# GPS/BDS RTK Positioning Based on Equivalence Principle Using Multiple Reference Stations
## 2 Methodology  
### SD Observation Equations  
As shown in Figure 1, there are four receivers tracking \(n\) satellites simultaneously, including three reference receivers \(r\) (\(r=a\), \(b\), _and_\(c\)) and one rover receiver \(u\). It is preferable that the precise positions of three reference receivers are obtained in advance. Three reference receivers can be used to enhance the strength of the kinematic positioning model. Without loss of simplicity, the SD observation equations of carrier phase and code observations for single satellite system can be presented as follows:  
\[\left\{ \begin{array}{l}\Lambda\Delta\varphi_{uu}^{s}=\Delta\rho_{au}^{s}+c \Delta t_{au}+\lambda\Delta N_{au}^{s}+\Delta T_{au}^{s}-\Delta I_{au}^{s}+ \Delta\varepsilon_{au}^{s}\\ \Delta P_{uu}^{s}=\Delta\rho_{au}^{s}+c\Delta t_{au}+\Delta T_{au}^{s}+\Delta S _{au}^{s}+\Delta\varepsilon_{au}^{s}\\ \Lambda\Delta\varphi_{bu}^{s}=\Delta\rho_{bu}^{s}+c\Delta t_{bu}+\lambda\Delta N _{bu}^{s}+\Delta T_{bu}^{s}-\Delta I_{bu}^{s}+\Delta\varepsilon_{bu}^{s}\\ \Delta P_{bu}^{s}=\Delta\rho_{bu}^{s}+c\Delta t_{bu}+\Delta T_{bu}^{s}+\Delta S _{bu}^{s}+\Delta\varepsilon_{bu}^{s}\\ \Lambda\Delta\varphi_{cu}^{s}=\Delta\rho_{cu}^{s}+c\Delta t_{cu}+\lambda \Delta N_{cu}^{s}+\Delta T_{cu}^{s}-\Delta I_{cu}^{s}+\Delta\varepsilon_{cu}^{ s}\\ \Delta P_{cu}^{s}=\Delta\rho_{cu}^{s}+c\Delta t_{cu}+\Delta T_{cu}^{s}+\Delta \varepsilon_{cu}^{s}+\Delta\varepsilon_{cu}^{s}\end{array} \right. \tag{1}\]  
where \(\Delta(\cdot)_{ru}=(\cdot)_{u}-(\cdot)_{r}\) is the SD operator between reference receiver \(r\) and rover receiver \(u\). The superscript \(s\) (\(s=j\), \(k\),..., \(n\)) denotes the satellite, and satellite \(j\) is set as the reference satellite. All satellites can be arranged with the descending order by satellite elevation angle. \(P\) and \(\varphi\) are the observations of code and carrier phase, respectively, and \(\lambda\) denotes the wavelength of the carrier. \(\rho\) is the geometric distance as a function of the receiver and satellite coordinates. \(c\) is the speed of light in vacuum. \(t\) represents user receiver clock error. \(N\) is the integer ambiguity of carrier phase. \(T\) and \(I\) are the tropospheric and ionospheric delay, whose difference can be ignored for short baseline. \(\varsigma\) and \(\varepsilon\) are the measurement noise of code and carrier phase, respectively. By combining SD observation equations of all satellites for single satellite system, the linearized equations in matrix formation can be represented as  
\[v=Ax+BN+Ct-L,\quad P\]  
Figure 1: Four receivers track \(n\) satellites. where \(x\) is a column vector of coordinate corrections. \(N\) is a column vector of SD ambiguities. \(t\) is a column vector of SD receiver clock errors. \(A\) and \(C\) are the corresponding coefficient matrices, respectively. \(L\) is a column vector of the observation minus computation (OMC) terms. \(v\) is a vector of SD residual errors. \(P\) is a weight matrix of SD observations. The sub-matrices of the above matrices can be calculated by Equation (1) and the specific expressions can be derived as  
\[A=\left[ \begin{array}{c}A^{j}\\ A^{k}\\ \ldots\\ A^{n}\end{array} \right]\text{, }B=\left[ \begin{array}{cccc}B^{j}&&&\\ &B^{k}&&\\ &&\ldots&\\ &&&B^{n}\end{array} \right]\text{, }C=\left[ \begin{array}{c}C^{j}\\ C^{k}\\ \ldots\\ C^{n}\end{array} \right]\text{, }L=\left[ \begin{array}{c}L^{j}\\ L^{k}\\ \ldots\\ L^{n}\end{array} \right]\text{, }v=\left[ \begin{array}{c}v^{j}\\ v^{k}\\ \ldots\\ v^{n}\end{array} \right] \tag{3}\]  
It is assumed that undifferenced observations are independent with elevation-dependent weights and their weight matrix is a diagonal matrix. The different SD observations among multiple receivers tracking the different satellite are not correlative. However, the different SD observations among multiple receivers tracking the identical satellites are correlative, which cannot be ignored. The SD weight matrix can be calculated exactly by the law of variance-covariance propagation.