Mono-Component Feature Extraction for Condition Assessment in Civil Structures Using Empirical Wavelet Transform

This paper proposes a methodology to process and interpret the complex signals acquired from the health monitoring of civil structures via scale-space empirical wavelet transform (EWT). The FREEVIB method, a widely used instantaneous modal parameters identification method, determines the structural characteristics from the individual components separated by EWT first. The scale-space EWT turns the detecting of the frequency boundaries into the scale-space representation of the Fourier spectrum. As well, to find meaningful modes becomes a clustering problem on the length of minima scale-space curves. The Otsu’s algorithm is employed to determine the threshold for the clustering analysis. To retain the time-varying features, the EWT-extracted mono-components are analyzed by the FREEVIB method to obtain the instantaneous modal parameters and the linearity characteristics of the structures. Both simulated and real SHM signals from civil structures are used to validate the effectiveness of the present method. The results demonstrate that the proposed methodology is capable of separating the signal components, even those closely spaced ones in frequency domain, with high accuracy, and extracting the structural features reliably.


Introduction
The data in Structural Health Monitoring (SHM) from civil structures contains essential information on their condition. Instantaneous features shall be desirable to be extracted for further structural condition assessment. However, under operating condition, the dynamic responses of civil structures are usually stored in time domain and are non-stationary because of complex excitation. They include complex components induced by distinct loads and intricate load-structure interactions. In addition, the acquired data includes unavoidable noises, spikes and trends, further challenging the extraction of useful information.
Time-frequency (TF) approaches were developed to obtain the instantaneous features from time-varying signals. These methods can provide information of signals in both time and frequency domains, becoming more competitive to process SHM data. Various TF methods including Short Time Fourier Transform (STFT) [1], Wigner-Ville Distribution (WVD) [2], Wavelet Transform (WT) [3], and Empirical Mode Decomposition (EMD) [4] were raised. Even these methods can determine some useful results, their limitations are also clear. For instance, the window length fixes the spectral resolution of the STFT. Despite the WVD can provide a good resolution, the cross-term inference restricts its application. With the merit of multi-resolution, the WT has been one of the most widely characteristics. Simulated signals with different levels of noise and real SHM signals from two civil structures, namely, a high-rise building and a footbridge, are used to demonstrate the effectiveness of the proposed signal processing procedure. The usefulness and accuracy of the scale-space EWT in signal decomposition is validated by the synthetic signals. To show the advantages of the scale-space EWT, the TF representations are compared with those from SWT and EMD. In the experimental study, the extracted instantaneous structural features are affirmed by comparing with results of the previous studies.

Empirical Wavelet Transform
The EWT aims to extract the individual modes from a signal by an adaptively designed wavelet filter bank. The modes extracted by EWT are amplitude modulated-frequency modulated (AM-FM) signals that have a compact support Fourier spectrum [17]. Separating different modes equals dividing the Fourier spectrum first and then to perform WT using the empirical wavelets constructed based on the detected supports.
Fast Fourier Transform (FFT) is implemented to the signal s(t) to obtain the frequency spectrum s(ω). Segmenting the Fourier spectrum is the crucial step for the adaptability of EWT. The local maxima of s(ω) is estimated first. Then, the Fourier axis is segmented to individual portions corresponding to different modes that are centered around a local maximum. The number of continuous segments the Fourier axis divided into is denoted as N. The limits between each segment are denoted as ω i (ω 0 = 0, and ω N = π), and each segment is denoted as Λ i = [ω i−1 , ω i ]. A transition phase T i of width 2τ i is defined surrounding each ω i . The simplest choice of τ i is Similar to constructing Littlewood-Paley and Meyer's wavelets [3], the empirical scaling function and the empirical wavelets can be defined by the following expressions of Equations (2) and (3), respectively.

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The signal is reconstructed by The empirical mode s k is given by

Methodology
The flowchart of the methodology is shown in Figure 1. The steps are illustrated as follows.

Definition of Meaningful Modes
The number of minima with respect to ω of L(ω, σ) is a decreasing function of the scale parameter σ [32]. In the scale-space plane, a curve is produced by each initial minima. Let us denote the number of initial minima as N0, and the 'scale-space curve' defined as Ci (i∈[1, N0]), with a length of Li. Li indicates the life span of the minimum i. A mode in a spectrum is defined as

Scale-Space Boundary Detection
As the most important step in EWT, boundary detection to build the wavelet filter bank provides adaptability to the analyzed signal. Gilles [17] estimated the boundaries based on the local maxima and minima of the signal's Fourier spectrum. However, the Fourier spectrum is very sensitive to noises, as is usual the case for SHM signals of civil structures. These noises may produce redundant local maxima, leading to false boundaries. The Gilles' method requires the predefined frequency-band number. Other spectra immune to noises [29,30] can be employed, but special attention should be paid to the probability of missing useful modes. This study takes an alternative way, to use a different method other than the original local-maxima-minima one to segment the Fourier spectrum. The scale-space approach, which automatically detects meaningful modes in a spectrum based on the behavior of local minima in a scale-space representation [32], is employed. Finding meaningful modes is equivalent to a binary clustering problem on the length of minima scale-space curves. To reduce the influence of noises, a wavelet-based approach [7] is employed for denoising before the EWT procedure.

Scale-Space Representation of a Spectrum
Let function f (ω) define over an interval [0, ω max ]. Its discrete scale-space representation is defined as where g(n; σ) = 1 √ 2πσ e −n 2 /2σ (11) which is a sampled Gaussian kernel, M is large enough so that the approximation error of the Gaussian is negligible. The scale parameter σ is sampled in the following manner where k = 1, . . . , k max are integers, √ σ 0 is set to be 0.5, and √ σ max equals ω max .

Definition of Meaningful Modes
The number of minima with respect to ω of L(ω, σ) is a decreasing function of the scale parameter σ [32]. In the scale-space plane, a curve is produced by each initial minima. Let us denote the number of initial minima as N 0 , and the 'scale-space curve' defined as C i (i ∈ [1, N 0 ]), with a length of L i . L i indicates the life span of the minimum i. A mode in a spectrum is defined as meaningful if its support is delimited by two local minima corresponding to two scale-space curves C i above a certain length T [32]. Consequently, detecting meaningful modes is a two-class clustering problem on The key point is to automatically determine a threshold T.

Determination of Threshold
The Otsu's method separates a spectrum into two classes of C 1 and C 2 , and finds T that maximizes the between class variance where and Details of the Otsu's method can be found in [34].

Time-Frequency Representation of Extracted Modes
The modes extracted by the EWT are AM-FM signals s j (t) = S j (t)cos(ϕ j (t)) (j = 0, 1, . . . , N). Following Hilbert-Huang transform (HHT), the HT of a function s j (t) is defined as where p.v. represents the Cauchy principle value. The analytical form s ja (t) of s j (t) can be derived by the HT In AM-FM signals the HT provides where the instantaneous amplitude S j (t) and frequency ϕ j (t) can be extracted. The TF representation of the signal is obtained by plotting each curve ϕ j (t) with intensity of S j (t) in the TF plane. The time varying of the frequency and amplitude of each mode can be observed from this TF representation.

Modal Characteristics
For a time-varying SDOF structure under free vibration, if its parameters vary slower than the dynamic response, both the natural frequency ω 0 (t) and the damping coefficient h 0 (t) are slowly varying functions of time. They can be evaluated by [33] and where A and ω are the instantaneous amplitude and frequency of the response, respectively. When it comes to a time-varying SDOF system under forced vibration, the instantaneous frequency of the vibration signal is For ambient vibration, the second term of Equation (21) is a zero mean fast time-varying function [35]. The instantaneous frequency and amplitude of the dynamic signals will obtain time-varying parameters.
The FREEVIB method [33] relies on HT. It can obtain the stiffness and damping characteristics, and identify the instantaneous modal parameters of free vibration SDOF systems. This method includes the following steps [36]: (1) Taking the HT of the measured dynamic responses and calculating the envelope and the instantaneous frequency; (2) identifying the instantaneous parameters; (3) low-pass filtering of the modal parameters, and scaling the smooth modal parameters; and (4) plotting the backbones of the frequency, damping curves, frequency response functions (FRF), and force static characteristics.

Backbone and Damping Curve
From Equations (19) and (20), the instantaneous modal parameters are functions of the first and second deviations of the signal envelop and the instantaneous frequency of the dynamic response. Linking the modal frequency and the envelope gets a skeleton curve or backbone. Similarly, linking the modal damping and the envelope obtains a damping curve. Backbones and damping curves are used as a traditional instrument in nonlinear vibration analysis [36].
For small and slow nonlinear variations, The instantaneous modal frequency of the system will be close to the instantaneous frequency of the dynamic response, and the instantaneous damping coefficient equals the ratio between the envelope and its derivative.

Numerical Study
A synthetic signal consisting of three frequency components of 1, 3 and 6 Hz is used in this section to investigate the advantages of EWT. White Gaussian noise included into the signal to study the noise effect. The exponential function is embedded to simulate the signal attenuation with time. This simulated signal is expressed as where n(t) is the white Gaussian noise. High-level noises with three different signal-to-noise ratios (SNRs), i.e., −2, −6 and −10 dB, are used to test the efficacy of the scale-space EWT method. The 20-s simulated signal with an SNR of −2 dB is shown in Figure 2a. The sampling frequency is 50 Hz. The boundaries for the spectrum segmentation are detected using the scale-space method stated in Section 3.1.1, and the result is shown in Figure 2b. The threshold for the boundary detection is calculated to be 14 by the Otsu's method. Thus, minima scale-space curves with length longer than 14 are the boundaries. The empirical wavelets are defined according to Equations (2) and (3) based on these boundaries. Using these basis functions, WT is applied to decompose the signal. The extracted signal components, x rec1 (t) to x rec3 (t) are displayed in Figure 3. All the three components are extracted without redundant modes. parameters; (3) low-pass filtering of the modal parameters, and scaling the smooth modal parameters; and (4) plotting the backbones of the frequency, damping curves, frequency response functions (FRF), and force static characteristics.

Backbone and Damping Curve
From Equations (19) and (20), the instantaneous modal parameters are functions of the first and second deviations of the signal envelop and the instantaneous frequency of the dynamic response. Linking the modal frequency and the envelope gets a skeleton curve or backbone. Similarly, linking the modal damping and the envelope obtains a damping curve. Backbones and damping curves are used as a traditional instrument in nonlinear vibration analysis [36].
For small and slow nonlinear variations, The instantaneous modal frequency of the system will be close to the instantaneous frequency of the dynamic response, and the instantaneous damping coefficient equals the ratio between the envelope and its derivative.

Numerical Study
A synthetic signal consisting of three frequency components of 1, 3 and 6 Hz is used in this section to investigate the advantages of EWT. White Gaussian noise included into the signal to study the noise effect. The exponential function is embedded to simulate the signal attenuation with time. This simulated signal is expressed as where n(t) is the white Gaussian noise. High-level noises with three different signal-to-noise ratios (SNRs), i.e., −2, −6 and −10 dB, are used to test the efficacy of the scale-space EWT method. The 20-s simulated signal with an SNR of −2 dB is shown in Figure 2a. The sampling frequency is 50 Hz. The boundaries for the spectrum segmentation are detected using the scale-space method stated in Section 3.1.1, and the result is shown in Figure 2b. The threshold for the boundary detection is calculated to be 14 by the Otsu's method. Thus, minima scale-space curves with length longer than 14 are the boundaries. The empirical wavelets are defined according to Equations (2) and (3) based on these boundaries. Using these basis functions, WT is applied to decompose the signal. The extracted signal components, xrec1(t) to xrec3(t) are displayed in Figure 3. All the three components are extracted without redundant modes.  Performing HT on each extracted component, their instantaneous frequencies are obtained. The TF plane is shown in Figure 4a, where the brightness of the instantaneous-frequency lines represents the amplitude of the corresponding components. The time varying of frequencies and amplitudes for each component can be noted. The added noise makes the frequencies fluctuate continuously. The brightness of these lines decays with time, meaning that the magnitudes of the components decrease with time. This phenomenon coincides with the fact that signal components attenuate with time due to the exponential function in Equation (23).
As comparison, SWT and EMD are also employed. In SWT, the analytical Morlet wavelet is used as the prescribed basis function. The EMD self-adaptively decomposes the signal into N+1 IMFs f k (t), which are AM-FM components under the assumption that S j (t) and ϕ (t) vary much slower than ϕ(t). TF planes obtained by these two methods are shown in Figure 4b,c. The three signal components are separated by SWT with satisfactory frequency resolution. However, in the EMD results the instantaneous frequencies of the second and third components are not so legible. Redundant modes below 1 Hz are produced. Applying the FREEVIB method to extracted signal components, the instantaneous frequencies and damping ratios for each mode are obtained. To compare the three methods of EWT, SWT, and EMD, the most probable values of these two parameters are selected as the indicators. The results of are shown in Table 1. Recognizing that the accuracy of these three methods is different, the coefficients of variation (CVs) are also listed in the table for comparison.  Performing HT on each extracted component, their instantaneous frequencies are obtained. The TF plane is shown in Figure 4a, where the brightness of the instantaneous-frequency lines represents the amplitude of the corresponding components. The time varying of frequencies and amplitudes for each component can be noted. The added noise makes the frequencies fluctuate continuously. The brightness of these lines decays with time, meaning that the magnitudes of the components decrease with time. This phenomenon coincides with the fact that signal components attenuate with time due to the exponential function in Equation (23).
As comparison, SWT and EMD are also employed. In SWT, the analytical Morlet wavelet is used as the prescribed basis function. The EMD self-adaptively decomposes the signal into N+1 IMFs fk(t), which are AM-FM components under the assumption that Sj(t) and φ'(t) vary much slower than φ(t). TF planes obtained by these two methods are shown in Figure 4b,c. The three signal components are separated by SWT with satisfactory frequency resolution. However, in the EMD results the instantaneous frequencies of the second and third components are not so legible. Redundant modes below 1 Hz are produced. It can be observed that by this means all the three methods can identify the modal frequency with a value very close to the theoretical one. Larger discrepancy is found between the analyzed and theoretical damping ratios. Nonetheless, EWT performs better than the other two methods in the value estimation, especially for the third mode. As to the CVs, the EWT is superior in those of the frequency, but not so excellent in the damping ratio. It should be admitted that estimating the damping ratio accurately is difficult for all the three methods. An appropriate processing of the noises may improve the performance of the methods.
To test the immunity of the method to noises, higher-level noises with an SNR of −6 dB and −10 dB are added to the signal, respectively. The thresholds for spectrum detection are 74 and 155 for each. The detected frequency boundaries are shown in Figure 5. As well, the TF planes for the extracted mono-components are shown in Figure 6. The accuracy of the EWT method degrades by the noises. The boundary detection shown in Figure 5 is still reliable. The instantaneous frequencies of the three modes can be well separated even when the SNR is −10 dB. the value estimation, especially for the third mode. As to the CVs, the EWT is superior in those of the frequency, but not so excellent in the damping ratio. It should be admitted that estimating the damping ratio accurately is difficult for all the three methods. An appropriate processing of the noises may improve the performance of the methods.
To test the immunity of the method to noises, higher-level noises with an SNR of −6 dB and -10 dB are added to the signal, respectively. The thresholds for spectrum detection are 74 and 155 for each. The detected frequency boundaries are shown in Figure 5. As well, the TF planes for the extracted mono-components are shown in Figure 6. The accuracy of the EWT method degrades by the noises. The boundary detection shown in Figure 5 is still reliable. The instantaneous frequencies of the three modes can be well separated even when the SNR is −10 dB.    the value estimation, especially for the third mode. As to the CVs, the EWT is superior in those of the frequency, but not so excellent in the damping ratio. It should be admitted that estimating the damping ratio accurately is difficult for all the three methods. An appropriate processing of the noises may improve the performance of the methods.
To test the immunity of the method to noises, higher-level noises with an SNR of -6 dB and -10 dB are added to the signal, respectively. The thresholds for spectrum detection are 74 and 155 for each. The detected frequency boundaries are shown in Figure 5. As well, the TF planes for the extracted mono-components are shown in Figure 6. The accuracy of the EWT method degrades by the noises. The boundary detection shown in Figure 5 is still reliable. The instantaneous frequencies of the three modes can be well separated even when the SNR is −10 dB.

A High-Rise Building
The Canton Tower, a 610 m high TV tower in Guangzhou, China, is another test bed for the

A High-Rise Building
The Canton Tower, a 610 m high TV tower in Guangzhou, China, is another test bed for the feasibility of EWT. It is composed of a 454 m high main tower and a 156 m high antenna mast, as shown in Figure 7. The main tower has a tube-in-tube geometry consisting of a reinforced concrete inner structure and a steel lattice outer structure. Its construction was completed in May 2009.

A High-Rise Building
The Canton Tower, a 610 m high TV tower in Guangzhou, China, is another test bed for the feasibility of EWT. It is composed of a 454 m high main tower and a 156 m high antenna mast, as shown in Figure 7. The main tower has a tube-in-tube geometry consisting of a reinforced concrete inner structure and a steel lattice outer structure. Its construction was completed in May 2009. A long-term SHM system was deployed on the tower for real-time monitoring of the structure [37]. As part of this system, 20 uniaxial accelerometers (Tokyo Sokushin AS-2000C) are installed on eight different cross-sections of the main tower ( Figure 7) to measure the structural dynamic responses. Accelerometers 01, 03, 05, 07, 08, 11, 13, 15, 17, and 18, are used to collect the response in the short-axis direction, while others measure in the long-axis direction. The sampling frequency for the acceleration data is 50 Hz 24-h (18:00 p.m. 19 January 2010to 18:00 p.m. 20 January 2010) data recorded during one construction stage are provided for a benchmark study [38]. A long-term SHM system was deployed on the tower for real-time monitoring of the structure [37]. As part of this system, 20 uniaxial accelerometers (Tokyo Sokushin AS-2000C) are installed on eight different cross-sections of the main tower ( Figure 7) to measure the structural dynamic responses. Accelerometers 01, 03, 05, 07, 08, 11, 13, 15, 17, and 18, are used to collect the response in the short-axis direction, while others measure in the long-axis direction. The sampling frequency for the acceleration data is 50 Hz 24-h (18:00 p.m. 19 January 2010 to 18:00 p.m. 20 January 2010) data recorded during one construction stage are provided for a benchmark study [38].
This study used the data collected by the accelerometer 11 from 05:00 a.m. to 05:10 a.m. on 20 January 2010. The time history of the data and the corresponding frequency boundaries detected by the scale-space approach are shown in Figure 8. The first five components extracted by EWT are displayed in Figure 9. This study used the data collected by the accelerometer 11 from 05:00 a.m. to 05:10 a.m. on 20 January 2010. The time history of the data and the corresponding frequency boundaries detected by the scale-space approach are shown in Figure 8. The first five components extracted by EWT are displayed in Figure 9.    Applying the FREEVIB method to the extracted mono-components, the instantaneous modal parameters of the tower are derived. The instantaneous frequencies of these modes are shown in Figure 10a. For comparison, SWT and EMD are also used to analyze this signal, the results of which are displayed in Figure 10b,c, respectively. The frequency resolution of EWT is much higher than that of the SWT and EMD. The instantaneous frequencies identified by EWT are obviously more concentrated. The second and the third modes close to each other are clearly discriminated by EWT. In contrast, the instantaneous frequencies obtained by the other two methods are relatively Applying the FREEVIB method to the extracted mono-components, the instantaneous modal parameters of the tower are derived. The instantaneous frequencies of these modes are shown in Figure 10a. For comparison, SWT and EMD are also used to analyze this signal, the results of which are displayed in Figure 10b,c, respectively. The frequency resolution of EWT is much higher than that of the SWT and EMD. The instantaneous frequencies identified by EWT are obviously more concentrated. The second and the third modes close to each other are clearly discriminated by EWT. In contrast, the instantaneous frequencies obtained by the other two methods are relatively scattered, especially for those determined by EMD. The performance of SWT is much better than EMD, but it cannot separate the two closely spaced modes clearly as EWT.
The histograms for the instantaneous frequencies and damping ratios of the extracted modes are shown in Figure 11a,b, respectively. The most probable value for each mode is considered as the modal parameter of the tower. They are listed in Table 1 with the corresponding CVs. In [39], the CVs for the modal frequencies are less than 0.005, and those for the damping ratios are less than 0.90. The CVs in Table 2 are much larger. The main reason is that in this study they are derived directly from a randomly selected signal with a duration of only 10 min, where there may be disturbances from environment, loads and so on. In contrast, the previous study uses data segment length long enough (one-hour) to reduce the noise effects. Moreover, in this study the 24 measurements of one-hour duration were not used directly but first decomposed into 70 overlapping data sets of one-hour duration with a 20-min shift. In Table 1, another observation is that the CVs of the damping ratio are much larger than those of the modal frequency. It is mainly because that the identified damping ratio of a structure is usually not as stable as the natural frequency.
are less than 0.90. The CVs in Table 2 are much larger. The main reason is that in this study they are derived directly from a randomly selected signal with a duration of only 10 min, where there may be disturbances from environment, loads and so on. In contrast, the previous study uses data segment length long enough (one-hour) to reduce the noise effects. Moreover, in this study the 24 measurements of one-hour duration were not used directly but first decomposed into 70 overlapping data sets of one-hour duration with a 20-min shift. In Table 1, another observation is that the CVs of the damping ratio are much larger than those of the modal frequency. It is mainly because that the identified damping ratio of a structure is usually not as stable as the natural frequency.    Figure 11. Histograms for the instantaneous modal parameters of Canton Tower identified by EWT: (a) Instantaneous frequency; (b) damping ratio.
As verification, the identified modal frequencies and damping ratios are compared with those obtained by the vector autoregressive (ARV) technique [40], the data-driven stochastic identification (SSI-DATA) method [40], the enhanced frequency domain decomposition (EFDD) algorithm [41], and an improved automatic modal identification method based on NExT-ERA [41], respectively. This comparison is shown in Figure 12 using bar plots. The modal frequencies identified by EWT agree quite well with those obtained by the other four methods. Though relatively big differences exist in the damping ratios identified by distinct methods, the values obtained from EWT have a satisfactory accordance with those determined from ARV and SSI-DATA. As verification, the identified modal frequencies and damping ratios are compared with those obtained by the vector autoregressive (ARV) technique [40], the data-driven stochastic identification (SSI-DATA) method [40], the enhanced frequency domain decomposition (EFDD) algorithm [41], and an improved automatic modal identification method based on NExT-ERA [41], respectively. This comparison is shown in Figure 12 using bar plots. The modal frequencies identified by EWT agree quite well with those obtained by the other four methods. Though relatively big differences exist in the damping ratios identified by distinct methods, the values obtained from EWT have a satisfactory accordance with those determined from ARV and SSI-DATA. The backbones estimated by the FREEVIB method for each mode and the corresponding damping curves are displayed in Figure 13a and Figure 13b, respectively. The identified instantaneous frequencies almost do not vary with the vibration amplitude. On the other hand, for the damping coefficients no obvious varying trends exist with the amplitude, though they are more scattered than the instantaneous frequencies. This implies that under the analyzed condition the The backbones estimated by the FREEVIB method for each mode and the corresponding damping curves are displayed in Figure 13a,b, respectively. The identified instantaneous frequencies almost do not vary with the vibration amplitude. On the other hand, for the damping coefficients no obvious varying trends exist with the amplitude, though they are more scattered than the instantaneous frequencies. This implies that under the analyzed condition the Canton tower can be considered as a linear system. This is affirmed by the identified elastic-force characteristics illustrated in Figure 13c. The backbones estimated by the FREEVIB method for each mode and the corresponding damping curves are displayed in Figure 13a,b, respectively. The identified instantaneous frequencies almost do not vary with the vibration amplitude. On the other hand, for the damping coefficients no obvious varying trends exist with the amplitude, though they are more scattered than the instantaneous frequencies. This implies that under the analyzed condition the Canton tower can be considered as a linear system. This is affirmed by the identified elastic-force characteristics illustrated in Figure 13c.

A Footbridge
A vibration-based continuous SHM system with 8 accelerometers and 10 thermocouples has been deployed on the Dowling Hall Footbridge (DHF) at Tufts University in Medford. This footbridge is 44 m long and 3.7 m wide. As shown in Figure 14a, it is a two-span continuous steel frame bridge. The eight uniaxial accelerometers were permanently installed on the underside of the bridge [42] to measure the structural vibration under ambient stimulations, as displayed in Figure 14b. More details about this bridge and its monitoring system can be found in [42,43].
Data of seventeen weeks have been released to the public. A set of 300-s data was recorded once an hour with a sampling frequency of 2048 Hz. This study analyzes acceleration data collected by accelerometer number 1 (Figure 11b) Figure 15. Figure 16 displays the first six components extracted by EWT.
A vibration-based continuous SHM system with 8 accelerometers and 10 thermocouples has been deployed on the Dowling Hall Footbridge (DHF) at Tufts University in Medford. This footbridge is 44 m long and 3.7 m wide. As shown in Figure 14a, it is a two-span continuous steel frame bridge. The eight uniaxial accelerometers were permanently installed on the underside of the bridge [42] to measure the structural vibration under ambient stimulations, as displayed in Figure 14b. More details about this bridge and its monitoring system can be found in [42,43].  Figure 15. Figure 16 displays the first six components extracted by EWT.
The instantaneous modal characteristics of the bridge corresponding to the six extracted modes are obtained by the FREEVIB method. The identified instantaneous frequencies are shown in Figure 17, compared with those from SWT and EMD. All the six modes have been clearly separated by EWT, even the two (modes 5 and 6) quite close to each other. However, no clear and reliable modes are extracted by SWT and EMD. That is to say, the frequency resolution of EWT is much higher than that of SWT and EMD, implying that it is better to separate the SHM signals of a civil structure into meaningful mono-components. Figure 18 shows histograms of the instantaneous frequencies and damping ratios for the extracted modes obtained by the FREEVIB method. As in the Canton tower case study, the most probable values corresponding to each mode is adopted as the modal parameter of the structure. The instantaneous modal characteristics of the bridge corresponding to the six extracted modes are obtained by the FREEVIB method. The identified instantaneous frequencies are shown in Figure 17, compared with those from SWT and EMD. All the six modes have been clearly separated by EWT, even the two (modes 5 and 6) quite close to each other. However, no clear and reliable modes are extracted by SWT and EMD. That is to say, the frequency resolution of EWT is much higher than that of SWT and EMD, implying that it is better to separate the SHM signals of a civil structure into meaningful mono-components. The modal parameters derived in this way are listed in Table 3 as well as the corresponding CVs. The CVs for the fifth and sixth modal frequencies are much larger than those for others. The reason is that these two modes are more sensitive to disturbances, and the corresponding instantaneous frequencies fluctuate relatively greatly with time, as shown in Figure 14a. Similar to the CVs of the damping ratio for Canton Tower, those of DHF are also very large because of the inherent instability.      Figure 16. The first six components extracted from the acceleration signal of DHF by EWT (The Acc. represents acceleration). Figure 18 shows histograms of the instantaneous frequencies and damping ratios for the extracted modes obtained by the FREEVIB method. As in the Canton tower case study, the most probable values corresponding to each mode is adopted as the modal parameter of the structure. The modal parameters derived in this way are listed in Table 3 as well as the corresponding CVs. The CVs for the fifth and sixth modal frequencies are much larger than those for others. The reason is that these two modes are more sensitive to disturbances, and the corresponding instantaneous frequencies fluctuate relatively greatly with time, as shown in Figure 14a. Similar to the CVs of the damping ratio for Canton Tower, those of DHF are also very large because of the inherent instability.
To verify the modal frequencies identified by the proposed methodology, they are compared with those from a preliminary dynamic test on 4 April 2009 [44], for which the modal parameters were identified by the SSI-DATA method. The comparison bars are shown in Figure 19. The EWT results are extremely close to those determined from the modal test. The closely spaced modes, i.e., the fifth and the sixth ones, are well identified by EWT. In contrast, the SSI-DATA method requires selecting the order carefully.
Using the extracted mono-components, the backbones, damping curves and elastic forces corresponding to each mode of the bridge are also analyzed by the FREEVIB method ( Figure 20). The backbones can be considered as vertical lines, implying that the identified instantaneous frequencies do not vary with the vibration amplitude. Though the damping coefficients are quite decentralized, they also do not change evidently with the amplitude. It is deduced that the DHF can be considered as a linear system in the analyzed condition, which is further validated by the identified elastic-force characteristics shown in Figure 20c.     Figure 18. Histograms for the instantaneous modal parameters of DHF identified by EWT: (a) Instantaneous frequency; (b) damping ratio.
To verify the modal frequencies identified by the proposed methodology, they are compared with those from a preliminary dynamic test on 4 April 2009 [44], for which the modal parameters were identified by the SSI-DATA method. The comparison bars are shown in Figure 19. The EWT results are extremely close to those determined from the modal test. The closely spaced modes, i.e., the fifth and the sixth ones, are well identified by EWT. In contrast, the SSI-DATA method requires selecting the order carefully.
Using the extracted mono-components, the backbones, damping curves and elastic forces corresponding to each mode of the bridge are also analyzed by the FREEVIB method ( Figure 20). The backbones can be considered as vertical lines, implying that the identified instantaneous frequencies do not vary with the vibration amplitude. Though the damping coefficients are quite decentralized, they also do not change evidently with the amplitude. It is deduced that the DHF can

Discussion
Due to the large-scale and complexity, together with the intricate structure-load interactions, SHM signals acquired from civil structures are complex but contain rich information about the structural condition. For structural evaluation, it is desirable to extract instantaneous structural features from these complex signals. EWT is a promising tool because it is an advanced TF method that combines the merits of both EMD and WT. However, in the processing of real SHM signals, the original EWT method proposed by Gilles [17] is very susceptible to noises, leading to false signal decomposition. Another problem is that after separating the signals, the traditional way is to convert the mono-components into free decaying vibration function subjected to ambient vibrations. Consequently, the time-varying features of the structure are ignored. This paper proposes a systematical procedure to extract the instantaneous structural features based on the improved EWT. The accuracy and boundary detection capability of the scale-space method using the Otsu's algorithm are demonstrated by both the numerical study and experimental study on real civil structures (Figures 2, 4-6, 8, 10, 15 and 17). Its immunity to noises is also tested by the synthetic signals with different levels of noises. Compared with EMD and SWT, the proposed scale-space EWT can extract more accurate modes. All the concerned modes were separated from the studied signals, without extra ones. However, from Figures 4, 10 and 17, some redundant modes that are difficult to be interpreted are produced by EMD. The reason is that EMD forces the extraction of IMFs through an ad-hoc process even if the initial components are not. The frequency resolution of EWT is much higher than that of traditional WT methods including the SWT, a smart utilization of the output of the classical WT (Figures 4, 10 and 17). Even the closely spaced modes can still be well discriminated by the scale-space EWT.
By applying FREEVIB to the mono-components extracted from the vibration signals of civil structures, the structural features, including the instantaneous modal parameters and the linearity characteristics, are derived. In the case study, the obtained modal parameters are proved to be reasonable by comparing with those identified by the traditional methods such as SSI-DATA and EFDD. In contrast with the usual way that considers the extracted modes as free decaying vibrations, the FREEVIB retains the instantaneous features of these parameters. It implies that the time-varying performance of structure can be tracked, which provides a new perspective for SHM-based structural condition assessment. For example, if there is any anomaly in the structure, these characteristics or their derivations such as the parameters for their statistical distributions may changes correspondingly. In addition, the linearity of the structural system can also be judged by the FREEVIB results.
Spectra instead of the Fourier one, such as the standardized autoregression power spectrum [29] and the pseudo-spectrum obtained by the multiple signal classification approach [30,31], have been used in some studies to reduce the disturbance of noises on the boundary detection in EWT. Nevertheless, it may not be so convinced that no useful structural information is overlooked. This study still uses the Fourier spectrum to build the wavelet filter bank, though it is susceptible to noises. For this study, the improvement of EWT lies in the boundary detection method, or the definition of boundaries. The Fourier spectrum is transformed to the scale-space, and the boundaries are those corresponding to scale-space curves above a certain length.
In this study, some noises embedded in the signal were removed at the beginning of the EWT procedure, and the results are as expected. However, the denoising method is trail-based. A thorough investigation on how to remove noises effectively is necessary in the future. The best spectrum segmentation method to extract different modes, including but not limited to spectra to be used, and algorithms for boundary definition, is still an open question to be addressed.
Though the instantaneous structural characteristics of the civil structures are obtained by the proposed methodology, how to take advantages of these results in the structural condition assessment is also another direction to explore.

Conclusions
This paper proposes a systematic methodology to extract instantaneous features about the structural condition from SHM signals of civil structures. The signal is decomposed into individual components using a scale-space EWT method first. Subsequently, the FREEVIB method is applied to the extracted mono-components to obtain the time-varying structural characteristics.
The scale-space EWT means an EWT method detecting the frequency boundaries for the wavelet filter bank in the scale-space representation of the traditional Fourier transform. The boundaries are defined as those with scale-space curves above a certain threshold. To find the threshold, the Otsu's method is adopted in this study. The scale-space EWT aims to improve the original EWT method proposed by Gilles [17] that is sensitive to noises and requires a predefined boundary number.
For modal identification of civil structures, the traditional way to utilize the decomposed signal components is to regard them as free decaying vibrations. However, the time-varying effect of the modal parameters is ignored consequently. This paper employs the FREEVIB method to process the EWT-extracted mono-components. By this means, the instantaneous modal parameters are obtained. Moreover, this method can analyze the linear characteristics of the structure based on backbones, damping coefficients and elastic forces.
Both numerical and experimental studies are conducted to validate the proposed method. Different levels of noises are added to the simulated signals to test its immunity. The performance of the scale-space EWT in boundary detection, and the accuracy of the extracted instantaneous frequencies are verified by comparing with the results of EMD and SWT. Real SHM signals from a high-rise building and a footbridge are analyzed. The instantaneous modal parameters are reasonable after being compared with results in the previous studies.
The proposed signal processing procedure is effective to identify the instantaneous modal parameters and obtain the linearity characteristic of civil structures. Moreover, this method can deal with the high-level noisy signals. Studies on the optimal spectrum segmentation, and the utilization of the obtained instantaneous features in structural condition assessment, will be carried out in the future.