## 4 MS-FARIMA Algorithm and Orbit Extrapolation Improvement with MSEC
### Theoretical Performance Analysis  
At time \(t\), we assume that the UE can obtain the broadcast ephemeris from the currently serving satellite \(\textit{satellite}_{\textit{service}}\) as \(\textit{eph}_{\textit{broadcast}}=\textit{eph}_{\textit{t},0}\); \(\textit{eph}_{\textit{t},0}\) represents the ephemeris broadcast by the satellite at time \(t\), and \(\textit{eph}_{\textit{t},J}\) represents the ephemeris that the UE can receive from other satellites. \(J\) is the orbital matrix, which can be expressed as  
\[J=\{j_{n},j_{k}\} \tag{36}\]  
where \(j_{n}\) represents \(n\)-th satellites before or after the current satellite \(\textit{satellite}_{\textit{service}}\) on the same orbit, \(j_{k}\) represents \(k\)-th satellites before or after the current satellite \(\textit{satellite}_{\textit{service}}\) on the adjacent orbit. Based on the previous analysis, \(\textit{eph}_{\textit{t},J}\) includes three types of errors: RE \(\Delta\textit{eph}_{\textit{t},k}(\textit{rand})\), TE \(\Delta\textit{eph}_{\textit{t},k}(\textit{trend})\), and PE \(\Delta\textit{eph}_{\textit{t},k}(\textit{period})\). The ephemeris satisfies the following matrix:  
\[C_{eph}= \begin{bmatrix}\textit{eph}_{\textit{t}-\tau_{t},k-1}&\textit{eph}_{ \textit{t}-\tau_{t},k}&\textit{eph}_{\textit{t}-\tau_{t},k+1}\\ \textit{eph}_{\textit{t},k-1}&\textit{eph}_{\textit{t},k}&\textit{eph}_{ \textit{t},k+1}\\ \textit{eph}_{\textit{t}-\tau_{t},k-1}&\textit{eph}_{\textit{t}-\tau_{t},k-1}& \textit{eph}_{\textit{t}-\tau_{t},k+1}\end{bmatrix}  \tag{37}\]  
where \(\textit{eph}_{\textit{t},k}\) is the broadcast ephemeris from the currently serving satellite. Simultaneously, the UE can look up the corresponding broadcast ephemeris \(\textit{eph}_{\textit{t}-\tau_{t},k}\) from the time \(t-T_{t}\). Currently, the broadcast ephemerides of the two satellites before and after the current one on the same orbit are \(\textit{eph}_{\textit{t},k-1}\) and \(\textit{eph}_{\textit{t},k+1}\). We assume the MSEC operator \(A(\cdot)\); this operator will eliminate the RE \(\Delta\textit{eph}_{\textit{t},k}(\textit{rand})\) coexisting among multiple satellites, which can be expressed as  
\[A(C_{eph})= \begin{bmatrix}\textit{eph}^{\prime}_{\textit{t}-\tau_{t},k-1}& \textit{eph}^{\prime}_{\textit{t}-\tau_{t},k}&\textit{eph}^{\prime}_{\textit {t}-\tau_{t},k+1}\\ \textit{eph}^{\prime}_{\textit{t},k-1}&\textit{eph}^{\prime}_{\textit{t},k}& \textit{eph}^{\prime}_{\textit{t},k+1}\\ \textit{eph}^{\prime}_{\textit{t}-T_{t},k-1}&\textit{eph}^{\prime}_{\textit{t }-T_{t},k-1}&\textit{eph}^{\prime}_{\textit{t}-T_{t},k+1}\end{bmatrix}  \tag{38}\]  
where \(\textit{eph}^{\prime}_{\textit{t},k}\) represents the broadcast ephemeris after the elimination of the RE \(\Delta\textit{eph}_{\textit{t},k}(\textit{rand})\) using MSEC:  
\[\textit{eph}^{\prime}_{\textit{t},k}=\Delta\textit{eph}_{\textit{t},k}( \textit{trend})+\Delta\textit{eph}_{\textit{t},k}(\textit{period}) \tag{39}\]  
For PE \(\Delta\textit{eph}_{\textit{t},k}(\textit{period})\), due to its periodic characteristics, spectral analysis can be used to extract the PE:  
\[SPE(\textit{eph}^{\prime}_{\textit{t},k})=\Delta\textit{eph}_{\textit{t},k}( \textit{period}) \tag{40}\]  
Then, TE \(\Delta\textit{eph}_{\textit{t},k}(\textit{trend})\) can be expressed as  
\[\Delta\textit{eph}_{\textit{t},k}(\textit{trend})=\textit{eph}^{\prime}_{ \textit{t},k}-\Delta\textit{eph}_{\textit{t},k}(\textit{period}) \tag{41}\]  
For TE \(\Delta\textit{eph}_{\textit{t},k}(\textit{trend})\), MS-FARIMA is used to eliminate the TE:  
\[Time_{\textit{predict}}(\Delta\textit{eph}_{\textit{t},k}(\textit{trend}))= \Delta\textit{eph}^{\prime}_{\textit{t},k}(\textit{period}) \tag{42}\]where \(\Delta eplt^{\prime}_{t,k}(trend)\) is the new TE after eliminating TE \(\Delta eplt_{t,k}(trend)\) using MS-FARIMA. The originally broadcast ephemeris from the initial analysis can be expressed as  
\[ \begin{array}{l}eph_{broadcast}(t,k)=eph_{0}+\Delta eph_{t,k}\\ =eph_{0}+\Delta eplt_{t,k}(rand)+\Delta eph_{t,k}(period)+\Delta eph_{t,k}(trend) \end{array}