## 2 System Principle
### Two-Dimensional Differential Positioning Algorithm for Tunnels with Dual T-Type LCX Cables Layout  
As shown in Figure 8, the signal propagation paths in the dual T-type LCX cables layout consist of two parallel LCX cables, both with a length of L meters, and the distance between them is H meters. The coordinates of the receivers in the tunnel coordinate system are (x, y), while the reference receiver R is positioned at (L/2, H/2). The B1I signal of the BDS system drives the port A1 of LCX_1 and radiates vertically to the receiver at points Ci (i = 1, 2,..., N). The G1 signal of the GLO system excites the B1 segment of LCX_1 and radiates vertically to the receiver at points Ci (i = 1, 2,..., N). The L1 signal of the GPS system excites the port A2 of LCX_2 and radiates vertically to the receiver at points D1 (i = 1, 2,..., N). Figure 8: Signal propagation path in the dual T-type LCX cables layout. Figure 7: Observation error of GNSS signal transmission distance in LCX by the receiver. In this layout, after applying the VEM correction, the geometric paths of signals from different GNSS systems transmitted from the LCX ports to the receiver are given by Equation (11):  
\[\left\{ \begin{array}{l}S_{BDS}=\frac{\mathbb{C}}{\mathbb{V}}\cdot A_{1}C_{i}+C_{ i}U\\ S_{GPS}=\frac{\mathbb{C}}{\mathbb{V}}\cdot A_{2}D_{i}+D_{i}U\\ S_{GLO}=\frac{\mathbb{C}}{\mathbb{V}}\cdot B_{1}C_{i}+C_{i}U\end{array} \right. \tag{11}\]  
In Equation (11), A\({}_{1}\)C\({}_{i}\), A\({}_{2}\)D\({}_{i}\), B\({}_{1}\)C\({}_{i}\) represent the propagation distance of the GNSS signal within the LCX, and C\({}_{i}\)U, D\({}_{i}\)U represent the propagation distance of the GNSS signal through the air from the LCX to the receiver. Based on the geometric relationship between the tunnel port and the receiver coordinates (x, y), the signal transmission paths are given by Equation (12):  
\[\left\{ \begin{array}{l}S_{BDS}=\frac{\mathbb{C}}{\mathbb{V}}\cdot x+y\\ S_{GPS}=\frac{\mathbb{C}}{\mathbb{C}}\cdot x+(H-y)\\ S_{GLO}=\frac{\mathbb{C}}{\mathbb{V}}\cdot(L-x)+y\end{array} \right. \tag{12}\]  
By substituting Equation (12) into Equation (8), the pseudorange observation equations for the receiver in this layout are obtained, as shown in Equation (13):  
\[\left\{ \begin{array}{l}\rho_{BDS}^{i}=r_{BDS}^{i}+C\cdot(\delta t_{BDS}- \delta t_{BDS}^{i})+C\cdot t_{BDS}+I_{BDS}^{i}+T_{BDS}^{i}+\frac{\mathbb{C}}{ \mathbb{V}}\cdot x+y+\dot{\xi_{BDS}}\\ \rho_{GPS}^{i}=r_{GPS}^{i}+C\cdot((\delta t_{BDS}+\tau_{1})-((1-A_{1 GPS}) \cdot\delta t_{BDS}^{i}-A_{0 GPS}))+C\cdot t_{GPS}+I_{GPS}^{i}+T_{GPS}^{i}+ \frac{\mathbb{C}}{\mathbb{V}}\cdot(L-x)+y+\dot{\xi_{GPS}^{i}}\\ \rho_{GLO}^{i}=r_{GLO}^{i}+C\cdot((\delta t_{BDS}+\tau_{2})-((1-A_{1 GLO}) \cdot\delta t_{BDS}^{i}-A_{0 GLO}))+C\cdot t_{GLO}+I_{GLO}^{i}+T_{GLO}^{i}+ \frac{\mathbb{C}}{\mathbb{V}}\cdot x+H-y+\dot{\xi_{GLO}^{i}}\end{array} \right. \tag{13}\]  
By subtracting Equation (a) from Equation (c) in Equation (13), the expression for calculating the receiver's vertical coordinate is obtained, as shown in Equation (14):  
\[ \begin{array}{rl}y=&\frac{1}{2}\left[(\rho_{BDS}^{i}-\rho_{GPS}^{i})-(r_{BDS }^{i}-r_{GPS}^{i})+C\cdot(A_{1 GPS}\cdot\delta t_{BDS}^{i}+A_{0 GPS}+\tau_{1}) \right.\\ &\left.-C\cdot(t_{BDS}^{i}-t_{GPS}^{i})-(I_{BDS}^{i}-I_{GPS}^{i})-(T_{BDS}^{i} -T_{GPS}^{i})+(\dot{\epsilon_{BDS}^{i}}-\dot{\epsilon_{GPS}^{i}})\right]+H \end{array}  \tag{14}\]  
By subtracting Equation (a) from Equation (b) in Equation (13), the expression for calculating the receiver's horizontal coordinate is obtained, as shown in Equation (15):  
\[ \begin{array}{rl}x=&\frac{\mathbb{V}}{2\mathbb{C}}\left[(\rho_{BDS}^{i}- \rho_{GLO}^{i})-(r_{BDS}^{i}-r_{GLO}^{i})+C\cdot(A_{1 GLO}\delta t_{BDS}^{i}+A_ {0 GLO}+\tau_{2})\right.\\ &\left.-C\cdot(t_{BDS}^{i}-t_{GLO}^{i})-(I_{BDS}^{i}-I_{GLO}^{i})-(T_{BDS}^{i} -T_{GLO}^{i})+(\dot{\epsilon_{BDS}^{i}}-\dot{\epsilon_{GLO}^{i}})+\frac{ \mathbb{C}}{\mathbb{V}}\cdot L\right]\end{array}