## 2 Methodology
### Improved SOBI  
The SOBI method uses joint approximation diagonalization to process the covariance matrix, which can solve the best estimation of the source signal and the mixing matrix and further obtain the separated signal components from different source signals. It is a relatively robust blind source separation method [26]. The SOBI method is based on the second-order statistics of the original observation data, which can estimate the source signal component and separate multiple Gaussian noise sources using relatively few data points [33]. By considering the linear mixed model together with the characteristics of the obtained time-series displacement using GBSAR, it is proposed in this study to select the obtained time-series displacements of three adjacent monitoring points in the same time domain as the original signals to obtain the de-noised signal of the middle point among the three adjacent monitoring points. The three time-series displacements can be regarded as a linear mixture of source signals of useful information and noise. The basic principle of the improved SOBI signal de-nosing method is based on a time correlation of the source signal and the joint approximation diagonalization of the covariance matrix [27]. When it is applied to signal de-noising, the noise information is separated and zeroed by the other two sets of data, and the mixed matrix is used to inversely reconstruct the signal to obtain the de-noised signal. Figure 2 shows the entire workflow of the improved SOBI signal de-noising method for the obtained time-series displacement of bridges using GBSAR. The workflow includes the following four key technologies: (1) the observation signal, \(X(t)\), is whitened to obtain a unit matrix covariance matrix, \(Z(t)\). The second-order correlation among the signal components is removed, and the sampling covariance matrix of the whitened data, \(Z(t)\), is calculated. (2) The orthogonal matrix is obtained using joint approximate diagonalization of the covariance matrix with different time delays. (3) The estimated value of the source signal, \(Y(t)\), and the mixing matrix, \(A\), are calculated using the orthogonal matrix. (4) The estimated value of the source signal, \(Y(t)\), is converted in the frequency domain using the fast Fourier transform (FFT), and the noise signal component is identified using the frequency characteristics. A new independent separated signal component matrix, \(Y_{Z}(t)\), is generated using a zeroing process for noise signal components, which is inversely reconstructed using a mixing matrix, \(A\), to recover the original amplitude of the de-noised signal, \(Y_{N}(t)\). Denote \(X(t)=\left[x_{1}(t)x_{2}(t)x_{3}(t)\right]^{T}\) as mixed signals of time-series displacements of three adjacent points in the same time domain, and denote \(S(t)=\left[s_{1}(t)s_{2}(t)s_{3}(t)\right]^{T}\) as the source signals, which are assumed to be independent of each other. In addition, denote \(Y(t)\) as the estimated value of the source signal. The detailed steps of the improved SOBI de-noising method for the obtained time-series displacement of bridges using GBSAR are as follows [26; 28; 34]:  
(1) Estimate the sample covariance, \(R(0)\), from the three sets of original observation signals (time-series displacements of three adjacent monitoring points), denote \(\lambda_{1},\lambda_{2},\lambda_{3}\) as the largest eigenvalues and \(h_{1},h_{2},h_{3}\) as the corresponding eigenvectors of \(R(0)\). (2) In the case where white noise is assumed to exist, the estimated value of the noise variance, \(\sigma^{2}\), is the average of the smallest eigenvalues of \(R(0)\). The whitening signal is \(Z(t)=\left[z_{1}(t)z_{2}(t)z_{3}(t)\right]^{T}\), where \(z_{i}(t)=\left(\lambda_{i}-\sigma^{2}\right)^{-\frac{1}{2}}h_{i}*x(t)\), \((i=1,2,3)\).