## 5 Two-Way Measurement Method in CCFD
### System Model of Two-Way Measurement Method in CCFD  
Based on the process of inter-satellite two-way measurement, the observation equation can be derived as follows:  
\[ \begin{split}\rho_{ij}=&\left|\mathbf{r}_{j}(t^{\prime 1})-\mathbf{r}_{i}(t^{ \prime 1})\right|+c[dt_{j}(t^{\prime 1})-dt_{i}(t^{\prime 1})]+c\delta_{ij}^{sys}+ \varepsilon_{ij}\\ \rho_{ji}=&\left|\mathbf{r}_{i}(t^{\prime 2})-\mathbf{r}_{j}(t^{ \prime 2})\right|+c[dt_{i}(t^{\prime 2})-dt_{j}(t^{\prime 2})]+c\delta_{ji}^{sys}+ \varepsilon_{ji}\end{split}  \tag{22}\]  
where \(\rho_{ij}\) and \(\rho_{ji}\) represent the locally measured pseudo-ranges of satellites \(j\) and \(i\), respectively. \(dt_{i}(t^{\prime 1})\) and \(dt_{i}(t^{\prime 2})\) are the clock bias between the local time of the satellite \(i\) and BDT at times \(t^{\prime 1}\) and \(t^{\prime 2}\), respectively. \(dt_{j}(t^{\prime 2})\) and \(dt_{j}(t^{\prime 1})\) are the clock bias between the local time of the satellite \(j\) and BDT at times \(t^{\prime 2}\) and \(t^{\prime 1}\), respectively. The clock bias can be expressed as follows:  
\[ \begin{split} dt_{i}(t^{\prime 1})&=t^{s}_{i}-t^{ \prime 1}\\ dt_{i}(t^{\prime 2})&=t^{t}_{i}-t^{\prime 2}\\ dt_{j}(t^{\prime 2})&=t^{s}_{j}-t^{\prime 2}\\ dt_{j}(t^{\prime 1})&=t^{r}_{j}-t^{\prime 1}\end{split}  \tag{23}\]  
\(\delta_{ij}^{sys}\) and \(\delta_{ji}^{sys}\) represent the systematic errors of the two-way measurement, while \(\varepsilon_{ij}\) and \(\varepsilon_{ji}\) denote the ranging noise. The systematic errors can be further expressed as follows:  
\[ \begin{split}\delta_{ij}^{sys}&=\tau_{i}^{ts}+\tau _{j}^{tr}+\tau_{ij}^{rel,g}+\tau_{ij}^{rel,p}+\tau_{ij}^{poo}\\ \delta_{ji}^{sys}&=\tau_{j}^{ts}+\tau_{i}^{tr}+\tau _{ji}^{rel,g}+\tau_{ji}^{rel,p}+\tau_{ji}^{poo}\end{split}  \tag{24}\]  
where \(\tau_{ij}^{rel,g}\) and \(\tau_{ji}^{rel,g}\) represent the gravitational time delay of the two-way measurement, respectively. \(\tau_{ij}^{rel,p}\) and \(\tau_{ji}^{rel,p}\) denote the time delay of periodic relativistic effects in the two-way measurement, respectively. \(\tau_{ij}^{poo}\) and \(\tau_{ji}^{poo}\) represent the time delay of phase center offset (PCO) in the two-way measurement, respectively. Therefore, the inter-satellite two-way observation equation can be expressed in detail as follows:  
Figure 6: Timing diagram of inter-satellite two-way ranging in CCFD. \[\rho_{ij}= \left|\mathbf{r}_{j}(t^{r1})-\mathbf{r}_{i}(t^{s1})\right|+c[dt_{j}(t^{r1})- dt_{i}(t^{s1})]+c(\tau_{i}^{ts}+\tau_{j}^{tr}+\tau_{ij}^{rel.g}+\tau_{ij}^{rel.p}+ \tau_{ij}^{po})+\epsilon_{ij} \tag{25}\] \[\rho_{ji}= \left|\mathbf{r}_{i}(t^{r2})-\mathbf{r}_{j}(t^{s2})\right|+c[dt_{i}(t^{r2} )-dt_{j}(t^{s2})]+c(\tau_{j}^{ts}+\tau_{i}^{tr}+\tau_{ji}^{rel.g}+\tau_{ji}^{rel.p }+\tau_{ji}^{po})+\epsilon_{ji}\]  
Given that the maximum communication distance of BDS-3 ISLs can reach 70,000 km and the relative velocity between satellites can reach several km/s, the effects of signal propagation delay and high-speed satellite motion cannot be overlooked [65]. ISL ranging aims to determine the geometric distance between two satellites at the same instant. Therefore, it is necessary to perform timestamp calculations on the observations to align the inter-satellite pseudo-range measurement to a unified system time.