# An Efficient BDS-3 Long-Range Undifferenced Network RTK Positioning Algorithm
## 2 Long-Range URTK Algorithm
### Reference Station Network Double-Difference Integer AR  
Due to the long range, of more than 100 km, between the long-range network RTK reference stations, the effect of ionospheric and tropospheric delays on the double-difference observations is much stronger than that of the 0.5 cycle. Even when the multi-frequency observation data and the reference station coordinates are known, the integer ambiguity and error are difficult to separate. In this study, the undifferenced observation equations of phase and pseudo-range linearization used in the long-range network RTK algorithm are respectively expressed as follows [22]:  
\[\lambda_{i}\varphi_{i}=H\cdot X+\rho_{r}^{s}+c(t_{r}-t^{s})+O^{s}-I_{r,i}^{s}+T ^{s}-\lambda_{i}(N_{r,i}^{s}+\sigma_{r}-\sigma^{s})+\epsilon_{r,i}^{s} \tag{1}\]  
\[P_{i}=H\cdot X+\rho_{r}^{s}+c(t_{r}-t^{s}+d_{r}-d^{s})+O^{s}+I_{r,i}^{s}+T^{s}+ \delta_{r,i}^{s} \tag{2}\]  
where \(\lambda_{i}\) is the phase observation wavelength; \(\varphi_{i}\) denotes the phase observation data; \(H\) is the coefficient of a position parameter \(X\); \(\rho_{r}^{s}\) is the distance between stations and satellites; \(c\) is the speed of light in a vacuum; \(t_{r}\) is the receiver clock error; \(t^{s}\) is the satellite clock error; \(O^{s}\) is the satellite orbit error; \(I\) denotes the ionospheric delay error; \(T\) represents the tropospheric delay error; \(N\) is the integer ambiguity; \(\sigma_{r}\) is the phase hardware delay of a receiver; \(\sigma^{s}\) is the phase hardware delay of a satellite; \(\epsilon_{r,i}^{s}\) is the phase observation noise; superscript \(s\) represents the satellite number, and subscript \(i\) represents different frequencies; \(P\) is the pseudo-range observation value; \(d_{r}\) is the pseudo-range hardware delay of the  
Figure 1: The flowchart of the long-range URTK algorithm. receiver; \(d^{s}\) is the pseudo-range hardware delay of the satellite; and \(\delta^{s}_{r,i}\) represents the pseudo-range measurement noise. Consider the synchronous observation of the first-frequency carrier phase observations of a reference satellite \(q\) and a satellite \(p\), obtained via the three reference stations denoted as \(A\), \(B\), and \(C\), where the coordinates of the reference station are accurately known; then, the observation equation for the carrier phase integer AR of the reference station is obtained as follows:  
\[ \begin{cases}\Delta\
abla L_{AB}^{pq}=(I_{A,i}^{p}-I_{B,i}^{p})-(I_{A,i}^{q}- I_{B,i}^{q})+(Map_{A}^{p}-Map_{A}^{q})RZTD+\lambda_{i}\Delta\
abla N_{AB}^{pq}\\ \Delta\
abla L_{BC}^{pq}=(I_{B,i}^{p}-I_{C,i}^{p})-(I_{B,i}^{q}-I_{C,i}^{q})+(Map _{B}^{p}-Map_{B}^{q})RZTD+\lambda_{i}\Delta\
abla N_{BC}^{pq}\\ \Delta\
abla L_{CA}^{pq}=(I_{C,i}^{p}-I_{A,i}^{p})-(I_{C,i}^{q}-I_{A,i}^{q})+( Map_{C}^{p}-Map_{C}^{q})RZTD+\lambda_{i}\Delta\
abla N_{CA}^{pq}\end{cases}  \tag{3}\]  
where,  
\[ \begin{cases}\Delta\
abla L_{AB}^{pq}=(\rho_{B}^{p}-\rho_{B}^{q}-\rho_{A}^{p} +\rho_{A}^{q})-(\varphi_{B}^{p}\lambda_{i}-\varphi_{B}^{q}\lambda_{i}-\varphi_ {A}^{p}\lambda_{i}+\varphi_{A}^{q}\lambda_{i})\\ \Delta\
abla N_{AB}^{pq}=(N_{B,i}^{p}-N_{B,i}^{p})-(N_{A,i}^{q}-N_{A,i}^{q}) \end{cases}  \tag{4}\]  
where \(\Delta\
abla\) is a double-difference operator, \(L\) is a constant term, \(Map\) is the tropospheric projection function, and \(RZTD\) denotes the relative zenith tropospheric delay error of two reference stations. The main errors in the observation equation are non-dispersive errors, which are dominated by tropospheric delay and dispersive errors [12]. The GMF projection function and the zenith tropospheric delay error are used to represent the tropospheric delay error in a reference station's observations [29]. The projection function values of the same satellite of two adjacent reference stations are close, and the zenith tropospheric delay error is included in the relative zenith tropospheric delay error parameter.