## 5 Impact Assessments on Estimated Parameters at Collocated Stations
### Carrier-Phase Post-Fit Residuals and Slant Delays  
Figure 15 shows the carrier-phase post-fit residuals when using the Kalman filter PPP (a) and the backward smoothing PPP (b) for multi-GNSS solutions supported with the COM (top) and GBM (bottom) MGEX products. The carrier-phase residuals are useful indicators of an overall performance of the solution including the quality of input products and models. Showing plots for the ZIM2 station only, below discussed characteristics are common to other stations too. First, we observe a common elevation-dependent pattern of characteristics of post-fit residuals when using elevation-dependent observation weighting, \(1/\sqrt{\sin^{2}(\mathrm{e})}\). Second, the backward smoothing does not change the distribution of the carrier-phase post-fit residuals significantly. The main effect of the backward smoothing is thus understood mainly as improved accuracy of the estimated parameters. The tropospheric slant delays reconstructed from the model parameters and post-fit residuals will thus benefit primarily from the improvement of the parameters. Third, the GPS residuals (black) are the smallest and compact compared to other systems indicating actual quality of precise models and products. Galileo shows the largest residuals, however, we had to substitute various precise models, in particular station antenna  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Station Pair** & **GNSS** & \begin{tabular}{c} **BIAS \(\pm\) SDEV** \\ **ZTD (mm)** \\ \end{tabular} & \
\begin{tabular}{c} **BIAS \(\pm\) SDEV** \\ **N-GRD (mm)** \\ \end{tabular} &
\begin{tabular}{c} **BIAS \(\pm\) SDEV** \\ **E-GRD (mm)** \\ \end{tabular} \\ \hline
**ZIM2-ZIMJ** & G & +2.7 \(\pm\) 1.1 & +0.11 \(\pm\) 0.12 & \(-\)0.02 \(\pm\) 0.10 \\
**ZIM2-ZIMJ** & GR & +2.3 \(\pm\) 1.0 & +0.06 \(\pm\) 0.11 & \(-\)0.02 \(\pm\) 0.09 \\
**ZIM2-ZIMJ** & GRE & +1.9 \(\pm\) 1.0 & +0.07 \(\pm\) 0.12 & \(-\)0.04 \(\pm\) 0.09 \\
**MAT1-MATE** & G & \(-\)1.3 \(\pm\) 1.6 & \(-\)0.04 \(\pm\) 0.15 & +0.22 \(\pm\) 0.19 \\
**MAT1-MATE** & GR & +0.6 \(\pm\) 1.4 & +0.00 \(\pm\) 0.12 & +0.16 \(\pm\) 0.17 \\
**MAT1-MATE** & GRE & +0.5 \(\pm\) 1.4 & -0.01 \(\pm\) 0.11 & +0.16 \(\pm\) 0.16 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Statistics (BIAS \(\pm\) SDEV) for backward smoothing using GPS (G), GLONASS (R), Galileo (E).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Station Pair** & **Products** & \begin{tabular}{c} **Kalman/Smoother** \\ **ZTD SDEV (mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Kalman/Smoother** \\ **N-GRD SDEV (mm)** \\ \end{tabular} &
\begin{tabular}{c} **Kalman/Smoother** \\ **E-GRD SDEV (mm)** \\ \end{tabular} \\ \hline
**ZIM2-ZIMJ** & IGS01 & 3.3/2.7 & 0.34/0.26 & 0.32/0.25 \\
**ZIM2-ZIMJ** & IGS03 & 2.3/1.9 & 0.25/0.23 & 0.25/0.21 \\
**ZIM2-ZIMJ** & COM MGEX & 1.4/1.1 & 0.17/0.12 & 0.14/0.10 \\
**ZIM2-ZIMJ** & GFZ MGEX & 1.4/1.1 & 0.18/0.12 & 0.14/0.10 \\
**MAT1-MATE** & IGS01 & 4.8/3.6 & 0.41/0.36 & 0.42/0.33 \\
**MAT1-MATE** & IGS03 & 3.4/2.6 & 0.31/0.33 & 0.39/0.32 \\
**MAT1-MATE** & COM MGEX & 2.5/1.6 & 0.18/0.15 & 0.25/0.19 \\
**MAT1-MATE** & GFZ MGEX & 2.5/1.6 & 0.19/0.15 & 0.24/0.19 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Summary statistics for dual stations using different products: real-time (IGS01, IGS03) and final MGEX (COM, GBM), the Kalman filter and, in addition, the backward smoothing.  
phase center offsets and variations by using the values from GPS models. Due to the same reason, we may notice systematic changes in the elevation-dependent redistribution of Galileo residuals (red) after applying the backward smoothing. Fourth, we notice post-fit residuals GLONASS (green) about double the size when using the GBM product compared to the COM product. As the characteristics are common to all the stations, it indicates a lower quality of GLONASS orbits and clocks from the GBM product or some inconsistent models used for the product generation and in the PPP software.