# Modal Parameters Identification of Bridge Structures from GNSS Data Using the Improved Empirical Wavelet Transform

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Improved EWT

_{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 [6].

#### 2.2. Modal Parameters Identification Based on the Improved EWT

_{R}, ${R}_{ijk}\left(\tau \right)$ denotes the correlation function, and moreover, ${x}_{ik}\left(t\right)$ and ${x}_{jk}\left(t\right)$ are the response of the measuring points $i$ and $j$, respectively. The core of NExT is that the correlation function between two points of a structure is similar to the free decay response under white noise excitation. Hence, the correlation function can be used instead of the free decay response to identify the modal parameters.

_{R}. The analytical signal of ${u}_{R}^{f}\left(t\right)$ can be written as follows:

_{R}, and b defines the decay rate of the exponential function.

## 3. Numerical Studies

#### 3.1. Numerical Study on a 4-Storey Steel Frame Model

#### 3.2. Numerical Study on Acceleration Response Data of a Suspension Bridge

## 4. Field Experiments

#### 4.1. Engineering Background

_{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, Figure 12 shows the time–history of the data observed by GNSS for a total of 14,400 epochs, and the accelerometer data.

#### 4.2. Data Processing and Analysis

_{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 Figure 14. The structural vibration effect caused by the jumping excitation on the time–history response is consistent with the time–frequency response of IMF2, indicating that the modal parameters of this component are related to the jumping excitation. The brightness for the instantaneous frequency lines of IMF3 at 540~570 s is very high, which is consistent with the fluctuation of the time–history curve, confirming that modal parameters of this component are related to environmental excitation.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Abbreviation | The Full Name |

GNSS | Global Navigation Satellite System |

EWT | Empirical Wavelet Transform |

NExT | Natural Excitation Technique |

HT | Hilbert Transform |

WT | Wavelet Transform |

CWT | Continuous Wavelet Transform |

LSWA | Least-Squares Wavelet Analysis |

WWA | Weighted Wavelet Analysis |

EMD | Empirical Mode Decomposition |

EEMD | Ensemble Empirical Mode Decomposition |

MEMD | Multivariate Empirical Mode Decomposition |

IMF | Intrinsic Mode Function |

MUSIC | Multiple Signal Classification |

NF | Natural Frequency |

DR | Damping Ratio |

SET | Synchro-Extracting Transform |

LWM | Local Window Maxima |

AM-FM | Amplitude Modulation-Frequency Modulation |

DOF | Degrees-of-Freedom |

FEA | Finite Element Analysis |

RTK | Real-Time Kinematic |

NRTK | Network Real-Time Kinematic |

PPK | Post-Processing Kinematic |

CORS | Continuous Operation Reference Station |

ACC | Accelerometer |

TF | Time–frequency |

SNR | Signal-to-Noise Ratio |

RMSE | Root Mean Square Error |

R | Correlation Coefficient |

## References

- Shen, N.; Chen, L.; Liu, J.B.; Wang, L.; Tao, T.Y.; Wu, D.W.; Chen, R.Z. A review of Global Navigation Satellite System (GNSS)-based dynamic monitoring technologies for structural health monitoring. Remote Sens.
**2019**, 11, 1001. [Google Scholar] [CrossRef] [Green Version] - Moschas, F.; Stiros, S. Dynamic deflections of a stiff footbridge using 100-Hz GNSS and accelerometer data. J. Surv. Eng.
**2015**, 141, 04015003. [Google Scholar] [CrossRef] - Yu, J.Y.; Meng, X.L.; Yan, B.F.; Xu, B.; Fan, Q.; Xie, Y.L. Global Navigation Satellite System-based positioning technology for structural health monitoring: A review. Struct. Control Health Monit.
**2020**, 27, e2467. [Google Scholar] [CrossRef] [Green Version] - Gaxiola-Camacho, J.R.; Bennett, R.; Guzman-Acevedo, G.M.; Gaxiola-Camacho, I.E. Structural evaluation of dynamic and semi-static displacements of the Juarez Bridge using GPS technology. Measurement
**2017**, 110, 146–153. [Google Scholar] [CrossRef] - Xi, R.J.; Chen, H.; Meng, X.L.; Jiang, W.P.; Chen, Q.S. Reliable dynamic monitoring of bridges with integrated GPS and BeiDou. J. Surv. Eng.
**2018**, 144, 04018008. [Google Scholar] [CrossRef] - Yu, L.N.; Xiong, C.B.; Gao, Y.; Zhu, J.S. Combining GNSS and accelerometer measurements for evaluation of dynamic and semi-static characteristics of bridge structures. Meas. Sci. Technol.
**2020**, 31, 125102. [Google Scholar] [CrossRef] - Zhou, W.; Feng, Z.R.; Liu, D.S.; Wang, X.J.; Chen, B.B. Modal parameter identification of structures based on short-time narrow-banded mode decomposition. Adv. Struct. Eng.
**2020**, 23, 3062–3074. [Google Scholar] [CrossRef] - Fan, Q.; Meng, X.L.; Nguyen, D.T.; Xie, Y.L.; Yu, J.Y. Predicting displacement of bridge based on CEEMDAN-KELM model using GNSS monitoring data. J. Appl. Geod.
**2020**, 14, 253–261. [Google Scholar] [CrossRef] - Kaloop, M.R.; Hussan, M.; Kim, D. Time-series analysis of GPS measurements for long-span bridge movements using wavelet and model prediction techniques. Adv. Space Res.
**2019**, 63, 3505–3521. [Google Scholar] [CrossRef] - Kaczmarek, A.; Kontny, B. Identification of the noise model in the time series of GNSS stations coordinates using wavelet analysis. Remote Sens.
**2018**, 10, 1611. [Google Scholar] [CrossRef] [Green Version] - Ji, K.; Shen, Y.; Wang, F. Signal extraction from GNSS position time series using weighted wavelet analysis. Remote Sens.
**2020**, 12, 992. [Google Scholar] [CrossRef] [Green Version] - Ghaderpour, E.; Ghaderpour, S. Least-squares spectral and wavelet analyses of V455 Andromedae time series: The life after the super-outburst. Publ. Astron. Soc. Pac.
**2020**, 132, 114504. [Google Scholar] [CrossRef] - Barbosh, M.; Singh, P.; Sadhu, A. Empirical mode decomposition and its variants: A review with applications in structural health monitoring. Smart Mater. Struct.
**2020**, 29, 093001. [Google Scholar] [CrossRef] - Civera, M.; Surace, C. A comparative analysis of signal decomposition techniques for structural health monitoring on an experimental benchmark. Sensors
**2021**, 21, 1825. [Google Scholar] [CrossRef] [PubMed] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.-C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Hu, Y.; Li, F.; Li, H.; Liu, C. An enhanced empirical wavelet transform for noisy and non-stationary signal processing. Digit. Signal Process.
**2017**, 60, 220–229. [Google Scholar] [CrossRef] - Gilles, J. Empirical wavelet transform. IEEE Trans. Signal Process.
**2013**, 61, 3999–4010. [Google Scholar] [CrossRef] - Gilles, J.; Heal, K. A parameterless scale-space approach to find meaningful modes in histograms—Application to image and spectrum segmentation. Int. J. Wavelets Multiresolut. Inf. Process.
**2014**, 12, 1450044. [Google Scholar] [CrossRef] - Akbari, H.; Sadiq, M.T.; Rehman, A.U. Classification of normal and depressed EEG signals based on centered correntropy of rhythms in empirical wavelet transform domain. Health Inf. Sci. Syst.
**2021**, 9, 1–15. [Google Scholar] [CrossRef] - Liao, Z.R.; Zhang, Y.F.; Li, Z.Y.; He, B.B.; Lang, X.; Liang, H.; Chen, J.H. Classification of red blood cell aggregation using empirical wavelet transform analysis of ultrasonic radiofrequency echo signals. Ultrasonics
**2021**, 114, 106419. [Google Scholar] [CrossRef] - Liu, H.; Yu, C.Q.; Wu, H.P.; Duan, Z.; Yan, G.X. A new hybrid ensemble deep reinforcement learning model for wind speed short term forecasting. Energy
**2020**, 202, 117794. [Google Scholar] [CrossRef] - Kalra, M.; Kumar, S.; Das, B. Seismic signal analysis using Empirical Wavelet Transform for moving ground target detection and classification. IEEE Sens. J.
**2020**, 20, 7886–7895. [Google Scholar] [CrossRef] - Yu, H.; Li, H.R.; Li, Y.L. Vibration signal fusion using improved empirical wavelet transform and variance contribution rate for weak fault detection of hydraulic pumps. ISA Trans.
**2020**, 107, 385–401. [Google Scholar] [CrossRef] - Amezquita-Sanchez, J.P.; Park, H.S.; Adeli, H. A novel methodology for modal parameters identification of large smart structures using MUSIC, empirical wavelet transform, and Hilbert transform. Eng. Struct.
**2017**, 147, 148–159. [Google Scholar] [CrossRef] - Amezquita-Sanchez, J.P.; Adeli, H. A new music-empirical wavelet transform methodology for time-frequency analysis of noisy nonlinear and non-stationary signals. Digit. Signal Process.
**2015**, 45, 55–68. [Google Scholar] [CrossRef] - Xin, Y.; Hao, H.; Li, J. Operational modal identification of structures based on improved empirical wavelet transform. Struct. Control Health Monit.
**2019**, 26, e2323. [Google Scholar] [CrossRef] - Xia, Y.X.; Zhou, Y.L. Mono-component feature extraction for condition assessment in civil structures using empirical wavelet transform. Sensors
**2019**, 19, 4280. [Google Scholar] [CrossRef] [Green Version] - Xin, Y.; Hao, H.; Li, J. Time-varying system identification by enhanced empirical wavelet transform based on synchroextracting transform. Eng. Struct.
**2019**, 196, 109313. [Google Scholar] [CrossRef] - Dong, S.; Yuan, M.; Wang, Q.; Liang, Z. A modified empirical wavelet transform for acoustic emission signal decomposition in structural health monitoring. Sensors
**2018**, 18, 1645. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhao, L.; He, Y. The power spectrum estimation of the AR model based on motor imagery EEG. In Mechatronics and Intelligent Materials Iii, Pts 1-3; Chen, R., Sung, W.P., Kao, J.C.M., Eds.; Advanced Materials Research; Trans Tech Publications Ltd.: Stafa-Zurich, Switzerland, 2013; Volumes 706–708, pp. 1923–1927. [Google Scholar]
- Xue, B.; Hong, H.; Zhou, S.; Chen, G.; Li, Y.; Wang, Z.; Zhu, X. Morphological filtering enhanced empirical wavelet transform for mode decomposition. IEEE Access
**2019**, 7, 14283–14293. [Google Scholar] [CrossRef] - Liu, X.L.; Jiang, M.Z.; Liu, Z.Q.; Wang, H. A morphology filter-assisted extreme-point symmetric mode decomposition (MF-ESMD) denoising method for bridge dynamic deflection based on ground-based microwave interferometry. Shock Vib.
**2020**, 2020, 8430986. [Google Scholar] [CrossRef] - Pei, Q.; Li, L. Structural modal parameter identification based on natural excitation technique. In Advanced Research on Civil Engineering, Materials Engineering and Applied Technology; Zhang, H., Jin, D., Zhao, X.J., Eds.; Advanced Materials Research; Trans Tech Publications Ltd.: Stafa-Zurich, Switzerland, 2014; Volume 859, pp. 167–170. [Google Scholar]
- Dyke, S.; Agrawal, A.K.; Caicedo, J.M.; Christenson, R.; Gavin, H.; Johnson, E.; Nagarajaiah, S.; Narasimhan, S.; Spencer, B. NEES: Database for Structural Control and Monitoring Benchmark Problems. 2015. Available online: https://datacenterhub.org/resources/257 (accessed on 12 May 2021).
- Johnson, E.A.; Lam, H.F.; Katafygiotis, L.S.; Beck, J.L. Phase I IASC-ASCE structural health monitoring benchmark problem using simulated data. J. Eng. Mech.
**2004**, 130, 3–15. [Google Scholar] [CrossRef] - Cheynet, E.; Daniotti, N.; Jakobsen, J.B.; Snæbjörnsson, J. Improved long-span bridge modeling using data-driven identification of vehicle-induced vibrations. Struct. Control Health Monit.
**2020**, 27, e2574. [Google Scholar] [CrossRef] - Yu, J.Y.; Meng, X.L.; Shao, X.D.; Yan, B.F.; Yang, L. Identification of dynamic displacements and modal frequencies of a medium-span suspension bridge using multimode GNSS processing. Eng. Struct.
**2014**, 81, 432–443. [Google Scholar] [CrossRef] - Yu, J.Y.; Yan, B.F.; Meng, X.L.; Shao, X.D.; Ye, H. Measurement of bridge dynamic responses using network-based real-time kinematic GNSS technique. J. Surv. Eng.
**2016**, 142, 04015013. [Google Scholar] [CrossRef] - Meng, X.L.; Dodson, A.H.; Roberts, G.W. Detecting bridge dynamics with GPS and triaxial accelerometers. Eng. Struct.
**2007**, 29, 3178–3184. [Google Scholar] [CrossRef]

**Figure 5.**Time–history response of sensor 9 in the x-direction and its power spectrum: (

**a**) time–history response and (

**b**) Yule–Walker power spectral density estimate.

**Figure 8.**Modal parameters identification of IMF2: (

**a**) time–history response of IMF2; (

**b**) free vibration response and envelop; (

**c**) logarithmic amplitude curve and (

**d**) phase angle curve.

**Figure 9.**Time–history response and power spectrum of Lysefjord suspension bridge: (

**a**) time–history response and (

**b**) Yule–Walker power spectral estimate.

**Figure 10.**Segmentation of the Fourier spectrum and the decomposed EWT components: (

**a**) segmentation of Fourier spectrum and (

**b**) decomposed EWT components.

**Figure 13.**Spectrum segmentation and correlation coefficient distribution diagram of ACC data: (

**a**) segmentation of power spectrum and (

**b**) correlation coefficient distribution map.

**Figure 15.**Spectrum segmentation and correlation coefficient diagram of NRTK-GNSS: (

**a**) segmentation of power spectrum and (

**b**) correlation coefficient distribution map.

**Figure 19.**Comparison of the dynamic displacement in E2 interval: (

**a**) dynamic displacement of NRTK-GNSS and ACC and (

**b**) the displacement difference between NRTK-GNSS and ACC.

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 |

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 |

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

1 | 1.6710 | 0.82 |

2 | 2.8434 | 0.48 |

3 | 5.2059 | 0.50 |

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 |

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**MDPI and ACS Style**

Fang, Z.; Yu, J.; Meng, X.
Modal Parameters Identification of Bridge Structures from GNSS Data Using the Improved Empirical Wavelet Transform. *Remote Sens.* **2021**, *13*, 3375.
https://doi.org/10.3390/rs13173375

**AMA Style**

Fang Z, Yu J, Meng X.
Modal Parameters Identification of Bridge Structures from GNSS Data Using the Improved Empirical Wavelet Transform. *Remote Sensing*. 2021; 13(17):3375.
https://doi.org/10.3390/rs13173375

**Chicago/Turabian Style**

Fang, Zhen, Jiayong Yu, and Xiaolin Meng.
2021. "Modal Parameters Identification of Bridge Structures from GNSS Data Using the Improved Empirical Wavelet Transform" *Remote Sensing* 13, no. 17: 3375.
https://doi.org/10.3390/rs13173375