## 2 Modeling and Analysis of Carrier-Phase Differential Positioning  
### Modeling of Carrier-Phase Differential Positioning  
The carrier-phase observation is the phase difference between the satellite carrier signal received by the receiver and the same reference signal of the receiver itself. Since the phase difference can only measure the phase value within one cycle, unknown cycles have passed since the satellite carrier signal propagates to the receiver in the actual measurement, i.e., there is an unknown number of full cycles. The carrier-phase observation equation is shown as Equation (1). It indicates the number of cycles the carrier travels from the satellite to the ground station multiplied by its wavelength is the distance from the satellite to the station. \[\left\{ \begin{array}{l}\varphi\lambda=\rho-cV_{tr}+cV_{ts}-N\lambda-V_{ion}- V_{prop}\\ \rho=\sqrt{\left(x^{s}-X_{ref}\right)^{2}+\left(y^{s}-Y_{ref}\right)^{2}+\left( z^{s}-Z_{ref}\right)^{2}}\end{array} \right. \tag{1}\]  
where \(\varphi\) is the observation of the carrier phase, \(\rho\) is the distance from the station to the satellite, \((x^{s},y^{s},z^{s})\) and \((X_{ref},Y_{ref},Z_{ref})\) are satellite coordinates and approximate coordinates of ground stations, respectively, \(c\) is the speed of light, \(N\) is the integer ambiguity, \(\lambda\) is the wavelength of the carrier, \(V_{tr}\) and \(V_{ts}\) are the receiver clock error and the satellite clock error, respectively, \(V_{ion}\) is the ionospheric delay error, \(V_{trop}\) is the tropospheric delay error. Equation (2) can be obtained by Taylor expansion at the approximate coordinate \((X_{ref},Y_{ref},Z_{ref})\) of the station  
\[\varphi\lambda=\rho-\frac{x^{s}-X_{ref}}{\rho}dX-\frac{y^{s}-Y_{ref}}{\rho}dY -\frac{z^{s}-Z_{ref}}{\rho}dZ-cV_{tr}+cV_{ts}-N\lambda-V_{ion}-V_{trop} \tag{2}\]  
where \(dX\), \(dY\) and \(dZ\) are the coordinate correction quantities of the station coordinates in the three directions of \(X\), \(Y\) and \(Z\). Let \(l=\frac{x^{s}-X_{ref}}{\rho},m=\frac{y^{s}-Y_{ref}}{\rho},n=\frac{z^{s}-Z_{ref}} {\rho}\). \(l\), \(m\), and \(n\) are the cosine of the approximate position of the station to the satellite in three directions, then, Equation (2) can be simplified to Equation (3). \[\varphi\lambda=\rho-ldX-mdY-ndZ-cV_{tr}+cV_{ts}-N\lambda-V_{ion}-V_{trop} \tag{3}\]  
Since differential positioning requires at least two receivers, one or more of which is set at a known location as the reference station, and one is used as a mobile station to measure unknown position. Simultaneous observation of satellite \(p\) on two stations \(i\) (reference station) and \(j\) (mobile station). Since station \(i\) is a reference station whose coordinates are known, there is no need for Taylor expansion. Hence, the carrier-phase observation equation Equation (4) can be obtained directly from Equation (1). Since station \(j\) is a mobile station, only the approximate value of its coordinates is required. Taylor expansion needs to be performed at the approximate coordinates. The carrier-phase observation equation Equation (5) can be obtained from Equation (3). \[\varphi_{i}^{p}\lambda=\rho_{i}^{p}-cV_{tr}^{i}+cV_{t}^{p}-N_{i}^{p}\lambda-(V _{ion})_{i}^{p}-\left(V_{trop}\right)_{i}^{p} \tag{4}\]  
\[\varphi_{j}^{p}\lambda=\rho_{j}^{p}-l_{j}^{p}dX-m_{j}^{p}dY-n_{j}^{p}dZ-cV_{tr }^{j}+cV_{t}^{p}-N_{j}^{p}\lambda-(V_{ion})_{j}^{p}-\left(V_{trop}\right)_{j}^{p} \tag{5}\]  
By a differential between stations i.e., Equation (4) minus Equation (5), we can obtain Equation (6). \[\left(\varphi_{j}^{p}-\varphi_{i}^{p}\right)\lambda=-l_{j}^{p}dX-m_{j}^{p}dY-n _{j}^{p}dZ+c\left(V_{tr}^{i}-V_{tr}^{j}\right)+\left(N_{i}^{p}-N_{j}^{p} \right)\lambda+\left(\rho_{j}^{p}-\rho_{i}^{p}\right) \tag{6}\]  
From Equation (6), we can know that the satellite-related error terms are eliminated, and only the receiver-related error terms are left. Similarly, if the two stations observe \(q\) satellite simultaneously, we can obtain Equation (7).