# Performance Analysis of Undifferenced NRTK Considering Time-Varying Characteristics of Atmosphere
## 2 Positioning Model of BDS Undifferenced NRTK
### Generation and Interpolation of UD Error Correction Value for Users  
The key to generating the UD error correction value is the estimation of UD integer ambiguity on the reference stations. After passing the strict DD ambiguity check conditions, the DD ambiguity has been successfully determined. When generating UD error correction values, it is necessary to convert the DD ambiguity to UD ambiguity to achieve the separation of UD error correction values and establish a UD error correction model. Starting from the combination relationship between the reference station and the satellite, the separation of DD integer ambiguity is achieved by setting the reference ambiguity of the reference station and reference satellite. The implementation process is shown in Figure 1.  
Taking the B1 frequency DD integer ambiguity as an example, the B1 UD integer ambiguity of the satellite \(p\), \(k\), and \(q\) on reference stations \(A\), \(B\), and \(C\) can be obtained:  
\[ \begin{cases}\Delta\
abla N_{AB,1}^{pq}=N_{A,1}^{p}-N_{A,1}^{q}+N_{B,1}^{q}-N_{B,1}^{p}\\ \Delta\
abla N_{BC,1}^{pq}=N_{B,1}^{p}-N_{B,1}^{q}+N_{C,1}^{q}-N_{C,1}^{p}\\ \Delta\
abla N_{CA,1}^{pq}=N_{C,1}^{p}-N_{C,1}^{q}+N_{A,1}^{q}-N_{A,1}^{p}\\ \end{cases}  \tag{9}\]  
\[ \begin{cases}\Delta\
abla N_{AB,1}^{kq}=N_{A,1}^{k}-N_{A,1}^{q}+N_{B,1}^{q}-N_{ B,1}^{k}\\ \Delta\
abla N_{BC,1}^{kq}=N_{B,1}^{k}-N_{B,1}^{q}+N_{C,1}^{q}-N_{C,1}^{k}\\ \Delta\
abla N_{CA,1}^{kq}=N_{C,1}^{k}-N_{C,1}^{q}+N_{A,1}^{q}-N_{A,1}^{k}\\ \end{cases}  \tag{10}\]  
In Equations (9) and (10), the left side of the equation is the known DD integer ambiguity and the right side is the UD integer ambiguity. The integer ambiguity related to reference station \(A\) and satellite \(q\) are selected as the reference ambiguity, which are known and to be set to an arbitrary constant value. Use Equations (11) and (12) to set the reference ambiguity. \[ \begin{cases}N_{A,1}^{p}=0\\ N_{A,1}^{q}=0\\ N_{A,1}^{k}=0\\ N_{A,1}^{q}=0\\ N_{B,1}^{q}=0\\ N_{C,1}^{q}=0\\ \end{cases}  \tag{11}\]  
Setting the ambiguity related to reference station \(A\) to \(0\) in Equation (11), and the ambiguity related to satellite \(q\) to \(0\) in Equation (12), we then can obtain the UD ambiguity of each satellite on a reference station:  
\[ \begin{cases}N_{A,1}^{p}=0\\ N_{B,1}^{p}=N_{A,1}^{p}-N_{A,1}^{q}+N_{B,1}^{q}-\Delta\
abla N_{AB,1}^{pq}=- \Delta\
abla N_{AB,1}^{pq}\\ N_{C,1}^{p}=N_{B,1}^{p}-N_{B,1}^{q}+N_{C,1}^{q}-\Delta\
abla N_{BC,1}^{pq}=- \Delta\
abla N_{AB,1}^{pq}-\Delta\
abla N_{BC,1}^{pq}\end{cases}  \tag{13}\]  
Figure 1: Undifferenced integer ambiguity conversion process. \[\left\{ \begin{aligned} & N_{A,1}^{k}=0\\ N_{B,1}^{k}=N_{A,1}^{k}-N_{A,1}^{q}+N_{1,B}^{q}-\Delta\
abla N_{AB,1}^{kq} =-\Delta\
abla N_{AB,1}^{kq}\\ N_{C,1}^{k}=N_{B,1}^{k}-N_{B,1}^{q}+N_{C,1}^{q}-\Delta\
abla N_{BC,1}^{ kq}=-\Delta\
abla N_{AB,1}^{kq}-\Delta\
abla N_{BC,1}^{kq}\\ \left\{\begin{aligned} N_{A,1}^{q}&=0\\ N_{B,1}^{q}&=0\\ N_{C,1}^{q}&=0\end{aligned}\right.\end{aligned} \right. \tag{14}\]  
By using Equations (13)-(15), the DD ambiguity can be converted into UD ambiguity. Due to the fact that each station contains the same reference ambiguity benchmark, the ambiguity benchmark remains integer state in user station with sum of interpolation coefficient of 1, which does not affect the ambiguity fixing and high-precision positioning of the user station. After estimating the UD ambiguity, independent UD error correction information can be generated for each reference station. In the process of user error correction, it is necessary to determine a reasonable error correction weighting coefficient and improve the accuracy of atmospheric error interpolation. The comprehensive error interpolation method combines ionospheric error, tropospheric error, satellite orbit error, and other errors. It is shown in Figure 2, where \(A\), \(B\), and \(C\) are three reference stations, and \(U\) is the user station.