Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran

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.

Keywords

ambient vibration test / high level noise / iterative signal enhancement / wavelet / cross and autocorrelation of wavelets

Introduction

Ambient vibration tests are usually used for: damage detections [¹,²]; modal parameter identifications in health monitoring studies [³–⁹]; assessment of real time conditions [³,⁴,¹⁰–²²]; and vibration controls [²³]. The ambient vibration tests are fast and cheap, and require no excitation equipment and can be used for different structures and infrastructures [²⁴,²⁵]. In these tests, the source of loading can be environmental or natural excitation; some common examples are: wind [³,¹⁶,¹⁷,¹⁹,²⁰], traffic loads [²⁶], and small to moderate earthquakes [²⁷]. The ambient vibration tests mainly estimate the linear response of (damaged) structures, since the magnitude of loading is small. The in situ tests are interesting [²⁸,²⁹] because they are directly performed on (partially damaged) full-scale structures to control their performance (deviations from initial conditions). However, the identification of system parameters (i.e., modal frequencies, service condition’s modal damping and mode shapes) is nearly challenging in ambient vibrations, due to non-stationary responses, nonlinear signals and high-level of noise in recorded data [³⁰,³¹]. The signal-to-noise ratios (SNRs) of these recorded signals have significant effects on the accuracy of parameter identification phase [³²]. For this reason, noise effects on modal identifications have been studied for different approaches [³³–³⁶].

For removing/curing noise effects in recorded data in a multiresolution framework, wavelet theories, which are fast with a strong mathematical background, have been developed. Wavelet transforms (WTs) study data in the time-frequency representation with different decomposition levels and resolution accuracy in a non-parametric approach using only the output data [³,⁴,¹⁰–¹³,¹⁸,¹⁹,²⁷,³⁷–⁴²]. For high level of noise, however, this powerful transform has its own shortcomings [⁴³].

Two general denoising approaches can be addressed in the framework of multiresolution analysis (MRA): 1) one-step [⁴⁴–⁵⁰] and iterative techniques [⁵¹–⁵⁷]. In the former, one stage of MRA allows for noise to be estimated and removed from the data. In the iterative approach, successive refinements of the data are used for estimation of the denoised data. This iterative scheme can filter noise in a step-by-step approach to clear or enhance the data. It should be noted that, in signals with high-level of noise, it is almost impossible to completely eliminate (filter) the noise.

The iterative denoising algorithm was originally proposed by Starck and Bijaoui [⁵⁷] and Coifman and Wickerhauser [⁵¹,⁵²]; and then it was followed by Hadjilleontiadis et al. [⁵³,⁵⁴] and Ranta et al. [⁵⁵,⁵⁶]. In each iteration, information with coherent structures in different resolution levels is selected and the remaining data are assumed as noise in the next iteration. Since the physical information is gathered layer by layer from the noise residual, the approach is also known as the “peeling off successive layers” scheme [⁵¹,⁵²]. The iterative method assumes that large wavelet coefficients contain physical information. These coefficients are detected based on a (predefined) threshold in each resolution level: known as the level-dependent thresholds. Each threshold in each resolution level can be in accordance with the deviation of wavelet coefficients in that resolution (a level-depending thresholding).

The thresholds for the iterative scheme can be chosen empirically or based on some criteria [⁵³]. In this study, a simple data-dependent criterion is explained based on peak signal-to-noise ratio (PSNR) and SNR. This approach leads to dynamic threshold values for different recorded data obtained even in an ambient vibration test. This leads to a more flexible noise estimation method in comparison to commonly used approaches employing one constant threshold for all data. To control the stochastic feature of the estimated noises, the time-frequency representation of estimated noises (by the iterative enhancing method) are presented and studied for some recorded data. This feature confirms effectiveness of the estimated dynamic thresholds and the iterative method.

For noisy information with small values of SNR, the importance of the iterative denoising approach is studied and confirmed by comparing the results with:

1) The one-step MRA-based denoising scheme: This comparison is performed for different decomposition levels and wavelet families. In computations, wavelet thresholds are those obtained by both theoretical and empirical approaches [⁵⁸].

2) The variational-based filtering approach with different constraints [⁵⁸].

Neither the one-step MRA-based denoising nor the variational-based one can properly enhance high-level noisy recorded data gathered during the ambient vibration tests.

After the signal-enhancement step, other MRA-based approaches are used here to improve the detection power of physical features (without stochastic properties). After the enhancement step, it is still a challenging task to determine physical features in the enhanced data. In this regard, detections are performed by:

1) The concept of the XWT [⁴⁵,⁵⁹], and the corresponding spectral power. The XWT analysis and the corresponding spectral power are simultaneously used for frequency detections. Responses with both the continuity pattern in time (controlled by XWT) and considerable energies in the spectral power can be assumed to include physical phenomena,

2) Autocorrelation of wavelet coefficients: by autocorrelation of wavelet coefficients, the repeating patterns of different scales can be detected for different resolution levels in time [⁶⁰,⁶¹]. This transform is especially important for capturing the repeating features of weakly excited responses (e.g., higher modes of a structure with small participations) which cannot be detected by the above-mentioned transforms.

For data with high-level of noise, in general, the noises affect all time-scales in the scalogram of CWTs as background information (especially in the high frequency ranges). This causes difficulties for the detection of continuously excited frequencies (especially those with small modal participation). The MRA-based iterative scheme can reduce the noise-effects; however, the deblurring effects can still be considerable in the enhanced data. Hence, the concepts of XWT and the autocorrelation of CWT coefficients are used as an additional step to reduce noise-effects. These effects, however, still remain in both the XWT and autocorrelation analysis. In this regard, in this study, the energies of wavelet-based results are presented for the range {\left|Z\right|}^{\mathrm{2}}\in [\mathrm{0.02},\mathrm{1}]\times \mathrm{M}\mathrm{a}\mathrm{x}{\left|Z\right|}^{\mathrm{2}} , to eliminate noise-effects.

In brief, to achieve more reliable results, it can be recommended to use independently different denoising-detecting approaches. And then the results are compared with each other, e.g., those from the wavelet, HHT and Fourier transforms [⁶²,⁶³].

Abovementioned identifications can also be classified as the output-only modal identification, since modal parameters are only determined by the output (recorded) data (caused by wind with stochastic feature) [⁶⁴].

At the last stage, the detected frequencies are compared with those of a linear undamaged finite element (FE) model. Most of the time, these two types of frequencies may not be the same and FE models cannot precisely predict modal frequencies even for undamaged structures [⁶⁵]. The main reasons are: 1) conservation in designs; 2) improper considering of the torsional eccentricity; 3) foundation compliance effects [⁶⁵]; 4) the effects of damaged parts and flaws (for damaged systems).

In brief, there are other methods for MRA-based studying, such as: 1) the Hilbert-Huang transform (HHT) [³⁸,⁶⁶] and the modified HHT method [⁶⁷–⁷⁰] (both developed for nonlinear signals); 2) the synchrosqueezing transform [⁷¹–⁷³] and its generalizations [⁷⁴,⁷⁵].

Finally, it should be mentioned that the aforementioned signal enhancement technique with the iterative denoising method can be integrated (as a pre-processor) with other wavelet-based methods developed for the modal parameter identification, see Refs. [¹³,¹⁹,⁷⁶].

After the signal processing stage, it is necessary to detect damages in the structure by the direct or inverse approaches to consider flaw or damage effects and then in the next step to consider them in simulations. Flaw detections are complex due to the fact that the proper solution of these inverse problems are challenging because of their inherent uncertainty, both in their spatial locations and material properties due to fracture and softening [⁷⁷]. This uncertainty leads to developing different detection algorithms [⁷⁸], stochastic modeling [⁷⁹] and sensitivity/uncertainty analysis [⁸⁰,⁸¹]. Forward simulations of cracks and flaws leads to major challenges to adjust the interfaces to the discretization, which requires efficient remeshing [⁸²–⁸⁴], division [⁸⁵], and adaptation [⁸⁶,⁸⁷] techniques. An efficient alternative approach is the extended finite element method (XFEM) [¹⁵,¹⁶] or modifications of it such as the smoothed XFEM [¹⁷,¹⁸], the phantom node method [⁸⁸], certain multi-scale methods for fracture [⁸⁹–⁹²], or phase-field problems [⁹³,⁹⁴]. Though XFEM has mostly applied to fracture problems, there are also several applications of XFEM related to this topic, i.e., inverse analysis and optimization [³²–³⁸]. Similar advantages are observed in peridynamics and dual-horizon peridynamics [⁹⁵,⁹⁶], meshfree methods [⁹⁷–⁹⁹], partition of unity [¹⁰⁰], cracking particles methods [¹⁰¹–¹⁰³], or extended meshfree methods [¹⁰⁴]. Another powerful tool to accurately capture complex geometries accurately is the isogeometric analysis (IGA) [¹⁰⁵–¹⁰⁸], applied to numerous problems including thin shells [¹⁰⁹–¹¹²], or optimization [¹¹³]. Furthermore, IGA can also be used in an adaptive analysis [⁷⁹–⁸¹]. There are also contributions of combining the advantages of XFEM and IGA [¹¹⁰,¹¹⁴].

This work is composed of seven parts. Section 2 explains the general features of the periling tower and the corresponding ambient vibration tests. Section 3 reviews different one-step denoising techniques by discrete wavelets. There, some common wavelet-based denoising approaches and corresponding effective parameters are also presented. Section 4 devotes to issues related to the iterative denoising and a simple criterion for the adaptive threshold selection. Section 5 surveys three wavelet-based signal processing and pattern recognition tools: XWT, spectral powers and autocorrelation analysis of WTs. Section 6 presents the signal enhancement and frequency detection of the recorded data from the ambient vibration test. The concluding remarks are presented in Section 7.

Ambient test on the Ammonium Nitrate periling tower

The petrochemical complex, constructed in 1959, is placed 45 Kilometres north of the city of Shiraz, Iran. The Ammonium Nitrate periling unites composed of two reinforced concrete towers, including the process tower and the elevator tower. The process tower is a cylindrical podium overtopped with a rectangular structure. Four steel chimneys with the height of 25 m diameter of 1.9 m and thickness of 0.5 cm are attached to the top of cylindrical part of the process tower between the heights of 55.7 and 80.7 m (Fig. 1). A schematic illustration of the tower and the corresponding cross-section are presented in Fig. 2. The towers are attached to each other at the heights of 47.7, 50.6, and 55.7m.

Considering degradation and deterioration from its original condition and increasing vibrational response (mainly due to stiffness reduction), motivated the owners to pursue safety evaluations and potential rehabilitation approaches. Figure 3 depicts some deteriorated parts of the structure of the process tower. The initial assessment of both towers revealed that:

1) Although, during visual tests many parts of reinforced concrete structures seem undamaged, but random petrographic test results provide owners with critical deterioration of cement gel, which extends up to 75 mm inside the outer surfaces.

2) Because of operational conditions and serious health consequences, it is somehow impossible to have access to many parts of process tower’s inner surface.

3) Considering the explosion-sensitivity of products.

First, the specifications of instruments and their installation details should be determined to obtain reasonable results. These specifications depend on the dynamic properties of structure, the dynamic characteristics of operational systems and specifications of environmental excitation. The frequency range of some operating machineries is from 30 up to 1500 RPM. The machineries include (but not limited to) stacks, hovers, compressors, condensers and elevator. These machineries could not be turned off during the ambient test, according to the owner’s operational protocols.

Regarding accelerometers, a variable capacitance is selected with nominal sensitivity threshold of 1V/g for the pre-test phase. Observations in the pre-test phase show that SNRs are extremely low; this means more sensitive accelerometers are needed for the main test. Hence, the Kinematics’ force balance accelerometer (FBA11) is nominated for the main test. This accelerometer has 100 Hz band-width, adjustable sensitivity ranging from+ 5 up to+ 80 V/g and a dynamic domain equivalent to 140 dB. Also, digital data loggers with ten-simultaneous channels of 24 bit resolution and effective band-width of 50 kHz are used to restore vibrational data.

Another crucial question is about “height-wise and/or plan-wise arrangements of the loggers” during the data accumulation phase. It is extremely preferable to install sensors at antinode locations to catch maximum amplitudes of the structure; however, other factors such as location accessibility, safety considerations and existing vibrational aliasing adjacent to heavy machineries put great limitations on locations of the sensors.

Spatial locations of sensors along the tower and recording directions “1” and “2” are illustrated in Fig. 2. Acceleration recorders on the tower are denoted by P_i where i\in \{\mathrm{1},\dots ,\mathrm{7}\}, and the recorder R is the reference one. Recording directions for the recorded data at the main tower are denoted by k as P_ik for k\in \{\mathrm{1},\mathrm{2}\}. The ambient vibration test is performed by the wind load and the corresponding responses are recorded by the sampling step \mathrm{\Delta }t=\mathrm{1}/\mathrm{100} s. Due to the limitation of sensors, signals are recorded in different times.

MRA-based denoising and enhancement of data

Denoising by discrete wavelets

In this subsection, wavelet-based denoising methods and their effective parameters are studied [⁴⁶].

MRA-based denoising

In general, wavelet-based denoising can be summarized as: (1) estimation of noise level in a process; (2) modification of detail (wavelet) coefficients \{d(j,k)\} (measuring local fluctuations), where the modified set is denoted by \left\{\widehat{d}(j,k)\right\} ; (3) reconstruction of the denoised signal by both \left\{\widehat{d}(j,k)\right\} and unchanged scale coefficients c(J_min,l) (reflecting overall smooth variations) [⁴⁶,⁴⁷], where j\in \{{{\displaystyle J}}_{\min },{{\displaystyle J}}_{\min }+\mathrm{1},\dots ,{{\displaystyle J}}_{\max }\} denotes the resolution level, wherein J_min (J_max) shows the coarsest (the finest) resolution level with the sampling step \mathrm{1}/{\mathrm{2}}^{{{\displaystyle J}}_{\min }}(\mathrm{1}/{\mathrm{2}}^{{{\displaystyle J}}_{\max }}); k denotes the location of the wavelet functions in the spatial domain, as (2k+1)/2^j+1\in [\mathrm{0},\mathrm{1}]; and l denotes the location of the scaling functions in the spatial domain, as l/{\mathrm{2}}^j\in [\mathrm{0},\mathrm{1}] [⁴⁶].

The modifying stage can be performed by the thresholding technique. For instance, for a pre-defined threshold \epsilon, detail coefficients below this threshold are set to zero. This simple kill-or-keep method is known as the hard thresholding [⁴⁶–⁴⁸]. To prevent sudden jump in modified (thresholded) detail coefficients, there is another simple and famous approach known as the soft thresholding method, defined as [⁴⁶–⁴⁸]:

(1)\widehat{d}(j,k)=\{\begin{array}{l}\mathrm{0},\left|d(j,k)\right|\le \epsilon ,\\ Sign[d(j,k)][d(j,k)-\epsilon ],\left|d(j,k)\right|>\epsilon .\end{array}

The threshold value can be independent or dependent to the resolution level j. The level independent one is known as the global thresholding, where a single threshold is used for all resolutions. For the level dependent case, a threshold value {{\displaystyle \epsilon }}_j is used for the level [⁴⁸].

Regarding the number of iterations for denoising (thresholding), two different approaches have been developed:

1) the one step [⁴⁴–⁴⁹];

2) the iterative method, also known as the peeling scheme [⁵¹–⁵⁷].

In the first approach, the noise is cleaned by the one step of thresholding (with either a global or level-dependent one), while in the iterative method, the noise is removed layer by layer (by repeating a denoising procedure on the denoised signal resulted from the previous step) until a convergence criterion is satisfied [⁵¹–⁵⁷]. In general, for estimation of the threshold values, two general approaches exist: theoretical [⁴⁶,⁴⁸], and empirical [⁵⁸].

For the theoretical case, some famous approaches are: Universal, SURE, and GCV (Generalize Cross Validation) [⁴⁶–⁴⁸]. They all try to minimize the mean squared error, MSE(\epsilon ). These three methods are available for both global and level-dependent thresholding methods (see Appendix A for more information). In the empirical approach, the threshold is determined by a criterion, e.g., the curve of SNR against \epsilon (threshold). In this case, the proper threshold value is the one maximizing SNR.

To quantify the performance of different denoising methods and wavelet family effects, two criteria are considered based on the MSE concept: SNR and PSNR [⁴⁷,⁴⁸]:

(2)SNR=\mathrm{10}{{\displaystyle \mathrm{L}\mathrm{o}\mathrm{g}}}_{\mathrm{10}}\left[{\displaystyle \frac{{\sigma }^{\mathrm{2}}(\widehat{s})}{{\sigma }^{\mathrm{2}}\left(s-\widehat{s}\right)}}\right],PSNR=\mathrm{10}{{\displaystyle \mathrm{L}\mathrm{o}\mathrm{g}}}_{\mathrm{10}}\left[{\displaystyle \frac{\mathrm{M}\mathrm{a}\mathrm{x}{{\displaystyle \left(\left|{{\displaystyle \widehat{s}}}_i\right|\right)}}^{\mathrm{2}}}{{\sigma }^{\mathrm{2}}\left(s-\widehat{s}\right)}}\right],

where s and \widehat{s} denote the original (noisy) and denoised signals, respectively; {\sigma }^{\mathrm{2}}(Z) is the variance of Z; {{\displaystyle \widehat{s}}}_i is the ith element of {\displaystyle \widehat{s}}; and {\sigma }^{\mathrm{2}}\left(s-\widehat{s}\right) measures the variance of the noise: s-\widehat{s}. Function R={\sigma }^{\mathrm{2}}\left(s-\widehat{s}\right) denotes the risk-function, which is equivalent to MSE [⁴⁸]. Both SNR and PSNR are measured in dB.

The SNR and PSNR of large values provide better denoised (or compressed) results, because they have inverse proportionality to the variance of noise ({\sigma }^{\mathrm{2}}\left(s-\widehat{s}\right)). For PSNR, values larger than 30 dB are sufficient in practical applications [¹¹⁵].

Effective parameters on wavelet-based denoising

To achieve the best denoised signal, especially for the one-step denoising methods, it is essential to select the best denoising method, the best wavelet family and a proper decomposition level. Wavelet functions may be data dependent and this can have significant effects on the denoising performance. For an optimal wavelet, several features should be considered: 1) orthogonality (or bi-orthogonality); 2) support length which may be equivalent to filter length; 3) vanishing order of moments (the ith moment is: {{\displaystyle M}}_j(\Psi )={\displaystyle \int x^j}\Psi (x)\mathrm{d}x , where \Psi (x) denotes the wavelet (detail) function); 4) symmetry. The third condition is also known as the smoothness, approximation, or the regularity condition.

To achieve the best denoised signal, especially for the one-step denoising methods, it is essential to select the best denoising method, the best wavelet family and a proper decomposition level. Wavelet functions may be data dependent and this can have significant effects on the denoising performance. For an optimal wavelet, several features should be considered: 1) orthogonality (or bi-orthogonality); 2) support length which may be equivalent to filter length; 3) vanishing order of moments (the jth moment is: {{\displaystyle M}}_j\left(\psi \right)={\displaystyle \int x^j}\psi \left(x\right)\mathrm{d}x , where \psi \left(x\right) denotes the wavelet (detail) function); 4) symmetry. The third condition is also known as the smoothness, approximation, or the regularity condition.

Iterative denoising by discrete wavelets

Wavelet one-step denoising methods are widely used for signal processing problems. However, such approaches may not lead to proper results for data with high-levels of noise (as will be shown in this study). To improve the wavelet-based one-step methods, iterative-based denoising scheme has been developed [⁵¹–⁵⁷]. In this approach, in each iteration, the data having coherent structures in different resolution levels are selected and the remaining information is used as the noise in the next iteration. Due to the high level of noise in the data, the noise is removed from the data as much as possible by decomposing signals to coherent structures and incoherent noise by localizing both in time and frequency domains, simultaneously.

An algorithm for the MRA-based iterative denoising

In this subsection, the iterative denoising approach proposed in [⁵³–⁵⁵] is followed. For a noisy signal Z=\{z[\mathrm{1}],\dots ,z[N]\}, at the first step of the iterative denoising algorithm, it is assumed that the signal Z is totally noise. Then, the coherence and significant features in the noise are gathered.

Detection is performed by decomposing Z by MRA, and then in each resolution level, those detail coefficients (d(j,k)), that represent significant phenomena, are detected. In a statistical approach, the distribution of d(j,k) is measured at each scale j and is shown by the variance of the detail coefficients, i.e., {({\sigma }^j)}^{\mathrm{2}}. Then, detail coefficients larger than C\times {\sigma }^j are assumed to belong to a physical phenomenon, where C={{\displaystyle C}}_n^j is a positive constant and C\ge \mathrm{1} (n denotes the iteration number). The selected detail coefficients along with the scale coefficients are reconstructed as the first estimation of the denoised signal, and the remaining detail coefficients are reconstructed and named as the updated noise. The above peeling procedure is repeated for the updated noise. At the end of each iteration, the estimated denoised signal is added to the previous one (obtained from the previous step). In brief, the iterative algorithm can be summarized as:

1) For the first iteration, n=1, the denoised signal is assumed to be zero: {{\displaystyle Z}}_n=\{{{\displaystyle z}}_n[\mathrm{1}],\dots ,{{\displaystyle z}}_n[N]\}=\{\mathrm{0},\dots ,\mathrm{0}\} , and the noise is \mathrm{\Delta }{{\displaystyle Z}}_{n=\mathrm{1}}=Z=\{{{\displaystyle z}}_n[\mathrm{1}],\dots ,{{\displaystyle z}}_n[N]\}.

2) Decompose the noisy signal \mathrm{\Delta }{{\displaystyle Z}}_n by the DWT to obtain scale coefficients {c(J_min,l)} and detail coefficients in different resolutions as {{d(j,k)};j=J_min,...,J_max-1}, where J_min and J_max denote the resolution level of the coarsest and finest resolutions, respectively.

3) Compute the variance of {d(j,k)} for each resolution level j as {({{\displaystyle \sigma }}_n^j)}^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{{{\displaystyle N}}_j}}{{\displaystyle \Vert d(j,k)\Vert }}_{\mathrm{2}}^{\mathrm{2}} , where{{\displaystyle \Vert d(j,k)\Vert }}_{\mathrm{2}}^{\mathrm{2}}={\displaystyle {\sum }_k{\left|d(j,k)\right|}^{\mathrm{2}}} , n denotes the iteration number; and N_j is the length of vector {d(j,k)} (notice that: \mu ={\displaystyle {\sum }_{k}d(j,k)}=\mathrm{0}).

4) For each resolution level j, compute the threshold as {{\displaystyle T}}_n^j=f({{\displaystyle \sigma }}_n^j) . It is assumed: f({{\displaystyle \sigma }}_n^j)={{\displaystyle C}}_n^j{{\displaystyle \sigma }}_n^j , where {{\displaystyle C}}_n^j\ge \mathrm{1} is a constant.

5) Detect the noise coefficients by the hard thresholding method. Coefficients d(j,k) can be decomposed as: d(j,k)=\widehat{d}(j,k)+\mathrm{\Delta }d(j,k), where \widehat{d}(j,k)\ge {{\displaystyle T}}_n^j and \mathrm{\Delta }d(j,k) denotes the noise coefficients with values less than {{\displaystyle T}}_n^j .

6) Reconstruct the modified detail coefficients \left\{\widehat{d}(j,k);j={{\displaystyle J}}_{\min },\dots ,{{\displaystyle J}}_{\max }-\mathrm{1}\right\} with {c(J_min,l)} (from the second step). The reconstructed signal is {{\displaystyle \widehat{Z}}}_n=\left\{{{\displaystyle \widehat{Z}}}_n[\mathrm{1}],\dots ,{{\displaystyle \widehat{Z}}}_n[N]\right\}.

7) Reconstruct the detail coefficients belonging to noise, i.e. \{\{\mathrm{\Delta }d(j,k)\};j={{\displaystyle J}}_{\min },\dots ,{{\displaystyle J}}_{\max }-\mathrm{1}\} with zero approximation coefficients, \overline{c}({{\displaystyle J}}_{\min },l)=\{\mathrm{0},\dots ,\mathrm{0}\}. The reconstructed signal is \mathrm{\Delta }{{\displaystyle Z}}_{n+\mathrm{1}}=\{\mathrm{\Delta }{{\displaystyle Z}}_{n+\mathrm{1}}[\mathrm{1}],\dots \mathrm{\Delta }{{\displaystyle Z}}_{n+\mathrm{1}}[N]\} .

8) Update the estimated (denoised) signal as: {{\displaystyle \mathit{Z}}}_{n+\mathrm{1}}=\{{{\displaystyle z}}_{n+\mathrm{1}}[\mathrm{1}],\dots ,{{\displaystyle z}}_{n+\mathrm{1}}[N]\}=\{{{\displaystyle z}}_n[\mathrm{1}],\dots {{\displaystyle ,z}}_n[N]\}+\left\{{{\displaystyle \widehat{z}}}_n[\mathrm{1}],\dots ,{{\displaystyle \widehat{z}}}_n[N]\right\}.

9) Update the noisy signal to be \mathrm{\Delta }{{\displaystyle Z}}_{n+\mathrm{1}}, and go back to the second step (to restart a new peeling procedure).

10) This iteration is terminated after a predefined loop-number or reaching a termination criterion, where {{\displaystyle STC}}_{n+\mathrm{1}}={{\displaystyle \Vert \Delta {{\displaystyle Z}}_n\Vert }}_{\mathrm{2}}^{\mathrm{2}}-{{\displaystyle \Vert \Delta {{\displaystyle Z}}_{n+\mathrm{1}}\Vert }}_{\mathrm{2}}^{\mathrm{2}} and \epsilon denotes a positive constant.

A simple criterion for choosing {{\displaystyle C}}_n^j

The heart of the iterative denoising method is the proper selection of the variance-dependent thresholds controlled by the coefficients {{\displaystyle C}}_n^j. While some constant values of {{\displaystyle C}}_n^j were empirically proposed for different applications (e.g., {{\displaystyle C}}_n^j=3 or {{\displaystyle C}}_n^j=2.5) [⁵³], few computational approaches were proposed based on extra information about data, such as their experimental distributions [⁵⁵,⁵⁶]. One drawback of the constant {{\displaystyle C}}_n^j values (in Refs. [⁵³,¹¹⁶]) is that they can vary for different signals recorded even in one study. In this regard, a flexible approach is needed for adaptive selection of {{\displaystyle C}}_n^j values.

In this study, a simple method is proposed based on measuring the PSNR and SNR values for several {{\displaystyle C}}_n^j values. Admissible values {{\displaystyle C}}_n^j for correspond to cases where PSNR\ge {PSNR}^{*} ; here, it is assumed that {PSNR}^{*}=\mathrm{30} [¹¹⁵]. The largest acceptable {{\displaystyle C}}_n^j can be used as the first estimation of {{\displaystyle C}}_n^j in the iterative denoising method. The present study shows that this selection can be affected by other important parameters:

1) The continuity feature of denoised signals to prevent unphysical gaps in denoised data.

2) Sufficiently large values of SNRs.

The selection procedure for {{\displaystyle C}}_n^j is explained in detail in Section 6 for different recorded data.

Signal processing of the enhanced data by continuous wavelet transforms (CWTs)

After the signal-enhancement stage, the signal processing step is performed by continuous wavelet transforms (CWTs) for detection of time-frequency information in the data.

In this study, the complex Morlet wavelet is used for detection of instantaneous frequencies and pattern recognitions. Parameters of the Morlet wavelet are: {{\displaystyle \upsilon }}_{\mathrm{b}}=2 ( is the bandwidth frequency) and {{\displaystyle \upsilon }}_c=1.10 ( denotes the central frequency). Hence, the inequality condition \sqrt{{{\displaystyle \upsilon }}_{\mathrm{b}}}{{\displaystyle \upsilon }}_{\mathrm{c}}\ge \sqrt{\mathrm{2}} is satisfied for this wavelet family (defined as \Psi (t)={\displaystyle \frac{\mathrm{1}}{\sqrt{\pi \times {{\displaystyle \upsilon }}_{\mathrm{b}}}}}{e}^{\mathrm{2}\mathrm{\pi }i{v}_{c}t\times e^{-t^2/{\upsilon }_{\mathrm{b}}}} ). For other possible values of {{\displaystyle \upsilon }}_{\mathrm{b}} and {{\displaystyle \upsilon }}_{\mathrm{c}} , some other approaches are reviewed in Appendix B.

Based on the CWTs, some effective signal processing tools are: XWT, the autocorrelation of wavelet coefficients and the spectral power. Initially, the concept of the spectral power of WT is reviewed. This helps to identify energy concentrations in the frequency (or scale) domain. The spectral power is defined as:

(3){{\displaystyle P}}_W(a)={\displaystyle \frac{\mathrm{1}}{T}}{\displaystyle {\int }_{{{\displaystyle t}}_{\mathrm{0}}}^{{{\displaystyle t}}_{\mathrm{0}}+T}{\left|{{\displaystyle W}}_{\Psi }f(a,b)\right|}^{\mathrm{2}}}\mathrm{d}b,

where a denotes the scaling (dilation) number controlling the width (support) of wavelet functions; b represents the spatial position of the scaled wavelets \Psi (t/a) ; T denotes the duration of data; and {{\displaystyle W}}_{\Psi }f(a,b) denotes the CWT of data f(t), defined as {{\displaystyle W}}_{\Psi }f(a,b)={\displaystyle \frac{\mathrm{1}}{\sqrt{\left|a\right|}}}{\displaystyle {\int }_{-\infty }^{+\infty }f(t)}{\Psi }^{*}\left({\displaystyle \frac{t-b}{a}}\right)dt, where the symbol “” shows the imaginary conjugate of a function.

XWTs, corresponding spectral powers and autocorrelation of wavelet coefficients

Based on the coefficients of WTs, different analyses can be performed. Three of them are:

1) XWTs: In practical computations, it is often useful to capture possible links between two existing processes (signals). For assessing energy coherencies, the concept of the XWT (denoted by ) is introduced. For signals f(t) and g(t), {{\displaystyle XW}}_{\Psi }(f,g) is defined as [⁴⁵]:

(4){{\displaystyle XW}}_{\Psi }(f,g)={{\displaystyle W}}_{\Psi }f(a,b)\times {{\displaystyle W}}_{\Psi }g{(a,b)}^{*},

where symbol “*” shows the imaginary conjugate of a function. The XWT detects zones in the time-frequency space where the two data show the coherency of the power (energy). Since the noise has a random feature, the two recorded data are expected to have uncorrelated energies of noise in different resolutions and times.

2) Spectral power of XWTs: To detect energy coherencies between two data in the frequency domain, the power of XWTs can also be defined, as:

(5){{\displaystyle P}}_{XW}(a)={\displaystyle \frac{\mathrm{1}}{T}}{\displaystyle {\int }_{{{\displaystyle t}}_{\mathrm{0}}}^{{{\displaystyle t}}_{\mathrm{0}}+T}\left|{{\displaystyle XW}}_{\Psi }(f,g)\right|}\mathrm{d}b.

Simultaneous study of XWTs and the corresponding spectral powers helps to identify physical (real) phenomena in data. The energy of WT reveals the continuous distribution/pattern of energies in the time-frequency representation, and at the same time the corresponding spectral power shows the concentration of energies in the frequency domain.

3) Autocorrelation of WTs: This operator measures the correlation of a signal with a shifted copy of itself. It assists similarities or repeating patterns in the signal as a function of time delay or time lag. One example of such feature is the detection of periodic signals covered with noise. For the signal {x_i}, where i=1,...,n, the autocorrelation R(\tau ) is defined as:

(6)R(\tau )={\displaystyle \frac{{\displaystyle {\sum }_{i=\mathrm{1}}^{n-\tau }\left({{\displaystyle x}}_i-\widehat{\mu }\right)\left({{\displaystyle x}}_{i+\tau }-\widehat{\mu }\right)}}{{\displaystyle {\sum }_{i=\mathrm{1}}^{n-\tau }{{\displaystyle \left({{\displaystyle x}}_i-\widehat{\mu }\right)}}^{\mathrm{2}}}}},

where \widehat{\mu }=({\displaystyle {\sum }_{i=\mathrm{1}}^n{{\displaystyle x}}_i})/n denotes the mean value of the signal {x_i}, and \tau is the time lag.

To integrate the MRA with the autocorrelation concept, at the scale a, the autocorrelation of {{\displaystyle XW}}_{\Psi }f(a,b) is evaluated over the translation parameter b. This procedure is repeated for different a values. For each scale, this helps capturing similarity features through time, or detecting the repeating patterns of data. This type of detection is nearly free from noise effects, since, stochastic coefficients due to noise do not have correlation with each other. In this study, it is shown that this transform is especially important for capturing weakly excited responses or vibration modes (in ambient tests) which cannot be detected by other transforms. Frequencies for modes with small participations have the wavelet coefficients of small values; hence, these frequencies may not be distinguishable from their wavelet energies, XWTs or their spectral densities.

It should be noted that, the scaling parameter a can be related to the corresponding frequency {{\displaystyle \upsilon }}_a (in Hz) based on the central frequency of the scaled- shifted wavelet {{\displaystyle \Psi }}_{a,b}(t), as:

(7){{\displaystyle \upsilon }}_a={\displaystyle \frac{{{\displaystyle \upsilon }}_{\mathrm{c}}}{a\times \mathrm{\Delta }}},

where {{\displaystyle \Psi }}_{a,b}(t)=\Psi ((t-b)/a)/\sqrt{\left|a\right|},\mathrm{\Delta }=\mathrm{d}t denotes the sampling period, and {{\displaystyle \upsilon }}_{\mathrm{c}} represents the central frequency of the wavelet function \Psi (t) in Hz.

Results: the signal enhancements and the frequency detection of the ambient vibration test

The effects of regularization- and wavelet-based denoising schemes are now studied for different data in this section. The recorded data P21, P31, P51, P12, and P72 are presented in Fig. 4 with the sampling time step \mathrm{\Delta }T=\mathrm{0.01}. The denoising or enhancing procedure is performed with the MRA and the regularization approaches. For the MRA-based denoising approach, both the one-step method and the iterative scheme are studied. For the regularization-based denoising scheme see Appendix C.

The one-step denoising approach

Denoising with the regularization approach

Let us consider the signal P12. At the first step, it is essential to estimate a proper value for the regularization parameter p (or \lambda). This can be performed empirically by the SNR- p curve [⁵⁸]: the proper value that maximizes SNR. These curves are presented in Fig. 5 for the constraints \Omega {{\displaystyle (f)}}_{\mathrm{T}\mathrm{V}} , \Omega {{\displaystyle (f)}}_{L^{\mathrm{2}}} and \Omega {{\displaystyle (f)}}_{\mathrm{S}\mathrm{o}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{v}}. It is obvious that the curves do not have any extrema. Due to the high-level of noise, the SNR values can even be negative, and for the large values of p , the SNR values increase rapidly. For checking the denoising performance, the residual noise (after the denoising stage) and the corresponding energy in the wavelet space (by the complex Morlet wavelet with parameters {{\displaystyle \upsilon }}_{\mathrm{b}}=\mathrm{2} and {{\displaystyle \upsilon }}_{\mathrm{c}}=\mathrm{1.10}) are presented in Fig. 6 for the constraints \Omega {{\displaystyle (f)}}_{L^{\mathrm{2}}} and \Omega {{\displaystyle (f)}}_{\mathrm{S}\mathrm{o}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{v}}, where p=0.98. The energies are presented for the range of \mathrm{0.02}\mathrm{M}\mathrm{a}\mathrm{x}[{\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}]\le {\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}\le \mathrm{M}\mathrm{a}\mathrm{x}[{\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}] . It is evident that the energies of the noises do not contain localized stochastic information; instead, they contain several instantaneous frequencies excited continuously through time. Such information can be physically justified as the modal frequencies of the structure. Hence, this denoising approach is not efficient.

DWT-based one-step denoising methods

The performance of different one-step DWT-based denoising approaches is studied for signals P12 and P51. Denoising approaches are mentioned in Table 1; the term “Level” indicates the level depending thresholding.

For the signal P51, the corresponding results based on Symlet[12] wavelet with N_d=13 (the number of decomposition levels) are presented in Fig. 7. For P12 and P51, the performance of different denoising methods, measured by the SNR and PSNR criteria, is also presented in Table 2. This table and Fig. 7 offer that the GCV denoising method leads to the best result for the Symlet[12] wavelet with N_d=13. However, in general, this conclusion could not be made. This is studied in terms of different denoising approaches, wavelet families and the number of decomposition levels (N_d ) in the following.

The effects of the wavelet family are studied in Tables 3–5 for the GCV method with N_d =13 and the considered wavelet families are: Db[N], Symlet[N], and BattleLemarie [N]. Accordingly, the wavelet-dependency of the results is clear even for the GCV method. The effects of N_d , presented in Table 6, confirm that the values of SNR and PSNR are in accordance with the values of N_d.

The performance of the empirical approach for the threshold selection is studied in the following. Let us consider the signal P12 with both the hard and the soft thresholding approaches. Variations of SNR against \epsilon /\sigma values are presented for different wavelet families in Fig. 8 for N_d=13; where \epsilon and \sigma denote the threshold and the deviation of a signal, respectively. In Fig. 8, the wavelets are those with the best performance according to Tables 3–5. The results offer that the curves do not have extremum values for SNRs; therefore, an empirical threshold cannot be recommended.

Results for the iterative denoising method

Selection of {{\displaystyle C}}_n^j values

For the iterative denoising, the largest possible value of {{\displaystyle C}}_n^j for each signal can be estimated by a simple algorithm based on measuring PSNR values.

Initially, the effects of {{\displaystyle C}}_n^j values on the iterative thresholding is studied for the signal P12. Two different values of {{\displaystyle C}}_n^j are chosen as {{\displaystyle C}}_n^j=1.8 and {{\displaystyle C}}_n^j=2. Enhanced signals by thresholds {{\displaystyle C}}_n^j=1.8 and {{\displaystyle C}}_n^j=2 are presented in Figs. 9 and 10, respectively. The wavelet is Symlet[12], with N_d=13 and N=10 (the number of iterations). In these figures, the captured information ({{\displaystyle \widehat{Z}}}_n) and estimated noise (\mathrm{\Delta }{{\displaystyle Z}}_n) are presented at each iteration. In the last row, both the denoised data (i.e.,: {{\displaystyle Z}}_{\mathrm{1}+\mathrm{10}}={\displaystyle {\sum }_{n=\mathrm{1}}^{\mathrm{10}}{{\displaystyle \widehat{Z}}}_n} ) and the estimated noises (that is: Z-{\displaystyle {\sum }_{n=\mathrm{1}}^{\mathrm{10}}{{\displaystyle \widehat{Z}}}_n}) are presented after ten iterations. The denoised data offer that:

1) For {{\displaystyle C}}_n^j=1.8: SNR=2.3072 and PSNR=28.145: an insufficient denoised signal as the SNR is small; the denoised signal has several gaps.

2) For {{\displaystyle C}}_n^j=2: SNR=-10.078 and PSNR=24.286: a false denoised signal; most parts of physical information are eliminated.

Therefore, choosing a proper value for {{\displaystyle C}}_n^j is crucial. For this reason, the simple approach, proposed in Section 4.2, is now explained in detail. Variations of the PSNR and SNR values against {{\displaystyle C}}_n^j are illustrated in Fig. 11 for different signals with parameters: Symlet[12], N_d=13 and N=10 (the number of iterations). Based on the PSNR values and PSNR^*=30, the initially estimated values of (the largest possible values) are: 1) For P12: {{\displaystyle C}}_n^j=1.78; 2) For P51: {{\displaystyle C}}_n^j=1.94; 3) For P 31: {{\displaystyle C}}_n^j=1.95; 4) For P 72: {{\displaystyle C}}_n^j=1.91; and 5) For P 21: {{\displaystyle C}}_n^j=1.96.

In this study, slightly smaller values of {{\displaystyle C}}_n^j are used: to prevent developing of artificial gaps, and to have a larger value for PSNR in denoised signals, as: 1) For P12: {{\displaystyle C}}_n^j=\mathrm{1.75}; 2) For P51: {{\displaystyle C}}_n^j=1.90; 3) For P31: {{\displaystyle C}}_n^j=1.90; 4) For P72: {{\displaystyle C}}_n^j=\mathrm{1.85}; and 5) For P21: {{\displaystyle C}}_n^j=1.90.

For the signal P12, with the threshold {{\displaystyle C}}_n^j=\mathrm{1.75}, the effectiveness of the enhancement (measured by SNR and PSNR) after each iteration is presented in Table 7. The assumptions are: Two Symlet wavelets of orders 4 and 12 with N_d=13 are utilized; at the end of ith iteration, for evaluation of SNR and PSNR, the enhanced results at the ith iteration (i.e., {{\displaystyle Z}}_{\mathrm{1}+i}={\displaystyle {\sum }_{n=\mathrm{1}}^i{{\displaystyle \widehat{Z}}}_n}) and the initial noisy (raw) data (Z) are used. It is evident that even for the Symlet[4] the results become sufficiently good after ten iterations, while with this wavelet the results obtained from the one-step denoising approach are not acceptable (see Table 4).

For the signal P12, both the captured information ({{\displaystyle \widehat{Z}}}_n ) and the estimated noise (\mathrm{\Delta }{{\displaystyle Z}}_n ) are presented in Fig. 12 at each iteration (or the peeling step), where {{\displaystyle C}}_n^j=\mathrm{1.75} and N_d=13. In this figure, the last row includes: the left column: the denoised signal after ten iterations, (i.e., Z₁₊₁₀), and the right column: the estimated noise (Z={\displaystyle {\sum }_{n=\mathrm{1}}^{\mathrm{10}}{{\displaystyle \widehat{Z}}}_n}).

The feature of the estimated noise by MRA

The effects of the iterative enhancement for different signals are discussed in more detail in this subsection. The energy densities of the final noises, obtained after ten iterations (Z={\displaystyle {\sum }_{n=\mathrm{1}}^{\mathrm{10}}{{\displaystyle \widehat{Z}}}_n}; see Fig. 10), are presented in Fig. 13 for different signals in the time-frequency representations (obtained by CWT). All computations are performed by the complex Morlet wavelet, where {{\displaystyle \upsilon }}_b=\mathrm{2} and {{\displaystyle \upsilon }}_c=\mathrm{1.10}. The plotted range for the energies is \mathrm{0.02}\mathrm{M}\mathrm{a}\mathrm{x}[{\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}]\le {\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}\le \mathrm{M}\mathrm{a}\mathrm{x}[{\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}] . The results confirm that the noises contain localized random-wise information at different times and resolution levels, especially at the high frequency ranges (as expected for noise). For short periods, i.e., T<0.5, stochastic localized information exists and for the range of 0.9<T<1.4, several considerable phenomena occur locally, which may represent the operations of mechanical systems in the tower. These local or stochastic-like data are not important in this study, because only the modal frequencies excited continuously in time are sought.

So far, some mathematical tools have been used to enhance signals. At the next step, it is tried to integrate the physical properties to extract more reliable information from the enhanced data. The main idea is that if some information is repeated in different data, they may include the physical responses of the structure. The MRA-based approaches for such detections are XWT and the autocorrelation of wavelet coefficients (since the noise is assumed to have random feature with zero coherence).

XWTs and autocorrelation of wavelet coefficients for frequency detections

Physical responses are tried to be detected by XWT, the corresponding spectral power and the autocorrelation of wavelet coefficients.

XWT is used for detection of energy coherencies in the enhanced pairs { P21, P31}, { P21, P51} and { P12, P72}. These signals are enhanced with the iterative algorithm with Symlet[12], where N_d=13, and N=10 (the number of iterations). Results presented in Figs. 14, 15, and 16 are for the enhanced pairs { P21, P31}, { P21, P51} and { P12, P72}, respectively. In each figure, the density of wavelet energies, the corresponding XWT and the spectral powers are provided. In these figures, plotted ranges for {\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}} and \left|{{\displaystyle W}}_{\Psi }\right| are \mathrm{0.02}\mathrm{M}\mathrm{a}\mathrm{x}[{\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}]\le {\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}\le \mathrm{M}\mathrm{a}\mathrm{x}[{\left|{{\displaystyle W}}_{\Psi }\right|}^{\mathrm{2}}]and \mathrm{0.02}\mathrm{M}\mathrm{a}\mathrm{x}[\left|{{\displaystyle W}}_{\Psi }\right|]\le \left|{{\displaystyle W}}_{\Psi }\right|\le \mathrm{M}\mathrm{a}\mathrm{x}[\left|{{\displaystyle W}}_{\Psi }\right|], respectively.

Based on XWTs and the corresponding powers, the detected frequencies are:

1) Form the pair { P21, P31} (Figs. 14(e) and 14(f)): 0.88, 0.92, 1.05, 1.09, 1.11, 1.22, 1.24, 1.29, 1.59, 1.64, 1.8, 1.84, 2.12, 2.16, and 2.3 s.

2) Form the pair { P21, P51} (Figs. 15(c) and 14(d)): 0.88, 0.92, 1.05, 1.09, 1.24, 1.29, 1.59, 1.64, 1.71, 1.78, 1.8, 1.84, 2.12, 2.16, and 2.3 s.

3) Form the pair { P12, P72} (Figs. 16(e) and 14(f)): 0.88, 0.92, 1.05, 1.09, 1.24, 1.29, 1.55, 1.59, 1.64, 1.8, 1.84, 1.89, 1.96, 2.12, 2.16, and 2.3 s.

Since signals are not recorded simultaneously, periods which are detected in all pairs are assumed to be the possible modal periods. They are: 0.88, 0.92, 1.05, 1.24, 1.29, 1.59, 1.64, 1.8, 1.84, 2.12, 2.16, and 2.3 s. The results also show that due to the semi-symmetric shape of the structure, there are several pairs of nearly excited periods.

In the following, by the autocorrelation of the wavelet coefficients (of CWTs), interrelationships are studied at different resolution levels of the enhanced data and the results are illustrated in Fig. 17 for P21, P31, P51, P12 and P72. It is evident that by the autocorrelation transform, it is possible to detect some short periods which are not captured by XWT. The detected short periods are: 0.12 and 0.4 s, which are common in all data. These short periods participate marginally in the ambient vibration test, due to: 1) the nature of wind loading: this type of load mobilizes mainly some first modes; 2) the rigidity of the concrete tower: the participation of higher modes is small for the wind load.

Based on the aforementioned detections, the possible modal periods are mentioned in Table 8. The tower was independently modeled by an undamaged FE model using 3-D continuum elements with linear shape functions [¹¹⁷]. The modal frequencies of the FE model reported in Ref. [¹¹⁷] are presented in Table 8.

Regarding the modal frequencies, the difference between the results may be attributed to: (1) damage existence in different parts of the concrete tower, which is not considered in the FE model; (2) the effects (mass and stiffness) of three operating fans on the top of the main tower, one Hooper and the operation of some mechanical systems in the main tower are not considered in the FE model; (3) the steel-concrete interaction due to the existence of four narrow and long steel towers on the top of the main concrete tower (Fig. 1) and their connection to the concrete tower. In the FE model, steel towers have rigid connections with the main tower, which may not be the case in real structure; (4) ignoring the foundation effects. These differences between theoretical and experimental results were also reported in some independent studies [⁶⁵].

Conclusions

In this study, the main novelty from the application-wise point of view is introducing the iterative enhancement approach. For this approach, then, a computational method is suggested to help in choosing proper values of the iterative algorithm.

Initially, two general MRA-based approaches have been examined for the enhancement of high-level noisy signals recorded from the ambient vibration tests: the one-step and the iterative denoising methods. The regularization-based denoiser is also studied as the one-step denoiser. For the data with small values of SNR, the results of the variational-based minimization (the regularization) and the MRA-based one-step denoising methods confirm that:

1) Outputs can be incorrect (Figs. 6 and 7).

2) For the variational-based denoising, there is no optimum value of SNR in the empirical approach for estimation of \lambda (Fig. 5). Moreover, denoised data (even with marginal effects of regularization) may contain physical information (Fig. 6).

3) For wavelets, both empirical and theoretical approaches are unsuccessful for estimation of threshold values (Figs. 7 and 8).

4) Results are very sensitive to wavelet families (Tables 3, 4, and 5).

5) Different thresholding methods lead to different results (Table 2).

6) The number of decomposition levels is also important (Table 6).

For the iterative denoising approach in high-level noisy data, the results offer:

1) Different {{\displaystyle C}}_n^j values are needed for different recorded data (Fig. 11); hence, it is essential to use an adaptive approach for selection of {{\displaystyle C}}_n^j.

2) It can enhance the signals (see captured noise in Fig. 12 and corresponding stochastic density of energy in Fig. 13).

3) The method can detect some stochastic-like data, which confirms the importance of such enhancements (Fig. 13).

4) Noise cannot completely be removed (see the localized features in Fig. 15).

5) The results are not so sensitive to the wavelet orders or families (Table 7).

6) While some criteria can be introduced (e.g., Fig. 11) for the threshold estimation, the trial and error method can also be recommended for determination of threshold (this is clear by comparing Figs. 9, 10, and 12).

After the enhancement of the high-level noisy signals (from ambient vibrations) by the iterative algorithm, it has been tried to detect excited frequencies by some MRA-based signal-processing approaches: the cross-wavelet analyses, corresponding spectral powers and the autocorrelations of wavelet coefficients. CWTs have been performed with the complex Morlet wavelets for capturing both instantaneous frequencies and corresponding excitation patterns in the time-frequency representations.

An excited frequency is known as a physical phenomenon if it is continuous through the time in the time-frequency representation, and has a local concentration of the spectral power in the frequency (scale) domain. This simultaneous study for different pairs of the enhanced data provides:

1) XWTs can improve the time-frequency representation for detection of excited frequencies (e.g., Fig. 15).

2) This transform, however, seems to be insufficient since there are several excited frequencies with continuous pattern through the time (see Figs. 14–16).

3) In this regard, the spectral power of WTs helps in capturing frequencies with considerable energies.

4) As the participation of higher modes is small (due to the stiffness of the structure), the multiresolution-based autocorrelation analysis can help distinguish such possible modal frequencies (Fig. 17).

5) XWTs can also decrease the effects of noise-like stochastic phenomena (see Fig. 15).

6) In detected periods, there are several couple of periods (near each other), because of the semi-symmetric plan of the structure (see Figs. 1 and 2).

Finally, the captured frequencies have been compared with those of a 3-D FE model; the results show a good agreement. Nevertheless, there are some frequencies which are not seen in the FE model, which can be justified by: (1) the absence of four narrow steel towers on the top of the main (concrete) tower in the FE model; (2) during the ambient vibration test, the main tower was in operation (this means some detected frequencies may correspond to the operating systems); (3) the operations of some equipment and fans on the main tower (which were not considered in the FE model); (4) the real tower has several localized damaged parts which were not considered in the FE model.

The detection algorithm used here is completely based on the signal processing approach. The results can be examined by other methods which take into account the structural properties (such as modal superposition in MDOFs) [¹³,¹⁹,⁷⁶]. The identification of modal shapes and the corresponding damping will be studied in an independent work.

For future works, the following tasks are recommended:

1) Updating the FE model to include the damages and operating systems (mass and stiffness),

2) Using wavelet packets for the iterative enhancement approach. One of the shortcoming of the common discrete wavelet transforms is that they may not suitable for the processing of high frequency signals with nearly narrow bandwidth. For this reason, the wavelet packet was developed. The full transform leads to a large number of subbands (time-scale cells) and so significant numbers of signal representation possibilities. Decomposition increasing may lead to a more flexible analysis capability, especially, for data including different slow and fast variations. To capture properly these responses in the time-scale representations, the concept of the best basis was proposed. The best basis can be evaluated by defining a cost function where different ones were developed [¹¹⁸–¹²⁰]. Hence, the best basis can depend on the cost definition. This dependency can also affect the iterative denoising results. This can also be considered for the future study.

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