## 2 System Principle
### Pseudorange Observation Equation for Receivers in Tunnels  
In the tunnel environment shown in Figure 1, the pseudorange observation equation for the receiver is given by Equation (2):  
\[\dot{p}_{j}^{i}=r_{j}^{i}+c\cdot\biggl{(}\delta tr_{j}-\delta t_{j}^{i}\biggr{)} +c\cdot t_{j}+I_{j}^{i}+T_{j}^{i}+S_{j}+\epsilon_{j}^{i} \tag{2}\]  
In the equation, j = BDS, GPS, GLONASS represents different GNSS systems, and I = 1, 2,..., N represents the satellite number under different GNSS systems. \(\rm p_{j}^{i}\) indicates the pseudorange observation; \(\rm r_{j}^{i}\) refers to the geometric distance from the satellite to the antenna; \(\rm\delta tr_{j}\) signifies the receiver clock bias when processing signals from different  
Figure 6: (**a**) Slotted structure of inclined-type LCX cable; (**b**) signal propagation of inclined-type LCX cable. systems; \(\delta\)t\({}_{\text{j}}\) stands for the satellite clock bias; \(\text{t}_{\text{j}}^{\text{i}}\) corresponds to the frequency division relay processing delay caused by the signal from different GNSS systems being received by the GNSS antenna and transmitted to the LCX cables port; \(\text{t}_{\text{j}}^{\text{i}}\) accounts for the ionospheric error; \(\text{T}_{\text{j}}^{\text{i}}\) signifies the tropospheric error; \(\text{S}_{\text{j}}\) describes the geometric path of GNSS signals propagating within the tunnel; and \(\text{\epsilon}_{\text{j}}^{\text{i}}\) represents the pseudorange observation noise of the receiver. The same receiver exhibits different processing delays when handling signals from different GNSS systems. Using the receiver clock bias for BDS signal processing as the reference, the receiver clock biases for the GPS and GLOASS systems are given by Equation (3):  
\[ \begin{cases}\delta tr_{GPS}=\ \delta tr_{BDS}+\tau_{1}\\ \delta tr_{GLO}=\delta tr_{BDS}+\tau_{2}\end{cases}  \tag{3}\]  
In Equation (3), \(\delta\)t\({}_{\text{rj}}(\text{j}=\text{BDS},\text{GPS},\text{GLO})\) represents the receiver clock bias when processing the corresponding GNSS signals; \(\tau_{1}\) denotes the processing delay of the receiver for GPS signals compared to BDS signals; and \(\tau_{2}\) denotes the processing delay for GLO signals compared to BDS signals. In practical applications, \(\tau_{1}\) and \(\tau_{2}\) can be measured to obtain specific values. Because BDS, GPS, and GLONASS use different time references, it is necessary to synchronize the satellite clock biases of different GNSS systems when the receiver calculates the position based on the pseudorange differences from the three systems. According to the official BDS ICD document, BDS Time (BDT) can be synchronized with other GNSS systems through the relevant parameters broadcast in the BDS navigation message. BDT can be synchronized with GPS time using Equation (4):  
\[\delta t_{GPS}=t_{E}-\Delta t_{GPS} \tag{4}\]  
In Equation (4), \(t_{E}\) refers to the BDT calculated by the user, and \(\Delta t_{GPS}\) represents the GPS time synchronization correction term, whose calculation formula is provided in Equation (5):  
\[\Delta t_{GPS}=A_{0 GPS}-A_{1 GPS}\cdot t_{E} \tag{5}\]  
In Equation (5), \(\text{A}_{0\text{GPS}}\) represents the clock bias of BDT relative to GPS system time, and \(\text{A}_{1\text{GPS}}\) represents the clock drift of BDT relative to GPS system time. Both are synchronization parameters between BDT and GPS time, broadcast through the BDS navigation message. The calculation formulas for synchronizing BDT with GLONASS time are provided in Equations (6) and (7):  
\[\delta t_{GLO}=t_{E}-\Delta t_{GLO} \tag{6}\]  
\[\Delta t_{GLO}=A_{0 GLO}-A_{1 GLO}\cdot t_{E} \tag{7}\]  
In Equation (7), \(\text{A}_{0\text{GLO}}\) represents the clock bias of BDT relative to GLONASS system time, and \(\text{A}_{1\text{GLO}}\) represents the clock drift of BDT relative to GLONASS system time. Both are broadcast through the BDS navigation message. By organizing the above equations, the pseudorange observation equations for the receiver in the tunnel environment are given in Equation (8):  
\[