## 3 System Model and Problem Formulation  
### System Model  
In the LEO-SIN, UEs receive ephemeris information not only from the current serving satellite at an observed time, but also from all visible satellites within the UE's line-of-sight (LOS) coverage at that moment and some previous moments, as shown in Figure 2. The following information can support orbit extrapolation performance enhancement: the ephemeris for the same satellite at different times, the ephemeris for adjacent satellites at the same time, and the correlation between the broadcast ephemeris and relatively accurate ephemeris for the same satellite. The analysis and use of this information is expected to further improve the ephemeris extrapolation errors. We assume a three-parameter set \(\{\mathrm{T},\ \mathrm{P},\ \mathrm{F}\}\) to define a Walker constellation, where \(\mathrm{T}\) is the total number of satellites in the constellation, \(\mathrm{P}\) is the number of orbital planes, and \(\mathrm{F}\) is the phase factor, which is an integer ranging from 0 to \(\mathrm{P}-1\). The phase difference \(\Delta\varphi_{i}\) between satellites with the same number in different orbital planes is determined by the phase factor \(\mathrm{F}\) and can be expressed as  
\[\Delta\varphi_{i}=\frac{2\pi}{\mathrm{T}}\times\mathrm{F},\ \mathrm{F}=0,1,2,3, \cdots,\mathrm{P}-1 \tag{1}\]  
The phase difference between satellites with the same number in the \(\mathrm{P}-1\) and 0 th orbital planes is given by  
\[\Delta\varphi_{P-1}=(\mathrm{P}-1)\Delta\varphi_{1} \tag{2}\]  
For a given orbital altitude, the orbital inclination, the initial position of the seed satellite, and the three-parameter set \(\{\mathrm{T},\ \mathrm{P},\ \mathrm{F}\}\) uniquely determines a Walker constellation. In this work, a series of related satellites within the UE's LOS coverage will be analyzed. These satellites form a group of satellites, which consists of satellites with the same orbit or adjacent orbits within the same constellation. Each satellite periodically broadcasts its ephemeris in TLE format to the UEs, while the GS periodically send measured and calculated accurate ephemeris data to the satellites. Figure 3 illustrates the scenario of satellite monitoring, ephemeris calculation, and broadcasting. We assume that a UE is considered in the satellite footprint, which can simultaneously receive ephemeris information from multiple different satellites. Orbit extrapolation with different satellites' ephemerides exhibits different levels of accuracy. At a given time \(t\), the corresponding ephemeris can be denoted as \(\mathbb{E}_{\mathbb{L},\infty,\mathbb{S}}\), where \(\mathbb{L}\) denotes the UE LOS area, which is typically the region covered by the satellite signals. Here, \(\mathbb{L}=\{\mathcal{L}_{0},\mathcal{L}_{1},\ldots,\mathcal{L}_{N}\}\), where \(N\) is the number of beam footprints of the serving satellite and \(\mathcal{L}_{i}\) represents the \(i\)th beam footprint. \(\approx\) is the set of times at which TLE data are received in region \(\mathbb{L}\), defined as \(\approx=\{\mathcal{T}_{\mathcal{L}_{0}},\mathcal{T}_{\mathcal{L}_{1}},\ldots, \mathcal{T}_{\mathcal{L}_{N}}\}\). \(\mathbb{S}\) is the set of correlated satellites, which is defined as \(\mathbb{S}=\{S_{0},S_{1},\ldots,S_{K}\}\). The set of ephemeris data \(\mathcal{T}_{\mathcal{L}_{i},t}\) received by the UE at position \(\mathcal{L}_{i}\) at time \(t\) is given by  
\[\mathcal{T}_{\mathcal{L}_{i},t}=\{T_{\mathcal{L}_{i},t,1}:T_{\mathcal{L}_{i}, K}\} \tag{3}\]  
where \(K\) is the number of correlated satellites and \(T_{\mathcal{L}_{i},t,K}\) represents the ephemeris data of satellite \(K\) received by the UE in this service area at the time \(t\). The proposed method, aimed at multi-satellite ephemeris extrapolation, can effectively improve the extrapolation accuracy. Figure 3 shows the flowchart of the proposed method. The left side of the flowchart represents the overall process of optimizing the satellite's ephemeris extrapolation accuracy.