# Considering Inter-Frequency Clock Bias for BDS Triple-Frequency Precise Point Positioning
## 4 Discussion  
The GNSS of the future will feature the combination of multi-constellation and multi-frequency signals. In this paper, we focus on triple-frequency integration. A key issue in triple-frequency PPP is to align the satellite clocks, namely IFCB consideration. The IFCB of GPS Block IIF satellites has been investigated by many researchers in terms of sources, dependence, periods, prediction, modeling, variations and amplitude limits [13; 14; 20; 21]. The GPS multipath, which can be extracted using the ionospheric-free geometric linear combination, is a function of satellite elevation angle and antenna height, etc. [25; 26; 27]. The multipath effects should be considered during the IFCB estimation with the use of DIF measurements consisting of two different ionospheric-free carrier phase combinations. Full consistency of three carriers can be ensured for QZSS and Galileo, and thus the IFCB in multi-frequency integrated precise positioning can be ignored for them [15; 16; 17]. Several studies also relate to the IFCB of BDS-2 satellites [18; 19]. Compared with their results, a wider range of IFCB variations is obtained because the datasets spanning a longer period of time are involved in the analysis. In addition, some initial results are presented, including the analysis of cross-correlation and prediction for real-time precise positioning users as well as the IFCB of BDS-3 satellites. [PERSON] et al. [5] implemented the triple-frequency PPP with BDS data for the first time. However, the additional frequency had a marginal effect on the positioning accuracy in their results, likely because they neglected the effects of IFCB. By contrast, the triple-frequency PPP solutions here achieve higher positioning accuracy in comparison to dual-frequency PPP solutions. The consideration of the triple-frequency measurements for PPP for any satellite system was also discussed in the literature [40], which shows that one can model the triple-frequency code biases, and rigorously provide them for the PPP users for positioning. The difference between the two literatures on triple-frequency PPP is that another S-basis (datum) has been used in [PERSON] et al. [40], but one can transform between any S-basis choices through an S-transformation. Moreover, the code observations were also rigorously taken into account so as to maximize the redundancy of the models in the literature [40], which is a limitation for triple-frequency PPP model developed in this paper. According to the results shown in Figure 6, the IFCB estimates of BDS-2 satellites are at a level of several centimeters and exhibit higher noise level, which may limit the applications of IFCB. There are two ways to solve this issue. One way is to increase the number of stations with the capability of offering \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Direction** & **Without \(P_{1}\)/\(P_{3}\)** & **With \(P_{1}\)/\(P_{3}\)** \\ \hline \multirow{2}{*}{Positioning Accuracy (cm)} & East & 1.5 & 1.5 \\  & North & 0.3 & 0.3 \\  & Vertical & 1.8 & 1.7 \\ \hline \multirow{3}{*}{Convergence Time (min)} & East & 60.0 & 57.6 \\  & North & 34.0 & 32.9 \\ \cline{1-1}  & Vertical & 87.8 & 84.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Positioning accuracy and convergence time for triple-frequency PPP ignoring \(P_{1}/P_{3}\) code observations as well as triple-frequency PPP using \(P_{1}/P_{3}\) code observations and code-specific IFCB corrections. B1/B2/B3 BDS-2 tracking since the accuracy and reliability of IFCB estimates are mainly determined by the station number in the network. The other way is to develop the efficient undifferenced estimation approach for IFCB because the ED strategy will generate accumulated error. The ED IFCB estimate of a satellite may be inaccurate at an epoch because of the effects of measurement noises when the number of stations that can track the satellite at the epoch is not sufficient.