# Estimation of Inter-System Biases between BDS-3/GPS/Galileo and Its Application in RTK Positioning
## 3 Datasets and Experiments
### Quality of the Signals
#### 3.2.2 Triple-Frequency Phase Combination  
The triple-frequency (TF) phase combination, which is extracted by the difference in two different ionosphere-free (IF) combinations, is available, but at least three different phase observations should exist in one satellite. Thus, the TF phase combination is both geometry-free and ionosphere-free and can be applied in the evaluation of the noise of phase observations, which is defined as [8]:  
\[\begin{array}{ll}TFP&=(\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}\phi_{1}-\frac{f_ {2}^{2}}{f_{1}^{2}-f_{2}^{2}}\phi_{2})-(\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}} \phi_{1}-\frac{f_{2}^{2}}{f_{1}^{2}-f_{3}^{2}}\phi_{3})\\ &=(\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}-\frac{f_{1}^{2}}{f_{1}^{2}-f_{3}^{2}} )\phi_{1}-\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}\phi_{2}+\frac{f_{3}^{2}}{f_{1} ^{2}-f_{3}^{2}}\phi_{3}\end{array} \tag{9}\]  
where \(TFP\) denotes the TF phase combination; \(f_{1}\), \(f_{2}\), and \(f_{3}\) are the frequencies of the phase observations \(\phi_{1}\), \(\phi_{2}\), and \(\phi_{3}\) in meters, respectively. Then, the residuals of the TF phase combination can be extracted by subtracting the mean value of the combination in the continuous observations when no cycle slips occur to eliminate the unknown ambiguity in the combination.  
To compute the TF phase combination in BDS-3/GPS/Galileo, Table 3 shows the sets of the frequencies in BDS-3/GPS/Galileo. Table 3 lists the corresponding coefficients of the phase observations in TF phase combination. Influenced by these coefficients, the noise of TF phase combination is magnified with respect to any original phase observations. Thus, the amplification factor of BDS-3/GPS/Galileo in TF phase combination is also listed in Table 3.  
Figure 3 displays the time series of the TF phase combination of C19, G01, and E01 in the left panel. The noise decreases as the elevation increases in C19 and E01. However, some bias variation occurs in the TF phase combination of G01, which is known as inter-frequency clock bias (IFCB) [34]. In past studies [8; 35], the presence of IFCB in GPS was identified, whereas no apparent bias variation appears in Galileo or BDS-3. To eliminate this kind of bias variation, the between-receiver difference is the method usually applied because the IFCB remains the same for a certain satellite in different stations. Figure 3 displays the RMS of the between-receiver difference in the TF phase combination against the elevation for every 2\({}^{\circ}\) with all satellites considered. The between-receiver difference in the TF phase combination further magnifies the noise, and we only considered the amplification factor influenced by the coefficients of the phase observations. It is clear that Galileo has the largest RMS over all elevations compared to BDS-3 or GPS after IFCB correction. When the elevation is lower than 40\({}^{\circ}\), BDS-3 has a lower RMS than GPS. However, with increasing elevation, the RMS of BDS-3 and GPS are at a similar level of about 0.5-1 mm. As for Galileo, the RMS remains at a high level, over 1 mm, even at a relatively high elevation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**System** & **Signal 1 1** & **Signal 2 1** & **Signal 3 1** & **Amplification Factor** \\ \hline BDS-3 & B1C (\(-\)0.1614) & B2a (\(-\)1.2606) & B2b (1.4220) & 1.9071 \\ GPS & L1 (0.2851) & L2 (\(-\)1.5457) & L5 (1.2606) & 2.0149 \\ Galileo & E1 (\(-\)0.1614) & E5a (\(-\)1.2606) & E5b (1.4220) & 1.9071 \\ \hline \hline \end{tabular} 1  
\end{table}
Table 3: The choice of the frequencies used to compute triple frequency (TF) phase combinations and the corresponding coefficient and amplification factor of phase observations.  
Figure 2: Root mean square (RMS) of multipath (MP) vs. elevation angle of all the observed signals in BDS-3, GPS, and Galileo. The RMS of MP of every 2\({}^{\circ}\) of elevation of all the signals were computed with the whole constellation that can be tracked.