Global Navigation Satellite System (GNSS) positioning technology, as an innovative monitoring method, features the provision of real-time 3D absolute displacements of monitoring structures; continuously autonomous operation, regardless of the weather and visibility conditions; and easy operation. Additionally, the GNSS positioning technology can overcome some shortcomings of traditional monitoring methods, as it easily identifies low-frequency structural vibration responses. It has been widely used in the bridge deformation monitoring in the last few years [1
]. However, due to the existence of the multipath effect and random errors, true dynamic displacements are often overflooded by strong noise, which limits the GNSS vibration monitoring in modal parameters identification [5
]. Hence, data processing methods should be used to eliminate GNSS measurement error before extracting the structural dynamic characteristics.
The data processing methods consist of the time domain methods, the frequency domain methods, and the time–frequency domain methods [7
]. Bridge vibration responses are usually nonlinear and non-stationary, and the local time-varying characteristic may be ignored by using methods with a single parameter of time or frequency [8
]. Consequently, time–frequency methods, which can provide instantaneous data information in both time and frequency domains, are becoming more competitive to process structural health monitoring (SHM) data [9
]. In recent years, some time–frequency domain methods have been widely used in the identification of structural and modal parameters, and have shown a good performance. Time–frequency domain methods are mainly divided into two groups. One group is wavelet transform (WT) and its variants, such as continuous wavelet transform (CWT), least-squares wavelet analysis (LSWA), and weighted wavelet analysis (WWA). The other group includes empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD) and multivariate empirical mode decomposition (MEMD), etc. [10
In terms of noise reduction, data processing methods based on WT and EMD have been widely studied. WT can provide a high time–frequency resolution, by selecting a suitable basis function and a decomposition scale. However, the non-adaptive binary frequency partition technique may cause modal aliasing and false modes. Huang et al. [15
] proposed empirical mode decomposition (EMD), a well-known adaptive approach, to adaptively decompose the oscillatory data into sets of Intrinsic Mode Functions (IMFs). This method was able to separate stationary and non-stationary components effectively. However, the major shortcomings of the EMD are a lack of mathematical theory, mode aliasing, and the end effect [16
A novel empirical wavelet transform (EWT) has been developed recently by Gilles [17
], combining the advantages of WT and EMD. The main idea of EWT is to determine the segmentation of spectrum and then build a wavelet filter bank to decompose the vibration responses into a series of IMFs. Furthermore, the modal parameters are derived from the EWT-extracted IMFs by using Natural Excitation Technique (NExT) and Hilbert transform (HT). The EWT method exhibits calculation efficiency, excellent adaptation, and consolidated mathematic foundation. It is highly favorable for the processing and interpretation of non-stationary and complex data. Therefore, EWT demonstrates outstanding performance in various applications of machine fault diagnosis, seismic data analysis, image processing, medical disease diagnosis, and so on [19
]. The vibration responses of engineering structures are usually complex due to the structural scale and intricate interactions between structures and dynamic loadings. There will be improper frequency band division and false modes when the EWT method is applied to the above structural health monitoring data.
Recently, several improvements or modifications have been proposed to overcome the shortcomings of traditional EWT. One way to improve the traditional EWT is employing a spectrum other than the Fourier, one which is used for an appropriate boundaries division, such as the pseudospectrum [24
], power spectrum [26
], scale–space representation [27
], and time–frequency representation [28
]. Amezquita-Sanchez et al. [24
] presented a pseudospectrum segment method based on multiple signal classification (MUSIC). The MUSIC-EWT method identified the first six natural frequencies (NFs) and damping ratios (DRs) of the 123-storey Lotte World Tower. Xin et al. [26
] used a standardized autoregression power spectrum calculated by the Burg algorithm and the time–frequency representation determined by Synchro-extracting Transform (SET) to define the boundaries for EWT analysis. The enhanced EWT methods reliably identified the loosely spaced modes of a real footbridge and the instantaneous frequencies of a time-varying highway bridge. Xia et al. [27
] separated the mono-components from the health monitoring data of the civil structure via scale–space EWT and obtained the instantaneous modal parameters by the FREEVIB method. Another way to improve the traditional EWT is optimizing the Fourier spectrum segmentation method. Hu et al. [16
] presented an enhanced EWT based on the envelope of the Fourier spectrum, calculated by the order statistic filter, and with criterions presented to pick out useful peaks. Dong et al. [29
] proposed an EWT algorithm of modified spectrum separation based on the local window maxima (LWM) method. The experimental results indicated that the proposed method performed better than the original EWT method in identifying different damage mechanisms of composite structures. EWT achieved some successful applications in the field of the modal identification of civil structures, but the measurements were mainly based on the accelerometer whose data features were simpler and clearer than GNSS. Moreover, fewer studies discussed the judgment criterion of effective IMFs among a series of EWT-extracted IMFs. With the continuous development of GNSS hardware and software, it was crucial to identify the structural modes and dynamic displacements from GNSS vibration monitoring data.
In this paper, an improved EWT-based method is presented to denoise data and identify the modal parameters of bridge structures. Three steps are involved in the proposed method. Firstly, an auto-power spectrum based on the Yule–Walker algorithm [30
] and the Fourier amplitude spectrum are jointly applied to build the appropriate boundaries. Secondly, the improved EWT algorithm is used to decompose the GNSS coordinate time series into a number of effective IMFs and reconstruct the dynamic response according to a correlation coefficient-based criterion. Thirdly, effective IMFs are further used to identify the structural modal parameters, NFs and DRs, by using Natural Excitation Technique (NExT) and Hilbert transform (HT). Numerical and experimental studies are conducted in this study to validate the feasibility and reliability of this proposed method.
The paper is organized as follows. In Section 2
, the basic principles of the improved EWT algorithm and modal parameters identification based on the improved EWT are explained briefly, and the flowchart of data denoising and modal parameters identification is provided. Section 3
verifies the feasibility and accuracy of the improved EWT-based method for the identification of structural and modal parameters. Numerical studies on the vibration responses of a four-storey steel frame model, and acceleration response data of a suspension bridge are provided. In Section 4
, field experiments with GNSS and an accelerometer on the Wilford pedestrian bridge located in Nottingham, UK, are conducted to further validate the capability of the proposed method. Finally, the conclusions are presented in Section 5
3. Numerical Studies
For the feasibility and effectiveness of the improved EWT-based method, the vibration responses of a four-storey steel frame model, and the acceleration response data of a suspension bridge are provided in this section.
3.1. Numerical Study on a 4-Storey Steel Frame Model
The Digital Environment for Enabling Data-Driven Science (NEEDS) datasets provide a 4-storey, 2
2 bay, 3D steel-frame structure benchmark model, as shown in Figure 4
, which is used for related research on building structural health monitoring under external excitations [34
]. A 12-DOFs finite element model code in MATLAB provided by Johnson et al. [35
] was employed to simulate the dynamic response. Each floor was subject to environmental excitation in the form of white noise, perpendicular to the central column. The vibration data were measured by 16 accelerometers placed in the x- and y-directions on each floor. These sensors recorded the data for a duration of 20 s using a sampling rate of 1000 Hz and the damping ratio was assumed as 1% for each mode of the frame model. Moreover, 10% of the largest structural response root mean square (RMS) was added as noise. Figure 5
a shows the time–history response of sensor 9 in the x-direction and the corresponding Yule–Walker power spectral density estimate is displayed in Figure 5
Considering that the inherent modes of the structure were mainly concentrated from 0~100 Hz, a 100 Hz lowpass filter was applied to the acceleration response for a better separation of the closely spaced modes. An auto-power spectrum, based on the Yule–Walker algorithm and the Fourier amplitude spectrum of the filtered simulation time–history response, were jointly adopted to calculate the boundaries. We fixed a prior number of segments, N = 10, and no global trend removal, as well as smoothing operations, for the improved EWT. The frequency segment results are shown in Figure 6
As shown in Figure 6
, in the range from 0~100 Hz, the Fourier spectrum is divided into 10 frequency bands. The frequency band division results obtained using the improved EWT method do not cause modal aliasing and have a good segmentation. Furthermore, IMF10 belongs to a noise component and IMF 1,5,7,8,9 are pseudo components whose correlation coefficient is less than 0.1, as displayed in Figure 7
. According to the judgment criteria, these IMFs do not participate in the subsequent identification of modal parameters.
Based on the defined boundaries, the corresponding wavelet filter bank is established, and the original data is decomposed into several mono individual components through EWT. Taking IMF2 as an example (see Figure 8
a), the free decay response is obtained through NExT, and then the HT is applied to determine the envelop of the free decay response (see Figure 8
b). Besides, the logarithmic amplitude and phase angle representation are fitted by the least square algorithm (see Figure 8
c,d). The NF and DR estimated by the HT are 9.4051 Hz and 0.92%, respectively. The DR, using the nonlinear exponential function, is 0.96%, which is closer to the theoretical value. The modal parameters information and corresponding theoretical values of the remaining meaningful IMF are shown in Table 1
In addition, the measured frequencies of the pseudo components IMF5, 7, and 8 are 38.6387, 56.7514, and 66.6769 Hz, respectively, which are within 1% of the FEA results. This phenomenon proves that the pseudo components also contain useful information about the structure. The DRs of three sets of pseudo components are 1.17%, 2.35%, and 2.71%, respectively; the error from the theoretical value increases with the frequency. In summary, the improved EWT-based methodology can accurately identify the modal parameters of the closely spaced modes, with an NF error of less than 1% and a DR error of less than 5%. Meanwhile, the pseudo components were found to contain useful information regarding the structure. However, due to the small value of the correlation coefficient, there is a certain error between the recognized modes and the FEA values.
3.2. Numerical Study on Acceleration Response Data of a Suspension Bridge
Cheynet et al. [36
] established a simple model of the Lysefjord suspension bridge based on the long-term monitoring data. The acceleration-response data were based on simulated displacement records which were recorded by a sampling frequency of 15 Hz for a duration of 2000 s. Figure 9
shows the time–history responses of the acceleration data and its power spectrum.
a shows the segmentation results of the Fourier spectrum determined by the improved EWT method, and the obtained EWT component is shown in Figure 10
b. According to the effective component judgment criteria, the reconstructed components IMF 2~7 are effective components. Using the improved EWT-based procedure, the modal parameters of the Lysefjord suspension bridge are identified. For a comparison with the results of the proposed method, the target values of corresponding modal parameters are also provided in Table 2
It is noted that the first five modal parameters identified by the proposed method are essentially consistent with the target values. The NF identification error is less than 0.1%, and the highest error of DR does not exceed 8%. However, with the increasing frequency, the NF error of mono individual components is increased to 2%, and the DR error of high-order modes exceeds 10%. This may be related to the interference of high-frequency noise.
In order to effectively reduce the noise of bridge GNSS monitoring data and identify the structural modal parameters, this paper proposed an improved EWT-based method. The vibration responses of a four-storey steel frame model, acceleration response data of the Lysefjord bridge, and a Wilford bridge experimental study were employed to illustrate the efficiency of the proposed method. Moreover, the denoising ability of the proposed method was evaluated in comparison with the EMD and WT algorithm. In the numerical examples, the improved EWT, building boundaries using the Yule–Walker algorithm-based auto-power spectrum, combined with the Fourier spectrum, could identify the structural low-order, closely spaced modes. The modal parameters error of NF and DR was less than 2% and 10%, respectively. However, the DR of high-order components could not be measured accurately because of the existence of high-frequency noise. In the field experiments, the first three modal parameters of the Wilford bridge were extracted from the accelerometer data using the improved EWT-based procedure. Due to the low sampling frequency of the GNSS receiver, only a group of the modal parameters of 1.6707 Hz and 0.84% were identified from the NRTK-GNSS monitoring data, which was less than 5% in the fundamental frequency error compared with the error detected by Meng et al. [39
]. Moreover, the DR error between the NRTK-GNSS and the accelerometer result was 2.38%. The maximum dynamic displacements (10.10 mm) of the Wilford bridge were successfully derived from the NRTK-GNSS.
The first contribution of this study was that the feasibility of using the improved EWT-based method for data denoising was validated. The power spectrum calculated by the Yule–Walker algorithm combined with the Fourier spectrum could divide the frequency band properly. The proposed judgment criteria could separate effective modes from a series of components. Moreover, the effect of data denoising and dynamic displacements reconstruction was superior to the EMD and WT method.
In addition, the feasibility of using the improved EWT-based procedure for the identification of modal parameters was proven in the experiment presented herein. The low-order NFs and DRs of a four-storey steel-frame model and the Lysefjord bridge model were identified accurately. Moreover, its DR identification results were better than the estimation of Zhou et al. [7
] in the first four modes of the Lysefjord bridge. However, the DR of high-frequency components was an uncertain parameter. According to this method, the fundamental frequency and first-order damping ratio of the Wilford footbridge were effectively identified from NRTK-GNSS monitoring data, which were verified by accelerometer identification results.
The improved empirical wavelet transform would therefore be a promising tool of denoising GNSS data, as well as identifying structural modal parameters. In this study, the proposed method is capable of accurately identifying the low-order, closely spaced modal parameters of bridge structures. However, the DR error of high-order modes is large since the effective components extracted by improved EWT still contained noise. Further research needs to be conducted on the combination of EWT with other algorithms to denoise data further.