It is difficult to accurately identify the dynamic deformation of bridges from Global Navigation Satellite System (GNSS) due to the influence of the multipath effect and random errors, etc. To solve this problem, an improved empirical wavelet transform (EWT)-based procedure was proposed to denoise GNSS data and identify the modal parameters of bridge structures. Firstly, the Yule–Walker algorithm-based auto-power spectrum and Fourier spectrum were jointly adopted to segment the frequency bands of structural dynamic response data. Secondly, the improved EWT algorithm was used to decompose and reconstruct the dynamic response data according to a correlation coefficient-based criterion. Finally, Natural Excitation Technique (NExT) and Hilbert Transform (HT) were applied to identify the modal parameters of structures from the decomposed efficient components. Two groups of simulation data were used to validate the feasibility and reliability of the proposed method, which consisted of the vibration responses of a four-storey steel frame model, and the acceleration response data of a suspension bridge. Moreover, field experiments were carried out on the Wilford suspension bridge in Nottingham, UK, with GNSS and an accelerometer. The fundamental frequency (1.6707 Hz), the damping ratio (0.82%), as well as the maximum dynamic displacements (10.10 mm) of the Wilford suspension bridge were detected by using this proposed method from the GNSS measurements, which were consistent with the accelerometer results. In conclusion, the analysis revealed that the improved EWT-based method was capable of accurately identifying the low-order, closely spaced modal parameters of bridge structures under operational conditions.

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 [

The data processing methods consist of the time domain methods, the frequency domain methods, and the time–frequency domain methods [

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. [

A novel empirical wavelet transform (EWT) has been developed recently by Gilles [

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 [

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 [

The paper is organized as follows. In

In order to further improve the identification accuracy of operational modal of bridge structures, this paper proposes an improved EWT-based method to denoise data and identify modal parameters, as shown in

The traditional EWT process contains three important aspects [

The spectrum segmentation is at the core of EWT for adaptively obtaining ideal frequency bands. However, the Fourier spectrum is very sensitive to noise, leading to spurious local maxima. The traditional EWT segmentation method may lead to an improper separation when the data are contaminated with significant noise and/or nonstationary components. The autoregression power spectrum is smoothed with a lower-level variance and can define the boundaries more appropriately than the Fourier spectrum. Li et al. [

According to the theory of spectral analysis, stationary random data

Based on the Yule–Walker algorithm, the regular equation of the AR model is obtained, and the auto-power spectral density estimation of the random data is calculated according to the solved

The theory of the Yule–Walker algorithm is very clear, simple, and easy to use based on the above discussion. Compared with the Fourier spectrum, the auto-power spectrum based on the Yule–Walker method is more robust and can identify significant spectral peaks even in the noisy data. It is more suitable to separate different portions in EWT analysis than using the Fourier spectrum for analyzing complex data. In this study, the measured response data are decomposed into a number of effective components by using the improved EWT. The specific procedure is described in

Traditionally, the judgment and reconstruction of effective IMFs are based on correlation coefficients [

Firstly, the Pearson coefficients, between each decomposed IMF and the original vibration data, is calculated and sorted by frequency, from high to low. Then, the correlation coefficient distribution diagram is obtained. Secondly, The IMF_{K} is denoted as the corresponding IMF component of K which mutates in the correlation coefficient diagram. The high-frequency IMFs after IMF_{K} are removed as the noise components, since noise is mainly concentrated in the high-frequency components of the data. The IMF_{K} and the low-frequency IMFs are retained. Thirdly, the remaining IMF components are regarded as information components to be further selected. The components with a value of a Pearson correlation coefficient of less than 0.1 are classified as pseudo components, otherwise they are classified as meaningful components [

The improved EWT method has been successfully utilized for decomposing the multi-frequency data of structure vibration into a series of mono-frequency IMFs. Therefore, the issue of modal parameters identification is transformed from multi-Degrees-of-Freedom (DOFs) system parameters identification to Single-Degree-of-Freedom (SDOF). With the individual IMF from the improved EWT method, NExT is applied to estimate the free decay response of each IMF. Once the free decay response of the mono component is estimated, HT is adopted to calculate the envelope of each free decay response to further determine the NF. Afterwards, a nonlinear exponential function is used to fit the envelope for computing the DR of each component. The modal parameters identification process based on the improved EWT is shown in

When the vibration responses of civil structures under environmental excitation are decomposed into various IMFs using the improved EWT, NExT [_{R},

Each estimated free decay response _{R}. The analytical signal of

The traditional HT estimates the instantaneous amplitude logarithmic curve and phase curve of _{R}

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.

The Digital Environment for Enabling Data-Driven Science (NEEDS) datasets provide a 4-storey, 2

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

As shown in

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

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.

Cheynet et al. [

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 this section, the proposed method is used to process the forced vibration data of the Wilford footbridge, which is a self-anchored suspension bridge with double main cable located in Nottingham, UK. The main span length of this bridge is 69 m, and the width is 3.7 m. In order to monitor the vibration responses of this bridge structure, the main instrumentation employed in the experiments included three sets of GS10 GNSS receivers with a sampling rate of 20 Hz, a Kistler 8392A2 tri-axial accelerometer with a sampling rate of 100 Hz, a precise time–data logger, a signal splitter, a Leica AR10 antenna (used on the monitoring site) and a Leica AT504 choke-ring antenna (used at the reference station). Five monitoring experiments, which mainly recorded the dynamic responses of the structure under the synchronous jumping excitation of the experimenters, were carried out on this suspension bridge. For more details about this field of experiments, refer to Yu et al. [

The kinematic solution methods of the GNSS data adopted three data-processing modes (real-time kinematic, network real-time kinematic, and post-processing kinematic). The real-time kinematic (RTK) mode received correction differences sent by an independent reference station (3# receiver) set up at the riverside near the bridge, about 60 m away. Whereas the network RTK (NRTK) used the correction differences from the Smart NET CORS system in United Kingdom. Both kinematic solution methods transmitted the corrections at the updating rate of 1 Hz. The monitoring site was located on the downstream side of the mid-span of the bridge. The receivers, 1# and 2#, were connected to the Leica AR10 antenna through a signal splitter, which enabled the receivers to synchronously acquire GNSS data (S_{RTK}, S_{NRTK}). The accelerometer was kept coaxial with the GNSS antenna and the center of the base through a cage monitoring device, by rotating the upper and lower plate, where one axis of the accelerometer was parallel to the longitudinal axis of the bridge. The forced vibrations were excited by three experimenters with a total weight of 180 kg jumping synchronously for 10 s every 3 min at the mid-span of the bridge. During the field experiments, the GNSS receivers received only GPS satellite signals with an elevation cutoff angle of 15 degrees. Using the above sensors, three groups of monitoring data were collected simultaneously (S_{RTK}, S_{NRTK}, S_{ACC}). To verify the proposed method, the GNSS and accelerometer (ACC) data covering approximately 12 min were selected. Taking the z-direction as an example,

As shown in _{ACC} of the bridge are divided into two parts. Part one is the random vibration caused by environmental excitation, and part two is the forced vibration under synchronous jumping excitation of three experimenters weighing 180 kg. In the first step, according to the improved algorithm, the meaningful dynamic displacement and various modal parameters are extracted from the ACC data.

_{ACC} is displayed in

NRTK obtains high-quality continuous observation data and establishes an accurate differential calculation model through the establishment of the continuous operation reference station (CORS) network system. It realizes the real-time dynamic high-precision relative positioning of the rover station, and its positioning accuracy can achieve the accuracy of the traditional RTK short baseline [

Due to the influence of long-period displacement and various noise, the structural vibration characteristics monitored by GNSS are cloaked by noise. As shown in

The maximum vibration displacement is an important index to evaluate the safety performance of bridge structure. The maximum dynamic displacements of the NRTK, RTK data, extracted using the improved EWT method, are 10.10 mm and 10.40 mm, respectively. Besides, as the verification group, the maximum dynamic displacement calculated by the accelerometer is 9.42 mm. The monitoring difference between two sensors is lower than 1.0 mm after data processing. As shown in

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. [

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. [

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.

Conceptualization, J.Y. and X.M.; methodology, J.Y. and Z.F.; formal analysis, Z.F.; data curation, J.Y.; writing—original draft preparation, Z.F.; writing—review and editing, J.Y. and X.M.; supervision, X.M.; project administration, J.Y. and X.M.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.

This research was funded by Changsha science and technology project (No. kq1907110), Hunan Water Resources Science and Technology project (No. XSKJ2021000-46), Hunan Provincial Natural Science Foundation of China (No. 2021JJ30102) and Hunan commercialization of research findings and industrialization plan (No. 2020GK2026).

The four-storey steel frame model and corresponding MATLAB codes are available on the web site at

The University of Nottingham is thanked for supplying instruments in field experiments.

The authors declare no conflict of interest.

Flowchart of data denoising and modal parameters identification.

Flowchart of the improved EWT.

Flowchart of modal parameters identification based on the improved EWT.

Scale model structure of steel frame.

Time–history response of sensor 9 in the x-direction and its power spectrum: (

Segmentation of the Fourier spectrum.

Correlation coefficient distribution diagram.

Modal parameters identification of IMF2: (

Time–history response and power spectrum of Lysefjord suspension bridge: (

Segmentation of the Fourier spectrum and the decomposed EWT components: (

The Wilford suspension bridge and the instrumentation layout.

Time–history of GNSS and the accelerometer in the z-direction.

Spectrum segmentation and correlation coefficient distribution diagram of ACC data: (

TF plane of the effective modes of ACC data.

Spectrum segmentation and correlation coefficient diagram of NRTK-GNSS: (

Correlation coefficient distribution diagram after reconstruction.

TF plane of the effective modes of NRTK-GNSS.

Dynamic displacements derived from the GNSS and ACC data.

Comparison of the dynamic displacement in E2 interval: (

Modal parameters identification results of the frame model.

IMF | FEA | Proposed Method | Difference | |||
---|---|---|---|---|---|---|

NF (Hz) | DR (%) | NF (Hz) | DR (%) | NF (%) | DR (%) | |

2 | 9.41 | 1.0 | 9.4051 | 0.96 | 0.05 | 4 |

3 | 16.38 | 1.0 | 16.3540 | 1.0 | 0.16 | 0 |

4 | 25.54 | 1.0 | 25.4529 | 1.02 | 0.34 | 2 |

6 | 48.01 | 1.0 | 47.9889 | 0.96 | 0.04 | 4 |

The modal parameters identification of Lysefjord bridge.

IMF | Target Value | Proposed Method | Difference | |||
---|---|---|---|---|---|---|

NF (Hz) | DR (%) | NF (Hz) | DR (%) | NF (%) | DR (%) | |

2 | 0.2046 | 0.50 | 0.2046 | 0.53 | 0 | 6 |

3 | 0.3189 | 0.50 | 0.3192 | 0.52 | 0.09 | 4 |

4 | 0.4391 | 0.50 | 0.4381 | 0.54 | 0.23 | 8 |

5 | 0.5852 | 0.50 | 0.5852 | 0.54 | 0 | 8 |

6 | 0.8643 | 0.50 | 0.8574 | 0.67 | 0.80 | 34 |

7 | 1.1944 | 0.50 | 1.1718 | 0.39 | 1.89 | 22 |

Modal parameters identification from ACC data of Wilford bridge.

Mode | NF (Hz) | DR (%) |
---|---|---|

1 | 1.6710 | 0.82 |

2 | 2.8434 | 0.48 |

3 | 5.2059 | 0.50 |

The denoising effect of different methods.

Method | SNR (dB) | RMSE (mm) | R |
---|---|---|---|

EMD | 2.0424 | 1.7 | 0.6145 |

WT | 2.4835 | 1.5 | 0.6628 |

Proposed method | 8.7773 | 0.52 | 0.9343 |