# Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations
## 3 Results
### Performance of EWL AR for BDS-3 and Galileo
Table 8 shows the mean and STD values of the WL ambiguity fractions. It can be found that, in experiment 1, the mean and STD values of the five WL ambiguity fractions are less than 0.03 and 0.2 cycles, respectively. The ambiguity fraction STD value of combination scheme No. 3 is almost equal to that of combination scheme No. 1. In general, the above discussion is still applied in experiment 2, although the STD values of experiment 2 are larger than that of experiment 1. Figure 6: Experiment 2: WL ambiguity fractions and the corresponding histograms of combination schemes No. 1, No. 2, No. 3, and No. 4 (from top to bottom) for Galileo: (**a**) WL ambiguity fractions, each color denotes one satellite pair; (**b**) WL ambiguity fraction statistics histograms. Figure 7: Experiment 2: WL ambiguity fractions and the corresponding histograms of combination schemes No. 1, No. 2, No. 3, and No. 4 (from top to bottom) for Galileo: (**a**) WL ambiguity fractions, each color denotes one satellite pair; (**b**) WL ambiguity fraction statistics histograms. It can be seen that the positioning precision of combination schemes No. 1 and No. 3 are almost the same, with a 22.5 cm plane precision. The positioning precision of combination scheme No. 4 is in the second place, with a 27.1 cm plane precision, and the precision of combination scheme No. 2 is the worst, which is about 31.1 cm. Table 10 shows the percentage of past ratio-test values with the given thresholds. Combination scheme No. 3 still has a 99% passing rate under the ratio-test value of 2.5, even better than the performance of combination scheme No. 1. However, combination schemes No. 2 and No. 4 are both lower than 90% under the same ratio-test value. Figure 9 shows the results of experiment 2 calculated using combination schemes No. 1, No. 2, No. 3, and No. 4. The corresponding RMS errors of the three directions E, N, and U and in 3D are shown in Table 11.  
It can be seen that the positioning deviation of Experiment 2 is significantly larger than that of experiment 1, which is consistent with the previous discussion. Similar to that in experiment 1, the positioning precisions of combination schemes No. 1 and No. 3 are almost the same as that in experiment 2, and the plane precision is about 43.2 cm. The positioning precision of combination scheme No. 4 is in second place, with a 50.4 cm plane precision. The precision of combination scheme No. 2 is the worst and is about 51.9 cm. Table 12 shows the percentage of past ratio-test values with the given thresholds. Combination scheme No. 3 still has a 99% passing rate under the ratio-test value of 2 and is close to that of combination scheme No. 1 under all given thresholds. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Combination Scheme (No.)**} & \multirow{2}{*}{\(R\) **> 1.5(\%)**} & \multirow{2}{*}{\(R\) **> 2(\%)**} & \multirow{2}{*}{\(R\) **> 2.5(\%)**} & \multirow{2}{*}{\(R\) **> 3(\%)**} \\  & & & & \\ \hline
1 & 100.0 & 99.9 & 98.9 & 96.0 \\
2 & 99.5 & 95.5 & 88.1 & 80.0 \\
3 & 100.0 & 99.9 & 99.0 & 96.2 \\
4 & 99.4 & 93.1 & 83.4 & 73.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The percentage of past ratio-test values with the given thresholds in experiment 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Combination Scheme (No.)**} & \multicolumn{4}{c}{**RMS (cm)**} \\ \cline{2-5}  & E & N & U & **3D** \\ \hline
1 & 14.5 & 17.2 & 37.4 & 43.6 \\
2 & 20.8 & 23.1 & 52.9 & 61.4 \\
3 & 14.5 & 17.2 & 37.3 & 43.6 \\
4 & 17.9 & 20.3 & 44.5 & 52.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The statistical RMS results of the positioning differences in experiment 1.