# GNSS-Based Vertically Monitoring of Super-Tall Buildings
## 5 Verticality Monitoring Result and Analysis of the W Tower
### Horizontal Motion Analysis of the W Tower
Referring to the statistics listed in Table 5, after processing, the corrections of all baseline vectors are less than 3 times the standard deviation \(\sigma\) of the baseline vector, and 96.27%, 91.59%, and 98.03% of corrections are within the range of \(\pm\)2.0 cm for all three directions (not including the 659 single baseline vectors). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Before Processing**} & \multicolumn{3}{c}{**After Processing**} \\ \hline  & \(V_{\Delta X}\) & \(V_{\Delta Y}\) & \(V_{\Delta Z}\) & \(V_{\Delta X}\) & \(V_{\Delta Y}\) & \(V_{\Delta Z}\) \\ \hline \(<\)1 cm & 78.81 & 63.38 & 80.17 & 80.64 & 65.27 & 82.08 \\
1–2 cm & 15.46 & 25.32 & 16.42 & 15.63 & 26.32 & 15.95 \\
2–3 cm & 3.81 & 8.40 & 2.94 & 3.73 & 8.41 & 1.97 \\ \(\geq\)3 cm & 1.92 & 2.90 & 0.47 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistics of baseline vectors’ corrections of the spatial GNSS monitoring network (%). Figure 6: GNSS monitoring network. Finally, the reference points' coordinates can be used as a constraint to then process the GNSS monitoring network via a least-squares adjustment at each epoch. Table 6 lists the statistics of the differences in position with respect to the coordinate time series of the monitoring point as shown in Figure 7 in the \(x\), \(y\), and \(H\) directions without and with processing. Referring to the statistics listed in Table 6, one may see that a small number of gross errors have been eliminated, especially in the \(x\) and \(y\) directions. After processing, the differences of 96.48% of the \(x\) coordinates and 98.57% of the \(y\) coordinates are within \(\pm\)2.0 cm, while the differences of 90.49% of the \(H\) coordinates is within \(\pm\)4.0 cm. However, there are some abnormal coordinates included in the result above (shown by the green points in Figure 7), which are calculated by just one reference point and one baseline vector instead of using a least-squares adjustment. In addition, the operation of a tower crane has an impact on the motion of the W Tower. Nevertheless, the coordinate time series of the monitoring point contain both the displacement information and the measurement-error information after the aforementioned processing. The displacement of the monitoring point is characterized by a systematic signal even though it may be small over a short time period, but the measurement-error is characterized by strong noise that can be removed by wavelet filtering [29; 30]. In order to extract the displacement signal of the W Tower, the wavelet filter is applied to further process the coordinate time series. Based on the understanding of the signal as discussed previously, a db4 wavelet filter at level 6 is applied to eliminate the background noise hidden in the coordinate time series. Figure 8 shows the coordinate variations before and after filtering. The standard deviations (STDs) of the background noises are \(\pm\)5.0, \(\pm\)4.5, and \(\pm\)13.0 mm in the \(x\), \(y\), and \(H\) directions, respectively. It is evident that, after the filtering, the background noises are almost eliminated and the GNSS time-series becomes smoother. After filtering, the differences of 89.34% of the \(x\) coordinates and 95.08% of the \(y\) coordinates are within \(\pm\)1.0 cm, while the differences of 95.24% of the \(H\) coordinates are within \(\pm\)4.0 cm. In addition, the results show that the displacement in the \(x\) direction is slightly larger than the displacement in the \(y\) direction, which may be related to wind direction. Therefore, one may conclude that the displacement of the W Tower is within \(\pm\)1.0 cm in horizontal direction at most epochs, so it has less impact on the vertical reference transmission.