Wavelet-Based Signal Processing of Large SHM Data · Thus, the signal processing should also include the source separation. In particular, for long-term data, which implies an extremely

7th Asia-Pacific Workshop on Structural Health Monitoring November 12-15, 2018 Hong Kong SAR, P.R. China

Wavelet-Based Signal Processing of Large SHM Data

Y. X. Xia 1*, Y. Q. Ni 2

1 School of Civil Engineering, Qingdao University of Technology, Qingdao, China

Email: [email protected]

2 Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

Email: [email protected] KEYWORDS: Signal Processing; Large Data; Structural Health Monitoring; Wavelet Transform ABSTRACT A wealth of structural health monitoring (SHM) data has been collected from the grand civil structures instrumented with long-term SHM systems. Nevertheless, the mining of information associated with the structural condition from the big data is still a great challenge. The signal processing is the first and essential step to exploit potentialities of the large SHM data. Collected by sophisticated systems from large-scale civil structures, which operate interacting with intricate loadings and environment, long-term SHM data are usually non-stationary and contaminated by noises, spikes and trends. Moreover, different signal sources are desired to be separated in some specific researches. Consequently, there is an immediate necessity to process the signals, in terms of de-noising, de-spiking and decomposing. Especially, for the large data that has an extraordinarily large volume automation and efficiency are particularly important. With the merit of time-frequency analysis and multi-resolution, the wavelet transform is a promising tool to process the long-term SHM data. A fast and unsupervised signal processing scheme is proposed in this paper to remove the noises, spikes, and trends embedded in the signals, and to separate different signal sources as well. To improve the computational speed, the algorithm is designed to be as simple as possible, so the procedures like constructing templates and searching anomalies from the beginning to end of a long signal have been avoided. The proposed methodology is demonstrated by applying to the long-term strain and displacement data measured from a long-span suspension bridge. 1. Introduction The structural health monitoring (SHM) technology is expected to lead to the next significant evolution of design, assessment and management of civil infrastructures, just like what computers and structural analysis software did. A rapid development of SHM technology in civil engineering has been witnessed in the recent two decades. Permanent SHM systems have been installed on some major civil structures such as bridges and high-rise buildings[1,2]. These SHM systems continuously collect data associated with the operational performances of the bridges, such as loads, environment, and structural responses. In consequence, massive data has been collected, which are expected to be converted to information about the structural condition. However, limited by the developing level of the SHM technology it is still a great challenge at present.

* Corresponding author.

Creative Commons CC-BY-NC licence https://creativecommons.org/licenses/by/4.0/

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The physical sizes of civil structures are grand, and the loadings and environmental factors are various and complicate. Besides, there are intricate interactions between the structures and operational loadings. In addition, there are occasional errors in the sensors, signal conditioning unit, analogue-to-digital converter (ADC), or digital communication network. As a result, the SHM signals are non-stationary and usually contaminated by noises, spikes, and trends. Signal processing is the first and fundamental step to mine information of structural condition from the long-term SHM data. In some specific studies, different signal sources are required to be separated. Thus, the signal processing should also include the source separation. In particular, for long-term data, which implies an extremely huge volume, the automation and efficiency of the signal processing technique deserve special concern. The wavelet transform (WT) is a promising tool to process non-stationary and complex signals. Except keeping information in both domains of time and frequency, the WT also has the merit of multi-resolution. It implies that this transform can characterize concerned event coefficients in multiple frequency bands. Thus, it has exhibited its notable capabilities in processing non-stationary signals of many fields, such as mechanical engineering [3–5], biomedical engineering[6–8], chemistry[9,10], and astronomy[11,12]. In the civil engineering field, the application of WT has mainly focused on damage detection by checking anomalies based on the wavelet coefficients [13–18]. However, to the best the authors’ knowledge there has been little literature about the application of WT to systematically processing the large SHM data with special attention to the speed and automation of the algorithm. Our interest in the signal processing method for the large SHM data arose from the current efforts to evaluate the condition of the Tsing Ma Bridge (TMB), a long-span suspension bridge in Hong Kong. This bridge was instrumented with a permanent SHM system after the completion of its construction in 1997. Therefore, a wealth of filed data is available for analysis and mining at present. An automatic and fast wavelet-based methodology is proposed in this paper to process the long-term SHM data, in terms of de-noising, de-spiking and decomposing. The noise throughout this paper means the Gaussian white noise. The data-adaptive wavelet thresholding method[19–23] is employed to eliminate the noises with specific threshold estimation and thresholding policies selected according to the unique characteristics of the signal. To improve the computational efficiency, in de-spiking the spikes are first identified fast and automatically using a spike detection method, and then removed robustly by a wavelet de-spiking algorithm focusing on the signal segment close to the spike. Finally, the signal sources are separated based on their characteristics of frequency and magnitude. The strain and displacement data measured by the long-term SHM system of the TMB are used to demonstrate the proposed signal processing scheme. The remainder of the paper is organized as follows. Section 2 presents the wavelet-based signal processing methodology for large SHM data. In section 3, the proposed method is demonstrated using the long-term strain and displacement data measured from the TMB, comparing with other approaches. Finally, conclusions are given in section 4. 2. Wavelet-Based Methodology to Process Large SHM Data 2.1 De-Noising Signal de-noising can be described as a process of extracting useful information from the raw data. In a broad sense, the noise is any signal of no interest. Thus, spikes, trends and any other signal sources irrelevant to the required signal all can be considered as noises. Nevertheless, each of them may embody unique characteristics; so different processing techniques are required. The noise throughout this paper is defined as the Gaussian white noise. The noises in the signal are modelled as stationary independent Gaussian variables [24,25]. Therefore, the noisy signal f(t) is represented by

( ) ( ) ( )f t s t t (1)

where s(t) is the real signal, ε(t) is standard white noise N(0,1), and δ is the intensity of the noise.

In this study, the wavelet thresholding method proposed by Donoho and his colleagues[19–23] is employed to remove the noises from the large SHM data. This approach is based on the discrete wavelet transform (DWT). The detail coefficients with absolute values below a certain threshold level are set to zero, and then the de-noised data is obtained by an inverse DWT (IDWT) on the modified coefficients. This de-noising approach is easy understanding and efficient. Because of the orthogonality of the set of wavelet basis in the DWT, wavelet coefficients of the white noise are also independent N(0,1) random variables. Therefore, the empirical approximation coefficients

0 ,ˆj ka and detail coefficients ,

ˆj kd of the noisy signal f(t) can be written as

0

0 0 0, , ,

, , , 0

ˆ 0,1,...,2 1ˆ ,..., 1 0,1,...,2 1

j

j k j k j k

j

j k j k j k

a a k

d d j j J k

(2)

where εj,k are independent N(0,1) variables; j0 is the primary resolution level, e.g. j0=log2(ln(n))+1, where n is the signal length[26]; and J=log2(n). The approximation coefficients

0j ,ka represent the low-frequency terms, which usually contain important information about the real signal s(t). Therefore, they are advised to be kept intact in the de-noising process. On the other hand, mostly only a few relatively large detail coefficients dj,k contain information of s(t), while the small coefficients dj,k are attributed to noise[27]. To remove the noises, thresholds are needed to define the “large” and “small” detail coefficients. The detail coefficients to be depressed for signal de-nosing are determined by the threshold estimation. Several threshold estimations, such as minimax[23], universal[23], translation invariant (TI)[28], and Stein’s unbiased risk estimate (SURE)[29], have been proposed. Many other scholars also proposed some different threshold estimations[30,31]. In practice, the threshold estimations of minimax, universal, TI and SURE are usually used. The minimax criterion estimates the threshold based on the minimum or maximum risk bound involved in the estimate function. The universal threshold is defined by

ˆ 2 ln( )U n (3)

where n is the signal length, and ̂ is the robust estimation of the noise level δ based on the median absolute deviation of the empirical detail coefficients ,

ˆj kd at the finest resolution level[23]

1, 1ˆmedian

ˆ , 0,1,...,2 10.6745

J k Jd

k (4)

The universal threshold is substantially larger than the minimax threshold. Therefore, a smoother de-noised signal would be obtained because fewer detail coefficients are included in the reconstruction. It has been noted that de-noising by either the minimax threshold or the universal threshold suffers from pseudo-Gibbs phenomena in the vicinity of discontinuities[27]. The TI thresholding scheme was proposed to suppress these phenomena. When there are several discontinuities in the function of interest, this approach will apply a range of shifts in the function and average the obtained results. The SURE is to obtain an unbiased estimate of variance between the filtered and unfiltered data[29]. When a threshold is available, what should be considered subsequently is how to apply the threshold to the detail coefficients, i.e. the thresholding policy. There are several policies, such as hard thresholding, soft thresholding, non-negative garrote thresholding, and firm thresholding[32]. The hard thresholding removes all detail coefficients with magnitude lower than the threshold. The soft thresholding removes all detail coefficients below the threshold as well as shrinks all those above it[19]. The hard thresholding may introduce discontinuities into the de-noised signal. Nonetheless, it has smaller mean square error (MSE) than the soft thresholding. The soft thresholding usually gives a smoother de-noised signal, but tends to over-smooth abrupt changes and broaden sharp peaks. Both the non-negative garrote and the firm thresholding policies attempt to moderate the limitations of hard thresholding and soft thresholding.

The non-negative garrote thresholding removes small detail coefficients and shrinks large coefficients by a nonlinear continuous function[33,34]. The firm thresholding has two thresholds, λ1 and λ2. The detail coefficients are partitioned into three treatments: (1) to retain those with magnitude larger than λ2, (2) to remove the small ones with magnitude lower than λ1, and (3) to linearly shrink the middle ones with magnitude between λ1 and λ2

[35]. However, the hard and the soft thresholding are most commonly applied because of their efficiency. The wavelet thresholding method can be summarized into three steps: (1) to forwardly transform the signal to the wavelet domain using the DWT; (2) to eliminate or shrink detail coefficients below a threshold; and (3) to reconstruct the signal using the approximation coefficients and the remaining detail coefficients. Throughout this procedure, four factors will have influence on the de-noising results: wavelet selection and decomposition level in step 1; and threshold estimation and thresholding policy in step 2. 2.2 De-Spiking The traditional wavelet de-spiking method removes the spikes by reversing the normal de-noising procedure as stated above, which means cancelling all wavelet coefficients that exceed the threshold. However, it has been proved that this approach significantly underestimates the initial amplitude in the case of spike series[36]. Incorporating algorithms of hypothesis testing and search of wavelet modulus maxima chain, a wavelet-based de-spiking method is designed in this paper to eliminate the diverse and complex spikes embedded in the large SHM data with a high computational efficiency. This method is implemented by two steps: (1) spike detection; and (2) spike removal. 2.2.1 Spike Detection The procedure is generally as follows. Firstly, the DWT with uniform scales is performed on the signal. Subsequently, the Bayesian hypothesis testing is conducted at each scale to detect the presence of spikes. Finally, the arrival time of individual spikes is estimated by combining the decisions of different scales. In the traditional DWT, dyadic scaling which means the scales of wavelet decomposition range from very fine to very coarse based on the “power of 2” restrictions is adopted. The coefficients at very fine scales mostly contain high frequency noise, which are irrelevant to the spike detection. On the other hand, at coarser scales the coefficients corresponding to relatively short spikes may be contaminated. To improve the efficiency, uniform scales are used in the DWT. These scales {s0, s1,…, sJ-1} are uniformly sampled between s0 and sJ with an arbitrary step, where s0 and sJ depend on the sampling rate of the signal, and the minimum and maximum durations of the spikes. For unsupervised spike detection, the wavelet coefficients should be separated by a threshold. In this paper, the threshold is estimated by hypothesis testing on the wavelet coefficients. The hypothesis testing rule for each wavelet coefficient W(j, k) is[37]

0

1

2ˆ ˆ( , ) log

ˆ2

H

j j

e j j

jH

W j k k B

(5)

In this equation, ˆj is determined by[23]

ˆ ( ,0) ,..., ( , 1) / 0.6745j j jM W j w W j N w (6)

where jw is the sample mean of wj, and M{ꞏ} means the sample median. ˆj in Eq. (5) is the sample

mean of the absolute value of the wavelet coefficients at scale sj under hypothesis H1; and γj is a parameter determined by the acceptable costs of false alarms and omissions, denoted by λFA and λOM and

the prior probabilities of the two hypotheses. More detail about the rules of this hypothesis testing can be found in Appendix II of Nenadic and Burdick’s paper[37]. Recognizing that a spike’s wavelet coefficients are neighbors in both time and scale, the continuities across different scales are combined by

11

H

j

H

js S

B B

(7)

where 1HB is the sets of acceptance of H1,

1H

jB is a subset of the translation set B corresponding to the acceptance of H1 at scale sj. And then, the set of H1 is organized into its contiguous constituents[37]

11

1

c HN

H

ii

B C

(8)

where 1H

iC are the contiguous regions of 1HB , and Nc is the number of contiguous regions.

The estimated location of the ith spike at scale sj is

1

arg max ( , ) : ( , ) 1,2,...,H ji

j

i j i c sk C

T W j k W j k N S

(9)

The arrival time candidate of the ith spike is estimated by averaging j

iT over the scales which accept H1 on the ith contiguous region

1

1

1 1,2,...,H

j i

j

i i i cHs Si

T T NS

(10)

where

1 1: ( , ) ,H H

i j j iS s S W j k k C (11)

and is the set size. 2.2.2 Spike Removal After the spikes are identified by the fast detection method stated above, the spikes are eliminated focusing on the signal segments close to them. In the spike removal process, the spikes are defined as chains of scale-invariant maximal or minimal wavelet coefficients. This process includes the following five key steps: (1) Signal decomposition; (2) Definition of maximal and minimal wavelet coefficients; (3) Maxima and minima chain search; (4) Maxima and minima chain removal; and (5) Signal reconstruction. The maximum overlap DWT(MODWT)[38] is used to transform the signal to the wavelet domain owing to the following advantages. In contrast with the traditional DWT, the length of signal to be analyzed has no restrictions of ‘power of two’, so it is applicable for all signal sizes. The wavelet coefficients at each scale are temporarily aligned based on the phase delay properties of the wavelet basis. Therefore, all wavelet coefficients Ws,t (s represents the scale, and t represents time) are redefined as

, , ss t s t TW W (12)

After the temporal alignment, the 2×2 neighborhood of each coefficient is searched for maximal or minimal wavelet coefficients[39]. The maximal or minimal coefficients are defined as those with values at least half the size of the local maximum or minimum. That is, for s={1,…,J} and t={2,…,N-3}[39]

max , , 2 , 2

min , , 2 , 2

0.5 max ,...,

0.5 min ,...,

s t s t s t

s t s t s t

W W W W

W W W W

(13)

For t={0,1,N-2,N-1} and s={1,…,J}, the coefficient Ws,N-1 is considered as maximal if

, 1 , 3 , 1 ,1 ,00.5 max ,..., , ,s N s N s N s sW W W W W (14)

and minimal if

, 1 , 3 , 1 ,1 ,00.5 min ,..., , ,s N s N s N s sW W W W W (15)

Among the maximal or minimal coefficients, a lenient threshold is set to identify those originated from spikes. The survivals are denoted as W'max and W'min, respectively. A search algorithm is employed to identify the spikes in the wavelet domain. It looks for the position and directionality of the maximal or minimal coefficients in the same or adjacent scales. The spikes are represented as chains of maximal and minimal wavelet coefficients at the same time point in multiple scales. The maxima or minima chains representing the spikes are set to zero. Subsequently, all wavelet coefficients are re-shifted back out of temporal alignment as follows

, , ss t s t TW W (1)

Finally, the signal piece with spikes can be reconstructed by the inverse MODWT (IMODWT). 2.3 Decomposing Based on the frequency and magnitude of the signal sources, the long-term SHM data collected from civil structures can generally be divided into the following two cases: (i) the signal sources can be discriminated by frequencies; and (ii) the signal sources overlap in frequency but can be discriminated by magnitudes. For the first case, if there is a prior knowledge about the frequency information, different signal sources can be reconstructed from the corresponding scales in the wavelet domain. Otherwise, the joint WT and Fourier transform approach[40], or the empirical WT (EWT)[41] can be employed. For the second case, the signal sources with obviously larger magnitudes can be considered as spikes. Consequently, the wavelet de-spiking algorithm can be used to achieve the source separation. 2.4 Wavelet-Based Signal Processing Scheme for Large SHM Data The signal processing scheme for the large SHM data collected from civil structures is briefly illustrated in Fig. 1. To avoid the influence of noises on de-spiking and decomposing, de-noising is set as the first step. And then, spikes embedded in the signals are removed using the algorithm sated in Section 2.2. Afterwards, the clean signal is separated into different components by the methods presented in Section 2.3. Finally, refined de-noising can be carried out for those signal sources still contaminated with noises. 3 Application to Real Data 3.1 Tsing Ma Bridge and Its SHM System The TMB which has a total length of 2,132 m and a main span of 1,377 m, is currently the world’s longest suspension bridge carrying both highway and railway traffic. The TMB was instrumented with a long-term SHM system called wind and structural health monitoring system (WASHMS) from 1997 to 1999, after its construction was completed. Thus, this system has operated continuously for more than 20 years up to now. The WASHMS is composed of 283 sensors in eight types[42], namely, 6 anemometers, 19 servo-type accelerometers, 115 temperature sensors, 110 welded foil-type strain gauges (or dynamic

strain gauges), 14 global positioning system (GPS) receivers, two displacement transducers, 10 level sensing stations, and seven dynamic weigh-in-motion (WIM) stations.

Figure 1. Signal processing scheme for large SHM data

The data measured by the strain gauges and the level sensing stations are selected to illustrate the proposed signal processing methodology. When there is no strong typhoon, the measured strain and displacement are mainly due to the highway traffic, railway traffic, and temperature. An obvious trend exists in the time history of the daily strain and displacement, as shown in Fig. 2. This trend is induced by the daily cycle of temperature. As shown in the close view, the salient peaks are due to railway traffic, whereas the oscillations with much smaller magnitudes are induced by the highway traffic. The signals are contaminated by noises due to the monitoring system itself and the operational environment (Fig. 3). Furthermore, it is found that spikes are embedded in the signals occasionally (Fig. 4).

(a) Strain

(b) Displacement

Figure 2. Time history of daily strain and displacement with close view of 2-hour data[43]

08:00 08:30 09:00 09:30 10:00-50

0

50

100

Time

Stra

in (μ

ε)

00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00-100

0

100

Time

Stra

in (μ

ε)

strain due to highway trafficstrain mainly due to railway traffic

00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00-200

0200400600

Time

Dis

plac

emen

t (m

m)

08:00 08:30 09:00 09:30 10:00-200

0

200

400

600

Time

Dis

plac

emen

t (m

m)

mainly due to railway traffic due to highway traffic

Figure 3. Noisy signal

Figure 4. Time history of daily strain with a spike embedded

3.2 De-Noising The wavelet basis selected for de-nosing is Symmlet 8. The decomposition level is set as seven. The four threshold estimations introduced in Section 2.1: minimax, universal, TI, and SURE, are used. To obtain smoother de-noised signals, which is required by the future research, soft thresholding was adopted. The approaches corresponding to the four threshold estimations are denoted as MINIMAX, UNI, TI and SURE, respectively. The performances of the wavelet de-noising methods are evaluated by the graphic outputs and computational time. There are two criteria to define a good graphic output: (1) the magnitudes of signals are attenuated by less than 1% after de-noising; and (2) the graphical output is as smooth as possible. The de-noised results of the signal segment in Fig. 3 are shown in Fig. 5. It can be observed that the UNI outperforms other approaches in smoothness. The computer employed in this study has a CPU of Intel Xeon E5-1620 with a basic frequency of 3.50 GHz, and a memory of 16 GB. The computation time it took to remove the noises in the strain data of one day using the approaches: MINIMAX, UNI, TI, and SURE is 0.89s, 0.54 s, 89.76 s, and 1.02 s respectively. Hence, in computational efficiency the UNI is also superior to others. The UNI approach is adopted as the de-noising method for the long-term strain data measured from the TMB. 3.3 De-Spiking The spike embedded in the signal in Fig. 4 is automatically detected first (Fig.6). The wavelet used is Haar, which is spike-like. The occurrence time of the spike is detected at 17:23:37:41, which is the same to that identified manually. And then, this spike is removed by the wavelet de-spiking algorithm stated in Section 2.2 focusing on the time domain close to the spike. The primary de-spiked result is not satisfactory, because there is still a valley trend (Fig. 7(b)) near the instant of the spike. To eliminate this trend, a DWT is further performed for on this de-spiked signal. Finally, the spike is completely removed, with other data points nearly unaffected (Fig. 7(c)).

Figure 5. Results of wavelet de-noising

Stra

in (

)St

rain

()

Stra

in (

)St

rain

()

Figure 6. Automatically detected spike

(a)

(b)

(c)

Figure 7. Wavelet de-spiking results: (a) raw data, (b) primary de-spiked result, and (c) de-spiked result after de-trending by the DWT

The proposed de-spiking algorithm can greatly improve the computational efficiency. It took the computer more than 400 s to remove the spike in Fig. 4 using the traditional wavelet de-spiking method which searches spikes from the beginning to the end of the signal; while the time consumed by the method with the fast spike-detection algorithm is only 3 s. 3.4 Decomposing

3.4.1 Extraction of Thermal Strain A 16-level decomposition based on the DWT was implemented for strain data measured in a day, which had been both de-noised and de-spiked. The approximation at level 16, A16, is shown in Fig. 8. In physical explanation, A16, which is the trend of the original signal, is the strain induced by temperature variation.

Figure 8. Approximation at level 16, A16

3.4.2 Separation of strain due to different traffic loads

With thermal strain extracted, the remaining signal is mainly induced by the highway traffic and railway traffic. However, it is difficult to discriminate them directly in the wavelet domain because of their

Stra

in (

)St

rain

()

Stra

in (

)

frequency overlapping. Fortunately, there is apparent difference between their amplitudes. The strain induced by the railway traffic can be regarded as spikes. Therefore, the wavelet de-spiking algorithm proposed in Section 2.2 is used to separate these two signal sources. An example of the separation results is shown in Fig. 9. Fig. 9 (a) is the strain due to traffic during 5:00-6:00 in the morning. Fig.9 (b)-(c) show the effectiveness of the signal decomposing method. In actual research, refined de-noising may be required for the strain due to highway traffic (Fig. 9(c)).

(a)

(b)

(c)

Figure 9. Wavelet decomposing results: (a) original strain, (b) strain induce by the railway traffic, and (c) strain induced by the highway traffic

4. Conclusions Motivated by the necessity to develop a fast and automatic algorithm to remove the noises and spikes embedded in the large SHM data, a wavelet-based signal processing scheme is proposed in this paper. The wavelet thresholding method is adopted to remove the spikes by selecting a specific threshold estimation method and thresholding policy based on the uniqueness of the signals. In the de-spiking process, a fast spike-detection algorithm is incorporated to improve the computational speed; and an MODWT-based minima and maxima chain search procedure is employed to eliminate the spikes robustly. The DWT is a filter bank, and different signal sources could be separated straightforwardly using DWT if there is an apparent gap between their frequencies. For those signal sources with overlapped frequencies but distinct magnitudes, they can be separated by the wavelet de-spiking algorithm. The proposed signal processing methodology is demonstrated using the SHM data collected from the TMB. The noises and spikes in the signals can be eliminated efficiently, and different signal sources can be separated as expected as well. Acknowledgement This work was supported by the grant from the National Natural Science Foundation of China (Grant No. 51708315), and the Provincial Key Research and Development Program of Shandong (2018GSF120017). References [1] JM Ko, and YQ Ni, “Technology developments in structural health monitoring of large-scale

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