# 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
It can be seen that the DD ranges of two configurations of triple-frequency signals (B1C,B3I,B2a) and (B1I,B3I,B2a) in BDS-3 are 0.335 m and 0.349 m, respectively, of which the values are close to the quad-frequency case; therefore, these two configurations of triple-frequency signals can be selected as the optimal triple-frequency signals of BDS-3. As the long baseline is additionally affected by unmodeled errors, when the DD ranges of the two triple-frequency configurations are similar, the configuration with a smaller ambiguity STD is selected. It is easy to determine that the best configuration for Galileo is (E1,E5a,E6) because the ambiguity STD of this configuration is the same as that of quad-frequency, of which the value is 0.316 m. Following Equation (13), the corresponding EWL/WL equation is expressed as follows:  
\[\left[ \begin{array}{c}\mathbf{v}_{P}^{C}\\ \mathbf{v}_{EWL}^{C}\\ \mathbf{v}_{WL}^{C}\\ \mathbf{v}_{P}^{E}\\ \mathbf{v}_{WL}^{E}\\ \mathbf{v}_{WL}^{E}\\ \end{array} \right]=\left[ \begin{array}{ccccc}\mathbf{e}_{4}\otimes\mathbf{B}^{C}&\bm {\eta}_{P}^{C}\otimes\mathbf{I}_{C}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{B}^{C}&-\eta_{EWL}\mathbf{I}_{C}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{B}^{C}&-\eta_{WL}\mathbf{I}_{C}&\lambda_{WL}^{C}\mathbf{I}_{C}&\mathbf{0}&\mathbf{0}\\ \mathbf{e}_{4}\otimes\mathbf{B}^{E}&\mathbf{0}&\mathbf{0}&\mathbf{\eta}_{P}^{E}\otimes\mathbf{I}_{C}& \mathbf{0}\\ \mathbf{B}^{E}&\mathbf{0}&\mathbf{0}&-\eta_{EWL}\mathbf{I}_{C}&\mathbf{0}\\ \mathbf{B}^{E}&\mathbf{0}&\mathbf{0}&-\eta_{WL}\mathbf{I}_{C}&\lambda_{WL}^{E}\mathbf{I}_{C}\\ \end{array} \right]\left[ \begin{array}{c}\mathbf{x}\\ \mathbf{ion}^{C}\\ \mathbf{I}_{WL}^{C}\\ \mathbf{I}_{P}^{E}\\ \mathbf{I}_{WL}^{E}\\ \mathbf{I}_{WL}^{E}\\ \end{array} \right] \tag{14}\]  
The quad-frequency combination scheme and the optimal triple-frequency combination scheme were selected, corresponding to the No. 1 and No. 3 combination schemes in Table 5. In addition, in Table 4, BDS-3 and Galileo have configurations of triple-frequency signals (B1C,B1I,B2a) and (E1,E5a,E5b) with the worst precision, respectively. It should be noted that 'worst' here refers to the worst of all the combination schemes that have been selected in this experiment, but it is the best in the corresponding triple-frequency combined signals. Coincidentally, they correspond to the quad-frequency combination scheme without the constraints of the fixed second EWL in the second step and directly use fixed EWL to constrain WL ambiguities. Therefore, we added this combination scheme to the comparison as No. 2. Finally, the DD range a priori precisions of the combination of BDS-3 (B1C,B1I,B3I) and the combination of Galileo (E1,E5b,E6) are medium, and they also \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **System** & **Frequencies** & **EWL/Cycle** & **WL/Cycle** & **DD Range/m** \\ \hline
1 & BDS-3 & B1C,B1I,B3I & 0.041 & 0.243 & 0.864 \\  & BDS-3 & B1C,B1I,B2a & 0.041 & 0.314 & 0.939 \\  & BDS-3 & B1C,B3I,B2a & 0.110 & 0.236 & 0.335 \\  & BDS-3 & B1I,B3I,B2a & 0.110 & 0.220 & 0.349 \\
2 & Galileo & E1,E5a,E5b & 0.043 & 0.358 & 0.635 \\  & Galileo & E1,E5a,E6 & 0.099 & 0.310 & 0.316 \\  & Galileo & E1,E5b,E6 & 0.070 & 0.300 & 0.425 \\
3 & GPS & L1,L2,L5 & 0.066 & 0.359 & 0.483 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Precision of parameters in BDS-3 and Galileo triple-frequency DD configurations. form a combination scheme for comparison, as No. 4. The corresponding coefficients of proposed combination schemes are shown in Table 5.  
According to Table 5, there are two EWL combinations in BDS-3; the corresponding wavelengths are, respectively, 20.93 m and 3.26 m, which are quite different. There are three EWL combinations in Galileo, corresponding to wavelengths of 9.77 m, 4.19 m, and 2.93 m. All coefficients are plus or minus 1 or 0.