## 2 BDS-3 SAIM Monitoring Method
### Enhanced Ephemeris Monitoring Methods
#### 2.2.3 Inter-Satellite Ranging  
The BDS-3 has inter-satellite link observation data, which can also be used as a reference for judging the integrity of ephemeris messages. Figure 4 shows a schematic diagram of the two-way ranging of the inter-satellite link. If we assume that \(A\) is the satellite to be monitored, and there is an inter-satellite link ranging between \(B\) and \(A\), the inter-satellite link ranging between \(A\) and \(B\) can be expressed as [21]:  
\[ \begin{split}\rho_{AB}(t_{1})&=\left|\stackrel{{ \rightarrow}}{{R}}_{B}(t_{1})-\stackrel{{\rightarrow}}{{R}}_{A}(t_ {1}-\Delta t_{1})\right|+c\left(clk_{B}(t_{1})-clk_{A}(t_{1})+\tau_{A}^{Send} +\tau_{B}^{Rx}\right)+\Delta\rho_{cor}^{AB},\\ \rho_{AB}(t_{2})&=\left|\stackrel{{ \rightarrow}}{{R}}_{B}(t_{2})-\stackrel{{\rightarrow}}{{R}}_{A}(t_ {2}-\Delta t_{2})\right|+c\left(clk_{A}(t_{2})-clk_{B}(t_{2})+\tau_{A}^{Rx}+ \tau_{B}^{Send}\right)+\Delta\rho_{cor}^{AB}\end{split}  \tag{1}\]  
In the Formula (1), \(t_{1}\) and \(t_{2}\) are the different moments; \(\stackrel{{\rightarrow}}{{R}}_{i}\) is the satellite's position vector; \(\Delta t_{i}\) is the transmission delay for ranging; \(clk_{i}\) is the satellite clock; \(\tau_{i}^{Send}\) and \(\tau_{i}^{Rx}\) are the delays of the transmitting and receiving equipment, respectively; \(\Delta\rho_{cor}^{AB}\) is the combined error of the ranging observation error, the satellite orbit error and the satellite clock error. Figure 4: Schematic diagram of the two-way ranging of the inter-satellite link. To obtain the two-way distance observations, the one-way observations need to be reduced to the same moment, and the reduction equation is shown in the following formula:  
\[ \begin{split}\rho_{AB}(t_{0})=\rho_{AB}(t_{1})+d\rho_{AB}& =\left|\stackrel{{\rightarrow}}{{R}}_{B}(t_{0})- \stackrel{{\rightarrow}}{{R}}_{A}(t_{0})\right|+c\left(clk_{B}(t_{0})-clk_{A}(t_{0})+\tau_{A}^{Send}+\tau_{B}^{Rx} \right)+\Delta\rho_{var}^{AB},\\ \rho_{AB}(t_{0})=\rho_{BA}(t_{2})+d\rho_{BA}&= \left|\stackrel{{\rightarrow}}{{R}}_{A}(t_{0})-\stackrel{{ \rightarrow}}{{R}}_{B}(t_{0})\right|+c\left(clk_{A}(t_{0})-clk_{B}(t_{0})+ \tau_{A}^{Rx}+\tau_{B}^{Send}\right)+\Delta\rho_{var}^{AB}\end{split}  \tag{2}\]  
where \(d\rho_{AB}\) and \(d\rho_{BA}\) represent the amount of reduction correction calculated from the observation time and the reduction time, which are related to the distance difference and the clock difference between the reduction time and the observation time; \(d\rho_{AB}\) and \(d\rho_{BA}\) are calculated by the following formula:  
\[ \begin{split} d\rho_{AB}&=\left|\stackrel{{ \rightarrow}}{{R}}_{B}(t_{0})-\stackrel{{\rightarrow}}{{R}}_{A}(t_ {0})\right|-\left|\stackrel{{\rightarrow}}{{R}}_{B}(t_{1})- \stackrel{{\rightarrow}}{{R}}_{A}(t_{1}-\Lambda_{1})\right|+c(clk_{B}(t_{1})-clk_{A}(t_{1}))-c(clk_{B}(t_{0})-clk_{A}(t_{0})),\\ d\rho_{BA}&=\left|\stackrel{{ \rightarrow}}{{R}}_{B}(t_{0})-\stackrel{{\rightarrow}}{{R}}_{A}(t_ {0})\right|-\left|\stackrel{{\rightarrow}}{{R}}_{B}(t_{2})- \stackrel{{\rightarrow}}{{R}}_{A}(t_{2}-\Lambda_{2})\right|+c(clk_{A}(t_{2})-clk_{B}(t_{2}))-c(clk_{A}(t_{0})-clk_{B}(t_{0}))\end{split}  \tag{3}\]  
\(d\rho_{AB}\) and \(d\rho_{BA}\) can be calculated by the differences in the satellite forecast orbit and forecast clock parameters, and its calculation accuracy determines the reduction accuracy of the time-division system's inter-satellite ranging data, which depend on the satellites' forecast clock speed accuracy and speed forecast accuracy. A pair of satellites generally completes a two-way measurement within 3 s. Therefore, in the previous algorithm, the target time is reduced to a distance \(t_{0}\) from the observation times \(t_{1}\) and \(t_{2}\) that is less than 3 s. At present, the BDS-3 satellite's speed forecast error is about 0.1 mm/s and the forecast clock speed error is less than \(1\times 10^{-13}\) s/s.