# An Efficient BDS-3 Long-Range Undifferenced Network RTK Positioning Algorithm
## 2 Long-Range URTK Algorithm
### Undifferenced Error Correction Value Calculation
\begin{cases}Cor^{p}_{A,i}=\lambda_{i}\varphi_{A,i}-\rho^{p}_{A}+\lambda_{i}N ^{p}_{A,i}=c(t_{A}-t^{p})+O^{p}-I^{p}_{A,i}+T^{p}-\lambda_{i}(\sigma_{A}- \sigma^{p})+\varepsilon^{p}_{A,i}\\ Cor^{p}_{B,i}=\lambda_{i}\varphi_{B,i}-\rho^{p}_{B}+\lambda_{i}N^{p}_{B,i}=c(t_ {B}-t^{p})+O^{p}-I^{p}_{B,i}+T^{p}-\lambda_{i}(\sigma_{B}-\sigma^{p})+ \varepsilon^{p}_{B,i}\\ Cor^{p}_{C,i}=\lambda_{i}\varphi_{C,i}-\rho^{p}_{C}+\lambda_{i}N^{p}_{C,i}=c(t_ {C}-t^{p})+O^{p}-I^{p}_{C,i}+T^{p}-\lambda_{i}(\sigma_{C}-\sigma^{p})+ \varepsilon^{p}_{C,i}\end{cases}  \tag{13}\]  
where \(Cor^{p}_{A,i}\), \(Cor^{p}_{B,i}\), and \(Cor^{p}_{C,i}\) are undifferenced error corrections of satellite \(p\) at reference stations \(A\), \(B\), and \(C\), respectively. According to Equation (13), the undifferenced error correction values mainly address the satellite and receiver clock errors, satellite and receiver hardware delays, ionospheric delay errors, tropospheric delay errors, and orbit errors. In network RTK, a comprehensive undifferenced error correction value can be directly used for the error correction of a rover station. However, in long-range situations, the correction value obtained via the comprehensive error processing method is not accurate. Therefore, it is necessary to classify the undifferenced error correction values according to the nature of the error. The schematic diagram of the classification region error interpolation method is shown in Figure 2, where \(A\), \(B\) and \(C\) denote reference stations, \(U\) is a rover station, and \(IPP\) represents an ionospheric puncture point. The comprehensive undifferenced error correction value contains mainly undifferenced dispersion error dominated by ionospheric delays and undifferenced non-dispersion error dominated by tropospheric delays. Due to the different error characteristics, it is necessary to classify the comprehensive undifferenced error correction value. The undifferenced dispersion error was calculated using the geometry-free (\(GF\)) model, and the non-dispersion error was calculated using the ionospheric-free (\(IF\)) model. The solution process is shown below:  
\[ \begin{cases}GF(Cor^{p}_{r,ij})=Cor^{p}_{r,i}-Cor^{p}_{r,j}=I^{p}_{r,j}-I^{p}_ {r,i}+\lambda_{i}(\sigma_{r}-\sigma^{p})-\lambda_{j}(\sigma_{r}-\sigma^{p})+ \varepsilon^{p}_{r,i}-\varepsilon^{p}_{r,j}\\ IF(Cor^{p}_{r,ij})=\alpha_{i,i,j}Cor^{p}_{r,i}-\alpha_{j,i,j}Cor^{p}_{r,j}=c( t_{r}-t^{p})+T^{p}+\lambda_{i}\alpha_{i,i,j}\sigma_{r}-\lambda_{j}\alpha_{j,i,j} \sigma_{r}\\ \hskip 28.452756 pt+\lambda_{j}\alpha_{j,j}\sigma^{p}-\lambda_{i}\alpha_{i,j,j }\sigma^{p}+\alpha_{i,j,j}\varepsilon^{p}_{r,i}-\alpha_{j,j,i}\varepsilon^{p} _{r,j}\end{cases}  \tag{14}\]  
where \(j\) is the other frequency, and \(\alpha_{i,i,j}=\frac{f_{i}^{2}}{f_{i}^{2}-f_{i}^{2}}\), \(\alpha_{j,i,j}=\frac{f_{i}^{2}}{f_{i}^{2}-f_{j}^{2}}\) is the ionosphere-free combination operator. It can be seen from Equation (14) that both dispersion and non  
Figure 2: The classification error interpolation diagram. dispersion error are influenced by other frequency phase hardware delays and observation noise. The error expressions after classification are provided below:  
\[\left\{ \begin{aligned} & Dis_{r,i}^{p}=-\alpha_{j,ij}(Cor_{r,i}^{p}- Cor_{r,j}^{p})\\ & UDis_{r,i}^{p}=\alpha_{i,ij}Cor_{r,i}^{p}-\alpha_{j,i,j}Cor_{r,j}^{ p}\\ & Cor_{r,i}^{p}=Dis_{r,i}^{p}+UDis_{r,i}^{p}\end{aligned} \right. \tag{15}\]  
where \(Dis_{r,i}^{p}\) is the undifferenced dispersion error, and \(UDis_{r,i}^{p}\) is the undifferenced non-dispersion error. The undifferenced non-dispersion error is interpolated on the Earth's surface, where the reference and rover stations are located, and the interpolation calculation of the undifferenced dispersion error of the rover station is performed on the interpolation plane at the height of the central ionosphere [24].