# Performance Analysis of Undifferenced NRTK Considering Time-Varying Characteristics of Atmosphere
## 2 Positioning Model of BDS Undifferenced NRTK
### Mathematical Model for Ambiguity Fixing of Reference Stations  
When reference station coordinates are known, the inter-station differential carrier phase and pseudo-range observation equations of reference stations \(A\) and \(B\) can be expressed as:  
\[\Delta L_{AB,j}^{s}=\Delta T_{AB}^{s}+\Delta dt_{AB}+\Delta\lambda_{j}\cdot \delta_{AB,j}^{s}-\mu_{j}\cdot\Delta t_{AB}^{s}+\lambda_{j}\cdot\Delta N_{AB, j}^{s}+\epsilon(\Delta L_{AB,j}^{s}) \tag{21}\]  
\[\Delta P_{AB,j}^{s}=\Delta T_{AB}^{s}+\Delta dt_{AB}+\Delta\lambda_{j}\cdot \delta_{AB,j}^{s}+\mu_{j}\cdot\Delta t_{AB}^{s}+\epsilon(\Delta P_{AB,j}^{s}) \tag{22}\]where \(\Delta\) is the inter-station differential operator. The index of the receiver is omitted, and the ionospheric delay parameter \(\Delta_{r}^{s}\) is expressed as the ionospheric delay parameter at the first frequency. The DD observation equation is as follows:  
\[\
abla\Delta L_{AB,j}^{sq}=\
abla\Delta T_{AB}^{sq}-\mu_{j}\cdot\left(\Delta_{ A}^{s}-\Delta t_{AB}^{q}\right)+\lambda_{j}\cdot\left(\Delta N_{AB,j}^{s}- \Delta N_{AB,j}^{q}\right)+\varepsilon(\
abla\Delta L_{AB,j}^{sq}) \tag{23}\]  
\[\
abla\Delta P_{AB,j}^{sq}=\
abla\Delta T_{AB}^{sq}+\mu_{j}\cdot\left(\Delta_{ AB}^{s}-\Delta t_{AB}^{q}\right)+\varepsilon(\
abla\Delta P_{AB,j}^{sq}) \tag{24}\]  
The superscript \(sq\) is the observed satellite and the selected reference satellite with highest altitude angle, respectively. Each satellite in the equation has independent integer ambiguity parameters and ionospheric parameters, avoiding correlation of the satellites and facilitating the analysis of independent satellites, which will make data processing simpler and more efficient. By adding reference constraints for ionospheric error and ambiguity in the subsequent solution process, it can be equivalent to the ordinary DD model [24]. The tropospheric error is expressed by the tropospheric mapping function \(m^{s,q}\) and the residual relative zenith tropospheric delay (RZTD) \(\
abla\Delta\tau_{AB}\)[25; 26]:  
\[\
abla\Delta T_{AB}^{s,d}\approx m^{s,q}\cdot\
abla\Delta\tau_{AB} \tag{25}\]  
Before the tropospheric delay error is estimated by the mapping function, most of the tropospheric delays can be corrected by the prior [PERSON] model [27] ([PERSON] et al. 1972), and the residual wet delay part is estimated with a random walk process. The functional model for solving integer ambiguity, RZTD, and ionospheric error can be expressed as:  
\[E(L)=BX\Rightarrow E \begin{cases}\
abla\Delta L_{AB,j}^{sq}\\ \
abla\Delta P_{AB,j}^{sq}\end{cases} = \begin{cases}m^{sq}&-\mu_{j}&\mu_{j}&\lambda _{j}&-\lambda_{j}\\ m^{sq}&\mu_{j}&-\mu_{j}&0&0\end{cases} \cdot \begin{cases}\
abla\Delta\tau_{AB} \\ \Delta_{AB}^{s}\\ \Delta t_{AB}^{s}\\ \Delta N_{AB,j}^{q}\\ \Delta N_{AB}^{q}\end{cases}  \tag{26}\]  
In Equation (26), the ionospheric parameters and ambiguity parameters remain single satellite. In order to avoid a situation where the equation cannot be solved, there is a strong constraint condition that the ionospheric and the ambiguity parameters of the reference satellite be set to 0. The stochastic model of observations can be represented as:  
\[D(L)=Q_{LL}\Rightarrow D \begin{cases}\
abla\Delta L_{i}^{,sq}-\
abla\Delta \rho^{sq}\\ \
abla\Delta P_{i}^{,sq}-\
abla\Delta\rho^{sq}\end{cases} =T\cdot \begin{cases} \sigma_{\varphi}^{2}(E)&0\\ 0&\sigma_{\mathbb{P}}^{2}(E)\end{cases} \cdot T^{T} \tag{27}\]  
In Equation (27), \(D\{\cdot\}\) represents the variance operator, \(T\) represents the transformation matrix from the original UD observation to the DD observation. In this paper, the elevation-dependent weighting mode of the UD observation is used [28; 29], and the elevation \(E\) is used to obtain the standard deviation \(\sigma_{\phi}(E),\sigma_{\mathbb{P}}(E)\) of the satellite signal propagation path, thereby obtaining the prior variance covariance matrix of the UD observation.