## 2 Submillimeter-Level Microwave Ranging  
### MWR Measurement Modeling  
MWR measures the biased distance between the antenna phase centers of two satellites through microwave carrier phase accumulation and Doppler drift, and then derives the distance variation and its rate between two satellites. Basically, the whole system is a synchronous dual one-way ranging (DOWR) microwave phase accumulation ranging system [16]. Three techniques are adopted by MWR that ensure ranging accuracy:  
* Correction of ionospheric influence by dual-frequency measurement during space flight around perigee;
* Synchronous bidirectional ranging comparison that cancels the long-term instability of the oscillator source; and,
* Carrier phase measurement by frequency difference to improve the accuracy. Figure 1 shows the principle of DOWR microwave phase measurement. The dual-way phase measurement is defined as the summation of one-way ranging:  
\[\Theta(t)\equiv\phi_{1}^{2}(t+\Delta t_{1})+\phi_{2}^{1}(t+\Delta t_{2}) \tag{1}\]  
and the carrier phase measurement received by satellite \(i\) at time \(t\) can be modeled as:  
\[ \begin{array}{c}\phi_{i}^{j}(t+\Delta t_{i})=\phi_{i}(t+\Delta t_{i})-\phi^ {j}(t+\Delta t_{i})+N_{i}^{j}+l_{i}^{j}+d_{i}^{j}+\varepsilon_{i}^{j}\\ i,j=1,2,\hskip 14.226378 pti\
eq j\end{array}  \tag{2}\]  
where \(\phi_{i}(t+\Delta t_{i})\) is the reference phase of satellite \(i\), \(\phi^{j}(t+\Delta t_{i})\) is the received signal phase from satellite \(j\), \(\Delta t_{i}\) is the oscillator instability error, \(N_{i}^{j}\) is the phase ambiguity, \(l_{i}^{j}\) is the phase bias due to ionosphere delay, \(d_{i}^{j}\) is the phase bias due to other reasons, and \(\varepsilon_{i}^{j}\) is the phase bias due to measurement noise. Substitute the receiving phase to sending phase  
\[\phi^{j}(t+\Delta t_{i})=\phi_{j}(t+\Delta t_{i}-\tau_{i}^{j}) \tag{3}\]  
Figure 1: Principle of dual one-way ranging. where \(\tau_{i}^{j}\) is the propagation time from satellite \(j\) to \(i\). Each phase \(\phi_{i}(t)\) is the summation of reference phase \(\ddot{\phi}_{i}\) and phase error due to oscillator \(\delta\phi_{i}\). Represent the phase rate \(\dot{\phi}_{i}(t)\) as reference frequency \(\ddot{f}_{i}\). Finally, the DOWR can be written as:  
\[ \begin{split}\Theta(t)=&(\ddot{f}_{1}\tau_{2}^{1}+ \ddot{f}_{2}\tau_{1}^{2})\\ &+\{[\delta\phi_{1}(t)-\delta\phi_{1}(t-\tau_{2}^{1})]+[\delta \phi_{2}(t)-\delta\phi_{2}(t-\tau_{1}^{2})]\}\\ &+(\ddot{f}_{1}-\ddot{f}_{2})(\Delta t_{1}-\Delta t_{2})\\ &+(\delta\dot{\phi}_{1}-\delta\dot{\phi}_{2})(\Delta t_{1}- \Delta t_{2})+N+I+d+\varepsilon\end{split}  \tag{4}\]  
where \(\ddot{f}_{1}\tau_{2}^{1}+\ddot{f}_{2}\tau_{1}^{2}\) denotes the real phase measurement;  
\(\{[\delta\phi_{1}(t)-\delta\phi_{1}(t-\tau_{2}^{1})]+[\delta\phi_{2}(t)- \delta\phi_{2}(t-\tau_{1}^{2})]\}\) denotes the phase error due to the oscillator, and the middle-term/long-term oscillator phase error can be eliminated if the propagation time is longer than one millisecond using DOWR;  
\((\ddot{f}_{1}-\ddot{f}_{2})(\Delta t_{1}-\Delta t_{2})\): time-tag error;  
\((\delta\dot{\phi}_{1}-\delta\phi_{2})(\Delta t_{1}-\Delta t_{2})\): coupling term of oscillator and time tag;  
\(N+I+d+\varepsilon\): other errors; and,  
\((\ddot{f}_{1}\tau_{2}^{1}+\ddot{f}_{2}\tau_{1}^{2})\approx(\ddot{f}_{1}+\ddot{ f}_{2})\tau+\Delta\Theta_{TOF}(t)\), with time of flight (TOF) defined as \(\tau=\rho(t)/c\). \(\Delta\Theta_{TOF}(t)\): correction of TOF, which can be calibrated using a precise orbit determination technique.