# Analysis of BDS-3 Onboard Clocks Based on GFZ Precise Clock Products
## 2 Methods and Data Collection
### Satellite Clock Offset Model  
The clock offset could be described as the coefficients of a quadratic polynomial in a given reference epoch \(t_{0}\)[20]:  
\[L_{i} =a_{0}+a_{1}(t_{i}-t_{0})+a_{2}(t_{i}-t_{0})^{2}+\delta_{i} \tag{3}\]  
where \(L_{i}\) is the clock offset of the satellite onboard clock at epoch \(t_{i}\); \(a_{0}\) denotes the bias parameter; \(a_{1}\) denotes the clock frequency, where variations can reflect clock stability;  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**PRN** & **Orbit** & **Satellite** & **Clock** & **Launch** & **PRN** & **Orbit** & **Satellite** & **Clock** & **Launch** \\  & **Type** & **Type** & **Type** & **Date** & **Type** & **Type** & **Type** & **Date** \\ \hline C01 & GEO & BDS-2 & Rb & 2019.05 & C26 & MEO & BDS-3 & PHM & 2018.08 \\ C02 & GEO & BDS-2 & Rb & 2012.10 & C27 & MEO & BDS-3 & PHM & 2018.01 \\ C03 & GEO & BDS-2 & Rb & 2016.06 & C28 & MEO & BDS-3 & PHM & 2018.01 \\ C04 & GEO & BDS-2 & Rb & 2010.11 & C29 & MEO & BDS-3 & PHM & 2018.03 \\ C05 & GEO & BDS-2 & Rb & 2012.02 & C30 & MEO & BDS-3 & PHM & 2018.03 \\ C06 & IGSO & BDS-2 & Rb & 2010.08 & C32 & MEO & BDS-3 & Rb & 2018.09 \\ C07 & IGSO & BDS-2 & Rb & 2010.12 & C33 & MEO & BDS-3 & Rb & 2018.09 \\ C08 & IGSO & BDS-2 & Rb & 2011.04 & C34 & MEO & BDS-3 & PHM & 2018.10 \\ C09 & IGSO & BDS-2 & Rb & 2011.07 & C35 & MEO & BDS-3 & PHM & 2018.10 \\ C10 & IGSO & BDS-2 & Rb & 2011.12 & C36 & MEO & BDS-3 & Rb & 2018.11 \\ C11 & MEO & BDS-2 & Rb & 2012.04 & C37 & MEO & BDS-3 & Rb & 2018.11 \\ C12 & MEO & BDS-2 & Rb & 2012.04 & C38 & IGSO & BDS-3 & PHM & 2019.04 \\ C13 & IGSO & BDS-2 & Rb & 2016.03 & C39 & IGSO & BDS-3 & PHM & 2018.06 \\ C14 & MEO & BDS-2 & Rb & 2012.09 & C40 & IGSO & BDS-3 & PHM & 2019.11 \\ C16 & IGSO & BDS-2 & Rb & 2018.07 & C41 & MEO & BDS-3 & PHM & 2019.12 \\ C19 & MEO & BDS-3 & Rb & 2017.11 & C42 & MEO & BDS-3 & PHM & 2019.12 \\ C20 & MEO & BDS-3 & Rb & 2017.11 & C43 & MEO & BDS-3 & PHM & 2019.11 \\ C21 & MEO & BDS-3 & Rb & 2018.02 & C44 & MEO & BDS-3 & PHM & 2019.11 \\ C22 & MEO & BDS-3 & Rb & 2018.02 & C45 & MEO & BDS-3 & Rb & 2019.09 \\ C23 & MEO & BDS-3 & Rb & 2018.07 & C46 & MEO & BDS-3 & Rb & 2019.09 \\ C24 & MEO & BDS-3 & Rb & 2018.07 & C59 & GEO & BDS-3 & PHM & 2018.11 \\ C25 & MEO & BDS-3 & PHM & 2018.08 & C60 & GEO & BDS-3 & PHM & 2020.03 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Information about BDS-2/3 satellites and clocks.  
denotes the drift rate, which is used to describe the frequency characteristic variation of the onboard clocks; and \(\delta_{i}\) denotes the random error. Therefore, the physical properties of the clock could be evaluated by the long-term series of clock drift rates and the daily fitting residuals. In this paper, these parameters are obtained by using the least square linear regression method. In addition, the root mean square (RMS) is taken as the statistical index for fitting residuals to evaluate the noise level of the satellite onboard clock. The computation of RMS is shown as follows:  
\[\text{RMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\delta_{i}{}^{2}} \tag{4}\]  
where \(N\) is the number of fitting residuals per day.