budget calculations, and also dependent on the elevation angle as mentioned in [13]. Furthermore, low values of \(\theta\) are also associated in practice with non-line-of-sight (NLOS) conditions, increasing ground interference [14], and increasing multipath fading in the land mobile satellite (LMS) channel [15; 16; 17]. Figure 3: LEO characteristics: (**a**) Graphical representation of the always-varying elevation angle, \(\theta\), and (**b**) an On-Off model showing the link availability for a LEO satellite vs a GEO satellite during a random day. There are several models available in the literature to characterize the received signal from satellite systems. Most of these works have been elaborated for the LMS channel, which describes the received signal at a land-moving ES; nonetheless, most of those works focus on GEO systems. Extensive reviews for the LMS channel can be found at [16; 18; 19]. Among the available channel models for LMS systems, a few of them focus on characterizations for LEO and non-GEO (NGEO) satellites; but, as noted by [4; 20], those are few and there are still many challenges.