# Earth Rotation Parameters Estimation Using GPS and SLR Measurements to Multiple LEO Satellites
## 4 ERP Estimation Based on the Combination of Multiple LEO Satellites
### Comparison of One-Step and Two-Step Methods  
Based on the observations of LAGEOS and seven LEO satellites, the one-step method is implemented and compared with the two-step method. Figure 14 shows the comparison of the ERP results derived from the two-step and one-step methods. When the one-step method is implemented, with the combination of seven LEO satellites and LAGEOS, an evident accuracy degradation is visible in the ERP result. Especially for the LOD parameter, a factor of 2.0 increase in RMS is observed. The similar degradation in simultaneously estimating the orbit and ERP also occurs in the ERP series derived from SLR observations to GNSS satellites, as reported by [PERSON] et al. [42]. This is possibly because of the introduction of orbit parameters. In the one-step solution, the orbit parameters of LEO satellites, such as the initial positions and velocities of LEO satellites, atmospheric drag scale parameter and empirical accelerations, have to be additionally estimated, which substantially increases the degree of freedom and weakens the stability of the solution.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Station**} & \multirow{2}{*}{**Solution**} & \multicolumn{2}{c}{**E (mm)**} & \multicolumn{2}{c}{**N (mm)**} & \multicolumn{2}{c}{**U (mm)**} \\ \cline{3-8}  & & **Median** & **IQR** & **Median** & **IQR** & **Median** & **IQR** \\ \hline \multirow{3}{*}{**All**} & **LAGEOS** & 1.2 & 23.1 & \(-\)4.9 & 25.6 & \(-\)5.6 & 24.9 \\ \cline{2-8}  & **7**LEO** & 1.9 & 16.4 & 0.3 & 11.5 & 1.9 & 17.1 \\ \cline{2-8}  & **7**LEO + LAGEOS** & 3.9 & 9.7 & \(-\)0.9 & 7.8 & 0.1 & 13.5 \\ \hline \multirow{3}{*}{**Core**} & **LAGEOS** & 0.2 & 14.4 & \(-\)1.6 & 19.0 & \(-\)4.3 & 14.5 \\ \cline{2-8}  & **7**LEO** & 3.0 & 15.0 & 0.4 & 10.6 & \(-\)0.4 & 11.3 \\ \cline{1-1} \cline{2-8}  & **7**LEO + LAGEOS** & 4.5 & 8.1 & \(-\)1.2 & 7.0 & \(-\)3.0 & 8.7 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Differences in estimated SLR station coordinates w.r.t. SLRF2014 for the LAGEOS solution, the 7-LEO solution and the combined solution. The statistical data are classified into all SLR stations and core stations decomposed into the east, north, and up components (in mm).  
Figure 13: Time series of station coordinate of a station in Monument Peak (7110) from LAGEOS solutions and the combination solutions.  
In addition, we also assess the quality of the LEO orbit derived from the two-step and one-step methods. Here, we take Sentinel-3A as an example, since the orbit product of Sentinel-3 from CNES is based on DORIS and GPS observations. Figure 16 shows the average RMS of orbit difference with respect to the orbit product of Sentinel-3A. This figure shows that, compared with the two-step solution, the orbit of the one-step solution can achieve a slight accuracy improvement at the level of sub-millimeter. A similar benefit is also reported by [PERSON] et al. [43], [PERSON] et al. [44], and [PERSON] et al. [45]. Such improvement can be attributed to the increased observations and strengthened observation geometry that benefit from additional SLR observations in the one-step method. It demonstrates that the inclusion of SLR observations can eliminate the inter-technique inconsistency between the GPS-only orbit and GPS-DORIS-based orbit product to some extent.