## 5 Two-Way Measurement Method in CCFD
### System Model of Two-Way Measurement Method in CCFD
\(t^{s1}\) can be selected as the reckoning time (\(t_{0}\)) in CCFD, and the inter-satellite two-way observation equation can be adjusted as follows:  
\[\rho_{ij}= \left|\mathbf{r}_{j}(t_{0})-\mathbf{r}_{i}(t_{0})\right|+\Delta D_{ij}+c \left\{\left[dt_{j}(t_{0})+dT_{j}(t^{r1},t_{0})\right]-\left[dt_{i}(t_{0})+dT_ {i}(t^{s1},t_{0})\right]\right\}+c(\tau_{i}^{ts}+\tau_{i}^{tr}+\tau_{ij}^{rel.g} +\tau_{ij}^{po})+\epsilon_{ij} \tag{26}\] \[\rho_{ji}= \left|\mathbf{r}_{i}(t_{0})-\mathbf{r}_{j}(t_{0})\right|+\Delta D_{ji}+c \left\{\left[dt_{i}(t_{0})+dT_{i}(t^{r2},t_{0})\right]-\left[dt_{i}(t_{0})+dT_ {j}(t^{s2},t_{0})\right]\right\}+c(\tau_{j}^{ts}+\tau_{i}^{tr}+\delta_{ji}^{rel. g}+\delta_{ji}^{po})+\epsilon_{ji}\]  
where \(\Delta D_{ij}\) and \(\Delta D_{ji}\) are the distance corrections for the inter-satellite two-way measurement, respectively. \(dT_{i}(t,t_{0})\) is the clock bias correction for satellite \(i\) from time \(t\) to \(t_{0}\), and \(dT_{j}(t,t_{0})\) is the clock bias correction for satellite \(j\) from time \(t\) to \(t_{0}\). Comparing Equations (25) and (26), the following equations can be derived as follows:  
\[\Delta D_{ij}= \left|\mathbf{r}_{j}(t^{r1})-\mathbf{r}_{i}(t^{s1})\right|-\left|\mathbf{r}_{ j}(t_{0})-\mathbf{r}_{i}(t_{0})\right| \tag{27}\] \[\Delta D_{ji}= \left|\mathbf{r}_{i}(t^{r2})-\mathbf{r}_{j}(t^{s2})\right|-\left|\mathbf{r}_{ i}(t_{0})-\mathbf{r}_{j}(t_{0})\right|\]  
\[ \begin{array}{l}dT_{i}(t^{s1},t_{0})=dt_{i}(t^{s1})-dt_{i}(t_{0})\\ dT_{i}(t^{r2},t_{0})=dt_{i}(t^{r2})-dt_{i}(t_{0})\\ dT_{j}(t^{s2},t_{0})=dt_{j}(t^{s2})-dt_{j}(t_{0})\\ dT_{j}(t^{r1},t_{0})=dt_{j}(t^{r1})-dt_{j}(t_{0})\end{array}  \tag{28}\]  
Analyzing the measurement process from satellite \(i\) to satellite \(j\), it can be concluded that:  
\[\left|\mathbf{r}_{j}(t^{r1})-\mathbf{r}_{i}(t^{s1})\right|=\left|\mathbf{r}_{j}(t^{s1})- \mathbf{r}_{i}(t^{s1})\right|+\int_{0}^{t^{s1}-t^{s1}}\mathbf{v}_{j}(t)\cdot\mathbf{e}_{ ij}(t)dt \tag{29}\]  
where \(\mathbf{v}_{j}(t)\) represents the velocity of satellite \(j\), and \(\mathbf{e}_{ij}(t)\) denotes the unit direction vector from satellite \(i\) to satellite \(j\). Let \(f(t)=\int_{0}^{t-t^{s1}}\mathbf{v}_{j}(t)\cdot\mathbf{e}_{ij}(t)dt\), and perform a first-order Taylor expansion at \(t=t^{s1}\), yielding as follows:  
\[ \begin{array}{l}f(t)=f(t^{s1})+\frac{d\prime}{dt}\left|t_{=t^{s1}}\cdot(t-t ^{s1})+R_{1}(t-t^{s1})\right.\\ =\mathbf{v}_{j}(t^{s1})\cdot\mathbf{e}_{ij}(t^{s1})\cdot(t-t^{s1})+R_{1}(t-t^{s1}) \end{array}  \tag{30}\]  
where  
\[R_{1}(t-t^{s1})=\frac{1}{2}(\mathbf{v}^{\prime}_{j}(\xi)\cdot\mathbf{e}_{ij}(\xi)+\mathbf{ v}_{j}(\xi)\cdot\mathbf{e}^{\prime}_{ij}(\xi))\cdot(t-t^{s1})^{2} \tag{31}\]  
where \(\xi\) is a value between \(t\) and \(t^{s1}\). Since the velocity of the satellite changes slowly during a single ranging process and the linked satellites are tens of thousands of kilometers apart, the acceleration of the satellite and the rate of change of the inter-satellite direction can be approximated as zero over a short period, and \(R_{1}(t-t^{s1})\) can be neglected. Letting \(t=t^{s1}\), Equation (30) can be written as follows:  
\[f(t^{r1})=\int_{0}^{t^{s1}-t^{s1}}\mathbf{v}_{j}(t)\cdot\mathbf{e}_{ij}(t)dt=\mathbf{v}_{j }(t^{s1})\cdot\mathbf{e}_{ij}(t^{s1})\cdot(t^{r1}-t^{s1}) \tag{32}\]Substituting Equation (32) into Equation (29), it can be concluded that:  
\[\Big{|}\tau_{j}(t^{r1})-\tau_{i}(t^{s1})\Big{|}{=}\Big{|}\tau_{j}(t^{s1})-\tau_{i }(t^{s1})\Big{|}{+}\upsilon_{j}(t^{s1})\cdot\mathbf{e}_{ij}(t^{s1})\cdot(t^{r1}-t^{ s1}) \tag{33}\]  
For the measurement process from satellite \(j\) to satellite \(i\), we can similarly derive the following result:  
\[\Big{|}\mathbf{r}_{i}(t^{r2})-\tau_{j}(t^{s2})\Big{|}{=}\Big{|}\mathbf{r}_{i}(t^{s2})- \mathbf{r}_{j}(t^{s2})\Big{|}{+}\upsilon_{i}(t^{s2})\cdot\mathbf{e}_{ji}(t^{s2})\cdot(t ^{r2}-t^{s2}) \tag{34}\]  
where \(\mathbf{v}_{i}(t)\) represents the velocity of the satellite \(i\), and \(\mathbf{e}_{ji}(t)\) denotes the unit direction vector from satellite \(j\) to satellite \(i\).