## 1 Introduction  
With the rapid deployment of transportation networks, an increased number of bridges have been built in the world. However, a combination of factors such as age, environment, overload, and geological activities have increased the deterioration of bridges, which can ultimately lead to a reduction in their load carrying capacity--or even sudden collapse--leading to a loss of life and property [1; 2]. Therefore, damage detection for bridges is of the utmost importance. Ground-based synthetic aperture radar (GBSAR) technology is a new type of ground-based microwave interferometry technology with the advantages of real-time monitoring, high-distance resolution, fast measurement speed, high precision, wide measurement coverage, easy operation, and all-weather and all-day measurements [3,4]. Since the first published paper (in 1999) on microwave interferometry for non-contact vibration measurements on large structures [5], GBSAR has been used extensively for dynamic deflection measurements of bridges in recent years [6,7]. However, during the data acquisition process of dynamic deflection measurements of bridges using GBSAR, the surrounding environment, human operation, and the equipment itself will inevitably increase noise in the obtained time-series displacements, which reduces the damage detection accuracy [8]. In particular, the used GBSAR equipment should be placed under the monitored bridge, and the instantaneous vibrations of the equipment itself will be inevitably caused by passing vehicles under the monitored bridge, which will reduce the accuracy of the obtained dynamic time-series displacement. Therefore, it is of great importance to reduce the influence of noise information in the time-series displacements of bridges obtained using GBSAR--especially for the instantaneous vibrations of the equipment itself. Currently, the primary signal de-noising methods for the time-series data include the filtering method [9, 10, 11, 12], the wavelet transform method [13,14], the singular value decomposition method [15,16], and the empirical mode decomposition (EMD) method [17, 18, 19]. Filtering methods use statistical features to derive some estimation algorithms, and further estimate the useful signals or filter the signals with certain statistical features from the mixed signal, which can improve the signal-to-noise ratio (SNR). However, they remove not only noise, but also the high frequency components of non-stationary signals, and normally require a priori information of the statistical characteristics of the signal and noise, which are not effective for non-stationary signals containing sharp edges and impulses of short duration [9,10,12]. Therefore, without any priori information of the obtained dynamic time-series displacement, filtering methods are not suitable for signal de-nosing of the non-stationary dynamic time-series displacement obtained using GBSAR. With the advantage of a good time-frequency localization characteristic, the wavelet transform can be used to obtain the wavelet coefficients of signal and noise with different properties on each scale. This method can further remove the wavelet coefficient of the noise portion to obtain the de-noised signal by reconstructing the remaining wavelet coefficients, which is suitable for analyzing non-stationary signals [13]. However, there are two main limitations of the wavelet transform: first of, the results are affected by the selection of the wavelet basis, and different noise frequency scales need different wavelet threshold criteria, which cause a lack of adaptability [15]; secondly, it is not suitable for non-linear signals [17]. In this study, the obtained time-series displacement should be a non-linear signal in case the monitored bridge was damaged. Moreover, it is also difficult to determine the suitable wavelet basis and wavelet threshold criteria as the instantaneous vibrations of the equipment itself caused by passing vehicles with different speed and weight.