# Frontiers of Structural and Civil Engineering

 Front. Struct. Civ. Eng.    2020, Vol. 14 Issue (2) : 446-472     https://doi.org/10.1007/s11709-019-0605-8
 RESEARCH ARTICLE
Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran
1. High Performance Computing Lab, School of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
2. School of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
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 Abstract The main purpose of the present study is to enhance high-level noisy data by a wavelet-based iterative filtering algorithm for identification of natural frequencies during ambient wind vibrational tests on a petrochemical process tower. Most of denoising methods fail to filter such noise properly. Both the signal-to-noise ratio and the peak signal-to-noise ratio are small. Multiresolution-based one-step and variational-based filtering methods fail to denoise properly with thresholds obtained by theoretical or empirical method. Due to the fact that it is impossible to completely denoise such high-level noisy data, the enhancing approach is used to improve the data quality, which is the main novelty from the application point of view here. For this iterative method, a simple computational approach is proposed to estimate the dynamic threshold values. Hence, different thresholds can be obtained for different recorded signals in one ambient test. This is in contrast to commonly used approaches recommending one global threshold estimated mainly by an empirical method. After the enhancements, modal frequencies are directly detected by the cross wavelet transform (XWT), the spectral power density and autocorrelation of wavelet coefficients. Estimated frequencies are then compared with those of an undamaged-model, simulated by the finite element method. Corresponding Authors: Hassan YOUSEFI Just Accepted Date: 23 February 2020   Online First Date: 08 April 2020    Issue Date: 08 May 2020
 Cite this article: Hassan YOUSEFI,Alireza TAGHAVI KANI,Iradj MAHMOUDZADEH KANI, et al. Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran[J]. Front. Struct. Civ. Eng., 2020, 14(2): 446-472. URL: http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0605-8 http://journal.hep.com.cn/fsce/EN/Y2020/V14/I2/446
 Fig.1  Ammonium Nitrate periling tower. Fig.2  Recording locations and directions: (a) locations of data recording; (b) recording directions “1” and “2”. (unit: mm) Fig.3  Some examples of concrete destruction. Fig.4  Recorded accelerations at different stations: (a) signal P21; (b) signal P31; (c) signal P51; (d) signal PP12; (e) signal P72. Fig.5  Empirical evaluation of p values based on SNR-p curves for data P12. (a) SNR-p curve by the TV regularization; (b) SNR-p curve by the L2 regularization; (c) SNR-p curve by the Sobolev regularization. Fig.6  Residual noise of the regularization-based denoising and corresponding energy in the wavelet space; plotted ranges are $0.02Max[ |W Ψ|2]≤ | WΨ|2≤M ax [ | WΨ| 2]$. (a) The residual noise obtained by the L2 regularization (by using $Ω (f) L2$); (b) the residual noise obtained by the Sobolev regularization (by using $Ω (f)Sobolev$); (c) the time-period representation in the wavelet space of the residual noise obtained by $Ω (f) L2$; (d) the time-period representation in the wavelet space of the residual noise obtained by $Ω (f)Sobolev$. Tab.1  Different denoising approaches with corresponding thresholding method and estimating noise level Fig.7  Denoising of the P51 signal with different one-step wavelet-based denoising methods using Symlet[12] wavelet where Nd =13. (a) denoising with GCV method; (b) denoising with GCVLevel method; (c) denoising with SURE method; (d) denoising with SURELevel method; (e) denoising with SUREhrink method; (f) denoising with Universal method; (g) denoising with UniversalLevel method; (h) denoising with VisuShrink method; (i) denoising with VisuShrinkLevel method. Tab.2  Effects of denoising with different thresholding methods by the Symlet[12] and Nd =13 Fig.8  Variations of SNR against for P12 signal with different one-step wavelet-based denoising methods, where Nd=13. (a) soft and hard denoising with Symlet [12]; (b) soft and hard denoising with Db [8]; (c) soft and hard denoising with BattleLemarie[8]. Tab.3  Effects of denoising with the GCV method for “Db” wavelets, where Nd=13 Tab.4  Effects of denoising with the GCV method for “Symlet” wavelets, where Nd=13 Tab.5  Effects of denoising with the GCV method for “BattleLemarie” wavelets, where Nd=13 Tab.6  Effect of decomposition levels (Nd ) with Symlet [12] wavelet Fig.9  The iterative thresholding for P12 where $Cnj$=1.8, Symlet [12] and Nd=13. The last row contains final denoised signal and estimated noise. Fig.10  The iterative thresholding for P12 where $Cnj$=2, Symlet [12] and Nd=13. The last row contains final denoised signal and estimated noise. Fig.11  Determination of thresholds for $Cnj$ for different enhanced data; SNRs and PSNRs are obtained by the iterative denoising method by the wavelet Symlet [12], Nd=13 and ten iterations. (a) SNR-$Cnj$ for P12; (b) PSNR-$Cnj$ for P12; (c) SNR-$Cnj$for P51; (d)PSNR-$Cnj$ for P51; (e)SNR-$Cnj$ for P31; (f)PSNR-$Cnj$ for P31; (g)SNR-$Cnj$ for P72; (h)PSNR-$Cnj$ for P72; (i)SNR-$Cnj$ for P21; (j)PSNR-$Cnj$ for P21. Tab.7  Iterative denoising of data P12 with parameters: $Cnj=1.75$and Nd=13 Fig.12  Iterative denoising of the signal P12 with ten iterations; both denoised signal and estimated noise are provided at each iteration (rows 1–5). The last row contains final denoised signal and estimated noise; evaluations are obtained with: Symlet[12], $Cnj=1.75$ and Nd=13. Fig.13  Energy Density for remaining noise after ten iterations of the peeling algorithm where Nd=13; the energies are evaluated by the complex Morlet wavelet with parameters $υ$b=2 and $υ$c=1.75. Energies are presented for the range $0.02Max[ |W Ψ|2]≤ | WΨ|2≤Max[ |W Ψ|2]$. (a) noise of P12 in the wavelet space where $Cnj=1.75$; (b) noise of P51 in the wavelet space where $C nj=1.90$; (c) noise of P31 in the wavelet space where $Cn j=1.90$; (d) noise of P72 in the wavelet space where $C nj=1.85$; (e) noise of P21 in the wavelet space where $Cn j=1.90$. Fig.14  Powers and energies of WTs of enhanced signals P21 and P31 and corresponding XWT and spectral power of XWT; plotted ranges are $0.02Max[ |W Ψ|2]≤ | WΨ|2≤M ax [ | WΨ| 2]$ and $0.02Max[| WΨ|] ≤|W Ψ| ≤Max [| WΨ|]$. (a) The spectral power of the enhanced data P21, evaluated by $|WΨ(P21) |2$ ; (b) the density of energy for the enhanced data P21, $| WΨ(P 21)| 2$; (c) the spectral power of the enhanced data P31, evaluated by $|W Ψ(P31 )|2$; (d) the density of energy of the enhanced data P31, $| WΨ (P31)|2$; (e) the spectral power of XWT for the enhanced data P21 and P31; (f) the density of $| XWΨ|$ for the enhanced data P21 and P31. Fig.15  Powers and energies of WTs of enhanced signals P21 and P51 and corresponding XWT and spectral power of XWT; plotted ranges are $0.02Max[ |W Ψ|2]≤ | WΨ|2≤M ax [ | WΨ| 2]$ and $0.02Max[| WΨ|] ≤|W Ψ| ≤Max [| WΨ|]$. (a) The spectral power of the enhanced data P51, evaluated by $|W Ψ(P51 )|2$; (b) the density of energy for the enhanced data P51, $| WΨ (P51)|2$; (c) the spectral power of XWT for the enhanced data P21 and P51; (d) the density of $| XWΨ|$ for the enhanced data P21 and P51. Fig.16  Powers and energies of WTs of enhanced signals P12 & P72 and corresponding XWT and spectral power of XWT; plotted ranges are $0.02Max[ |W Ψ|2]≤ | WΨ|2≤M ax [ | WΨ| 2]$ and $0.02Max[| WΨ|] ≤|W Ψ| ≤Max [| WΨ|]$. (a) The spectral power of the enhanced data P12, evaluated by $|W Ψ(12)| 2;$(b) the density of energy for the enhanced data P12, $| WΨ (12)| 2;$; (c) The spectral power of the enhanced data P72, evaluated by $| WΨ (72)| 2;$; (d) the density of energy for the enhanced data P72, $|W Ψ(72)| 2;$; (e) the spectral power of XWT for the enhanced data P12 and P72; (f) the density of $| XWΨ|$ for the enhanced data P12 and P72. Fig.17  Autocorrelations of wavelet coefficients of the enhanced data. (a) results for the enhanced P21; (b) results for the enhanced P31; (c) results for the enhanced P51; (d) results for the enhanced P12; (e) results for the enhanced P72. Tab.8  Comparison of modal periods from the ambient vibration test and the FE model [117]