# Evaluation of Network RTK Positioning Performance Based on BDS-3 New Signal System
## 2 Network RTK Positioning Algorithm  
### Ambiguity Resolution of Reference Stations  
The fixed baseline ambiguity between reference stations is the premise for network RTK to realize atmospheric modeling and generate high precision VRS observations. For the baseline between network RTK long-distance stations, the commonly used solution model is the ionosphere-free combined model, which combines the observation domain and the ambiguity domain [16; 17; 19; 20]. When using this model, the wide lane ambiguity should be fixed after the moving serial average based on the Hatch-Melbourne-Wubbena (HMW) combination, and then the ionosphere-free combination ambiguity can be combined to determine the narrow lane ambiguity. Since the solution of MW combination is highly dependent on pseudo-range observations, it takes a long time for smoothing to achieve the fixed ambiguity of wide lane [21]. The non-combined model can assist the rapid convergence of long-distance baseline ambiguity parameters by adding ionospheric and tropospheric variance constraints, and the algorithm is relatively flexible with low noise of observation [21; 22]. In this paper, the non-combined model is used to process the baseline data between network RTK reference stations. After correcting the errors of antenna phase center, phase winding, and tropospheric dry component delay, the non-combined double-difference pseudo-range, and phase observation equations are as follows:  
\[P^{pq}_{rb,k}=\rho^{pq}_{rb}+\mu_{k}1^{pq}_{rb,1}+m^{pq}_{r}w_{r}-m^{pq}_{b}w_{ b}+\varepsilon_{P,k} \tag{1}\]  
\[L^{pq}_{rb,k}=\rho^{pq}_{rb}-\mu_{k}L^{pq}_{rb,1}+m^{pq}_{r}w_{r}-m^{pq}_{b}w_{ b}+\lambda_{k}N^{pq}_{rb,k}+\varepsilon_{Lk} \tag{2}\]where \(P_{rb}^{pq}\) and \(L_{rb}^{pq}\) are the pseudo-range and phase observation by distance; \(\rho_{rb}^{pq}\) represents the double-difference geometric distance between satellite and receiver; \(k\) is the frequency signal; \(l_{rb,1}^{pq}\) represents the oblique path ionospheric delay at \(f_{1}\); \(\mu_{k}\) represents the frequency-dependent ionospheric delay coefficient; \(\mu_{k}=f_{1}^{2}/f_{k}^{2}\); \(m\) and \(w\) represent the wet delay mapping function of the single-difference between the satellites and zenith tropospheric wet delay of the corresponding stations respectively; \(N\) is double-difference ambiguity for the corresponding frequency; \(\lambda\) is the wavelength for the corresponding frequency; \(b\) and \(r\) represent the reference station and the rover station, respectively; \(p\) and \(q\) refer to the reference satellite and the non-reference satellite, respectively; \(\varepsilon_{P}\) and \(\varepsilon_{L}\) represent the observed noise of pseudo-range and phase, respectively. It should be pointed out that single-difference and double-difference are mathematical solution methods. It is essentially the Gaussian elimination method in linear algebra. The main purpose is to eliminate or weaken the various error terms in the observation equation so as to facilitate the solution of the equation system [17]. As the coordinates of the reference station in network RTK mode are accurately known, \(\rho_{rb}^{pq}\) is the known value, and the observation equation of the baseline calculation error is as follows:  
\[E\left[ \begin{array}{c}\delta P\\ \delta L\end{array} \right]=\left[ \begin{array}{ccc}\mu_{k}&m_{r}&-m_{b}&0\\ -\mu_{k}&m_{r}&-m_{b}&\lambda_{k}\end{array} \right]\left[ \begin{array}{c}I_{ 1}\\ w_{r}\\ w_{b}\\ N_{k}\end{array} \right] \tag{3}\]  
where \(E[\cdot]\) is expected value, \(\delta P\) and \(\delta L\) are the residual vectors of the double-difference pseudo-range and phase observations. \(I_{1}\), \(w_{r}\), \(w_{b}\) and \(N_{k}\) represent the double-difference oblique path ionospheric delay, zenith troposphere wet delay of rover station, zenith troposphere wet delay of the base station, and double-difference ambiguity, respectively. \(\mu_{k}\), \(m_{r}\), \(m_{b}\), and \(\lambda_{k}\) refer to the corresponding coefficient matrix.