# Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations
## 2 Materials and Methods
### Double-Differenced BDS-3 and Galileo Mathematical Model  
The basic DD observation equations of pseudo-range and carrier phase measurements for a system '\(sys\)' (denotes BDS-3 and Galileo) and frequency band \(i\) (\(i=1,\ 2,\ 3,\ 4\)) can be expressed as follows:  
\[\left\{\begin{array}{c}\Delta\
abla P_{i}^{sys}=\Delta\
abla\rho+\eta_{i}^{ sys}\Delta\
abla I_{1}+\Delta\
abla T+\Delta\
abla\varepsilon_{P}\\ \Delta\
abla\phi_{i}^{sys}=\Delta\
abla\rho-\eta_{i}^{sys}\Delta\
abla I_{1}+ \Delta\
abla T+\lambda_{i}^{sys}\Delta\
abla N_{i}^{sys}+\Delta\
abla \varepsilon_{\phi}\end{array}\right. \tag{1}\]  
where the symbol '\(\Delta\
abla\)' represents the DD operator; \(P_{i}^{sys}\) and \(\phi_{i}^{sys}\) represent the pseudo-range and carrier phase observation values of the corresponding system, respectively; \(\rho\) is the geometric distance between a satellite and a receiver; \(I_{1}\) is the first-order ionospheric delay at the first frequency; \(\eta_{i}^{sys}\) is the first-order ionospheric scalar factor of the corresponding system; \(T\) is the tropospheric delay; \(\varepsilon_{P}\) and \(\varepsilon_{\phi}\) are the measurement noise of the pseudo-range and carrier phase, respectively; and \(N_{i}^{sys}\) is the DD ambiguity of the corresponding system.  
The corresponding linearized DD pseudo-range and carrier phase measurement equations can be expressed as follows:  
\[\left\{\begin{array}{c}\Delta\
abla P_{i}^{sys}=AX+\Delta\
abla\varepsilon _{P}\\ \Delta\
abla\phi_{i}^{sys}=AX+\lambda_{i}^{sys}\Delta\
abla N_{i}^{sys}+ \Delta\
abla\varepsilon_{\phi}\end{array}\right. \tag{2}\]  
where \(A\) is the linear coefficient matrix and \(X\) denotes the baseline parameters and atmospheric parameters.  
Omitting the system superscript, the combined DD observation equations of quad-frequency signals can be written as follows [16]:  
\[\Delta\
abla\phi_{(i,j,k,m)}=\Delta\
abla\rho-\eta_{(i,j,k,m)}\Delta\
abla I_{ 1}+\Delta\
abla T+\lambda_{(i,j,k,m)}\Delta\
abla N_{(i,j,k,m)}+\Delta\
abla \varepsilon_{\phi(i,j,k,m)} \tag{3}\]  
where  
\[\Delta\
abla\phi_{(i,j,k,m)}=\frac{i\cdot f_{1}\cdot\Delta\
abla\phi_{1}+j \cdot f_{2}\cdot\Delta\
abla\phi_{2}+k\cdot f_{3}\cdot\Delta\
abla\phi_{3}+m \cdot f_{4}\cdot\Delta\
abla\phi_{4}}{i\cdot f_{1}+j\cdot f_{2}+k\cdot f_{3} +m\cdot f_{4}} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Satellite Systems** & **Order Number** & **Signals** & **Frequencies (MHz)** \\ \hline BDS-3 & 1 & B1C & 1575.420 \\  & 2 & B1I & 1561.098 \\  & 3 & B3I & 1268.520 \\  & 4 & B2a & 1176.450 \\ Galileo & 1 & E1 & 1575.420 \\  & 2 & E5a & 1176.450 \\  & 3 & E5b & 1207.140 \\  & 4 & E6 & 1278.750 \\ \hline \end{tabular}
\end{table}
Table 1: Quad-frequency signals of BDS-3 and Galileo.  
\[\eta_{(i,j,k,m)} =\frac{f_{1}^{2}\Big{(}\frac{i}{f_{1}}+\frac{i}{f_{2}}+\frac{k}{f_{3 }}+\frac{m}{f_{4}}\Big{)}}{i\cdot f_{1}+j\cdot f_{2}+k\cdot f_{3}+m\cdot f_{4}} \tag{5}\] \[\lambda_{(i,j,k,m)} =\frac{c}{i\cdot f_{1}+j\cdot f_{2}+k\cdot f_{3}+m\cdot f_{4}} \tag{6}\]  
where the coefficients \(i\), \(j\), \(k\), \(m\) of the combinations are represented by \(i_{1}\), \(j_{1}\), \(k_{1}\), \(m_{1}\) and \(i_{2}\), \(j_{2}\), \(k_{2}\), \(m_{2}\) in BDS-3 and Galileo, respectively; \(\Delta\
abla\phi_{(i,j,k,m)}\) is a DD carrier phase observation of the combined signal; the expression of \(\Delta\
abla P_{(i,j,k,m)}\) is similar to \(\Delta\
abla\phi_{(i,j,k,m)}\); \(\eta_{(i,j,k,m)}\) is the ionospheric scalar factor of the combined signal; \(\lambda_{(i,j,k,m)}\) is the wavelength of the combined signal; and \(c\) is the velocity of light.