# BDS-3 Triple-Frequency Timing Group Delay/Differential Code Bias and Its Effect on Positioning
## 2 Data and Methods
### Triple-Frequency Satellite TGD/DCB Correction Model for BDS-3
#### 2.3.1 TGD/DCB Correction Model with Broadcast Satellite Clock  
The partial UCDs are included when correcting the satellite clock offset using the broadcast ephemeris, as shown in Equation (14). This implies that there is no need to consider the TGD/DCB correction if the B3I is used for the single-frequency positioning. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Combination** & **Observation** & \(e_{i}\) & \(e_{j}\) & \(e_{k}\) & **Ionosphere** & **Noise** \\ \hline \multirow{4}{*}{IF123} & 0 & B1I/B2a/B3I & 2.343 & \(-\)1.254 & \(-\)0.089 & 0 & 2.659 \\  & 1 & B1C/B2a/B3I & 2.290 & \(-\)1.196 & \(-\)0.094 & 0 & 2.586 \\  & 2 & B1I/B2b/B3I & 2.566 & \(-\)1.229 & \(-\)0.338 & 0 & 2.865 \\  & 3 & B1C/B2b/B3I & 2.497 & \(-\)1.168 & \(-\)0.330 & 0 & 2.777 \\ \hline \multirow{4}{*}{IF1213} & 0 & B1I/B2a & 2.314 & \(-\)1.314 & 0 & 0 & 2.662 \\  & 0 & B1I/B3I & 2.944 & 0 & \(-\)1.944 & 0 & 3.527 \\  & 1 & B1C/B2a & 2.261 & \(-\)1.261 & 0 & 0 & 2.588 \\  & 1 & B1C/B3I & 2.844 & 0 & \(-\)1.844 & 0 & 3.389 \\  & 2 & B1I/B2b & 2.487 & \(-\)1.487 & 0 & 0 & 2.898 \\  & 2 & B1I/B3I & 2.944 & 0 & \(-\)1.944 & 0 & 3.527 \\  & 3 & B1C/B2b & 2.422 & \(-\)1.422 & 0 & 0 & 2.808 \\  & 3 & B1C/B3I & 2.844 & 0 & \(-\)1.844 & 0 & 3.389 \\ \hline \multirow{4}{*}{UC123} & \multirow{4}{*}{0} & B1I & 1 & 0 & 0 & 1 & 1 \\  & 0 & B2a & 0 & 1 & 0 & 1.761 & 1 \\  & & B3I & 0 & 0 & 1 & 1.514 & 1 \\  & & B1C & 1 & 0 & 0 & 1 & 1 \\  & 1 & B2a & 0 & 1 & 0 & 1.793 & 1 \\  & & B3I & 0 & 0 & 1 & 1.542 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: BDS-3 triple-frequency model characteristics. The IF1213 model is essentially two independent IF combinations. The TGD/DCB correction formula for the dual frequency IF combination with the frequencies \(i\) and \(j\), after correcting for the satellite clock offset using the broadcast ephemeris, can be expressed as:  
\[\overline{b}_{IF_{ij}}^{s,brdc}=\frac{f_{i}^{2}\cdot b_{i}^{s}-f_{j}^{2}\cdot b _{j}^{s}}{f_{i}^{2}-f_{j}^{2}}-b_{B3I}^{s} \tag{16}\]  
CAS provides BDS-3 DCB products with the following definitions:  
\[\left\{ \begin{array}{l}DCB_{C1X-C5X}=b_{B1C}^{s}-b_{B2n}^{s}\\ DCB_{C2I-C6I}=b_{B1I}^{s}-b_{B3I}^{s}\\ DCB_{C1X-C6I}=b_{B1C}^{s}-b_{B3I}^{s}\\ DCB_{C1X-C7Z}=b_{B1C}^{s}-b_{B2B}^{s}\end{array} \right. \tag{17}\]  
In addition, TGD and DCB can be converted as follows:  
\[\left\{ \begin{array}{l}TGD_{1}=DCB_{C2I-C6I}\\ TGD_{B2n}=DCB_{C1X-C6I}-DCB_{C1X-C5X}\\ TGD_{B1 Cp}=DCB_{C1X-C6I}\end{array} \right. \tag{18}\]  
Substituting the corresponding frequencies into Equations (16) and (17), the DCB correction of IF1213 can be expressed as:  
\[\left\{ \begin{array}{l}\overline{b}_{IF_{11224}}^{s,brdc}=\frac{f_{B1I}^{2} \cdot DCB_{C2I-C6I}-f_{B2n}^{2}\cdot(DCB_{C1X-C6I}-DCB_{C1X-C5X})}{f_{B1I}^{s} -f_{B2n}^{2}}\\ \overline{b}_{IF_{11231}}^{s,brdc}=\frac{f_{B1I}^{2}\cdot DCB_{C2I-C6I}}{f_{B1I }^{2}-f_{B3I}^{2}}\\ \overline{b}_{IF_{1123}}^{s,brdc}=\frac{f_{B1C}^{2}\cdot DCB_{C1X-C6I}-f_{B2n}^ {2}\cdot(DCB_{C1X-C6I}-DCB_{C1X-C7Z})}{f_{B1C}^{s}-f_{B2n}^{2}}\\ \overline{b}_{IF_{1123}}^{s,brdc}=\frac{f_{B1C}^{2}\cdot DCB_{C1X-C6I}-f_{B2n}^ {2}\cdot(DCB_{C1X-C6I}-DCB_{C1X-C7Z})}{f_{B1C}^{s}-f_{B2n}^{2}}\\ \overline{b}_{IF_{1123}}^{s,brdc}=\frac{f_{B1C}^{2}\cdot DCB_{C1X-C6I}-f_{B2n}^ {2}\cdot(DCB_{C1X-C6I}-DCB_{C1X-C7Z})}{f_{B1C}^{s}-f_{B2n}^{2}}\end{array} \right. \tag{19}\]  
After correcting for the satellite clock offset, the DCB correction of IF123 can be expressed as follows:  
\[\overline{b}_{IF_{ijk}}^{s,brdc}=e_{i}\cdot b_{i}^{s}+e_{j}\cdot b_{j}^{s}+e _{k}\cdot b_{k}^{s}-b_{B3I}^{s} \tag{20}\]  
then:  
\[\left\{