## 2 Methodology
### I-SOBI Method  
The traditional SOBI algorithm aims to recover independent source signals from their mixtures without prior knowledge of the sources or the mixing process. Its algorithm progression is shown in the following:  
1. Collect \(n\) independent source signals \(\mathbf{S}(t)=[s_{1}(t),s_{2}(t),\cdots,s_{n}(t)]^{T}\). This can be related to an \(m\times 1\) observed signals \(\mathbf{X}(t)=[x_{1}(t),x_{2}(t),\cdots,x_{m}(t)]^{T}\) through a transformation matrix \(\mathbf{A}\) as \(\mathbf{X}(t)=\mathbf{A}\cdot\mathbf{S}(t)+\Phi(t)\). In the application of bridge dynamic-deflection measurement by GB-SAR, time-series data of several close proximity points  
Figure 1: Workflow of the gradual de-noising algorithm for GB-SAR bridge deflection measurement. will be treated as the mixed signals \(\mathbf{X}(t)\) where \(\Phi(t)\) is the noise vector, \(n\) is the number of source signals, and \(m\) is the number of sensors. 2. Center the data by subtracting the mean to ensure that the signals have zero mean and estimate the covariance matrix \(\mathbf{R}_{x}\) of the zero-mean signals: \(\mathbf{R}_{x}=\mathbf{E}\Big{[}\mathbf{X}(t)\mathbf{X}^{T}(t)\Big{]}\), where \(\mathbf{E}[\cdot]\) denotes the expectation operation. 3. If \(\mathbf{R}_{x}\) is not a diagonal matrix, it can be transformed into a diagonal matrix \(\mathbf{R}_{z}\) through a whitening transformation matrix \(\mathbf{W}\). And \(m\times 1\) source signals estimates \(\mathbf{Z}(t)\) can be obtained. That is, \(\mathbf{Z}(t)=\mathbf{W}\cdot\mathbf{X}(t)\) and \(\mathbf{R}_{z}=\mathbf{W}\cdot\mathbf{R}_{x}\cdot\mathbf{W}^{T}\). 4. Find the joint diagonalization matrix \(\mathbf{B}\) for the two covariance matrices \(\mathbf{R}_{x}\) and \(\mathbf{R}_{z}\) through \(\mathbf{B}^{T}\cdot\mathbf{R}_{x}\cdot\mathbf{B}=\mathbf{\Lambda}\) and \(\mathbf{B}^{T}\cdot\mathbf{R}_{z}\cdot\mathbf{B}=\mathbf{I}\), where \(\mathbf{\Lambda}\) is a diagonal matrix and \(\mathbf{I}\) is the identity matrix. 5. Obtain the transformation matrix \(\mathbf{A}\) that will separate the sources through joint diagonalization matrix \(\mathbf{B}\) as \(\mathbf{A}=\mathbf{W}\cdot\mathbf{B}\cdot\mathbf{\Lambda}^{-1/2}\), where \(\mathbf{\Lambda}^{-1/2}\) is the inverse square root of matrix \(\mathbf{\Lambda}\). 6. Apply the transformation matrix to the estimated source signals to acquire the source signals with noises processed: \(\hat{\mathbf{S}}(t)=\mathbf{A}^{T}\cdot\mathbf{Z}(t)\)  
The SOBI algorithm is effective in separating sources when the source signals are statistically independent and have distinct second-order properties. It is important to note that the algorithm's performance can be affected by the presence of noises and the accuracy of the covariance and cumulant estimations [49]. To prevent the issue of indeterminate solutions that may arise in the SOBI de-noising process when utilizing dynamic-deflection data from a single-channel GB-SAR system, time-series data of several close proximity points will be collected and constructed as the mixed signals \(\mathbf{X}(t)\). This means that the source signals are not statistically independent. The noise information from adjacent signals will affect the de-noising effect of the current signal and may even introduce new noises in the current signal. As stated above, the diagonal elements of the transformation matrix \(\mathbf{A}\) represent the degree of association between each separated signal and the original signal source. Specifically, the element value in the \(i\)th row and \(j\)th column indicates the influence weight of the \(j\)th separated signal source on the \(i\)th output result. The off-diagonal elements reflect the weight of one signal source in the output corresponding to another signal source, representing the cross-components of different separated signal sources in the output data. These are key indicators for measuring the effect of signal source separation.