## 2 Methodology
### I-SOBI Method
Ideally, the closer the value of the diagonal elements to 1 and the closer the value of the off-diagonal elements to 0, the better the independence between the separated signal sources. If the value of the diagonal elements deviates from 1 and the value of the off-diagonal elements is large, this may indicate a certain correlation between the original signals, implying that the algorithm has not been able to effectively separate the signal sources. Traditionally, when using the SOBI algorithm for de-noising bridge deformation monitoring signals collected by GB-SAR, only the first separated signal component in separated signals \(\mathbf{Z}(t)\) is retained, while the rest of the separated signals are considered as noises and directly deleted. That is, the traditional SOBI method typically retains only the value of the first column in each row in transformation matrix \(\mathbf{A}\), setting all other elements to 0, in order to achieve the purpose of noise reduction. This approach ignores the weight of different separated signal sources in the outputs. Since the original signal sources composed of dynamic-deformation time-series data of several close proximity points have a certain degree of correlation, our goal is to identify the similar change trends of these dynamic deformation measurements, that is, the useful information, and separate out the noise information. Therefore, all separated signal components may contain useful information. Simply deleting all components except the first separated signal component will result in the loss of some useful information and reduce the accuracy of the reconstructed signal. Thus, the I-SOBI algorithm is proposed in this study in order to mitigate the issues caused by the linear correlation of original signals. The objective of this improvement is to significantly boost the separation efficacy between the signal sources. The fundamental concept of the proposed I-SOBI algorithm is to optimize the transformation matrix \(\mathbf{A}\) and separated signals \(\mathbf{Z}(t)\). The main improvements are expressed as follows:  
1. Traverse each row of the mixing matrix \(\mathbf{A}\) in a row-wise priority order. Take the \(i\)th row as an example and identify the minimum value of the elements in the row, denoted as \(a_{ij}\), which represents the influence weight of the \(j\)th separated signal on the \(i\)th output signal. This weight is typically small and close to 0, hence directly setting \(a_{ij}=0\). The transformation matrix can be optimized as \(\mathbf{A}^{\prime}\). 2. Keep the remaining separated signals with information close to the principal frequency in the first separated component, expressed as \(\mathbf{Z}^{\prime}(t)\). As analyzed previously, the separated signals, apart from the first component, may contain useful information in the remaining parts. This is an improvement because it replaces the traditional SOBI method of simply deleting all of the separated signal components except for the first one, which leads to the loss of some useful information. 3. Acquire the optimized outputs with noises processed through the means of the following: \(\mathbf{\hat{S}}^{\prime}(t)=\left(\mathbf{A}^{\prime}\right)^{\mathrm{T}} \cdot\mathbf{Z}^{\prime}(t)\). The improved SOBI method will keep the useful information that has been removed in the remaining separated signals and solve the problem of reduced accuracy in signal reconstruction to some extent. Through the I-SOBI algorithm, the impact of non-zero off-diagonal elements in the transformation matrix \(\mathbf{A}\) on the noise reduction results can be mitigated to a certain extent. In practical applications, the values of off-diagonal elements may also be affected by noises and interference. If the algorithm fails to correctly handle these disturbances, it may result in larger values for the off-diagonal elements. Considering that the residual noises are often presented in the high-frequency part of the signals, this study employs the FFT method further for post-processing of the signals processed by the I-SOBI algorithm.