## 3 Two-Way Ranging Formulations  
Before the launch of GRACE, [PERSON] provided the dual one-way ranging (DOWR) formulations for KBR in his thesis [23]. In this section, the two-way ranging formulations for LRI with similar expressions are described. The cumulated carrier phase measurement between two satellites at the specified nominal time \(t_{1}\) and \(t_{2}\) can be modeled as follows:  
\[ \begin{split}\varphi_{1}(t_{1}+\delta t_{1})&=-( \varphi_{1}^{\mathrm{TX}}(t_{1}+\delta t_{1})-\varphi_{1}^{\mathrm{RX}}(t_{1}+ \delta t_{1})+N_{1}+\varepsilon_{1}),\\ \varphi_{2}(t_{2}+\delta t_{2})&=+(\varphi_{2}^{ \mathrm{TX}}(t_{2}+\delta t_{2})-\varphi_{2}^{\mathrm{RX}}(t_{2}+\delta t_{2} )+N_{2}+\varepsilon_{2}),\end{split}  \tag{2}\]  
where  
\[ \begin{split}\varphi_{1}(t_{1}+\delta t_{1})&=\text{ carrier phase measurement at SAT1},\\ \varphi_{2}(t_{2}+\delta t_{2})&=\text{carrier phase measurement at SAT2},\\ \varphi_{1}^{\mathrm{TX}}(t_{1}+\delta t_{1})&=\text{ cumulated phase of TX/LO light at SAT1},\\ \varphi_{2}^{\mathrm{TX}}(t_{2}+\delta t_{2})&=\text{ cumulated phase of TX/LO light at SAT2},\\ \varphi_{1}^{\mathrm{RX}}(t_{1}+\delta t_{1})&=\text{ cumulated phase of RX light at SAT1},\\ \varphi_{2}^{\mathrm{RX}}(t_{2}+\delta t_{2})&=\text{ cumulated phase of RX light at SAT2},\\ t_{1},\ t_{2}&=\text{nominal time of SAT1 and SAT2},\\ \delta t_{1},\ \delta t_{2}&=\text{time tag error of SAT1 and SAT2},\\ N_{1},\ N_{2}&=\text{phase measurement bias of SAT1 and SAT2},\\ \varepsilon_{1},\ \varepsilon_{2}&=\text{phase readout noise of SAT1 and SAT2},\end{split} \]  
where SAT1 and SAT2 represent the master satellite and the transponder satellite, respectively. \(\delta t_{1}\) and \(\delta t_{2}\) represent the difference between the actual time and the nominal time of phase measurement. The sign of the phase expression of SAT2 in front of the brackets on the right-hand side should be negative to obtain positive phase measurement because the phasemeter interprets the frequency of each input signal as positive and \(\varphi_{2}^{\mathrm{TX}}(t_{2}+\delta t_{2})\) is assumed to be larger than \(\varphi_{2}^{\mathrm{RX}}(t_{2}+\delta t_{2})\) due to the frequency offset \(+10\,\mathrm{MHz}\). Figure 2 shows the phase observables of the two satellites at the actual time and their combinations \(\varphi_{\mathrm{LRI}}(t_{1})\). The light (red line) from the master (SAT1 with a dashed border) arrives at the transponder (SAT2 with a solid border) after a travel time \(\tau_{12}\). Similarly,  
the light (blue line) from the transponder arrives at the master (SAT1 with a solid border) after a travel time \(\tau_{21}\). The time-of-flight (TOF) \(\tau_{12}\) and \(\tau_{21}\) can be precisely calculated with an iteration process [24]. The nominal time \(t_{1}\) and \(t_{2}\) satisfy the equation  
\[t_{2}=t_{1}-\tau_{21}. \tag{3}\]  
The cumulated phase of RX light is equal to the phase of the corresponding TX light at the transmitting time, i.e.,  
\[ \begin{split}\varphi_{1}^{\text{RX}}(t_{1}+\delta t_{1})& =\varphi_{2}^{\text{TX}}(t_{1}+\delta t_{1}-\tau_{21}),\\ \varphi_{2}^{\text{RX}}(t_{2}+\delta t_{2})&= \varphi_{1}^{\text{TX}}(t_{2}+\delta t_{2}-\tau_{12}).\end{split}  \tag{4}\]  
Due to the frequency offset introduced in the OPLL, the phase of interference light in SAT2 increases fast as a ramp, the slope of which is roughly \(+10\,\text{MHz}\). By writing the OPLL phase contributions as \(\varphi_{2}^{\text{OPLL}}\), the phase of the TX light in SAT2 can be expressed as  
\[\varphi_{2}^{\text{TX}}(t_{2}+\delta t_{2})=\varphi_{2}^{\text{OPLL}}(t_{2}+ \delta t_{2})+\varphi_{2}^{\text{RX}}(t_{2}+\delta t_{2}). \tag{5}\]  
Substituting Equations (3)-(5) into Equation (2) yields  
\[