## 4 Precise Orbit Determination (POD)
### Orbit Validation
#### 4.2.2 SLR Validation  
SLR is an independent optical system and is treated as an external validation method, in contrast to the GNSS L band. Errors related to the propagation path can be eliminated accurately with a ranging accuracy of less than 1 cm, which benefits from the high signal frequency. As the optical system is located on the ground surface, values measured by SLR are mainly assigned to the radial component of the orbit. The precise coordinates and velocities of the tracking stations are provided by International Laser Ranging Service (ILRS) [41]. SLR observations can be used to check the geometric distances from satellites to SLR tracking stations. SLR supplies effective methods for diagnosing defects in SRP models [42], or for combined orbit determination of GNSS [43] and low Earth orbit satellites [44]. All the BeiDou-2 satellites are equipped with a laser retroreflector array (LRA), but only C01, C08, C10, C11, and C13 among the BeiDou-2 satellites and all the BeiDou-3e satellites are tracked by ILRS,  
Figure 5: Comparison of RMS values of orbit overlap before and after ambiguity fixing. The dark yellow and red bars represent real-valued ambiguity (float) and ambiguity fixed (fixed) solutions, respectively. where C31 has not been tracked by ILRS since 18 December 2016. The LRA offsets supplied by the OCC are listed in Table 6.  
Amongst all the SLR tracking stations, during test data period in our analysis, from DOY 021 to DOY 079 in 2017, 20 stations have been tracking the BeiDou-2 satellites, whereas only nine stations have been tracking the BeiDou-3e satellites. Thus, the number of normal points (NPs) for BeiDou-3e satellites is much fewer than that for BeiDou-2 satellites: three SLR stations tracked C32 (Yarragadee: 65 NPs, Shanghai: 10 NPs, Changchun: 10 NPs); four stations for C33 (Yarragadee: 13 NPs, Matera: 8 NPs, Monument Peak: 6 NPs, Herstmonceux: 6 NPs); and nine stations for C34 (Yarragadee: 15 NPs, Shanghai: 9 NPs, Changchun: 4 NPs, Matera: 6 NPs, Monument Peak: 7 NPs, Herstmonceux: 7 NPs, Tahiti: 5 NPs, Beijing: 3 NPs, Graz: 2 NPs). However, orbit precision can still be validated approximately by time series and statistical information of SLR. The results of SLR validation are listed in Table 7. It is noted that some abnormal data have been removed from the raw dataset, but the data utilization rates were all greater than 95%. The RMS values of the SLR validation results are also shown in Table 7. As seen from the results, the RMS values were slightly higher than the orbit overlap results for the radial component. As listed in Table 7, the GEO satellite C01 exhibited the worst performance, with a standard deviation (STD) of about 20 cm and a bias of about \(-19\) cm. The STDs for all the IGSOs and MEOs among the tracked satellites were smaller than 10 cm, and there was no obvious bias for C08, C10, C11, and C13. With respect to the BeiDou-3e satellites, number of the tracking points was significantly fewer than those for the BeiDou-2 satellites. For BeiDou-3e satellites, C32 exhibited the worst performance, with an STD of about 10 cm and an obvious bias of approximately \(-10\) cm in this calculation, which \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**SLR Offsets**} \\ \cline{2-4}  & **X** & **Y** & **Z** \\ \hline C01 & \(-54.38\) & \(-57.04\) & 109.30 \\ C08 & \(-40.03\) & \(-57.30\) & 109.34 \\ C10 & \(-40.23\) & \(-57.30\) & 110.00 \\ C11 & \(-37.54\) & \(-53.80\) & 110.00 \\ C13 & \(-40.23\) & \(-57.30\) & 110.00 \\ C31 & \(-95.91\) & 18.13 & 63.76 \\ C32 & 18.51 & 68.54 & 196.02 \\ C33 & 61.21 & \(-7.17\) & 122.90 \\ C34 & 61.01 & \(-7.12\) & 124.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Satellite laser ranging (SLR) offsets for BeiDou-2 and BeiDou-3e satellites (cm).