# An Efficient BDS-3 Long-Range Undifferenced Network RTK Positioning Algorithm
## 2 Long-Range URTK Algorithm
### Phase Integer AR and Rover Station Positioning  
Consider observation \(U\) of satellites \(p\) and \(q\), where satellite \(q\) is a reference satellite; then, the phase undifferenced observation equation of a rover station at frequency \(i\) is obtained as follows:  
\[\left\{ \begin{aligned} & L_{U,i}^{q}=H^{q}\cdot X+\rho_{U}^{q}- \lambda_{i}N_{U,i}^{q}+O^{q}-I_{U,i}^{q}+T_{U}^{q}+ct_{U}-ct^{q}-\lambda_{i} \sigma_{U}+\lambda_{i}\sigma^{q}+\epsilon_{U,i}^{q}\\ & L_{U,i}^{p}=H^{p}\cdot X+\rho_{U}^{p}-\lambda_{i}N_{U,i}^{p}+O^{p}- I_{U,i}^{p}+T_{U}^{p}+ct_{U}-ct^{p}-\lambda_{i}\sigma_{U}+\lambda_{i}\sigma^{p}+ \epsilon_{U,i}^{p}\end{aligned} \right. \tag{18}\]  
The satellite clock error and satellite hardware delay in an observation can be eliminated after the error correction of a rover station through undifferenced error correction. Inthis way, the tropospheric delay error, ionospheric delay error, and satellite orbit error are significantly weakened. The error equation is defined as follows:  
\[ \begin{cases}L^{\text{q}}_{U,j}-OMC^{\text{q}}_{U,i}=H^{\text{q}}\cdot X+\rho^{ \text{q}}_{U}-\lambda_{i}N^{\text{q}}_{U,i}+\Delta ct_{U}-\Delta\lambda_{i} \sigma_{U}\\ L^{\text{p}}_{U,i}-OMC^{\text{p}}_{U,i}=H^{\text{p}}\cdot X+\rho^{\text{p}}_{U} -\lambda_{i}N^{\text{p}}_{U,i}+\Delta ct_{U}-\Delta\lambda_{i}\sigma_{U}\end{cases}  \tag{19}\]  
where \(OMC^{q}_{U,i}\) and \(OMC^{\text{p}}_{U,i}\) are the sums of the undifferenced error correction values of the rover station obtained at multiple reference stations; \(\Delta ct_{U}\) is the residual receiver clock error; and \(\Delta\lambda_{i}\sigma_{U}\) denotes the residual of the carrier phase hardware delay at the receiver side. In order to eliminate the residual error of the receiver clock error and receiver hardware delay, the inter-satellite difference of Equation (19) can be obtained via:  
\[ \begin{split} V^{pq}_{U,i}&=(-H^{p}+H^{\text{q}}) \cdot X+\lambda_{i}(N^{\text{p}}_{U,i}-N^{\text{q}}_{U,i})\\ &+\Big{\{}[(L^{\text{p}}_{U,i}-OMC^{\text{p}}_{U,i})-(L^{\text{q} }_{U,i}-OMC^{\text{q}}_{U,i})]-(\rho^{\text{p}}_{U}-\rho^{\text{q}}_{U})\Big{\}} \end{split}  \tag{20}\]  
where,  
\[ \begin{cases}h^{pq}_{U,i}=-H^{p}+H^{\text{q}}\\ L^{pq}_{U,i}=\Big{\{}[(L^{\text{p}}_{U,i}-OMC^{\text{p}}_{U,i})-(L^{\text{q} }_{U,i}-OMC^{\text{q}}_{U,i})]-(\rho^{\text{p}}_{U}-\rho^{\text{q}}_{U})\Big{\}} \end{cases} \end{split} \tag{21}\]  
The error equation of the phase ambiguity and position parameters of the rover station is defined as:  
\[V^{pq}_{U,i}=h^{pq}_{U,i}\cdot X+\lambda_{i}(N^{\text{p}}_{U,i}-N^{\text{q}}_ {U,i})+L^{pq}_{U,i} \tag{22}\]  
Suppose that the BDS-3 multi-frequency observation data relate to \((s+1)\) satellites' data (the reference satellite number is \"1\") collected at a rover station; then, the matrix form of the error equation is defined as follows:  
\[ \begin{bmatrix}V^{S}_{U,i}\\ \vdots\\ V^{S}_{U,n}\end{bmatrix} = \begin{bmatrix}H^{S}&\lambda^{S}\end{bmatrix}\begin{bmatrix} X\\ N^{S}\end{bmatrix} + \begin{bmatrix}L^{S}\end{bmatrix}  \tag{23}\]  
where,  
\[V^{S}_{U,i}= \begin{bmatrix}V^{S_{1,2}}_{U,i}\\ \vdots\\ V^{S_{i,n+1}}_{\begin{subarray}{c}U,i\\ n\times\times\end{subarray}}\end{bmatrix} H^{S}= \begin{bmatrix}l^{S_{1,2}}_{1}&m^ {S_{1,2}}_{1}&e^{S_{1,2}}_{1}\\ \vdots&\vdots&\vdots\\ l^{S_{1,n+1}}_{1}&m^{S_{1,n+1}}_{1}&e^{S_{1,n+1}}_{1}\\ \vdots&\vdots&\vdots\\ l^{S_{1,n+1}}_{n}&m^{S_{1,n+1}}_{1}&e^{S_{1,n+1}}_{n}\end{bmatrix} \lambda^{S}= \begin{bmatrix}\lambda^{S}_{1}b&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\lambda^{S}_{n}b\\ (n\times s)\times(n\times(s+1))&e^{S_{1,n+1}}_{n}\end{bmatrix} \lambda^{S}= \begin{bmatrix}\lambda^{S}_{1}b&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\lambda^{S}_{n}b\\ (n\times s)\times(n\times(s+1))&\end{bmatrix} N^{S}= \begin{bmatrix}\lambda^{S_{1, 1}}_{1}\\ \vdots\\ N^{S_{1,n+1}}_{1}\\ \vdots\\ N^{S_{1,n+1}}_{n}\\ (n\times s)\times 1\end{bmatrix} L^{S}= \begin{bmatrix}L^{S_{1,2}}_{U,1}\\ \vdots\\ L^{S_{1,n+1}}_{U,1}\\ \vdots\\ L^{S_{1,n+1}}_{U,1}\\ \vdots\\ L^{S_{1,n+1}}_{U,1}\\ (n\times s)\times 1\end{bmatrix}