# Evaluation of BDS/GPS Multi-Frequency RTK Positioning Performance under Different Baseline Lengths
## 2 Multi-Frequency RTK Positioning Mathematical Model  
### RTK Double-Differenced Measurement Model  
Under the short baseline RTK positioning mode, the DD observation model can effectively attenuate the satellite ephemeris error, atmospheric delay error, receiver clock bias, receiver hardware delay, and preserve the integer value of the DD ambiguity [10]. The BDS/GPS DD carrier phase and pseudorange observations can be expressed as:  
\[\left\{\begin{array}{l}\lambda\cdot\
abla\Delta\Phi^{jk}_{rb}=\
abla\Delta\rho^ {jk}_{rb}-\lambda N^{jk}_{rb}+\epsilon^{jk}_{rb}\\ \
abla\Delta p^{jk}_{rb}=\
abla\Delta\rho^{jk}_{rb}+\epsilon^{jk}_{rb}\\ \end{array}\right. \tag{1}\]  
In Equation (1), \(b\), \(r\) represent the base-station and the rover station; \(j\), \(k\) represent the reference and non-reference satellites; \(\
abla\Delta\) is the DD symbol; \(P\) and \(\Phi\) are the code and carrier phase observations; \(\rho\) is the geometry distance between the receiver and satellite; \(\lambda\) is the wavelength; \(N\) denotes the carrier phase ambiguity; \(\epsilon^{jk}_{rb}\) and \(\epsilon^{jk}_{rb}\) denote the observation noise of DD carrier and pseudorange.  
For the long baseline (>20 km) RTK positioning mode, the tropospheric and ionospheric errors are difficult to eliminate. Therefore, the BDS/GPS DD carrier-phase and pseudorange observations are expressed as:  
\[\left\{\begin{array}{l}\lambda\cdot\
abla\Delta\Phi^{jk}_{rb}=\
abla\Delta \rho^{jk}_{rb}-\lambda N^{jk}_{rb}-\gamma_{f}\
abla\Delta I^{jk}_{rb}+\
abla \Delta T^{jk}_{rb}+\epsilon^{jk}_{rb}\\ \
abla\Delta p^{jk}_{rb}=\
abla\Delta\rho^{jk}_{rb}+\gamma_{f}\
abla\Delta I^ {jk}_{rb}+\
abla\Delta T^{jk}_{rb}+\epsilon^{jk}_{rb}\\ \end{array}\right. \tag{2}\]  
In Equation (2), \(T\) is the slant troposphere delay; \(I\) represents the slant ionosphere delay.  
By using the triple-frequency signal, more combined observations can be obtained to enhance the strength of observation equations [11]. Considering the ionospheric and tropospheric efforts, the DD observation equations of combined triple-frequency pseudorange and phase are:  
\[\left\{\begin{array}{l}\
abla\Delta P_{1}=\
abla\Delta\rho^{jk}_{rb}+\
abla \Delta T^{jk}_{rb}+\gamma_{1}\
abla\Delta I^{jk}_{rb}+\epsilon^{jk}_{rb,1}\\ \
abla\Delta P_{2}=\
abla\Delta\rho^{jk}_{rb}+\
abla\Delta T^{jk}_{rb}+\gamma _{2}\
abla\Delta I^{jk}_{rb}+\epsilon^{jk}_{rb,2}\\ \
abla\Delta P_{3}=\
abla\Delta\rho^{jk}_{rb}+\
abla\Delta T^{jk}_{rb}+\gamma _{3}\
abla\Delta I^{jk}_{rb}+\epsilon^{jk}_{rb,3}\\ \
abla\Delta\phi_{1}=\
abla\Delta\rho^{jk}_{rb}+\
abla\Delta T^{jk}_{rb}- \gamma_{1}\
abla\Delta I^{jk}_{rb}-\lambda_{1}\
abla\Delta N_{1}+\epsilon^{jk }_{rb,1}\\ \
abla\Delta\phi_{2}=\
abla\Delta\rho^{jk}_{rb}+\
abla\Delta T^{jk}_{rb}- \gamma_{2}\
abla\Delta I^{jk}_{rb}-\lambda_{2}\
abla\Delta N_{2}+\epsilon^{jk }_{rb,2}\\ \
abla\Delta\phi_{3}=\
abla\Delta\rho^{jk}_{rb}+\
abla\Delta T^{jk}_{rb}- \gamma_{3}\
abla\Delta I^{jk}_{rb}-\lambda_{3}\
abla\Delta N_{3}+\epsilon^{jk }_{rb,3}\\ \end{array}\right. \tag{3}\]  
In Equation (3), \(\gamma_{i}=f_{1}^{2}/f_{i}^{2}\). The triple-frequency observations enhance the strength of the observation model, avoid noise amplification, and shorten the convergence time, which greatly improves the success rate and reliability of ambiguity fixing.