## 4 Results and Discussion
### Uncertainty Estimation  
The results of the uncertainty estimation as introduced in Section 3.5 are illustrated in Figure 6, and summarized in Table 5 (3-month mean and standard deviation range), for the RO LEO satellites under investigation and two representative GPS transmitter satellites possibly involved in an occultation event (one GPS delivering good estimates and one slightly degraded). The component estimates in Table 5 use plausible fractional weighting of the 3D variance (\(u_{\rm 3D}^{2}\)) [69, 89]. In case of a typical day, when the geometry modeling system (Figure 1) delivers results within highest-quality demands for RO processing, the 3D-RMS uncertainties are generally obtained near the conservative bound of 5 cm in position and 0.05 mm/s in velocity for the LEO satellite.  
_LEO satellites._ The estimated random uncertainties for all missions (Figure 6, panels (a) to (d)) show consistent behavior over time, with lowest values found for GRACE. On days where the uncertainty calculation failed due to missing input (i.e., the POD of one of the control orbit runs was not successful) the estimated random uncertainty is set to a conservative estimate of 2 cm, which leads to a slightly increased combined uncertainty estimation on those days for GRACE. For the majority of days, the raw-estimated systematic uncertainty component stays within the conservative bound. In particular this applies to most days for GRACE and the Metop satellites. For CHAMP some more days are found that moderately exceed the conservative-bound threshold.  
_GPS satellites._ With regard to the GPS transmitter satellites (Figure 6, panels (e) and (f)), the estimated random uncertainties are fixed to the predefined conservative value of 1 cm (cf. Figure 3). The estimated systematic uncertainty, as derived from the accuracy codes, reveals some days of decreased orbit accuracy. Since the estimated uncertainties are empirically mapped with a conversion factor of \(1/3000\) from position to velocity (Section 4.3), it is well visible that the contribution of GPS velocity uncertainties is quite minor compared to the velocity uncertainties derived for the LEO satellites.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Satellite**} & \multicolumn{4}{c}{**Random (cm)**} & \multicolumn{4}{c}{**Systematic (cm)**} \\ \cline{2-10}  & **3D** & **Radial** & **Along** & **Cross** & **3D** & **Radial** & **Along** & **Cross** \\ \hline CHAMP & \(1.44\pm 0.42\) & \(0.72\pm 0.21\) & \(0.85\pm 0.25\) & \(0.91\pm 0.27\) & \(5.28\pm 1.39\) & \(2.64\pm 0.69\) & \(3.13\pm 0.82\) & \(3.34\pm 0.88\) \\ GRACE-A & \(1.00\pm 0.23\) & \(0.50\pm 0.11\) & \(0.59\pm 0.13\) & \(0.64\pm 0.14\) & \(5.00\pm 0.00\) & \(2.50\pm 0.00\) & \(2.96\pm 0.00\) & \(3.16\pm 0.00\) \\ Metop-A & \(1.90\pm 0.05\) & \(0.95\pm 0.03\) & \(1.13\pm 0.03\) & \(1.20\pm 0.03\) & \(5.00\pm 0.00\) & \(2.50\pm 0.00\) & \(2.96\pm 0.00\) & \(3.16\pm 0.00\) \\ Metop-B & \(1.93\pm 0.06\) & \(0.96\pm 0.03\) & \(1.14\pm 0.03\) & \(1.22\pm 0.04\) & \(5.02\pm 0.07\) & \(2.51\pm 0.04\) & \(2.97\pm 0.04\) & \(3.17\pm 0.05\) \\ \hline GPS-23 & \(1.00\pm 0.00\) & \(0.50\pm 0.00\) & \(0.59\pm 0.00\) & \(0.63\pm 0.00\) & \(3.11\pm 0.50\) & \(1.55\pm 0.25\) & \(1.84\pm 0.29\) & \(1.97\pm 0.31\) \\ GPS-24 & \(1.00\pm 0.00\) & \(0.50\pm 0.00\) & \(0.59\pm 0.00\) & \(0.63\pm 0.00\) & \(4.75\pm 3.19\) & \(2.37\pm 1.60\) & \(2.81\pm 1.89\) & \(3.00\pm 2.02\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Random and systematic position uncertainty estimates (3D) and decomposed with 3D-variance weighting factors \(w\) into radial (\(w_{\rm r}=0.25\)), along-track (\(w_{\rm a}=0.35\)), and cross-track (\(w_{\rm c}=0.4\)) components.