## 4 POD and the Atomic Clock's Frequency Stability Performance
The ranging accuracy of SLR is better than 1 cm, which indicates that it is an effective measuring method for orbit validation. Considering that the maximum nadir angle measured by SLR is less than 15 deg, the main SLR residual of the orbit is the error in the radial direction. Four satellites (C20, C21, C29 and C30) of the BD3 MEO satellites are tracked by the International Laser Ranging Service (ILRS) [33] and the LRA offsets (expressed in satellite coordinate system) from the satellite center of the mass used in the paper are listed in Table 7. The SLR residuals are shown in Figure 9.  
Seen from Figure 9, the absolute values of residuals greater than 20 cm have been removed and only a few normal points are deleted. It should be noted that more than 98% of the original data are retained and used, as shown in the last three columns of Table 8. The corresponding statistical results are also listed in Table 8, including the mean, Standard Deviation (STD) and RMS values of \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**SVN**} & \multirow{2}{*}{**PRN**} & \multicolumn{3}{c}{**LRA Offset Reference to the Center of Satellite Mass**} \\ \cline{3-5}  & & **X** & **Y** & **Z** \\ \hline C202 & C20 & 0.6021 & –0.0768 & 1.2367 \\ C206 & C21 & 0.5986 & –0.0866 & 1.2650 \\ C207 & C29 & 0.6646 & 0.4249 & 0.6427 \\ C208 & C30 & 0.6646 & 0.4249 & 0.6427 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The BD3 MEO laser retroreflector array (LRA) offset reference to the center of the satellite mass (units: m)  
Figure 9: The satellite laser ranging (SLR) validation residuals of the BD3 MEO satellites, where C20/C21 are made by CAST and C29/C30 are made by SECM. the residual series. The RMS values of SLR validation series of C20/C21/C29/C30 are 5.1 cm, 6.4 cm, 4.7 cm, and 4.3 cm, and a tiny constant bias of about \(-2\sim 4\) cm can be found in the CAST satellites (C20, C21) and SECM satellites (C29, C30), while there is no systematic bias. Fortunately, the SLR validation accuracy is consistent with the results of the BD2 MEO satellites. In addition, high-performance atomic clocks are very essential for satellite navigation and positioning systems. In general, clock bias can be estimated together with orbit parameters. The estimated clock bias parameters are the deviation from the system reference time, which are actually a kind of phase data. Different from the previous BD2 satellites, improved rubidium atomic frequency standard (RAFS) and the Passive Hydrogen Maser (PHM) made in China are installed on BD3 satellites. To evaluate the atomic clock frequency stability, Allan Deviation is an effective method, which can be expressed as:  
\[\sigma(\tau)=\sqrt{\frac{1}{2(N^{\prime}-2)\tau^{2}}\sum_{i=1}^{N^{\prime}-2} \left[x_{i+2m}-2x_{i+m}+x_{i}\right]^{2}},N^{\prime}=\text{int}(N/m)+1 \tag{4}\]  
where \(\tau\) represents the averaging time interval, \(N^{\prime}\) is the number of clock parameters within \(\tau,x\) means the clock bias parameters. To avoid the influence of gross errors, the trend term in the clock bias parameters is removed by quadratic linear fitting. The outliers greater than three times that of the STD are then marked and removed. In order to obtain more statistical samples, the overlapping Allan Deviation is adopted in the study and the corresponding results are presented in Figure 10.  
As Figure 10 shows, the clock's frequency stability of BD3 satellites improves obviously. This is mainly because the atomic clocks installed on BD2 satellites are the first generation RAFS made in China and the frequency accuracy and stability are relatively low. According to our analysis, the frequency stability of BD3 is approximately \(2.43\times 10^{-14}\) at an interval of 10,000 s, and \(2.51\times 10^{-15}\) at an interval of 86,400 s. Two distinctly sharp minima can be found in the vicinity of 86,400 s and 172,800 s, which is consistent with the orbital period. This indicates that the clock bias may be contaminated by orbit errors.