# Calibration and Impact of BeiDou Satellite-Dependent Timing Group Delay Bias
## 3 BeiDou Timing Group Delay Bias from Network Solution
### Estimation Method and Data Collection  
In Equation (4), the station coordinates can be fixed with a known position. The satellite coordinates, satellite clock, and TGD can be derived from the broadcast ephemeris. The troposphere delay can be corrected by the GPT2w and VMF models [18]. The ionosphere delay can be corrected by the Global Ionosphere Maps (GIM) model for single-frequency data [19] or eliminated by ionosphere-free combination for dual-frequency data. Hence, the estimated parameters are the receiver clocks and the TGD bias at each frequency. For a specific frequency, Equation (4) can be simplified as:  
\[y=\textit{bias}+t_{rcv}+\delta_{P} \tag{6}\]  
where _bias_ is the code bias correction and \(y\) is the pre-fit residual of the observation minus the correction. \(\delta_{P}\) is the observation noise with the stochastic model:  
\[\sigma^{2}=\sigma^{2}_{SISRE}+\sigma^{2}_{Trop}+\sigma^{2}_{Iono}+\sigma^{2}_{ \textit{Meas}} \tag{7}\]  
where \(\sigma\) is the variance of the code observation, while \(\sigma_{SISRE}\), \(\sigma_{Trop}\), \(\sigma_{Iono}\), and \(\sigma_{\textit{Meas}}\) represent the SISRE of the BDS satellites, the troposphere error after model correction, the ionosphere error after model correction, and the pseudo-range measurement noise, respectively.  
For each satellite tracked by a station, the observation model can be formed as shown in Equation (6). Therefore, the network-based observation model can be simplified as:  
\[\textbf{y}=\textbf{G}\textbf{x} \tag{8}\]  
where **G** is the design matrix of the observation model, and **x** is the estimated parameters:  
\[\textbf{G}=\left[\begin{array}{cccccc}1&0&\cdots&1&0&\cdots\\ 0&1&\cdots&1&0&\cdots\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots\\ 1&0&\cdots&0&1&\cdots\\ 0&1&\cdots&0&1&\cdots\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots\end{array}\right]\textbf{, }\textbf{x}=\left[\begin{array}{c}\textit{bias}_{1}\\ \textit{bias}_{2}\\ \vdots\\ t_{rcv1}\\ t_{rcv2}\\ \vdots\end{array}\right] \tag{9}\]  
The normal equation of Equation (9) is rank-defected, as no bias basis is chosen. Similar to DCB determination, in this study, we adopted the conventional zero-mean condition of satellite bias for a better comparison.  
By applying the least squares method, the bias on each satellite can be estimated together with the receiver clock, in which the satellite bias is estimated as a constant parameter while the receiver clock is different at each epoch and estimated with white noise. Thereafter, the TGD bias can be derived from Equation (5).  
For the estimation of the BDS TGD bias, 87 globally distributed MGEX stations were selected, as shown in Figure 4. The receiver information of these stations is summarized in Table 4. Although all the stations are able to track the B1I and B3I signals of BDS-2 and BDS-3, some BDS-3 satellites or the B3I signal of some satellites cannot be tracked by Trimble and Sepentrio, which may due to a firmware problem. For a statistical result, the observation data of each first day in a week from GPS weeks 2035\(-\)2059 were collected at these stations.