## 3 Satellite Geometries  
### Uplink Satellite Geometry for DoA Estimation  
This paper considers different satellite systems, i.e., GEO and LEO satellites. This section first discusses the uplink system geometry of the GEO satellite system and how it is modified for use in the LEO satellite system by adjusting the relevant parameters. The illustration shown in Figure 1 depicts the spherical geometric approximation utilized for the transmitting antenna of the Earth station, which is situated on the Earth's surface and communicates with the GEO satellite station equipped with receiving antennas during uplink communication. The GEO satellite station is positioned to travel eastward at the same rotational speed as the Earth, maintaining a stationary position relative to the Earth. The satellite station has a circular orbit and zero inclination [2]. The Earth station is labeled as ES, the satellite station is labeled as SP, and the center of the Earth is indicated by the symbol O. The azimuth and elevation angles of the ES are denoted as A\({}_{x}\) and EI, respectively. The A\({}_{x}\) is the horizontal angle from true north in a clockwise direction of the satellite with a range of (0 to 360)\({}^{\circ}\). On the other hand, EI is the angle between the satellite and the observer's horizon plane and has a range of (0 to 90)\({}^{\circ}\)[3]. The ES is defined by its geographic coordinate system (GCS) (i.e., a spherical coordinate system), denoted as \((\delta_{E},\lambda_{E})\), where \(\delta_{E}\) is the longitude measured in the positive east direction, and \(\lambda_{E}\) is the latitude of the ES measured in the positive north direction. The sub-satellite point is denoted as SS, and its longitude is represented by \(\delta_{ss}\) and its latitude by \(\lambda_{ss}\). The latitude of the sub-satellite point is ideally 0\({}^{\circ}\), as it is positioned directly above the equator. The slant distance from the ES to the SP is denoted as \(\mathrm{d}_{\mathrm{SPES}}\), while \(\alpha\) is the critical angle that needs to be determined [2; 21]. Figure 1a shows a spherical triangle with angles \"abc\" as arcs of great circles, while Figure 1b demonstrates a plane triangle \"OESSP\". The angle I shown in Figure 1a, which is the angle between the plane containing c and the plane containing b, is commonly employed to calculate the \(\mathrm{A}_{\mathrm{z}}\) angle [2]. Our model focused only on determining El. Thus, calculations related to \(\mathrm{A}_{\mathrm{z}}\) are not included, and are assumed to be 0\({}^{\circ}\). Thus, the information is summarized as follows: (i) \(\mathrm{a}=90^{\circ}\), the angle between the distribution for the range to the north pole and the radius to the SS (i.e., sub-satellite point), (ii) \(\mathrm{c}=90^{\circ}-\lambda_{E}\), the angle between the radius to the ES and the radius to the north pole, and (iii) \(\mathrm{B}=\delta_{E}-\delta_{SS}\), the angle between the plane having c and the plane containing a. Therefore, the angle b between the radius of ES and the radius of SS is determined using Napier's rules as below:  
\[\mathrm{b}=\arccos(\cos\mathrm{B}\cos\lambda_{E}), \tag{1}\]  
where B is negative when ES is west of the sub-satellite point and positive when east. To calculate El, the sine rule for plane triangles is applied to the triangle shown in Figure 1b [2] as below:  
\[\mathrm{El}=\arccos(\frac{\mathrm{a}_{\mathrm{SPO}}}{\mathrm{d}_{\mathrm{SPES }}}\sin\mathrm{b}), \tag{2}\]  
In Equation (2), \(\mathrm{a}_{\mathrm{SPO}}\) is the distance from the SP to O and is given by:  
\[\mathrm{a}_{\mathrm{SPO}}=\mathrm{a}_{\mathrm{E}}+\mathrm{h}_{\mathrm{SPO}} \tag{3}\]  
where \(\mathrm{a}_{\mathrm{E}}\) represents the equatorial radius of the Earth, typically considered to be 6378 km. In general, the average radius of the Earth (R) is 6371 km. Thus, (\(\mathrm{a}_{\mathrm{E}}\approx\mathrm{R}\)). Moreover, the altitude for a GEO satellite is denoted as \(\mathrm{h}_{\mathrm{SPO}}\), which is 35,786 km. Thus, the value of \(\mathrm{a}_{\mathrm{SPO}}\) sums to 42,164 km.