## 2 Measurement Model of RDSS  
BDS adopts an active positioning mode in the RDSS system, which adopts 4-way ranging observations. There are C-band uplink signals and S-band downlink signals in the out-station beam. There are L-band uplink signals and C-band downlink signals in the in-station beam. The out-station C-band uplink signals are transmitted by the master control centre (MCC) to a GEO satellite, the S-band downlink signals are transformed by GEOs to the RDSS user, the L-band uplink signals are transformed by the RDSS user to all observable GEO satellites, and the C-band downlink signals are transformed by GEOs to the MCC. The MCC accomplishes the 4-way ranging observations and estimates the position of the RDSS user within 1 s [5,6]. Figure 1 shows the schematic of the RDSS. The RDSS observati  
Figure 1: Schematic of the RDSS 4-way ranging observations. \[ \begin{split}&\rho_{S_{1}}=D_{Cl}+c\cdot\delta t_{CIO}+c\cdot\delta t_{ SiO}+D_{U\Pi}+c\cdot\delta t_{U\mu}+d_{U\mu}+c\cdot\delta t_{S1\Pi}+c\cdot \delta t_{CI\Pi}+\Delta E_{All\Pi}+\epsilon\\ &\rho_{S_{2}}=D_{Cl}+c\cdot\delta t_{CIO}+c\cdot\delta t_{SiO}+D_{ U\Pi}+c\cdot\delta t_{U\mu}+d_{U\mu}+c\cdot\delta t_{S2\Pi}+d_{C2}+c\cdot \delta t_{CI2\Pi}+\epsilon\\ &\tau_{all}=\delta t_{CIO}+\delta t_{SIO}+\delta t_{U}+\delta t_{ SI1}+\delta t_{CI\Pi}\end{split}  \tag{1}\]  
\[ \begin{split}& D_{Cl}=\sqrt{\left(X^{S1}(t_{1})-X_{C}\right)^{2}+ \left(Y^{S1}(t_{1})-Y_{C}\right)^{2}+\left(Z^{S1}(t_{1})-Z_{C}\right)^{2}}\\ & D_{U\Pi}=\sqrt{\left(X_{U}(t_{2})-X^{S1}(t_{1})\right)^{2}+ \left(Y_{U}(t_{2})-Y^{S1}(t_{1})\right)^{2}+\left([PERSON]_{U}(t_{2})-[PERSON]^{S1}(t_{1}) \right)^{2}}\\ & d_{U\Pi}=\sqrt{\left(X^{S1}(t_{3})-X_{U}(t_{2})\right)^{2}+ \left(Y^{S1}(t_{3})-Y_{U}(t_{2})\right)^{2}+\left([PERSON]^{S1}(t_{3})-[PERSON]_{U}(t_{2}) \right)^{2}}\\ &\qquad d_{C1}=\sqrt{\left([PERSON]^{S1}(t_{3})-X_{C}\right)^{2}+\left([PERSON] ^{S1}(t_{3})-[PERSON]_{C}\right)^{2}+\left([PERSON]^{S1}(t_{3})-[PERSON]_{C}\right)^{2}}\\ & d_{U2}=\sqrt{\left([PERSON]^{S2}(t_{4})-[PERSON]_{U}(t_{2})\right)^{2}+\left( [PERSON]^{S2}(t_{4})-[PERSON]_{U}(t_{2})\right)^{2}+\left([PERSON]^{S2}(t_{4})-[PERSON]_{U}\right)^{2}}\\ &\qquad d_{C2}=\sqrt{\left([PERSON]^{S2}(t_{4})-[PERSON]_{C}\right)^{2}+\left( [PERSON]^{S2}(t_{4})-[PERSON]_{C}\right)^{2}+\left([PERSON]^{S2}(t_{4})-[PERSON]_{C}\right)^{2}}\end{split}  \tag{2}\]  
where \(\rho_{S_{1}}\), \(\rho_{S_{2}}\) are the RDSS observation data for GEO01 and GEO02, respectively. \(D_{C1}\) is the geometric range between MCC and GEO01 satellite in the out-station C-band uplink; \(D_{U1}\) is the geometric range between GEO01 satellite and RDSS user in the out-station S-band downlink; \(d_{U1}\) is the geometric range between GEO01 satellite and RDSS user in the in-station L-band uplink; \(d_{C1}\) is the geometric range between MCC and GEO01 satellite in the in-station C-band downlink; \(d_{U2}\) is the geometric range between GEO02 satellite and RDSS user in the in-station L-band uplink; \(d_{C2}\) is the geometric range between MCC and GEO02 satellite in the in-station C-band downlink. \(\epsilon\) is the observation noise. \(c\) is the speed of light. \(\delta t_{CIO}\) is the transmission time delay in MCC in out-station downlink; \(\delta t_{SIO}\) is the time delay of the out-station transferring link for GEOs; \(\delta t_{U\mu}\) is the transmission time delay of the RDSS user; \(\delta t_{SI1}\), \(\delta t_{S2\Pi}\) are the time delay of the in-station transferring link for GEO01 and GEO02, respectively; \(\delta t_{C1I}\) and \(\delta t_{C2I}\) are the time delay of the receiver in MCC from GEO01 and GEO02, respectively. \(\tau_{all}\) is the sum of the time delays of the out-station beam and the in-station beam, including the transmission time delay in MCC and the time delay of the out-station transferring link for GEOs, the transmission time delay of the RDSS user, the time delay of the in-station transferring link for GEOs and the time delay of the receiver in MCC.