## 3 Space Wave Propagation for LEO Downlink Analysis  
In the LEO satellite data transmitting scenario, EM waves propagate through the troposphere, stratosphere, and ionosphere to reach ground stations. Thus, losses are affected and increased by the phenomena of refraction, attenuation, and reflection between each layer. To predict EM wave propagation in space environments, it is necessary to calculate the refractive indices of the troposphere and the stratosphere. The reflection and transmission of EM waves are then obtained at the interfaces of multi-layered spheres. The ray tracing technique and geometrical optics are employed to calculate EM wave propagation at interfaces among the troposphere, stratosphere, and ionosphere in a space environment. Figure 3: Receiving antenna of the ground station and the far-zone radiation pattern: (**a**) receiving antenna of the ground station; (**b**) far-zone radiation pattern of the ground station. Figure 4 shows the GO model when a ray passes through multi-layered atmospheric spheres [28]. The wave propagation characteristics can be determined by calculating the transmission and reflection at each interface of the multi-layered spheres. The polarization of the incident wave is also considered when calculating the reflection and transmission coefficients at each layer [29]. \(n_{0}\), \(n_{1}\),..., \(n_{I}\) are the effective refractive index in the divided layers of ionosphere, and \(n_{I+1}\),..., \(n_{T}\) are the effective refractive index in the divided layers of troposphere. \(\theta_{oi}\), \(\theta_{li}\), and \(\theta_{Ti}\) are incident angles on each layer of ionosphere and troposphere. \(E_{i0}\) is the electric field incident on the multi-layered ionosphere, \(E_{t0}\) is the electric field transmitted through each interface of the multi-layered ionosphere, and \(E_{i1}\) is the electric field incident on the multi-layered troposphere. \(E_{t1}\) is the electric field transmitted through the multi-layered troposphere, and \(E_{T}\) is the electric field reaching the observation point which is presented the blue circle marker. The effective refractive index according to the altitude in the troposphere and the stratosphere can be approximated by the equations in [30; 31]. The refractive index can be determined by temperature, pressure, and water vapor pressure obtained using daily updated weather data from the University of Wyoming [32]. Since the ionosphere has a plasma ion layer caused by solar radiation, it affects the refraction, attenuation, and reflection of EM waves [33]. The characteristics of the ionosphere and prediction methods are employed in [34] to investigate wave propagation in the ionosphere. The relative permeability of the atmosphere is assumed to be 1, and the anisotropy of the electrical conductivity is not considered. Figure 5a,b show the bore-sight error and atmospheric attenuation that occur when transmitting signals from the LEO satellite to the ground station on Earth. The bore-sight error increases as the satellite elevation angle \(\theta_{gs}\) decreases, as shown in Figure 5a. In particular, the bore-sight error significantly increases when considering the space environments, as shown in the blue line. At a low elevation angle (\(\theta_{gs}\) = 10\({}^{\circ}\)), the difference in bore-sight error is 0.04\({}^{\circ}\) and at a high elevation angle (\(\theta_{gs}\) = 90\({}^{\circ}\)), the difference is 0\({}^{\circ}\). The reason for the large difference, especially at the low elevation angle, is the increased ray refraction. Figure 5b shows the atmospheric attenuation, and again, how the attenuation considerably increases as the satellite elevation angle \(\theta_{gs}\) decreases. This result shows that at the low elevation angle, the total path is increased by the refraction, resulting in greater attenuation. Figure 6 shows the normalized received power at the Earth's surface according to latitude and longitude when transmitting signals from the LEO satellite.