## 4 MS-FARIMA Algorithm and Orbit Extrapolation Improvement with MSEC
### MS-FARIMA Algorithm for Orbit Extrapolation Based on MSEC  
#### 4.2.1 MSEC for Elimination of Ephemeris Errors  
As shown in Figure 6, in the LEO-SIN, the UE has a certain LOS range for satellites based on the strength of the received signals. Within this LOS range, the UE can fully receive the ephemeris information broadcast by the satellites. Satellites in the same constellation have similar orbital altitudes, satellite characteristics, and corresponding orbital characteristics. This is especially true for satellites on the same or adjacent orbits. Therefore, the ephemeris broadcast by these satellites also exhibits a certain degree of similarity. Based on this characteristic, research into the collaborative use of multiple satellites' ephemeris information can optimize the accuracy of ephemeris estimation. Figure 6: MS-FARIMA algorithm for orbit extrapolation based on MSEC. The algorithm first uses the ephemeris data from satellites preceding and following on the same orbit and adjacent satellites on different orbits to eliminate the RE through MSEC error elimination. At any given time, the UE simultaneously receives ephemerides from several satellites. This includes satellites on the same orbit as the current satellite, as well as those on different orbits. The UE integrates and corrects these ephemeris data to eliminate the RE. At time \(t\), we assume that the UE can receive \(K\) visible satellites' ephemeris data \(\mathcal{T}_{\mathcal{L}_{i}}\), at the beam position \(\mathcal{L}_{i}\). Using the reference time \(t_{\omega}\) and the SKOEs in the TLE ephemeris data, the SGP4 algorithm is used to extrapolate the three-dimensional coordinate components of the position vector \(\mathbf{r}\), as shown in Equation (27). Based on the broadcast ephemeris received from the \(K\) different satellites, the spatial position errors for the \(K\) different positions can be calculated:  
\[ \begin{bmatrix}\delta x_{i}(t)\\ \delta y_{i}(t)\\ \delta z_{i}(t)\end{bmatrix} = \begin{bmatrix}\frac{\partial x(P_{i},t_{\omega },t)}{\partial n},\cdots,\frac{\partial x(P_{i},t_{\omega },t)}{\partial t}\\ \frac{\partial y(P_{i},t_{\omega},t)}{\partial n},\cdots,\frac{\partial y(P_{ i},t_{\omega},t)}{\partial t}\\ \frac{\partial z(P_{i},t_{\omega},t)}{\partial n},\cdots,\frac{\partial z(P_{ i},t_{\omega},t)}{\partial t}\end{bmatrix}\begin{bmatrix}\delta n\\ \delta i\\ \vdots\\ \delta\Omega\end{bmatrix}  \tag{27}\]  
where \(\delta n,\delta i,\cdots,\delta\Omega\) represent the errors in the SKOEs, \(i=1,2,\cdots,K\) denotes the \(K\) satellites where TLE data are received by the UE at time \(t\). \(\delta x_{i}(t),\delta y_{i}(t),\delta z_{i}(t)\) represent the three-dimensional coordinate position errors of the \(i\)-th ephemeris data. Therefore, the error for the \(K\) sets of ephemerides can be expressed as  
\[ \begin{split}\Delta e^{1\frac{d\ell}{\Xi}}(\delta x_{1}(t),\delta y _{1}(t),\delta z_{1}(t))\\ \vdots\\ \Delta e^{K\frac{d\ell}{\Xi}}(\delta x_{K}(t),\delta y_{K}(t),\delta z _{K}(t))\end{split}  \tag{28}\]  
where \(\Delta e^{K}\) represents the error vector of the ephemeris for the \(K\)-th satellite, and \(\delta x^{K}(t)\), \(\delta y^{K}(t),\delta z^{K}(t)\) represent the error vectors of the \(K\)-th satellite's ephemeris in the X, Y, and Z directions, respectively. For error assessments in a single direction, error elimination can be performed using the ephemeris currently received from the \(K-1\) adjacent satellites. The error equations for the ephemeris of the \(K-1\) satellites in the X direction can be calculated  
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