## 3 Simulated Experiment and Analysis  
To validate the ability of the improved SOBI signal de-noising method to distinguish the useful signal and noise, a simulation example was conducted in this study. The application of the time-series displacements of three adjacent monitoring points used to obtain the de-noised time-series displacement of the middle monitoring point among three adjacent monitoring points, can be regarded as a linear mixture of source signals of useful signal and noise signal. Therefore, three simulated sub-signals, which included two useful sub-signals, were constructed; \(s_{1}(t)\) (Figure 3a) and \(s_{2}(t)\) (Figure 3b), and a noise signal, \(s_{3}(t)\) (Figure 3c), of which the waveforms are shown in Figure 3.  
\[s_{1}(t)=0.3 sin(3\pi t), \tag{9}\]  
\[s_{2}(t)=(0.2+0.3 sin(4\pi t))\ +0.5 cos(2\pi t)+cos(10\pi t), \tag{10}\]  
\[s_{3}(t)=0.1 randn\ (1,\ N), \tag{11}\]  
where the sampling point \(N\) is 600 for the noise signal \(S_{3}(t)\), and the sampling frequency is 200 HZ. Moreover, the mixing matrix \(A\) is a random matrix, which meets the requirement of a column full rank matrix. In this study, the mixing matrix \(A\) is defined as follows:  
\[A=\left[\begin{array}{ccc}0.2&0.7&0.5\\ 0.4&0.3&0.4\\ 0.1&0.9&0.8\end{array}\right]. \tag{12}\]  
Figure 3: Three sets of simulation source signals. (**a**) The curve of signal \(S_{1}(t)\), (**b**) the curve of signal \(S_{2}(t)\), and (**c**) the curve of signal \(s_{3}(t)\).  
As shown in Figure 4, the three mixed signals were linearly mixed using three sets of simulation source signals. Each mixed signal contained the characteristic information of the three sets of source signals. Without more prior information, the characteristic information of the source signals could not be obtained intuitively from the mixed signals. Using the improved SOBI method, three sets of signals were separated from the three mixed signals.  
As shown in Figure 5, the three separated sub-signals are consistent with the three source signals in the waveform characteristics. To further evaluate the accuracy of the separated results, correlation coefficients between the separated signal and the source signal were calculated using Equation (13):  
\[C=\sum_{n=1}^{N}\left(Y_{n}(t)-\overline{Y}_{n}(t)\right)\left(S_{n}(t)- \overline{S}_{n}(t)\right)/\sqrt{\sum_{n=1}^{N}\left(Y_{n}(t)-\overline{Y}_{n }(t)\right)^{2}*\sum_{n=1}^{N}\left(S_{n}(t)-\overline{S}_{n}(t)\right)^{2}}, \tag{13}\]  
where \(S_{n}(t)\) is the source signal, \(\overline{S}_{n}(t)\) represents the average value of the source signal, \(Y_{n}(t)\) is the separated signal, and \(\overline{Y}_{n}(t)\) represents the average of the separated signals.  
As shown in Table 1, the correlation coefficients between each simulated source signal and the corresponding separated signal component are all greater than 0.99, which indicates that the improved SOBI method effectively separated different source signals from the mixed signals. In particular, for the noise source signal and the corresponding separated signal component, the correlation coefficient is 0.9984, which can effectively distinguish the useful and noise signal and further perform signal de-noising. In this study, for the obtained time-series displacements using GBSAR, the useful and noise signals were regarded as independent source signals. Therefore, the noise in the obtained time-series displacements was effectively reduced using the improved SOBI signal de-noising method.  
Figure 4: The original signal generated from the source signals using a mixing matrix. (**a**) The curve of the first mixed signal, (**b**) the curve of the second mixed signal, and (**c**) the curve of the third mixed signal.