## 2 Methodology
### Ka-Band Observation Model  
As a pair of one-way pseudoranges, ISL measurement can be used alone or along with L-band data to determine satellite orbits and clock offsets. Suppose that satellite A receives the Ka-band signal transmitted by satellite B at time \(t_{1}\) and generates the observation \(\rho_{BA}(t_{1})\) and satellite B receives the Ka-band signal transmitted by satellite A at time \(t_{2}\) and generates the observation \(\rho_{AB}(t_{2})\). Then, the Ka-band observation model can be formulated as  
\[ \begin{split}\rho_{BA}(t_{1})=\left|\overset{\rightarrow}{r}_{A} (t_{1})-\overset{\rightarrow}{r}_{B}(t_{1}-\frac{\rho_{BA}}{\text{C}})\right| +\text{C}\cdot\left(dt_{A}(t_{1})-dt_{B}(t_{1}-\frac{\rho_{BA}}{\text{C}}) \right)+\text{C}\\ \left(\tau_{A}^{Rcv}+\tau_{B}^{Send}\right)+\Delta\delta_{BA}+ \varepsilon_{BA}\end{split}  \tag{3}\]  
\[ \begin{split}\rho_{AB}(t_{2})=\left|\overset{\rightarrow}{r}_{B} (t_{2})-\overset{\rightarrow}{r}_{A}(t_{2}-\frac{\rho_{AB}}{\text{C}})\right| +\text{C}\cdot\left(dt_{B}(t_{2})-dt_{A}(t_{2}-\frac{\rho_{AB}}{\text{C}}) \right)+\text{C}\\ \left(\tau_{B}^{Rcv}+\tau_{A}^{Send}\right)+\Delta\delta_{AB}+ \varepsilon_{AB}\end{split}  \tag{4}\]  
where \(\overset{\rightarrow}{r}_{A}\) and \(\overset{\rightarrow}{r}_{B}\) is the Ka-band satellite antenna phase center vectors in Cartesian coordinates at the certain observation epoch; \(dt_{A}\) and \(dt_{B}\) are satellite clock offset; \(\tau_{A}^{Send}\) and \(\tau_{B}^{Send}\) are the transmitting hardware delay; \(\tau_{A}^{Rcv}\) and \(\tau_{B}^{Rcv}\) are receiving hardware delay; C represents the speed of light; \(\Delta\delta_{AB}\) and \(\Delta\delta_{BA}\) refer to corrections that can be easily modeled for the measurement, such as relativistic effects and Ka-band phase center offsets; and \(\varepsilon_{BA}\) and \(\varepsilon_{AB}\) are the measurement noises. We can treat them as two independent one-way observations and model them directly similar to the L-band pseudoranges [17], or we can transform a pair of one-way observations to two observations at the same time for processing, as  
\[\rho_{BA}(t_{0})=\rho_{BA}(t_{1})+\Delta\rho_{BA}, \tag{5}\]  
\[\rho_{AB}(t_{0})=\rho_{AB}(t_{2})+\Delta\rho_{AB}, \tag{6}\]  
where  
\[\rho_{BA}(t_{0})=\left|\overset{\rightarrow}{r}_{A}(t_{0})-\overset{ \rightarrow}{r}_{B}(t_{0})\right|+\text{C}\cdot\left(dt_{A}(t_{0})-dt_{B}(t_{ 0})\right)+\text{C}\cdot\left(\tau_{A}^{Rcv}+\tau_{B}^{Send}\right)+\Delta \delta_{BA}+\varepsilon_{BA}, \tag{7}\]  
\[\rho_{AB}(t_{0})=\left|\overset{\rightarrow}{r}_{B}(t_{0})-\overset{ \rightarrow}{r}_{A}(t_{0})\right|+\text{C}\cdot\left(dt_{B}(t_{0})-dt_{A}(t_{ 0})\right)+\text{C}\cdot\left(\tau_{B}^{Rcv}+\tau_{A}^{Send}\right)+\Delta \delta_{AB}+\varepsilon_{AB}, \tag{8}\]  
and the correction \(\Delta\rho_{BA}\) and \(\Delta\rho_{AB}\) can be computed with broadcast messages following:  
\[ \begin{split}\Delta\rho_{BA}=\rho_{BA}(t_{0})-\rho_{BA}(t_{1})\\ =\left|\overset{\rightarrow}{r}_{A}(t_{0})-\overset{\rightarrow}{r }_{B}(t_{0})\right|-\left|\overset{\rightarrow}{r}_{A}(t_{1})-\overset{ \rightarrow}{r}_{B}\left(t_{1}-\frac{\rho_{BA}}{\text{C}}\right)\right|+\text{ C}\\ \cdot(dt_{A}(t_{0})-dt_{B}(t_{0}))-\text{C}\cdot\left(dt_{A}(t_{1} )-dt_{B}(t_{1}-\frac{\rho_{BA}}{\text{C}})\right)\end{split}  \tag{9}\]  
\[ \begin{split}\Delta\rho_{AB}=\rho_{AB}(t_{0})-\rho_{AB}(t_{2})\\ =\left|\overset{\rightarrow}{r}_{B}(t_{0})-\overset{\rightarrow}{r }_{A}(t_{0})\right|-\left|\overset{\rightarrow}{r}_{B}(t_{2})-\overset{ \rightarrow}{r}_{A}\left(t_{2}-\frac{\rho_{AB}}{\text{C}}\right)\right|+\text{ C}\\ \cdot(dt_{B}(t_{0})-dt_{A}(t_{0}))-\text{C}\cdot\left(dt_{B}(t_{2}) -dt_{A}(t_{2}-\frac{\rho_{AB}}{\text{C}})\right)\end{split}  \tag{10}\]  
The accuracy of \(\Delta\rho_{BA}\) and \(\Delta\rho_{AB}\) depends on the changing rate of satellite orbit and clock offsets computed by broadcast messages, and the accuracy loss is much smaller than the ISL residuals, which can be ignored in a short time [11].