# Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations
## 2 Materials and Methods
### Theoretical Precision of Triple- and Quad-Frequency EWL/WL Signals of BDS-3 and Galileo
In addition, it can be seen that using the fixed EWL observations as accurate pseudo-range observations to constrain second EWL ambiguities cannot greatly improve the precision of second EWL ambiguities. Moreover, the DD range is not improved by much after fixing the second EWL ambiguities compared with EWL, especially for Galileo. However, no matter for BDS-3 or Galileo, the precision of WL ambiguities has the same characteristics: the fixed EWL observations have little restriction on WL ambiguities, and the WL ambiguity STDs of BDS-3 and Galileo are 0.314 cycles and 0.358 cycles, respectively. Compared to without EWL constraints, the a priori success rate only increased by 2.3 and 2.5 percentage points. Nevertheless, the fixed second EWL can significantly improve the precision of WL ambiguities. After using the fixed second EWL constraints, the WL ambiguity STDs of BDS-3 and Galileo are 0.209 cycles and 0.298 cycles, respectively, and the a priori success rates are 98.3% and 90.7%, respectively. Compared with the fixed EWL, it has been increased by 9.5 and 6.9 percentage points, respectively. Correspondingly, the quad-frequency WL AR of BDS-3 and Galileo in Section 2.3 can be expressed as follows:  
\[\left[ \begin{array}{c}\
u_{P}^{C}\\ \
u_{EWL}^{C}\\ \
u_{SEVL}^{C}\\ \
u_{VVL}^{C}\\ \
u_{P}^{C}\\ \
u_{EWL}^{C}\\ \
u_{VEVL}^{C}\\ \
u_{VILL}^{C}\\ \end{array} \right]=\left[ \begin{array}{ccccc}\epsilon_{4}\otimes B^{C}&\eta_ {P}^{C}\otimes I_{C}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ B^{C}&-\eta_{EWL}I_{C}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ B^{C}&-\eta_{ESWL}I_{C}&\lambda_{WI}^{C}I_{C}&\mathbf{0}&\mathbf{0}\\ \epsilon_{4}\otimes B^{E}&\mathbf{0}&\mathbf{0}&\eta_{P}^{C}\otimes I_{e}& \mathbf{0}\\ B^{E}&\mathbf{0}&\mathbf{0}&-\eta_{EWL}I_{e}&\mathbf{0}\\ B^{E}&\mathbf{0}&\mathbf{0}&-\eta_{ESWL}I_{e}&\mathbf{0}\\ B^{E}&\mathbf{0}&\mathbf{0}&-\eta_{WIL}I_{e}&\lambda_{WI}^{E}I_{e}\\ \end{array} \right]\left[ \begin{array}{c}\mathbf{x}\\ \mathbf{i}_{P}^{C}\\ \mathbf{i}_{PWL}^{C}\\ \mathbf{i}_{P}^{C}\\ \mathbf{i}_{P}^{C}\\ \mathbf{i}_{PWL}^{C}\\ \mathbf{i}_{EWL}^{C}\\ \mathbf{i}_{EWL}^{C}\\ \end{array} \right] \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Wavelength/m**} & \multicolumn{5}{c}{**Number of Fixed Ambiguities**} \\ \cline{3-5}  & & **0** & **1** & **2** & **3** \\ \cline{3-5}  & & **0** & **1** & **2** & **3** \\ \hline E5a-E5b/(cycle) & 9.77 & 0.043 & 0 & 0 & 0 \\ E5b-E6/(cycle) & 4.19 & 0.070 & 0.060 & 0 & 0 \\ E1-E5a/(cycle) & 0.75 & 0.379 & 0.358 & 0.298 & 0 \\ DD Range / m & N/A & 1.249 & 1.173 & 1.169 & 0.316 \\ \hline \end{tabular}
\end{table} 
Table 3: Precision of parameters in the Galileo quad-frequency (E1, E5a, E5b, and E6) DD case. where \(\mathbf{v}_{P}^{C}=\left[\mathbf{v}_{P_{1}}^{C},\mathbf{v}_{P_{2}}^{C},\mathbf{v}_{P_{3}}^{C},\mathbf{ v}_{P_{4}}^{C}\right]^{T}\); \(I_{P}^{C}=\left[I_{P_{1}}^{C},I_{P_{2}}^{C},I_{P_{3}}^{C},I_{P_{4}}^{C}\right]^{T}\); \(v_{P}^{E}\) and \(I_{P}^{E}\) have the same expression; the superscript capitals 'C' and 'E' indicate BDS-3 and Galileo, respectively; \(\mathbf{e}_{n}\) is a column matrix vector with n rows of elements, all being 1; and the subscripts '\(c\)' and '\(e\)' of the identity matrix are the dimensions of matrix, corresponding to the number of satellite pairs of BDS-3 and Galileo, respectively. It should be noted that the first two steps of the EWL and second EWL AR processes cannot be ignored. In the above process, some suboptimal second EWL combinations have been eliminated, for example, the Galileo signal with a wavelength of 2.91 m. However, no suboptimal second EWL signal is eliminated in BDS-3. Therefore, in Table 4, optimal EWL/WL combinations in different triple-frequency configurations are selected in BDS-3 and Galileo, respectively. The corresponding ambiguity precisions of EWL/WL and the theoretical DD ranges are given. As a comparison, the optimal triple-frequency configuration of GPS is added.