# Propospheric Attenuation in GeoSurf Satellite Constellations
## 4 LEO Probability Distributions of Rain Attenuation  
In this section, we show the expected tropospheric attenuation affecting LEO satellites. To this end, we have considered, as an example, the Metop\(-\)C satellite (launched on 7 November 2018), which is part of the EUMETSAT Polar System Constellation [30], as seen from Spino d'Adda. The constellation altitude is 817 km, and the satellites provide data for both operational meteorology and climate studies. The instruments on board can observe the Earth day and night, as well as under cloudy conditions. The satellite visibility was calculated by means of the simplified general perturbation model, which allows propagating the satellite position and velocity along the orbit [31]. The input to the model is given by NORAD and also stored on the EUMETSAT website (see [[https://service.eumetsat.int/tle/](https://service.eumetsat.int/tle/)]([https://service.eumetsat.int/tle/](https://service.eumetsat.int/tle/)), accessed on 5 November 2021). As an example, Figure 8 shows the orbits and satellite track on the ground for 6 h (20 October 2021, starting time 20:07:26 UTC). Figure 9 shows two visibility windows, in terms of satellite elevation and azimuth angles, as seen from Spino d'Adda. Considering only elevation angles larger than 20\({}^{\circ}\), there are typically three daily passes with an average duration of approximately 6 minutes and a maximum duration of roughly 7.6 min when reaching the zenith. Figure 7: The difference \(A_{GEO}-A_{Z}\), calculated at fixed annual probabilities 1% (**a**), 0.1% (**b**), and 0.01% (**c**), between the rain attenuation exceed in the GEO path, \(A_{GEO}\), and the rain attenuation exceed in the GeoSurf path, \(A_{Z}\), i.e., at the zenith in Pasadena; 39.6 GHz, circular polarization. We have simulated the Metop\(-\)C orbits for two consecutive months for providing a stable histogram and have calculated the probability of the elevation angle seen from Spino d'Adda, whose histogram is shown in Figure 10 (5\({}^{\circ}\) bins), for values large than 20\({}^{\circ}\). This elevation angle is the minimum elevation tolerated in most LEO constellations, because at lower elevation angles, tropospheric facies, scintillation, and obstructions can produce severe outages. Afterwards, for each elevation angle up to zenith, we have calculated the probability distribution of rain attenuation predicted in each path and finally merged all probability distributions in the final curve by weighing each probability sample according to the probability shown in Figure 10. Moreover, we have supposed that sampling slant paths with an elevation angle larger than 20\({}^{\circ}\) is also equivalent to sampling all possible rain events of the area, which, of course, produce the local rain rate probability distribution. This is a strong hypothesis but the only one that can be done at this stage for comparing LEO tropospheric attenuation with GEO or GeoSurf attenuation at equal probability. Figure 8: Orbits and tracks on ground of Metop\(-\)C satellite. 20 October 2021; starting time: 20:07:26 UTC. Figure 9: Sample visibility windows: Metop\(-\)C satellite seen from Spino d’Adda. (**a**): LEO satellite elevation angle. (**b**): LEO satellite azimuth. Figure 11 shows (left panel) the probability distributions of rain attenuation predicted for the LEO paths at the reference sites with the Global SST obtained by weighing the probability distribution of rain attenuation predicted for each elevation angle larger than 20\({}^{\circ}\) according to the probability of the elevation angle shown in Figure 10. For comparing LEO and GeoSurf attenuation at equal probabilities, in Figure 11 (right panel), we show the difference \(A_{LEO}-A_{Z}\) versus the annual probability (%). From these curves, it is evident that GeoSurf \"gains\" several dBs compared to LEO. For example, at 0.1%, \(A_{LEO}-A_{Z}\approx 5\) dB for Pasadena and Canberra, and \(A_{LEO}-A_{Z}\approx 9\) dB for Holmdel.