## 3 Real GNSS Position Time Series Analysis
After CME was filtered out, the average errors of residual time series of each station were calculated with Equation (12) for PCA and Equation (13) for IPCA, as follows:  
\[\hat{\sigma}_{p}(i)=\sqrt{\frac{1}{m_{i}}\sum_{k=1}^{m_{i}}\Bigl{[}\left( \mathbf{X}(t_{k},i)-\hat{\mathbf{S}}_{p}(t_{k},i)\right)^{2}\Bigr{]}}\qquad i=1,2,\cdots,n \tag{12}\]  
Figure 2.0  
where \(\hat{\mathbf{S}}_{p}\) is the CME extracted with PCA; \(\hat{\mathbf{S}}_{I}\) is the CME extracted with IPCA; and \(\boldsymbol{P}\) is the weight for each epoch in each site in Equation (3). Since the weight is determined by Equations (4) and (5), average errors computed by Equations (12) and (13) are comparable. A smaller average error means that CME is better filtered out. Figure 5 presents the average errors of all 25 stations before and after spatiotemporal filtering via PCA and IPCA, as well as the relative improvement of IPCA with respect to PCA. It revealed that after spatiotemporal filtering, average error was reduced for all stations and all coordinate components, and IPCA perform better than PCA for all coordinate components at all stations, especially for East component at YNJP station, which had an improvement of up to 13%. For PCA, the mean reductions of 25 stations were 15.84%, 10.86% and 14.32% for North, East and Up components, respectively. While for IPCA, the mean reductions were 18.23%, 15.42% and 18.25%. Thereby, the reductions of IPCA relative to PCA were about 2.39%, 4.56% and 3.93% for North, East and Up components, respectively. The correspondent relative improvements for North, East and Up components were 2.99%, 4.97% and 4.54%. In addition, the rank of the three components in relative improvements was consistent with that of the mean standard deviation of the weight factor in Figure 2.1 The correlations of improvement and standard deviation of the weight factor were 0.29, 0.76 and 0.47 in North, East and Up, respectively. The relative improvement was dependent on the standard deviation of the weight factor to some extent. In fact, it can verify the necessity of consideration of formal errors in spatiotemporal filtering, especially when formal errors are significantly inhomogeneous. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**White Noise**} & \multicolumn{2}{c}{**Flicker Noise**} \\ \cline{2-5}  & **PCA** & **IPCA** & **PCA** & **IPCA** \\ \hline North & 0.212 & 0.211 & 0.788 & 0.789 \\ East & 0.329 & 0.315 & 0.671 & 0.685 \\ Up & 0.100 & 0.094 & 0.900 & 0.906 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The proportion of white and flicker noise in Common Mode Error (CME). Since flicker noise was dominant in CME [28], flicker noise before and after spatiotemporal filtering is shown in Figure 6. We can see from Figure 6 that flicker noise reduced significantly after the removal of CME, regardless of whether PCA or IPCA employed. Average reductions were 34.93, 24.43, and 36.99% for PCA, and 34.96, 24.62 and 37.01% for IPCA. Hence, IPCA performed slightly better than PCA in removing common mode flicker noise. Figure 5: Average error and relative improvement for three components. Figure 6: Flicker noise magnitude for three components.