## 4 Synthetic Time Series Analysis  
To further test and verify the performance of the IPCA approach, simulations were carried out in this section. Firstly, eight stations were randomly selected from the above 25 stations. The CME of Up component extracted with PCA in Section 3 and the formal errors series of the selected eight stations were employed. In synthetic time series, the extracted CME of the selected stations is regarded as the true signal \(S_{0}\), and noise \(e_{0}\) is generated according to formal errors \(\sigma(t_{k},i)\), where \(i\) and \(k\) denote the \(i\)th station and the \(k\)th epoch. Therefore, the simulated time series \(\mathbf{X}_{0}\) can be expressed as:  
\[\mathbf{X}_{0}=\mathbf{S}_{0}+\mathbf{e}_{0} \tag{14}\]  
Thus, in the simulated region with eight stations, the signals were spatial dependent, although noises were not. Since the true CME is known, we can compute the RMS of the extracted CME with:  
\[RMS=\sqrt{\frac{\sum\left(\mathbf{S}_{0i}-\hat{\mathbf{S}}_{0i}\right)^{2}}{m_{i}}} \tag{15}\]  
where \(\mathbf{S}_{0}(t_{k},i)\) and \(\hat{\mathbf{S}}_{0}(t_{k},i)\) denote the true CME and extracted CME of the \(i\)th station at the \(k\)th epoch; and \(m_{i}\) is the number of epochs of the \(i\)th station. The smaller the RMS computed from Equation (15), the better the extracted CME. The RMS improvement of IPCA relative to PCA is thus:  
\[\text{Imp}=\frac{RMS_{P}-RMS_{I}}{RMS_{P}}\times 100\% \tag{16}\]  
where \(RMS_{p}\) and \(RMS_{I}\) denote the RMS derived by PCA and IPCA, respectively. As follows, four cases were considered and compared. To avoid dependence on selected stations, eight stations were named as Group 1 and were randomly selected from 25 stations as follows:  
* Group 1: YNDC, YNGM, YNJP, YNNMZ, YNSD, YYA, YNYM, YYYS  
Taking the station YNDC for example, the CME and formal errors of the Up component are shown in Figure 7. We can see that the magnitude of formal errors was about 6 mm, which was even larger than that of CME. To avoid the signal being lost in the strong noise, a scale factor 0.3 was applied to the Up formal errors of all the selected stations. Figure 7: CME (top panel) and formal errors (bottom panel) of YNDC site in Up. Case 1: White noise and power law noise combination  
Many studies indicate that not only white noise, but also colored noise exists in GNSS position time series [29; 30] and that the combination of white noise and power law noise is quite common [7; 8; 9]. Therefore, the combination of white and power law noise was employed in noise simulation. Here, the fraction of white noise (\(\alpha\)) and power spectral index (\(\kappa\)) was chosen as 0.3 [9] and \(-0.4\)[31], respectively. The amplitude of white noise (\(\sigma_{wh}\)) and power law noise (\(\sigma_{pl}\)) was determined with [32]:  
\[\sigma_{wh}(t_{k},i)=\sqrt{\alpha}\,\sigma(t_{k},i) \tag{17}\]  
\[\sigma_{pl}(t_{k},i)=\frac{\sqrt{1-\alpha}}{dt^{-\kappa/4}}\,\sigma(t_{k},i) \tag{18}\]  
Since daily time series were investigated, \(dt\) was \(1/365.25\) in Equation (18). The program \"simulatenoise\" in Hector software [33] was used to generate the white and power law noise. Figure 8 shows the RMS values of the extracted CME signals of YNDC for each of the 500 simulations, where the RMS values for the CME derived by IPCA were smaller than those by PCA, meaning that CME by IPCA was closer to the true CME than that by PCA. The relative improvement of IPCA, with respect to PCA, was \(1.41\%\) for the mean RMS values of 500 simulations. Case 2: Impact of different formal errors  
Since the CME of Up component was selected as true value, it is natural that formal errors were used to simulate noise in Case 1, although it was scaled with a factor 0.3. From Equation (4), it is apparent that formal errors, scaled or not, do not change the weight factor. In order to investigate the impacts of different formal errors, the formal errors of North and East components were also employed in the simulation of Case 2, while the same true CME was used.