# Calibration and Impact of BeiDou Satellite-Dependent Timing Group Delay Bias
## 2 BeiDou Timing Group Delay Bias from Precise Products
### Calibration of the Timing Group Delay Bias  
For the pseudo-range measurement of the BDS, the observation equations at B1I, B3I, and B1I/B3I can be written as follows:  
\[\left\{\begin{array}{c}P_{1}=\rho+t_{rcv_{1}}-t^{sat}+\delta_{trop}-\frac{ \delta_{ismo}}{f_{1}^{2}}+\delta_{tgd_{1}}+\delta_{P_{1}}\\ P_{3}=\rho+t_{rcv_{3}}-t^{sat}+\delta_{trop}-\frac{\delta_{ismo}}{f_{3}^{2}}+ \delta_{P_{3}}\\ P_{IF}=\rho+t_{rcv_{IF}}-t^{sat}+\delta_{trop}+\frac{f_{1}^{2}}{f_{1}^{2}-f_{3 }^{2}}\delta_{tgd_{1}}+\delta_{P_{IF}}\end{array}\right. \tag{3}\]  
where \(P_{i}\) is the raw measurement of the pseudo-range at frequency \(i\) and \(IF\) denotes the ionosphere-free combination. \(t_{rcv_{i}}\) is the receiver clock including the receiver hardware delay at frequency \(i\); \(t^{sat}\) is the satellite clock on B3I; \(\delta_{trop}\) and \(\delta_{ismo}\) are the troposphere and ionosphere delays, respectively; \(\delta_{tgd_{1}}\) is the TGD correction from B3I to B1I; and \(\delta_{P_{i}}\) is the pseudo-range observation noise at frequency \(i\).  
Assuming that the DCB is a more accurate product compared with the TGD, it can be inferred that the abovementioned \(\alpha\cdot R-Clk\) bias may come from the TGD bias according to Equation (2). For the analysis of the B1I and B3I frequencies, let \(dtgd_{0}\) be the TGD bias at B3I frequency, and \(dtgd_{1}\) be the residual TGD bias of the TGD\({}_{1}\). If a TGD bias does exist in the broadcast ephemeris, Equation (3) can then be rewritten as:  
\[\left\{\begin{array}{c}P_{1}=\rho+t_{rcv_{1}}-t^{sat}+\delta_{trop}-\frac{ \delta_{ismo}}{f_{1}^{2}}+\left(\delta_{tgd_{1}}+dtgd_{1}\right)+dtgd_{0}+ \delta_{P_{1}}\\ P_{3}=\rho+t_{rcv_{3}}-t^{sat}+\delta_{trop}-\frac{\delta_{ismo}}{f_{3}^{2}}+ dtgd_{0}+\delta_{P_{3}}\\ P_{IF}=\rho+t_{rcv_{IF}}-t^{sat}+\delta_{trop}+\frac{f_{1}^{2}}{f_{1}^{2}-f_{3 }^{2}}\delta_{tgd_{1}}+dtgd_{0}+\delta_{P_{IF}}\end{array}\right. \tag{4}\]  
Figure 3: Averaged weekly \(\alpha\cdot R-Clk\) bias at B1I (**a**), B3I (**b**), and B1I/B3I (**c**) frequencies. The error bar stands for the standard deviation of the bias from GPS weeks 2035–2059. The BDS-2 and BDS-3 satellites are separated by the dashed line.  
Therefore, the relationship between \(\alpha\cdot R-Clk\) and the TGD bias is:  
\[\begin{split} dtgd_{0}+dtgd_{1}=bias_{B1I}\\ dtgd_{0}=bias_{B3I}\\ dtgd_{0}+\frac{f_{1}^{2}}{f_{1}^{2}-f_{3}^{2}}dtgd_{1}=bias_{B1I/ B3I}\end{split} \tag{5}\]  
where \(bias_{B1I}\), \(bias_{B3I}\), and \(bias_{B1I/B3I}\) are the \(\alpha\cdot R-Clk\) biases at B1I, B3I, and B1I/B3I frequencies, as discussed in Figure 3.  
By applying the least squares method, \(dtgd_{0}\) and \(dtgd_{1}\) can be easily derived for each satellite. The calibrated TGD bias is listed in Table 2. Meanwhile, given that the apparent systematic bias is observed between BDS-2 and BDS-3, we can also compute the TGD bias and the corresponding STD of BDS-2 and BDS-3 accordingly, which is listed in Table 3. It can be seen that \(dtgd_{0}\) and \(dtgd_{1}\) vary within \(\pm 4\) ns at different satellites and different frequencies. The overall \(dtgd_{1}\) between BDS-2 and BDS-3 is \(3.23\) ns according to Table 3, which is almost the same as the average TGD bias by direct comparison of the TGD and DCB bias. This is reasonable, as the \(\alpha\cdot R-Clk\) bias partly results from the bias between the TGD and DCB. Additionally, the STD of \(dtgd_{1}\) of BDS-3 is larger than that of BDS-2, which may be due to the fact that the BDS monitoring stations are limited in China and do not have good coverage of the Medium Earth Orbit (MEO) satellites [9]. For \(dtgd_{0}\), the overall bias between BDS-2 and BDS-3 is \(1.89\) ns, which also proves that there exists a systematic clock bias between BDS-2 and BDS-3. However, compared to the STD of less than \(0.3\) ns at each satellite, the STD of \(dtgd_{0}\) and \(dtgd_{1}\) is much larger. This indicates that, despite the distinct TGD bias between BDS-2 and BDS-3, it is better to calibrate this bias by each independent satellite.