## 4 Discussion  
When monitoring bridges using GB-SAR, the primary signals acquired are the true bridge movement signals. However, these signals are affected by various types of noise, including Gaussian noise, nonlinear noise, and background noise, which arise from hardware noise, environmental factors, and human-induced disturbances. These noise sources not only degrade the quality of the signals but also reduce the accuracy of the monitoring data. This study is dedicated to exploring effective strategies for complex environmental noise reduction to enhance the accuracy of GB-SAR technology in bridge deflection measurements. First, we propose a two-stage noise reduction approach. In the first stage, we employ the proposed I-SOBI method, which aims to preserve the contribution of practical \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Original Data** & **SOBI** & **I-SOBI** & **Proposed Method** \\ \hline STD & 0.0252 & 0.0195 & 0.0194 & 0.0190 \\ RMSE & 0.0254 & 0.0220 & 0.0221 & 0.0218 \\ RVR & \(\backslash\) & 0.7741 & 0.7730 & 0.7561 \\ NCC & \(\backslash\) & 0.9452 & 0.9460 & 0.9515 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results from the practical experiment. information in separated signals while recovering pure signals from noisy ones that exhibit nonlinear characteristics or non-Gaussian distributions. In the second stage, we apply the FFT, combined with the A-CFS method, to address the residual high-frequency noise in the signals. Second, the validation process, which utilizes multiple evaluation metrics, adds complexity by requiring robust and accurate performance across different test conditions, ensuring the reliability and effectiveness of the proposed approach. Finally, the method is validated using simulation signals and data collected from the Qixia Bridge. This adds further complexity by demonstrating the applicability of this method in synthetic and real-world environments. Our findings indicate that the two-stage gradual noise reduction method offers several advantages, such as its effectiveness in reducing noise in nonlinear time-series signals and its successful application in GB-SAR-based bridge deflection monitoring. Through simulation and real-world experiments, the following highlights can be derived:  
1. The proposed I-SOBI method is adept at preserving the valuable information within separated signals throughout the reconstruction process. Figures 10 and 11a illustrate that traditional SOBI's de-noising performance is suboptimal, often introducing additional noise, as indicated by the red line in the figures. This limitation arises from the traditional SOBI's difficulty in precisely separating nonlinear signals, which results in each separated signal retaining useful information. The straightforward discarding of noise-representing signals and subsequent reconstruction [20] inevitably results in inadequate noise-reduction. In contrast, the I-SOBI method yields a signal that is closer to the true value. By taking into account the dominant frequencies in the separated signals and optimizing the transformation matrix **A**, the I-SOBI method maximizes the retention of non-noise information, thus effectively preventing the introduction of extraneous noise. Nonetheless, the reconstructed signals still exhibit some random fluctuations indicative of residual high-frequency noise effects. 2. To address the residual high-frequency noise in signals after initial processing while retaining as much useful information as possible, the second stage of de-noising employs the FFT with A-CFS for post-processing signals that have been de-noised by the I-SOBI algorithm. It can determine an appropriate cutoff frequency and ensure the quality of the de-noised output for a given dataset. Figures 11b and 14 show that the signals further refined by the FFT with the A-CFS method are much closer to the true signal and exhibit greater smoothness. This method outperforms other techniques, with the SNR showing a 30% increase and RMSE reduced by 54%, compared to the original data.