# Performance Analysis of Undifferenced NRTK Considering Time-Varying Characteristics of Atmosphere
## 2 Positioning Model of BDS Undifferenced NRTK
### User Error Correction and Positioning  
From the error correction values provided for reference stations, it can be seen that the UD error correction values include the clock errors of receiver and satellite, the hardware delay errors of receiver and satellite, and the correction information for tropospheric and ionospheric errors. This section will analyze the user errors elimination process. Compared with the reference station, user stations require estimation of the user's position parameters, and the UD carrier observation equation for user stations is:  
\[ \begin{array}{ll}L_{r,j}^{s}(i)&=\rho_{r}^{s}(i)+H_{r}^{s}(i)x_{r}(i)+T_{r}^ {s}(i)+dt_{r}(i)+\lambda_{j}\delta_{r,j}(i)\\ &-d^{s}(i)-\lambda_{j}\delta_{j}^{s}(i)-\mu_{j}\cdot\epsilon_{r}^{s}(i)+\lambda _{j}N_{r,j}^{s}+\varepsilon_{L_{r,j}^{s}}\end{array}  \tag{17}\]  
where \(H\) is the coefficient corresponding to the station position parameters \(x_{r}\), other symbols have the same meaning as in Equation (1). From Equation (17), it can be seen that the observations contain phase delay parameters related to integer ambiguity, which will affect the integer characteristics of ambiguity in parameter estimation. The user station needs to receive error correction information provided by the reference station. On the one hand, it is used to restore the ambiguity integer characteristics of the user station, and on the other hand, it is used to eliminate atmospheric errors for the user, and to accelerate the convergence speed and positioning accuracy of user stations. The OSR correction information provided by the reference station and the error correction process of the user station are shown in Table 1:  
From Table 1, it can be seen that the satellite errors are consistent between the reference station and the user, that is, the satellite errors are directly eliminated through the error correction value of the reference station. On the basis of the error correction of the reference station, the user can eliminate the receiver errors through inter-satellite differential. The atmospheric errors of the user station corrected by a single reference station are: \(T_{r}^{s}-T_{A}^{s}\), \(t_{r}^{s}-t_{A}^{s}\). When the distance between the single reference station and the user station is very close, most of the errors can be eliminated by the error correction value. If the user station has satellites \(p\) and \(q\), the atmospheric errors after inter-satellite differential are: \(T_{rA}^{pq}:\left(T_{r}^{p}-T_{A}^{p}\right)-\left(T_{r}^{q}-T_{A}^{q}\right)\), \(r_{rA}^{pq}:\left( \begin{array}{c}p_{r}^{p}-p_{A}^{q}\end{array} \right)- \left( \begin{array}{c}q_{r}^{q}-q_{A}^{q}\end{array} \right)\); at this time, the atmospheric errors are equivalent to those in conventional ITK, both in a PD form. When three reference stations provide error correction information, \(T_{UA}^{pq}:\left\{T_{r}^{pq}-\left(\alpha_{A}\cdot T_{A}^{p}+\alpha_{B}\cdot T _{B}^{p}+\alpha_{C}\cdot T_{C}^{p}\right)\right\}-\left\{\tau_{r}^{q}-\left( \alpha_{A}\cdot T_{A}^{q}+\alpha_{B}\cdot T_{B}^{q}+\alpha_{C}\cdot T_{C}^{q} \right)\right\}\), \(r_{rA}^{q}:\left\{t_{r,j}^{p}-\left(\alpha_{A}\cdot t_{A,j}^{p}+\alpha_{B} \cdot t_{B,j}^{p}+\alpha_{B}\cdot t_{B,j}^{p}\right)\right\}-t_{r,j}^{q}\) (\(\alpha_{A}\cdot t_{A,j}^{q}+\alpha_{B}\cdot t_{B,j}^{q}+\alpha_{B}\cdot t_{B,j}^{q}+\alpha_{B}\cdot t_{B,j}^{q}\)), which is different from a single station by using the spatial correlation of the atmosphere to eliminate errors.