## 5 Impact Assessments on Estimated Parameters at Collocated Stations
### Propospheric Parameters from Multi-GNSS Analyses  
Improved solutions using multi-constellations and the backward smoothing are demonstrated in the time series of ZTD and horizontal gradient differences obtained from the two collocated stations, ZIM2 and ZIMJ, Figure 14. Results of the single-/multi-constellations are visualized by different colors: (1) standalone GPS in red, (2) GPS + GLO in green, and (3) GPS + GLO + GAL in blue. A positive effect is visible for all parameters, and is similar using both the Kalman filter (a) and the backward smoothing (b). The scatters for multi-constellations are smaller compared to the standalone GPS solution. An even more significant effect of the smoothing is visible for the gradient parameters. Theoretically, zero differences are expected for the collocated stations with the same antenna height. However, a vertical difference between ZIM2 and ZIMJ is about 2 m which can cause about 0.5 mm difference in ZTDs  
Figure 13: Standard deviations for ZTDs (**a**) and east gradients (**b**) using different delays in backward smoothing at two dual-stations—Zimmerwald (**top**) and Matera (**bottom**).  
when considering the pressure decreases approximately by 11.3 Pa per meter near the geoid and the 100 Pa difference in the atmospheric pressure causes a 2.27 mm difference in ZHD [51]. As we observe a ZTD difference about 2-3 mm, it can be still attributed to remaining station-specific systematic errors, e.g., such as phase center offset and variation models.  
Numerical statistics (biases and standard deviations) characterize an impact of single- and multi-constellation solutions on the estimated ZTDs and horizontal tropospheric gradients when using the Kalman filter, Table 5. It should be noted that GLONASS (R) and Galileo (E) observations were down-weighted by a factor of 2 with respect GPS (G) to reflect less accurate precise products and models. A positive effect of multi-constellation is visible at all parameters, and particularly in terms of the standard deviation, while the impact of GLONASS is more significant compared to Galileo. That is expected, as the GLONASS is the operational system longer supported by precise models and products in the scientific community as well as due to a lower number of operational Galileo satellites. As already discussed, ZIM2-ZIMJ differences indicate a bias of about 2-3 mm in ZTD. The improvements in all parameters reached 15-30% in terms of RMSE at both dual-stations. Table 6 then shows the impact of the backward smoothing on all solutions using single- or multi-constellation data. All the above-mentioned characteristics are similar to the Kalman filter, and the backward smoothing then improved mainly standard deviations (by about 25%).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Station Pair**} & \multirow{2}{*}{**GNSS**} & **BIAS \(\pm\) SDEV** & **BIAS \(\pm\) SDEV** & **BIAS \(\pm\) SDEV** \\  & & **ZTD (mm)** & **N-GRD (mm)** & **E-GRD (mm)** \\ \hline
**ZIM2-ZIMJ** & G & +2.8 \(\pm\) 1.4 & +0.08 \(\pm\) 0.17 & \(-\)0.02 \(\pm\) 0.14 \\
**ZIM2-ZIMJ** & GR & +2.4 \(\pm\) 1.3 & +0.02 \(\pm\) 0.14 & \(-\)0.02 \(\pm\) 0.12 \\
**ZIM2-ZIMJ** & GRE & +2.0 \(\pm\) 1.3 & +0.03 \(\pm\) 0.14 & \(-\)0.04 \(\pm\) 0.13 \\
**MAT1-MATE** & G & \(-\)0.5 \(\pm\) 2.4 & \(-\)0.03 \(\pm\) 0.18 & +0.18 \(\pm\) 0.25 \\
**MAT1-MATE** & GR & +0.1 \(\pm\) 2.3 & +0.01 \(\pm\) 0.15 & +0.14 \(\pm\) 0.22 \\
**MAT1-MATE** & GRE & +0.1 \(\pm\) 2.2 & +0.00 \(\pm\) 0.15 & +0.13 \(\pm\) 0.21 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistics (BIAS \(\pm\) SDEV) for Kalman filter using GPS (G), GLONASS (R) and Galileo (E).  
Figure 14: Time series of ZTD (**top**), north gradient (**middle**) and east gradient (**bottom**) differences at Zimmerwald dual-station when using the Kalman filter (**a**) and the backward smoothing (**b**).