Application of Wavelet and EEMD Joint Denoising in Nonlinear Ultrasonic Testing of Concrete

/e health state of concrete is deteriorating during its service. Nonlinear ultrasonic detection based on the amplitude of the fundamental and the second harmonic is considered to be a powerful tool for the discovery of the microcrack in concrete. However, the research on processing the nonlinear ultrasonic signal is still insufficient. In order to highlight the real frequency domain components in the nonlinear ultrasonic signal, wavelet and ensemble empirical mode decomposition (EEMD) were joined to denoise the numerical and measured signal. /e optimal wavelet base and the decomposition level were determined by the signal-to-noise ratios (SNRs). /en, the wavelet threshold denoising signal was decomposed by EEMD, omitting the highfrequency components and ultimately achieving the desired denoising effect. /e denoising result of the test signals demonstrates that this method is effective in denoising the details of the ultrasonic signal and improving the reliability and adaptability of the nonlinear ultrasonic testing. In this experiment, the concrete with the microcrack was tested by linear and nonlinear ultrasonic methods. Based on the variation regularity of the nonlinear ultrasonic coefficient β and velocity v, we can conclude that the nonlinear ultrasonic parameter β is more sensitive to the microcrack in concrete than the traditional wave velocity v. /e nonlinear ultrasonic testing can be an important supplement to the current nondestructive testing technique of the concrete.


Introduction
Nonlinear ultrasonic technique has been considered as a promising method for detecting the microdamage of materials [1].Lissenden's research shows that when singlefrequency ultrasonic waves pass through the damaged area of the medium, the tensile-compression asymmetry, shearnormal coupling, and deformation-induced compression asymmetry are the reasons for the generation of high-order harmonics [2].At present, the research on the nonlinear ultrasonic technique is mainly focused on the following aspects: (1) Experimental study on dislocation slip damage for metal material (microscale) [3,4].(2) Experimental and numerical simulation study on closed crack damage for metal or nonmetallic materials (mesoscale) [5][6][7].(3) Exploration of the generation and propagation mechanism of nonlinear ultrasound [8][9][10].However, the amplitude of higher order harmonic in signals tends to be smaller and differ by one to two orders from the fundamental.erefore, it is difficult to accurately measure the amplitude of the high-order harmonic in the experiment, and an effective signal processing method is needed.
Wavelet-based denoising and compression have been widely used in engineering, biology, and so on.Staszewski compared the wavelet compression effects of the periodic, continuous nonstationary and transient nonstationary vibration signals, providing data compression selecting basis for different types of signals [11].In addition, the wavelet transform also has the ability to recognize the damage of the material or structure.Using orthogonal wavelet transform and setting the appropriate threshold value, the reflected wave features caused by defect can be effectively displayed [12].Another important application of the wavelet-based technique is signal denoising.e combined denoising method based on wavelet entropy was proposed by Hou and Gui, which processes the high-frequency wavelet decomposition coefficients with wavelet entropy thresholds under different scales [13].Cunha et al. determined the wavelet functions based on the SNR computed from the wavelet coefficients, and the wavelet decomposition level was chosen according to the energy spectral density principle [14].e proposed method provides a new thought and approach for the selection of wavelet basis and decomposition layer in wavelet denoising.
Unlike the wavelet transform, the empirical mode decomposition (EMD) can be decomposed adaptively according to the trend of the signals.Wu and Huang added the Gaussian white noise to the original signal to supplement the missing frequency components and obtained desired decomposition results [15].A major application of EEMD is to extract intrinsic mode functions (IMFs) from the vibration signals to identify the mechanical faults.Žvokelj et al. proposed the EEMD-based multiscale ICA (EEMDMSICA) method.e rotating machinery vibration signals are analyzed by this method, and the bearing fault can be detected [16].In addition, Ridder et al. compared the SNR acquired by EEMD, wavelet, and FIR denoising and considered the wavelet denoising method is the best [17].Wei et al. denoised the IMF components using 3σ criteria, singular value decomposition, and SG filtering.e superiority of this method was verified by processing the simulated and experimental data [18].
In order to accurately extract the amplitude of the fundamental and the second harmonic, this paper uses wavelet and EEMD joint method to denoise the numerical signal containing higher order harmonics, which tend to be more effective than a single step denoising.en, the nonlinear ultrasonic experiment was carried out.And the wavelet and EEMD joint denoising method was applied to the experimental signal, gaining the correlation between the cracked angle and nonlinear coefficient β.

Wavelet
reshold Denoising eory.Wavelet theory originated in the late 19th century and has been widely applied in various fields.Wavelet threshold denoising is one of the important applications of the wavelet theory proposed by Donoho and Johnstone.e main idea is comparing each level of wavelet coefficients with the selected threshold.When the wavelet coefficient is greater than the selected threshold, it is completely preserved or shrunk.ese wavelet coefficients are considered as the original components of the signal.Otherwise, the wavelet coefficients are zero setting.en, the detail and approximation coefficients are reconstructed, and the wavelet threshold denoising is accomplished [19].
e traditional hard and soft threshold functions are shown in the following equations: where ω j,k is the detail coefficient, ω j,k ′ is the detail coefficient after denoising, λ is the wavelet threshold, and sgn() is the symbolic function.According to (1) and ( 2), there are smoothness and continuity problems when denoising by the hard threshold function.When using the soft threshold function, it is very likely to lose important information in the original signal.In order to preserve the frequency domain characteristics of the ultrasonic signal as much as possible, this paper uses the hard threshold method.If the wavelet coefficients are greater than the selected threshold, keep them intact, otherwise 0. In addition, the smoothness problem caused by the hard threshold method will be solved in the next step of EEMD.e threshold selection rule is based on Stein's unbiased risk estimation: where σ n is the standard deviation of the noise signal and w b is the risk function.e unbiased risk threshold method needs to find the wavelet coefficient c i corresponding to the minimum risk value, and then the threshold value λ is calculated according to it.Although the thresholds determined by this way are conservative and may result in incomplete noise removal, it is easier to separate useful weak components from noise when the high-frequency bands are confusing with noise.
Nonlinear ultrasonic testing requires using highfrequency information of the signal.However, the highfrequency band of the signal is more susceptible to noise pollution than low-frequency components.In view of the characteristics of the nonlinear ultrasonic experimental signal, it is appropriate to determine the threshold value by Stein's unbiased risk estimation to eliminate the noise in the high-frequency coefficients.

EEMD Denoising Method.
In order to overcome the illusive components and mode mixing problems existing in EMD, additive Gaussian white noise is added to the original signal to supplement the missing frequency scale of the signal.
e specific EEMD decomposition process is as follows [20]: (1) Adding the Gaussian white noise n(t) to the original signal s(t) and obtaining a new signal S(t): (2) e signal S(t) is decomposed by EMD to obtain the intrinsic modal function (IMF) C 1,j (t): (3) Adding the Gaussian white noise to the original signal s(t) again and repeating the above steps: 2 Advances in Materials Science and Engineering (4) Due to the entire frequency spectrum of the Gaussian white noise is zero, the time-frequency e ect after adding the noise signal can be neglected.Average the superimposed intrinsic modal function: e noise signal can be decomposed from high frequency to low frequency adaptively by the EEMD method.By ignoring the high-order IMFs and reconstructing the remaining IMFs, the denoising signals are obtained.

Wavelet and EEMD Joint Denoising Method.
e ultrasonic noise mainly comes from the ultrasonic machine, environment, and crystal scattering of materials, which is usually distributed in the high-frequency eld of the received signal.Figure 1 is a ow chart of the denoising method.e appropriate wavelet base and the decomposition layer are selected to denoise the noise signal.
en, the signal is decomposed by EEMD.Because the noise is mainly distributed in the high-frequency range, the high-frequency components imf1∼im are discarded and the remaining components are reconstructed [21].

Numerical Signal Denoising
3.1.Numerical Signals of Nonlinear Ultrasonic.According to the fundamentals of the nite amplitude method, the nonlinear ultrasonic signal can be simulated by two superimposed signals: e signal contains two parts: the center frequency is 50 kHz and the second harmonic is 100 kHz.Besides, the sampling frequency is 2000 Hz.Using the fast Fourier transform (FFT) to compute the numerical signal, the amplitude of the fundamental and the second harmonic are 9408 mV and 1633 mV, respectively.Calculated by (8), the nonlinear coe cient β 0 is about 1.845 * 10 −5 (Figure 2).Adding the Gaussian white noise to the numerical signal, according to the frequency domain curve of the noise signal, the amplitude of the fundamental and the second harmonic is 9293 mV and 2187 mV, respectively.e calculated nonlinear parameter β is about 2.532 * 10 −5 (Figure 3).
Compared with Figures 2 and 3, it can be found that the smoothness of the time-domain curve is signi cantly decreasing after adding the Gaussian white noise.However, there are obvious features of burrs in the time-domain curve.In addition, the amplitude of the second harmonic is greatly increased, which leads to the increase of the nonlinear ultrasonic parameter β.
Table 1 shows the characteristic indicators of the numerical signal after noise reduction.From the data in Table 1, it can be seen that the characteristic indicators of the frequency domain have changed greatly compared with the numerical signals before adding the noise, and the error rate of the nonlinear coe cient β is even 37.26%.erefore, it is necessary to nd a more e ective method to reduce the noise pollution of the signal for the sake of extracting more accurate nonlinear parameter β.

Wavelet reshold Denoising.
e e ect of the wavelet threshold denoising depends largely on the selection of the wavelet basis and the decomposition level.In this paper, three kinds of wavelet families (symN, dbN, and coifN) are used to denoise the signal with di erent decomposition levels.e corresponding SNR obtained by denoising with di erent wavelet basis functions and decomposition levels are shown in Figure 4. e distribution of the root-meansquare error (RMSE) is exactly the same as that of the SNR and is not described here.
e line chart shown in Figure 4 shows that the denoising e ect is best when the three-level decomposition is performed on the wavelet decomposition.While the e ect of the wavelet threshold denoising by two-level decomposition is not obvious, most of the noise is not ltered.For the wavelet denoising of four-layer decomposition, because the wavelet coe cients of each level are ltered, there occurs a phenomenon of excessive denoising.In addition, the e ect of Coif wavelet family denoising on the ultrasonic signal is unsatisfactory, while the denoising e ect of Db and Sym wavelet family is better.Speci cally, for the Db and the Sym wavelet families, the denoising e ect is superior and their Advances in Materials Science and Engineering di erences between them are small when sequence numbers of the wavelet basis are 4∼7.e SNR achieves best when using Sym6 wavelet base.In summary, we selected the Sym6 wavelet base and three-layer decomposition in wavelet threshold denoising.e detailed denoising indicators are shown in Table 2.
Figure 5 is the time-frequency domain curve of the denoising signal.Compared with the noise signal, the SNR has been greatly improved after wavelet threshold denoising, and the smoothness has also been signi cantly improved.However, there are still a few burrs in the signal, which need to be further ltered out.6).en, the rst two order high-frequency IMF components are discarded and the remaining components are reconstructed.After denoising, the time-domain curve is pretty smooth and no burr can be observed.In this way, the desired denoising e ect is achieved.
Comparing the data in Tables 1-3, it can be seen that the e ect of the wavelet threshold denoising is limited and the calculated SNR is hardly ideal.Actually, a large part of the   Advances in Materials Science and Engineering nonlinear ultrasonic parameter β is caused by noise so that the test accuracy cannot be guaranteed.e proposed method in this paper can greatly improve the accuracy of fundamental and the second harmonic.Furthermore, the error of the nonlinear parameter β is reduced dramatically.It can be seen from Tables 1-3 that the relative error of the nonlinear coe cient β reaches 37.26% after the Gaussian white noise added.e relative error of the nonlinear coe cient β decreases to 13.09% after the wavelet denoising method.After that, applying the EEMD denoising method to the wavelet denoising signal, the relative error of the nonlinear coe cient decreased to 7.32%.
Compared to the single wavelet denoising method, the relative error of the nonlinear coe cient β reduced 5.77% after using the joint denoising method.Furthermore, by observing Figures 5 and 7, we can conclude that the signal after wavelet denoising still has obvious noise.However, after the EEMD denoising, the time-domain curve of the signal is much smoother.Comparing the changes of the SNR, RMSE, and other indicators, it is con rmed that the denoising

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Advances in Materials Science and Engineering method proposed in this paper is more e ective than just using the wavelet denoising method.

Preparation of Concrete Samples and Test System.
In this experiment, microcrack samples and intact samples are prepared for the nonlinear ultrasonic test.e geometric size of the specimen is 20 cm * 20 cm * 20 cm. e introduction of the microcrack in concrete: Firstly, place the concrete mixture on the vibrating table and vibrate evenly.en, insert 2 mm thick steel into the mixture vertically at a certain angle.By clamping the steel sheet out of the samples with a plier after an hour, the low part of the open crack will gradually close due to the geostatic and the uidity of mixture.Eventually, microcracks are formed inside of the concrete (Figure 8).After the concrete completely hardened, the iron wire with a diameter of 0.8 mm is extended into the crack vertically, and the maximum extension length is 4.5 cm.us, we can consider that the formation region of the microcrack in concrete is 10-15 cm high.e angles of preexisting cracks are 30 °, 45 °, and 90 °(Figure 9).e nonlinear ultrasonic testing system constructed in this experiment is shown in Figure 10.In this experiment, the GSC-1 engineering ultrasonic imager was used to excite the high-voltage pulse signal and collect the data.e highvoltage electrical pulse signal of one thousand volts is generated by the ultrasonic machine, and the electric signal was converted to the ultrasonic signal by a 50 kHz center frequency compressional wave contact transducer.e ultrasonic signal passed through the concrete specimen and was received by a 100 kHz center frequency transducer.e height of the measuring point is 12 cm.Nonlinear ultrasonic testing was carried out on the same cracked sample at di erent measuring surfaces, and the nonlinear ultrasonic testing results of ve kinds of cracked angles (0 °, 30 °, 45 °, 60 °, and 90 °) were obtained.In order to minimize the error caused by coupling, Vaseline was evenly coated between the sample surface and transducer and then wound with elastic band tightly.e nonlinear ultrasonic signals were denoised by the wavelet and EEMD joint method.

Ultrasonic Signal Denoising.
Figure 11 shows the timedomain curve of the ultrasonic signal and its details.Comparing with the numerical ultrasonic signal, the test signal is relatively smooth, and the SNR is higher.e reason    Advances in Materials Science and Engineering for these di erences between the two signals is that the test system and environment and the conditions of laboratory ultrasonic testing are more stable.However, a great number of burrs can be observed at a large scale.Because the uctuation of the noise is tiny and evenly distributed, it can be speculated that the noise is caused by the ultrasonic instrument, voltage, and other system errors.e wavelet and EEMD joint denoising method is used to denoise the nonlinear ultrasonic signal, and the result is shown in Figure 12.It can be seen from Figure 12(b) that the improvement of the minutiae of the signal is obvious, and the burrs are completely removed.e smoothness and SNR of the signal are both better than before.

Testing Results of Nonlinear Ultrasonic Testing.
e nonlinear ultrasonic signal was denoised by the EEMD and wavelet joint method, and the variation law of the parameter β and wave velocity v was obtained.e conclusions of the experiment are as follows: (1) From the height of the two curves in Figure 13, we can conclude that the nonlinear parameter β of the cracked samples is larger than that of the intact ones, and the di erence between them is obvious.Moreover, it proves that the nonlinear ultrasonic testing has good e ect in detecting the microcrack in concrete.
(2) With the increase of the crack angle, the nonlinear parameter β we calculated tends to rise as a whole.Notably, the corresponding nonlinear coe cient of the 45 °crack is slightly lower than the previous one and becomes an outlier.e existing linear ultrasonic experiments and numerical simulations show that the wave velocity and the amplitude of the fundamental become smaller due to the strong scattering e ect when the angle between the incident wave and crack extension direction increases [22].Since the nonlinear ultrasonic parameter β increases with the crack angle, it can be inferred that the discontinuous forced vibration between the cracks is stronger when the incident wave approaches the vertical direction into the crack contact interface, causing more intense "clapping" e ect.
(3) By comparing the wave velocity between intact and cracked samples (Figure 14), it is obvious that the Advances in Materials Science and Engineering traditional wave velocity is not sensitive to the microcrack in concrete.e maximum change rate is only 2.02%.

Conclusions
In this paper, the wavelet and EEMD joint denoising method is used to process the nonlinear ultrasonic signal.In this way, the numerical signal is denoised, and the relative error of the nonlinear ultrasonic coe cient β decreased from 37.26% to 7.32%.In addition, the denoising method is carried out on the nonlinear ultrasonic testing data.e main conclusions are as follows: (1) e denoising results of the numerical and testing data show that the method of wavelet and EEMD joint denoising has good e ect on the details of signal.In the actual project, the source of noise (electronic components, environment, etc.) is more extensive, and the SNR will be much lower.erefore, this method has great potential in engineering applications.
(2) e nonlinear parameter β is very sensitive to the microcrack in concrete.e larger the angle between the incident wave and the crack, the larger the nonlinear parameter β can be measured.Comparing with the linear ultrasonic parameters v to the nonlinear parameter β, we can infer that the latter is more sensitive and accurate to characterize the initial damage in concrete.(3) In this experiment, the ultrasonic signals are denoised to eliminate the part of nonlinear interference caused by regular error and accidental error.However, except the microcrack, there are still many other sources of nonlinear e ects, such as coupling agent, ultrasonic instrument, and macrocrack.e experimental study and theoretical analysis on other main nonlinear sources are positive to construct a complete nonlinear ultrasonic evaluation system for concrete materials.Advances in Materials Science and Engineering

Figure 2 :Figure 3 :
Figure 2: Time-frequency curve of the numerical signals.

Figure 4 :
Figure 4: e SNR obtained by denoising with di erent wavelet bases and decomposition levels.Denoising by (a) Coif wavelet family, (b) Db wavelet family, and (c) Sym wavelet family.

Figure 12 :
Figure 12: Time-domain curve (a) and minutiae (b) of nonlinear ultrasonic signals after denoising.

Figure 13 :Figure 14 :
Figure 13: Comparison of the nonlinear parameter β between intact and cracked samples.

Table 1 :
Characteristic indicators of the numerical signals after adding noise.
3.3.Wavelet and EEMD Joint Denoising.By the method of EEMD, IMF components are obtained (Figure

Table 2 :
Characteristic indicators of the signal after wavelet threshold denoising.

Table 3 :
Characteristic indicators of the signal after wavelet and EEMD joint denoising.