# An Efficient BDS-3 Long-Range Undifferenced Network RTK Positioning Algorithm
## 2 Long-Range URTK Algorithm
### Undifferenced Error Correction Value Calculation  
After the double-difference ambiguity of the reference station is determined, the double-difference integer ambiguity of frequency \(i\) in the BDS system is analyzed. The relationship between the first undifferenced integer ambiguity and the corresponding double-difference ambiguity of satellites \(p\), \(q\), and \(k\), where \(q\) is a reference satellite, at reference stations \(A\), \(B\), and \(C\), is obtained as follows:  
\[ \begin{cases}\Delta\
abla N_{AB,i}^{pq}=N_{A,i}^{p}-N_{A,i}^{q}+N_{B,i}^{q}-N_ {B,i}^{p}\\ \Delta\
abla N_{BC,i}^{pq}=N_{B,i}^{p}-N_{B,i}^{q}+N_{C,i}^{q}-N_{C,i}^{p}\\ \Delta\
abla N_{CA,i}^{pq}=N_{C,i}^{p}-N_{CA,i}^{q}+N_{A,i}^{q}-N_{A,i}^{p}\\ \end{cases}  \tag{9}\]  
\[ \begin{cases}\Delta\
abla N_{AB,i}^{kj}=N_{A,i}^{k}-N_{A,i}^{q}+N_{B,i}^{q}-N_ {B,i}^{k}\\ \Delta\
abla N_{BC,i}^{kj}=N_{B,i}^{k}-N_{B,i}^{q}+N_{C,i}^{q}-N_{C,i}^{k}\\ \Delta\
abla N_{CA,i}^{kj}=N_{C,i}^{k}-N_{C,i}^{q}+N_{A,i}^{q}-N_{A,i}^{k}\\ \end{cases}  \tag{10}\]  
There are only two linearly independent double-difference integer ambiguity values in Equations (9) and (10). Therefore, in order to filter the undifferenced integer ambiguity of a single satellite at a single station, the reference station undifferenced reference integer ambiguity and satellite undifferenced reference integer ambiguity are defined and used to convert double-difference integer ambiguity into undifferenced integer ambiguity. The reference station \(A\) and satellite \(q\) are selected for undifferenced reference ambiguity. An integer value can be set as a constant, which will not affect the positioning of a user station. The undifferenced integer ambiguity related to the reference station and reference satellite is set to zero. Then, the ambiguity calculation process of the other satellites is performed as follows:  
\[ \begin{cases}N_{A,i}^{p}=0\\ N_{B,i}^{p}=N_{A,i}^{p}-N_{A,i}^{q}+N_{B,i}^{q}-\Delta\
abla N_{AB,i}^{pq}=- \Delta\
abla N_{AB,i}^{pq}\\ N_{C,i}^{p}=N_{B,i}^{p}-N_{B,i}^{q}+N_{C,i}^{q}-\Delta\
abla N_{BC,i}^{pq}=- \Delta\
abla N_{AB,i}^{pq}-\Delta\
abla N_{BC,i}^{pq}\\ \end{cases}  \tag{11}\]  
\[ \begin{cases}N_{A,i}^{k}=0\\ N_{B,i}^{k}=N_{A,i}^{k}-N_{A,i}^{q}+N_{B,i}^{q}-\Delta\
abla N_{AB,i}^{kq}=- \Delta\
abla N_{AB,i}^{kq}\\ N_{C,i}^{k}=N_{B,i}^{k}-N_{B,i}^{q}+N_{C,i}^{q}-\Delta\
abla N_{BC,i}^{kq}=- \Delta\
abla N_{AB,i}^{kq}-\Delta\
abla N_{BC,i}^{kq}\\ \end{cases}  \tag{12}\]  
where \(N_{A,i}^{p}\), \(N_{A,i}^{k}\), \(N_{A,i}^{q}\), \(N_{B,i}^{q}\), and \(N_{C,i}^{q}\) relate to the undifferenced reference ambiguity, and \(\Delta\
abla N_{AB,i}^{pq}\), \(\Delta\
abla N_{BC,i}^{pq}\), \(\Delta\
abla N_{AB,i}^{kq}\), and \(\Delta\
abla N_{BC,i}^{kq}\) relate to the known double-difference integer ambiguity. Through the above process, the double-difference integer ambiguity can be converted into undifferenced integer ambiguity. After determining the undifferenced integer ambiguity of each satellite corresponding to a particular station, the undifferenced error correction value of each reference station corresponding to the satellite can be calculated. The inverse distance weighted interpolation algorithm is used to establish a regional high-precision undifferenced error correction model. The undifferenced error correction value of a user station is calculated using the undifferenced error correction value of the reference station and used to correct the error of the user station. The calculation process of the undifferenced error correction value of a satellite \(p\) corresponding to the three reference stations \(A\), \(B\), and \(C\) is as follows:  
\[