# GPS + BDS Network Real-Time Differential Positioning Using a Position Domain Estimation Method
## 2 Materials and Methods  
### Differential Corrections Estimation  
Here the pseudorange correction from one reference station \(b\) to one satellite \(q\) can be expressed as ([PERSON] et al., 2018)  
\[PRC_{b}^{q}=\bar{R}_{b}^{q}-P_{b}^{q}=c\cdot\delta\tau_{b}-c\cdot\delta\tau^ {q}-\delta Q_{b}^{q}-\delta T_{b}^{q}-\delta l_{b}^{q}, \tag{1}\]where \(R\) is the distance between the reference station and the satellite, \(P\) is the original pseudorange observable, \(c\) denotes the speed of light in vacuum and \(\delta\tau_{b}\) and \(\delta\tau^{q}\) represent the receiver's and satellite's clock errors, respectively. \(\delta O\), \(\delta T\) and \(\delta I\) stand for the orbit error, the tropospheric delay and the ionospheric delay, respectively. Since the noise of the original pseudorange observation is very large, the pseudorange observable smoothed by the DFree Hatch filter [19] is usually used as an alternative. The model provided by [PERSON] et al. [28] is used to correct the BDS satellite-induced code bias, but it is negligible for the new-generation BDS-3 satellites [29]. The antenna phase centers of the satellite and receiver are corrected according to the information provided by the National Geodetic Survey [30]. It should be noted that the uncorrelated errors, such as hardware delay, multipath and pseudorange measurement noise, are not shown in Equation (1). Since the hardware delay can be calibrated [17], the multipath can be reduced by a choke ring antenna at the reference station [31], and the measurement noise can be weakened by a linear interpolation of pseudorange corrections [14], so only a small portion will remain in the differential positioning. In order to ensure the consistence of the correction, the receiver's clock error is estimated. The estimate can be obtained as follows  
\[c\cdot\delta\tau_{b}=\frac{1}{n}\sum_{i=1}^{n}\left(-P_{b}^{i}+R_{b}^{i}+\delta O _{b}^{i}+\delta T_{b}^{i}+\delta I_{b}^{i}+c\cdot\delta\tau^{i}\right), \tag{2}\]  
where \(n\) is the number of all valid satellites at a certain epoch. Herein the \"valid satellite\" means the observable is not an outlier and the satellite elevation angle is larger than the pre-set cut-off angle (\(10^{\circ}\)). \(\delta O\), \(\delta T\), \(\delta I\) and \(\delta\tau^{i}\) in Equation (2) are computed according to [PERSON] et al. [32]. Then, a modified pseudorange correction can be expressed as  
\[PRC_{b,m}^{q}=PRC_{b}^{q}-c\cdot\delta\tau_{b}=-c\cdot\delta\tau^{q}-\delta O _{b}^{q}-\delta T_{b}^{q}-\delta I_{b}^{q}, \tag{3}\]  
Although there is residual receiver's clock error in \(PRC_{b,m}^{q}\), it can be absorbed by the user receiver's clock error. After removing the receiver's clock error, the range of \(PRC_{b,m}^{q}\) is limited, and a cross comparison of the pseudorange corrections from each reference station can be made [33]. For the real-time differential positioning, corrections from only one reference station will be directly sent to the user. Whereas, for the network real-time differential positioning, at least three reference stations will be used to interpolate the user corrections. The adopted linear interpolation model can be expressed as [34]:  
\[\left[ \begin{array}{c}PRC_{b_{2},m}^{q}-PRC_{b_{1},m}^{q}\\ PRC_{b_{3},m}^{q}-PRC_{b_{1},m}^{q}\\ \vdots\\ PRC_{b_{n},m}^{q}-PRC_{b_{1},m}^{q}\end{array} \right]=\left[ \begin{array}{ ccc}x_{b_{2}}-x_{b_{1}}&y_{b_{2}}-y_{b_{1}}\\ x_{b_{3}}-xb_{1}&y_{b_{3}}-y_{b_{1}}\\ \vdots&\vdots\\ x_{b_{n}}-x_{b_{1}}&y_{b_{n}}-y_{b_{1}}\end{array} \right]\left[ \begin{array}[] {c}\alpha_{1}\\ \alpha_{2}\end{array} \right] \tag{4}\]  
where \(b_{1},\dots,b_{n}\) are the reference stations, \(x\) and \(y\) are the plane coordinates of the reference stations. In general, the reference station \(b_{1}\) is chosen as the reference station nearest to the user. The \(PR\)Cs used in the interpolation are all verified by the multiple reference consistency check [35]. Then the user correction can be calculated as follows  
\[PRC_{u}^{q}=PRC_{b_{1},m}^{q}+\left[