# Calibration and Impact of BeiDou Satellite-Dependent Timing Group Delay Bias
## 2 BeiDou Timing Group Delay Bias from Precise Products
### BeiDou Broadcast Ephemeris Error  
The combined orbit and clock SISRE of the BDS broadcast ephemeris can be expressed as follows [12]:  
\[\text{SISRE}=\left\{ \begin{array}{c}\sqrt{\left(0.99\cdot\text{R}-\text{Clk} \right)^{2}+\frac{1}{126}\left(\text{A}+\text{C}\right)^{2}},\;for\;\text{GEO},\text{IGSO}\\ \sqrt{\left(0.98\cdot\text{R}-\text{Clk}\right)^{2}+\frac{1}{54}\left(\text{A} +\text{C}\right)^{2}},\;for\;\text{MEO}\end{array} \right. \tag{1}\]  
where GEO (geostationary orbit), IGSO (inclined geosynchronous orbit), and MEO (medium Earth orbit) stand for different satellite constellation of BDS. R, A, and C denote the orbit errors in the radial, along-track, and cross-track directions, while Clk denotes the clock offset error. For the assessment of SISRE, the post-processed precise products including the orbit, clock, and DCB provided by MGEX analysis centers are commonly used as a reference. As an MGEX analysis center, Wuhan University started to generate precise orbit and clock (WUM) of BDS-3 from 2019 ([[ftp://igs.ensg.ign.fr/pub/igs/products/mgex/](ftp://igs.ensg.ign.fr/pub/igs/products/mgex/)]([ftp://igs.ensg.ign.fr/pub/igs/products/mgex/](ftp://igs.ensg.ign.fr/pub/igs/products/mgex/))). When comparing the precise products with the BDS broadcast ephemeris, the difference in the time and coordinate reference system, satellite antenna offset correction, and clock offset correction should be considered [15]. It should be pointed out that, different from the traditional B1I/B2I ionosphere-free (IF) combination, the WUM products apply the B1I/B3I IF combination for precise orbit determination, as the B2I signal is not supported by BDS-3 satellites [16]. Considering that the BDS satellite clock transmitted in the broadcast ephemeris is based on B3I frequency, when comparing the satellite clock difference, the TGD or DCB should be applied to correct the clock frequency difference. For instance, as B1I and B3I are the common frequencies adopted in BDS-2 and BDS-3, when aligning the clock to B1I, B3I, and B1I/B3I IF frequencies, the corresponding corrections are as follows [11; 17]:  
\[ \begin{split}\text{Corr}_{\text{B1I}}=\text{TGD}_{1}+\frac{\beta _{2}^{2}}{\sqrt{1_{1}^{2}-f_{3}^{2}}}\text{DCB}_{\text{B1I/B3I}}\\ \text{Corr}_{\text{B3I}}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{3}^{2}} \text{DCB}_{\text{B1I/B3I}}\\ \text{Corr}_{\text{B1I/B3I}}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{3}^{2}} \text{TGD}_{1}\end{split}  \tag{2}\]  
where TGD\({}_{1}\) is the TGD correction from B3I to B1I, which can be derived from the broadcast ephemeris. \(f1\) and \(f3\) represent the frequencies of the BDS at B1I and B3I, and DCB\({}_{\text{B1I/B3I}}\) stands for the DCB correction from B3I to B1I. It can be seen from Equation (1) that the integrated satellite clock and radial orbit error, i.e., \(\alpha\cdot R-Clk\), contributes the main part of the SISRE. To investigate the BDS broadcast ephemeris error, we computed the weekly \(\alpha\cdot R-Clk\) of each satellite. According to Equation (2), the \(\alpha\cdot R-Clk\) can be derived at different frequencies. Using the data from GPS weeks 2035-2059 (DOY 006-180 in 2019), the weekly average \(\alpha\cdot R-Clk\) with the corresponding standard deviations (STD) for each satellite at B1I, B3I, and B1I/B3I frequencies are illustrated in Figure 3. As can be seen, the values of \(\alpha\cdot R-Clk\) on B1I and B3I for most of the satellites are within 3 ns. For B1I/B3I, it can reach 8 ns at most, due to the amplification of the TGD error according to Equation (2). This implies that the \(\alpha\cdot R-Clk\) bias is different for each satellite and each frequency. Therefore, it is not proper to attribute the bias to clock bias as pointed by [PERSON] et al. [10]. Meanwhile, for B3I and B1I/B3I, a noticeable systematic bias between BDS-2 and BDS-3 can be found.