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Rayleigh wave velocity computation using principal wavelet-component analysis
NDT&E International 44 (2011) 47–56
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Rayleigh wave velocity computation using principal wavelet-component analysis Jae Hong Kim 1, Hyo-Gyoung Kwak n Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea
a r t i c l e in fo
abstract
Article history: Received 20 September 2009 Received in revised form 23 July 2010 Accepted 14 September 2010 Available online 21 September 2010
Many researchers utilize Rayleigh wave velocity for nondestructive evaluation of concrete structures. Nondestructive testing using Rayleigh wave requires only one side of a member, offering the advantage in applicability. After measuring surface waveforms on one side, a signal processing technique is required to compute the time-of-flight of the propagating Rayleigh wave. This paper proposes a technique to this end using a principal wavelet-component analysis. Through testing with numerical and experimental samples, it is found that the new technique is accurate like other conventional methods and has better reliability less obstructed by various noise and reflection waves. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Concrete Rayleigh wave Wavelet transform Principal component analysis
1. Introduction In order to assess the structural health of existing infrastructures, it is necessary to investigate the current status of the structure and constitutive materials. There are various methods to evaluate the stiffness of concrete materials in a nondestructive manner [1]. Among them, stress-wave methods based on the theory of wave propagation are favorable because the stress-wave velocity has a direct relation with the elastic modulus of the materials. For instance, a well-known nondestructive testing (NDT) method for concrete is the ultrasonic pulse velocity method, also known as UPV. UPV’s procedure is standardized by ASTM C597 [2]. The UPV method determines the longitudinal wave (P-wave) velocity by using two transducers placed on the opposite sides of a concrete specimen. ASTM C1383 [3] also describes a procedure to determine the propagation velocity of P-wave that was generated by an impact on a concrete surface. Both standard methods calculate P-wave arrival time with the signal-rise time above the background noise level. The concept of the rise time is based on the fact that the P-wave is the fastest among all types of stress waves. The P-wave velocity can be finally evaluated with the arrival time and the known distance between two sensors. Because of the advantages of one-sided approach and subsurface investigation, Rayleigh wave (R-wave) has also been adopted for NDT of concrete structures [1,4]. The rise time concept may apply even to the R-wave, generally slower than P-wave n
Corresponding author. Tel.: + 82 42 350 3621; fax: + 82 42 350 3610. E-mail address: [email protected] (H.-G. Kwak). 1 Present address: Center for Advanced Cement-Based Materials, Northwestern University, 2145 Sheridan Road, Suite A130, Evanston, IL 60208, USA. 0963-8695/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2010.09.005
propagation, because the body wave attenuates relatively fast on concrete surface. However, the wave attenuation could distort the rise time of the R-wave. The R-wave velocity is determined in another way. The R-wave velocity is calculated by using the arrival time of the maximum energy density, because guided waves such as R-wave occupy the most energy in a propagating surface waveform. In the case of concrete structures, peak-to-peak and crosscorrelation methods are useful in computing the R-wave velocity with the maximum energy density. The peak-to-peak method is a heuristic method in the time domain, and it determines the timeof-flight (TOF) with the arrival time of the peak amplitude in the measured waveforms [5]. The cross-correlation technique determines the TOF by calculating the time when the maximum correlation between the two waveforms appears [6]. However, the peak amplitude and the cross-correlation in the time domain are easily interfered by scattered and reflected body waves. Moreover, the deviation of the determined R-wave velocity increases when the surface wave includes many body wave components in near field of a wave source [7]. On the other hand, in order to compute the velocity of dispersive waves, short-time Fourier transform, continuous wavelet transform, Wigner–Ville distribution, and harmonic wavelet transform can be applied [8,11,12]. Multi-layered pavements, poor curing on the outer surface of concrete and deterioration can stratify the stiffness of a concrete member and thereby cause surface waves dispersive. Spectral analysis of surface waves is the most widely used method to determine the phase delay of dispersive R-wave on concrete. This analysis adopts fast Fourier transform (FFT) to calculate the phase of the R-wave [13]. Even though spectral analysis is efficient and simple
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for computation, it is susceptible to errors on the step of time windowing (tapering in the time domain), which precedes the FFT in order to extract the R-wave component having the maximum energy density. The case described above reports the difficulty on extracting the main wave component (R-wave) in a measured surface waveform. In addition, reflection waves generated at the boundary of concrete members should be excluded from the computation of the R-wave velocity. Concrete members have finite dimensions of a few hundred millimeters, and then the reflection waves are always included at the rear of the measured surface waveform. This paper proposes a new technique to compute the R-wave velocity using principal wavelet-component analysis. The proposed method robustly extracts the front part of a transient R-wave in order to exclude the reflection waves accompanying at the rear. Its accuracy was verified with numerical examples, and experimental application with concrete samples was also performed. Finally, the reliability of the proposed technique is demonstrated through a comparison of the results with other conventional techniques.
Eq. (2) develops a scalogram Wy(u,s), where it has the abscissa of time-shift (u) and the ordinate of scale (s). The scalogram means the distribution of the wavelet components corresponding to the measured signal y(t). An extracted wave component is considered as the dominant carrier having the maximum energy density only if the reconstruction error between the extracted component and the original signal is minimized. The reconstruction error Jy can be defined as the time integration of the squared error between the extracted ˆ(t) and the original signal y(t): component y Z ð3Þ Jy ¼ ðyðtÞy^ ðtÞÞ2 dt
In the above equation, the extracted component can be composed of the product of amplitude, a and unit basis function, ˆ(t) ¼a b(t). Given that the original transient signal has the b(t): y average amplitude of 0, which is true in general except the case of displacement measurement, the reconstruction error is rearranged with respect to the coefficient a: Z Z Jy ¼ a2 2a bðtÞyðtÞdt þ y2 ðtÞdt ð4Þ
2. Principal wavelet-component analysis Principal wavelet-component analysis (PWCA) is a signal processing technique to extract a pure R-wave component in a transient waveform. The previous study used premature PWCA for a feature extraction in order to develop an artificial intelligence algorithm for NDT [8]. It was verified that the maximum energy density in the measured waveform could be found via selecting the peak on the wavelet transformed signal. In this paper, the derivation for computing the R-wave TOF with the maximum energy density is described. The derivation begins with continuous wavelet transform for time–frequency domain analysis. Continuous wavelet transform uses a real mother wavelet, also known as real wavelet transform, which is good for detecting sharp signal transition [9]. This peculiar character is the most helpful in extracting the R-wave component by excluding other unwanted waves. A single-cycle mother wavelet is thought to be the best because it is sensitive to a wave fragment having short duration. Gaus1 wavelet, the first-order derivative of Gaussian distribution [10], is a reasonable applicant, as shown in Fig. 1, expressed as follows: 1=4 32 cðtÞ ¼ t expðt 2 Þ ð1Þ
Fig. 2. Measurement of an impact-generated surface wave.
p
Real wavelet transform for a signal y(t) is accomplished by using u-shifting and s-scaling wavelet c(t): Z Z 1 tu dt ð2Þ Wyðu,sÞ ¼ yðtÞcðt; u,sÞdt ¼ yðtÞ pffiffi c s s
Fig. 3. Finite element model for ideal surface waveforms.
Table 1 Material properties of numerical examples. Layer
Propertiesa
NES (sound)
NED (degraded)
MAT1
Elastic modulus (GPa) Rayleigh wave velocity (m/s) Elastic modulus (GPa) Rayleigh wave velocity (m/s)
30 2081 30 2081
27 1974 30 2081
MAT2
Fig. 1. Gaus1 wavelet function.
a
The density and Poisson’s ratio are 2400 kg/m3 and 0.2, respectively.
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By setting partial derivatives to zero (qJy/qa ¼0), the error is minimized. Accordingly, the coefficient and the error criterion become a¼
Z
bðtÞyðtÞdt
ð5:1Þ
Z 2 Z Juy ¼ bðtÞyðtÞdt þ y2 ðtÞdt
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ð5:2Þ
As observed in Eq. (5.2), the basis function that maximizes R ( bydt)2 minimizes the reconstruction error. On the other hand,
Fig. 4. Surface waveforms of numerical examples: (a) NES and (b) NED.
Fig. 5. Time-of-flight of NES case: determined by (a) peak-to-peak, (b) cross-correlation, and (c) PWCA. (d) The extracted components by PWCA.
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when Eq. (2) is compared with Eq. (5.2), the basis function becomes one of the shifted-and-scaled mother wavelets: b(t)’c(t;u,s). The arguments (u and s) maximizing the squared R scalogram Wy2 ¼( bydt)2 decide the basis function that minimizes the reconstruction error. Therefore, arg max Wy2(u,s) corresponds to the location of the maximum energy density (R-wave). Selecting the basis function as arg max Wy2(u,s) is the key issue for PWCA reconstruction. Given that the medium is homogeneous, the R-wave is non-dispersive and its velocity could be expressed as a single value. Hence, a single basis function is chosen for the representative R-wave velocity. The single basis function is determined as the extracted component (umax, smax) ¼arg max Wy2(u,s), which arrives at umax. Therefore, the TOF between two locations of X and Y can be computed based on the arrival time of the extracted component: TOF ¼umax,Y umax,X. ˆ(t)¼ab(t) could be composed with The extracted component y a ¼ Wyðarg maxWy2 ðu,sÞÞ
ð6:1Þ
bðtÞ ¼ cðt; arg max Wy2 ðu,sÞÞ
ð6:2Þ
In the case of a stratified system, it is necessary to compute the R-wave velocity according to frequency because the depthvarying stiffness brings on dispersion. Thus, the extracted components are separately composed at each scale, where the scale is an independent variable describing the frequency of R-wave contents: umax(s)¼arg max Wy2(u,s). In the same manner, except for the argument of scale (s), the extracted components ˆ(t;s)¼a(s) b(t;s) with become y aðsÞ ¼ Wyðarg max Wy2 ðu,sÞ,sÞ
ð7:1Þ
bðt; sÞ ¼ cðt; arg max Wy2 ðu,sÞ,sÞ
ð7:2Þ
Investigating wave dispersion requires a spectral analysis of the extracted components. The apparent frequency for each scale Table 2 Rayleigh wave velocity of numerical examples. Technique
Peak-to-peak Crosscorrelation PWCA
NES
NED
Velocity (m/s)
Error (%)a
Velocity (m/s)
Error (%)a
2105 2064
+ 1.15 0.82
2035 1965
2.21 5.57
2077
0.19
1986
4.57
a The relative errors are computed with 2081 m/s corresponding to the density of 2400 kg/m3, Poisson’s ratio of 0.2, and Young’s modulus of 30 GPa.
is defined as the center frequency that has the maximum amplitude of the Fourier transform of the s-scaled mother wavelet: f ðsÞ ¼ arg max FTfcðt; 0,sÞg
ð8Þ
The phase of the extracted components is simply computed as Eq. (9) using its time-shift umax(s) ¼arg max Wy2(u,s), where each extracted component has no phase difference. Furthermore, their spectrum is formulated as follows:
yy ðsÞ ¼ 2pf ðsÞarg max Wy2 ðu,sÞ YðsÞ ¼ aðsÞexpðiyy ðsÞÞ
ð9Þ ð10Þ
Even though the above equations are expressed as the functions of scale (s), they could be easily converted to frequency in Eq. (8). By using the principle of a rotating vector, the TOF for each frequency can be calculated from the phase delay between two sensor locations: TOF(f) ¼(yY(f) yX(f))/2pf. Therefore, the phase velocity of the extracted component can be computed with the phase delay.
3. Verification 3.1. Accuracy with numerical examples 3.1.1. Sample preparation Ideal surface waveforms were prepared via finite element analysis in order to confirm the accuracy of the proposed technique. The example signals did not have any reflection waves and noises. Therefore, the proposed technique had expected to give the exact R-wave velocity. Considering the general experimental setup for measurement [4–8], as shown in Fig. 2, the axissymmetric model was designed as shown in Fig. 3. The model was composed with 250,000 2 mm 2 mm rectangular elements. It was assumed that the measurements for surface waves were performed with two accelerometers located 100 and 200 mm apart from an impact source. The time–force function of the impact source was a fourth-order half-cycle sine function, and the contact time for the impact was set to 40 ms. All degrees of freedom at the bottom of the model were fixed for analysis. This boundary condition does not affect the resulting surface waveform because the model is large enough to exclude the reflection wave (1 m 1 m). Details for finite element simulation can be found in the previous study [8,15]. The simulation included two cases: the first case (NES) implied sound condition of a concrete member, but the second case (NED) simulated stiffness degradation due to damage or deterioration. NES assumes the constant material properties for the entire
Fig. 6. Spectrum by SASW: (a) amplitude and (b) complex phase of NES signals.
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Fig. 7. Spectrum by CWRA: (a) normalized scalogram of channel-X, (b) that of channel-Y, (c) amplitude, and (d) complex phase of the ridges of NES signals.
Fig. 8. Spectrum by PWCA: (a) the scalogram of channel-X, (b) that of channel-Y, (c) amplitude, and (f) complex phase of NES signals.
Fig. 9. Dispersion curves of numerical examples: (a) NES and (b) NED.
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model, but 10% degraded modulus was assigned for the top layer of NED case. Table 1 shows the assumed material properties of the model. The simulation results are shown in Fig. 4, where it was hard to figure out the difference between the two surface waveforms. 3.1.2. Verification results The proposed technique in Eq. (6) was applied in order to evaluate a single representative R-wave velocity. The proposed PWCA found the peak on the squared scalogram, which resulted in determining the TOF. Fig. 5 schematically shows the TOF determination using two conventional methods (peak-to-peak [5] and cross-correlation [6]) in addition to the proposed technique. In special, Fig. 5(d) shows the extracted component that was used for computing the R-wave velocity. As reported in Table 2, all three methods, including the proposed technique, show an acceptable performance with the error bound of about 1% in the case of NES example. The R-wave velocity for NED example decreases, where the results indicate the R-wave velocity of the upper damaged layer (MAT1). If relatively low-frequency R-wave is investigated, the estimated velocity would be closed to the bottom layer (MAT2). The frequency dependency of the NED degradation was deeply analyzed by evaluating the dispersion curve of the surface waveforms. Two other conventional methods were adopted for comparison: spectral analysis of surface waves (SASW) and continuous wavelet ridge analysis (CWRA). SASW converts the signals from the time domain to the frequency domain via Fourier transform [13], as shown in Fig. 6, where the signals were zero padded in order to refine the frequency resolution up to 0.1 kHz. CWRA is one of the time–frequency domain techniques [14] that give the amplitude and phase of the ridges, as indicated in the white points in Fig. 7(a and b). In this study, CWRA was performed using the Gabor wavelet that had the shaping factor of 3.5 and a time increment of 0.1 ms. The proposed PWCA also determined the Table 3 Dynamic properties of concrete specimens. Specimen
B-specimen
D-specimen
Mix-proportion (W:C:S:G)a Age (months) Density r [kg/m3] Young’s modulus E [GPa] Shear modulus G [GPa] Calculated Poisson’s ratiob Calculated Rayleigh wave velocity (m/s)b
0.54:1:2.13:2.96 18 2330 31.8 12.8 0.246 2152
0.38:1:1.55:2.11 32 2310 32.9 13.4 0.232 2205
a Mix proportion is expressed as the weight ratio of water, cement, sand, and gravel. b Poisson’s ratio is calculated as n ¼ E/2G 1 and the Rayleigh wave velocity is calculated with Viktorov’s approximate solution: VR ¼(0.8+ 1.12n)/(1+ n) O(G/r).
amplitude and phase of the extracted wave component indicated as the white lines in Fig. 8(a and b), where the time increment for the real wavelet transform was 0.1 ms. Finally, Fig. 9 shows the computed phase velocity of the R-wave. As shown in Fig. 9, the three techniques yield the same results for R-wave velocities within the frequency band of interest (30–50 kHz that correspond to 3 dB). Large error on the SASW result is found at frequencies of around 80 kHz, and it is thought to be due to spectral errors, such as spectral leakage by truncating the transient wave. This corresponds to the observation in Fig. 6(a), where a side lobe appears at around 80 kHz. Spectral leakage by the rectangular window (without time windowing) diverts some energy of the main lobe into the side. In addition, SASW seems to evaluate Lamb mode at frequencies lower than 20 kHz. The lowfrequency contents of the surface waveform have too small energy to be analyzed by neither CWRA nor PWCA. On the other hand, as shown in Fig. 9(b), the dispersive wave velocity declines when the frequency increases. This means that the R-wave components with shorter wavelengths transfer more slowly than those with longer wavelengths because of surface degradation. Therefore, via the test with the numerical examples, it is concluded that the proposed PWCA can compute the accurate R-wave velocity.
3.2. Reliability with experimental examples 3.2.1. Experimental sample preparation Two parallelepiped specimens that have a dimension of 400 mm 400 mm 150 mm were prepared for experimental validation. Table 3 shows the mix-proportion and age of the concrete specimens. The material properties of the used concrete were experimentally measured by the impact-resonant frequency method (ASTM C215) [16], where a cylindrical specimen of 150 mm diameter and 300 mm height was prepared. The determined dynamic moduli of elasticity and rigidity, as well as the density of the concrete, are reported in Table 3. The concrete samples including the parallelepiped specimens were cured carefully in order for the surface or interior of the specimen not to be damaged. Thus, it was thought that each specimen had a uniform stiffness and did not contain any internal damage. Their dispersion curves can be determined using the Rayleigh–Lamb frequency relation [17]. Fig. 10 shows the calculation results with the measured material properties in Table 3. As shown in Fig. 10, the wave velocities of fundamental Lamb modes converge to the R-wave velocity when the product of frequency and thickness is higher than about 3000 kHz mm. Considering that the concrete members are generally thicker than 150 mm, a frequency band higher than 20 kHz should be used to generate and measure R-wave. In this paper, a 6 mm-diameter plastic projectile fired by an air-shot gun was utilized as an impact source, and it generated transient wave of which
Fig. 10. Calculated dispersion curves: (a) B-specimen and (b) D-specimen.
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ques. Table 4 shows the computation results and their statistical analysis. The cross-correlation technique gave the largest confidence interval, indicating a wide distribution of the R-wave velocity results. This is because the R-waves were contaminated with the noise that followed. The peak-to-peak method and the proposed technique analyzed the front part of the waveforms having little noise, which led to more reliable results. However, the confidence interval by the heuristic peak-to-peak method is wider because the first positive peak in the time domain depends not only on the R-wave velocity, but also on attenuation. The proposed technique using the energy peak automatically reduces the influence of noise and attenuation, which results in giving better reliability. The three techniques for dispersion curve were also applied to the experimental examples. Fig. 13 shows several interim-stage graphs of a typical waveform. As previously described in the introduction, prior to using SASW, a time windowing shown in Fig. 13(a and b) was required to exclude the effects of unwanted reflection waves in the surface waveforms. The time windowing is a cumbersome and tricky part in the Fourier transform of a transient wave because the window center and length need to be manually assigned to respond to the wave’s characteristics. In this study, Hann window was used for centering of the second positive peak in the time domain, and its length was determined case-bycase by looking at the shape of the resultant windowed signals; this was the thumb rule based on various trials. On the other hand, the time–frequency domain analysis, such as CWRA and PWCA, robustly detected a group that has major wave energy. The scalograms shown in Fig. 13(c–f) are the wave-energy distribution based on the used mother wavelet: Gabor wavelet for CWRA and Gaus1 wavelet for PWCA. Fig. 14 shows the computed R-wave phase velocities and the coherence functions. The overall pattern for the dispersion curves looks similar, regardless of the used computing technique, but there is meaningful difference shown in Fig. 14(b). The SASW and CWRA results show that the R-wave velocity for 20–60 kHz decreases over frequency, while the proposed PWCA gave non-dispersive output. Which is correct? As previously described, the concrete specimens for the experiments were carefully cured with no damage.
acceleration had a frequency band of 20–40 kHz with high repeatability. Two miniature accelerometers (PCB 352A60) that has the resonant frequency higher than 95 kHz and the frequency range of 5 Hz to 60 kHz for73 dB were adopted for measuring surface waves on concrete. The analog signal was recorded with a signal conditioner (PCB 480B12) and an analog-to-digital converter (NI USB5133) that had 100 MS/s sampling rates and 8 bit resolution, as shown in Fig. 11. The impact was applied at the middle point of an edge. The accelerometers mounted with petro-wax were located 150 and 250 mm apart from the impact, on the center of the specimens. A total of 10 surface waves per specimen were measured to investigate reliability. Fig. 12 shows the measured surface waveforms. Contrary to the numerical examples in Fig. 4, it was observed that the measured waves have noise and scattered/ reflected waves that follow the main R-wave component.
3.2.2. Experimental validation In the same manner with the numerical examples, the representative R-wave velocity was computed using the three techni-
Fig. 11. Apparatus for measuring surface waveforms.
Table 4 Rayleigh wave velocity of the measured surface waves. Technique
Peak-to-peak Cross-correlation PWCA
11 waveforms on B-specimen
11 waveforms on D-specimen
Mean
S.D.
C.I.
Mean
S.D.
C.I.
2265 2310 2242
13.82 21.08 5.039
7 13.21 7 20.14 7 4.815
2315 2335 2307
17.14 17.12 5.759
7 16.38 7 16.36 7 5.503
The values of the sample mean, the standard deviation (S.D.), and the 99% confidence interval (C.I.) computed based on Student’s t-distribution are in the unit of (m/s).
Fig. 12. Measured surface waveforms: (a) B-specimen and (b) D-specimen.
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Fig. 13. Typical waveforms and their scalograms: (a) intact and windowed signals for channel-X, (b) those for channel-Y, (c) CWRA scalogram having the ridges for channel-X, (d) that for channel-Y, (e) PWCA scalogram having the principal components for channel-X, and (f) that for channel-Y.
Fig. 14. Evaluation results of the experimental samples: (a) coherence function of B-specimen, (b) its dispersion curve, (c) coherence function of D-specimen, and (b) its dispersion curve.
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Therefore, the surface wave having frequency higher than 20 kHz (R-wave) must be non-dispersive. In order to confirm the nondispersion explicitly, an impact-logging method was applied to measure sample’s modulus over thickness. The detailed procedure and results are reported in the next section, but the answer is reported in advance: non-dispersion is correct. Only the proposed PWCA could compute the precise R-wave velocity. The reason for this is described in the discussion section.
3.2.3. Impact-logging test result The interest frequency band of 20–40 kHz is high enough to get a stable R-wave velocity, excluding the fundamental Lamb mode, as previously stated. Hence, the only proof to show nondispersion is by examining whether the elastic property along the thickness of the B-specimen is consistent or not. The impactlogging method resembles the cross-hole test for deep foundations [4], but an impact hammer and a miniature accelerometer was used to determine the P-wave TOF. Fig. 15 shows the
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apparatus for the method. The procedure is as follows: (1) an impact hammer generates an impulse obtaining a force signal and (2) a sensor then measures the vibration on the opposite side of the specimen. In this study, the used impact hammer (B&K8204) had the sensitivity of 24.55 mV/N and a contact diameter of about 1 mm. The miniature accelerometer (PCB353 B15) that had the contact diameter of 10 mm was used as a sensor. Fig. 16(a) shows one of the measured signals at both sides, and 50 measurements were stacked at each depth from 10 to 140 mm. The TOF was computed with the rise time at both channels and the apparent P-wave velocity was calculated with a known specimen width of 401 mm. Fig. 16(b) shows the apparent velocity profile, where the B-specimen has an almost consistent property within73 percentage of the mean. Therefore, via the impact-logging test, it is confirmed that the concrete sample has a constant modulus over thickness. The non-dispersive R-wave should be obtained for the frequency of interest.
4. Discussion
Fig. 15. Impact-logging method for measuring thickness-varying properties.
In the last experimental example, the proposed PWCA computed the correct R-wave velocity. The reason why it gave better performance was due to the robust time windowing on the fore part of the surface waveforms. As shown in Fig. 13(e and f), the extracted components, displayed as white lines, are located on the fore part. Even though CWRA executed a robust time windowing, the determined ridges were based on a relatively wide district and centered in the middle part of the surface waveforms, as shown in Fig. 13(c and d). Moreover, in the case of the first channel-X in Fig. 13(c), two peaks (around 25 kHz) on the scalogram approach each other and then the arrival time of the ridges is delayed. Finally, the delay on the channel-X increased the R-wave velocity of around 25 kHz-frequency contents. This interference also happened on the SASW results. SASW used tapering in the time domain, and then it could not take out the low-frequency contents from the windowed surface waveforms, as shown in Fig. 13(a and b). Therefore, CWRA and PWCA gave rapid R-wave velocity for around 25 kHz-frequency contents showing the fake dispersion. Robust discovery of the fore part of a transient wave, by PWCA, also improves the reliability because the fore part of the surface waveform is less prone to having noise and reflection waves. The white line composed of the PWCA components in Fig. 13(e and f) is smoother than that composed of the CWRA ridges in Fig. 13(c and d). Consequently, the coherence function calculated with the PWCA components is also superior, as it always has the value over 0.97, as shown in Fig. 14. In addition, as previously described, the short confidence intervals (Table 4) are another evidence for the high-quality reliability of the proposed PWCA.
Fig. 16. Results of the impact-logging method: (a) measured force signal at channel-X and the measured acceleration at channel-Y and (b) apparent velocities along thickness.
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5. Conclusion R-wave is very efficient for NDT application to concrete structures, because it requires only one side of the concrete. However, a measured surface waveform on concrete can be easily contaminated by various noise and reflection waves. A technique named PWCA was proposed to compute R-wave velocity. First, the accuracy of the proposed technique was verified with the numerical examples. In addition, with the measured surface waves on concrete, the proposed PWCA’s reliability was tested via statistical investigation for multiple measurements. Better performance of the proposed technique was found due to its robust time windowing of the fore part of the surface waveforms, where relatively less noise and reflections potentially appear. Therefore, the proposed technique is recommended for computing the R-wave velocity in situations where a surface waveform has potential noise, for example, propagation in concrete. References [1] Malhotra VM, Carino NJ. Handbook on nondestructive testing of concrete. 2nd ed.. CRC Press; 2004. [2] ASTM International. Standard test method for pulse velocity through concrete. American Society for Testing and Materials. ASTM C597-02, 2002. [3] ASTM International. Standard test method for measuring the P-wave speed and the thickness of concrete plates using the impact-echo method. American Society for Testing and Materials. ASTM C1383-04, 2004.
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Rayleigh wave velocity computation using principal wavelet-component analysis Jae Hong Kim 1, Hyo-Gyoung Kwak n Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea
a r t i c l e in fo
abstract
Article history: Received 20 September 2009 Received in revised form 23 July 2010 Accepted 14 September 2010 Available online 21 September 2010
Many researchers utilize Rayleigh wave velocity for nondestructive evaluation of concrete structures. Nondestructive testing using Rayleigh wave requires only one side of a member, offering the advantage in applicability. After measuring surface waveforms on one side, a signal processing technique is required to compute the time-of-flight of the propagating Rayleigh wave. This paper proposes a technique to this end using a principal wavelet-component analysis. Through testing with numerical and experimental samples, it is found that the new technique is accurate like other conventional methods and has better reliability less obstructed by various noise and reflection waves. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Concrete Rayleigh wave Wavelet transform Principal component analysis
1. Introduction In order to assess the structural health of existing infrastructures, it is necessary to investigate the current status of the structure and constitutive materials. There are various methods to evaluate the stiffness of concrete materials in a nondestructive manner [1]. Among them, stress-wave methods based on the theory of wave propagation are favorable because the stress-wave velocity has a direct relation with the elastic modulus of the materials. For instance, a well-known nondestructive testing (NDT) method for concrete is the ultrasonic pulse velocity method, also known as UPV. UPV’s procedure is standardized by ASTM C597 [2]. The UPV method determines the longitudinal wave (P-wave) velocity by using two transducers placed on the opposite sides of a concrete specimen. ASTM C1383 [3] also describes a procedure to determine the propagation velocity of P-wave that was generated by an impact on a concrete surface. Both standard methods calculate P-wave arrival time with the signal-rise time above the background noise level. The concept of the rise time is based on the fact that the P-wave is the fastest among all types of stress waves. The P-wave velocity can be finally evaluated with the arrival time and the known distance between two sensors. Because of the advantages of one-sided approach and subsurface investigation, Rayleigh wave (R-wave) has also been adopted for NDT of concrete structures [1,4]. The rise time concept may apply even to the R-wave, generally slower than P-wave n
Corresponding author. Tel.: + 82 42 350 3621; fax: + 82 42 350 3610. E-mail address: [email protected] (H.-G. Kwak). 1 Present address: Center for Advanced Cement-Based Materials, Northwestern University, 2145 Sheridan Road, Suite A130, Evanston, IL 60208, USA. 0963-8695/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2010.09.005
propagation, because the body wave attenuates relatively fast on concrete surface. However, the wave attenuation could distort the rise time of the R-wave. The R-wave velocity is determined in another way. The R-wave velocity is calculated by using the arrival time of the maximum energy density, because guided waves such as R-wave occupy the most energy in a propagating surface waveform. In the case of concrete structures, peak-to-peak and crosscorrelation methods are useful in computing the R-wave velocity with the maximum energy density. The peak-to-peak method is a heuristic method in the time domain, and it determines the timeof-flight (TOF) with the arrival time of the peak amplitude in the measured waveforms [5]. The cross-correlation technique determines the TOF by calculating the time when the maximum correlation between the two waveforms appears [6]. However, the peak amplitude and the cross-correlation in the time domain are easily interfered by scattered and reflected body waves. Moreover, the deviation of the determined R-wave velocity increases when the surface wave includes many body wave components in near field of a wave source [7]. On the other hand, in order to compute the velocity of dispersive waves, short-time Fourier transform, continuous wavelet transform, Wigner–Ville distribution, and harmonic wavelet transform can be applied [8,11,12]. Multi-layered pavements, poor curing on the outer surface of concrete and deterioration can stratify the stiffness of a concrete member and thereby cause surface waves dispersive. Spectral analysis of surface waves is the most widely used method to determine the phase delay of dispersive R-wave on concrete. This analysis adopts fast Fourier transform (FFT) to calculate the phase of the R-wave [13]. Even though spectral analysis is efficient and simple
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for computation, it is susceptible to errors on the step of time windowing (tapering in the time domain), which precedes the FFT in order to extract the R-wave component having the maximum energy density. The case described above reports the difficulty on extracting the main wave component (R-wave) in a measured surface waveform. In addition, reflection waves generated at the boundary of concrete members should be excluded from the computation of the R-wave velocity. Concrete members have finite dimensions of a few hundred millimeters, and then the reflection waves are always included at the rear of the measured surface waveform. This paper proposes a new technique to compute the R-wave velocity using principal wavelet-component analysis. The proposed method robustly extracts the front part of a transient R-wave in order to exclude the reflection waves accompanying at the rear. Its accuracy was verified with numerical examples, and experimental application with concrete samples was also performed. Finally, the reliability of the proposed technique is demonstrated through a comparison of the results with other conventional techniques.
Eq. (2) develops a scalogram Wy(u,s), where it has the abscissa of time-shift (u) and the ordinate of scale (s). The scalogram means the distribution of the wavelet components corresponding to the measured signal y(t). An extracted wave component is considered as the dominant carrier having the maximum energy density only if the reconstruction error between the extracted component and the original signal is minimized. The reconstruction error Jy can be defined as the time integration of the squared error between the extracted ˆ(t) and the original signal y(t): component y Z ð3Þ Jy ¼ ðyðtÞy^ ðtÞÞ2 dt
In the above equation, the extracted component can be composed of the product of amplitude, a and unit basis function, ˆ(t) ¼a b(t). Given that the original transient signal has the b(t): y average amplitude of 0, which is true in general except the case of displacement measurement, the reconstruction error is rearranged with respect to the coefficient a: Z Z Jy ¼ a2 2a bðtÞyðtÞdt þ y2 ðtÞdt ð4Þ
2. Principal wavelet-component analysis Principal wavelet-component analysis (PWCA) is a signal processing technique to extract a pure R-wave component in a transient waveform. The previous study used premature PWCA for a feature extraction in order to develop an artificial intelligence algorithm for NDT [8]. It was verified that the maximum energy density in the measured waveform could be found via selecting the peak on the wavelet transformed signal. In this paper, the derivation for computing the R-wave TOF with the maximum energy density is described. The derivation begins with continuous wavelet transform for time–frequency domain analysis. Continuous wavelet transform uses a real mother wavelet, also known as real wavelet transform, which is good for detecting sharp signal transition [9]. This peculiar character is the most helpful in extracting the R-wave component by excluding other unwanted waves. A single-cycle mother wavelet is thought to be the best because it is sensitive to a wave fragment having short duration. Gaus1 wavelet, the first-order derivative of Gaussian distribution [10], is a reasonable applicant, as shown in Fig. 1, expressed as follows: 1=4 32 cðtÞ ¼ t expðt 2 Þ ð1Þ
Fig. 2. Measurement of an impact-generated surface wave.
p
Real wavelet transform for a signal y(t) is accomplished by using u-shifting and s-scaling wavelet c(t): Z Z 1 tu dt ð2Þ Wyðu,sÞ ¼ yðtÞcðt; u,sÞdt ¼ yðtÞ pffiffi c s s
Fig. 3. Finite element model for ideal surface waveforms.
Table 1 Material properties of numerical examples. Layer
Propertiesa
NES (sound)
NED (degraded)
MAT1
Elastic modulus (GPa) Rayleigh wave velocity (m/s) Elastic modulus (GPa) Rayleigh wave velocity (m/s)
30 2081 30 2081
27 1974 30 2081
MAT2
Fig. 1. Gaus1 wavelet function.
a
The density and Poisson’s ratio are 2400 kg/m3 and 0.2, respectively.
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By setting partial derivatives to zero (qJy/qa ¼0), the error is minimized. Accordingly, the coefficient and the error criterion become a¼
Z
bðtÞyðtÞdt
ð5:1Þ
Z 2 Z Juy ¼ bðtÞyðtÞdt þ y2 ðtÞdt
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ð5:2Þ
As observed in Eq. (5.2), the basis function that maximizes R ( bydt)2 minimizes the reconstruction error. On the other hand,
Fig. 4. Surface waveforms of numerical examples: (a) NES and (b) NED.
Fig. 5. Time-of-flight of NES case: determined by (a) peak-to-peak, (b) cross-correlation, and (c) PWCA. (d) The extracted components by PWCA.
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when Eq. (2) is compared with Eq. (5.2), the basis function becomes one of the shifted-and-scaled mother wavelets: b(t)’c(t;u,s). The arguments (u and s) maximizing the squared R scalogram Wy2 ¼( bydt)2 decide the basis function that minimizes the reconstruction error. Therefore, arg max Wy2(u,s) corresponds to the location of the maximum energy density (R-wave). Selecting the basis function as arg max Wy2(u,s) is the key issue for PWCA reconstruction. Given that the medium is homogeneous, the R-wave is non-dispersive and its velocity could be expressed as a single value. Hence, a single basis function is chosen for the representative R-wave velocity. The single basis function is determined as the extracted component (umax, smax) ¼arg max Wy2(u,s), which arrives at umax. Therefore, the TOF between two locations of X and Y can be computed based on the arrival time of the extracted component: TOF ¼umax,Y umax,X. ˆ(t)¼ab(t) could be composed with The extracted component y a ¼ Wyðarg maxWy2 ðu,sÞÞ
ð6:1Þ
bðtÞ ¼ cðt; arg max Wy2 ðu,sÞÞ
ð6:2Þ
In the case of a stratified system, it is necessary to compute the R-wave velocity according to frequency because the depthvarying stiffness brings on dispersion. Thus, the extracted components are separately composed at each scale, where the scale is an independent variable describing the frequency of R-wave contents: umax(s)¼arg max Wy2(u,s). In the same manner, except for the argument of scale (s), the extracted components ˆ(t;s)¼a(s) b(t;s) with become y aðsÞ ¼ Wyðarg max Wy2 ðu,sÞ,sÞ
ð7:1Þ
bðt; sÞ ¼ cðt; arg max Wy2 ðu,sÞ,sÞ
ð7:2Þ
Investigating wave dispersion requires a spectral analysis of the extracted components. The apparent frequency for each scale Table 2 Rayleigh wave velocity of numerical examples. Technique
Peak-to-peak Crosscorrelation PWCA
NES
NED
Velocity (m/s)
Error (%)a
Velocity (m/s)
Error (%)a
2105 2064
+ 1.15 0.82
2035 1965
2.21 5.57
2077
0.19
1986
4.57
a The relative errors are computed with 2081 m/s corresponding to the density of 2400 kg/m3, Poisson’s ratio of 0.2, and Young’s modulus of 30 GPa.
is defined as the center frequency that has the maximum amplitude of the Fourier transform of the s-scaled mother wavelet: f ðsÞ ¼ arg max FTfcðt; 0,sÞg
ð8Þ
The phase of the extracted components is simply computed as Eq. (9) using its time-shift umax(s) ¼arg max Wy2(u,s), where each extracted component has no phase difference. Furthermore, their spectrum is formulated as follows:
yy ðsÞ ¼ 2pf ðsÞarg max Wy2 ðu,sÞ YðsÞ ¼ aðsÞexpðiyy ðsÞÞ
ð9Þ ð10Þ
Even though the above equations are expressed as the functions of scale (s), they could be easily converted to frequency in Eq. (8). By using the principle of a rotating vector, the TOF for each frequency can be calculated from the phase delay between two sensor locations: TOF(f) ¼(yY(f) yX(f))/2pf. Therefore, the phase velocity of the extracted component can be computed with the phase delay.
3. Verification 3.1. Accuracy with numerical examples 3.1.1. Sample preparation Ideal surface waveforms were prepared via finite element analysis in order to confirm the accuracy of the proposed technique. The example signals did not have any reflection waves and noises. Therefore, the proposed technique had expected to give the exact R-wave velocity. Considering the general experimental setup for measurement [4–8], as shown in Fig. 2, the axissymmetric model was designed as shown in Fig. 3. The model was composed with 250,000 2 mm 2 mm rectangular elements. It was assumed that the measurements for surface waves were performed with two accelerometers located 100 and 200 mm apart from an impact source. The time–force function of the impact source was a fourth-order half-cycle sine function, and the contact time for the impact was set to 40 ms. All degrees of freedom at the bottom of the model were fixed for analysis. This boundary condition does not affect the resulting surface waveform because the model is large enough to exclude the reflection wave (1 m 1 m). Details for finite element simulation can be found in the previous study [8,15]. The simulation included two cases: the first case (NES) implied sound condition of a concrete member, but the second case (NED) simulated stiffness degradation due to damage or deterioration. NES assumes the constant material properties for the entire
Fig. 6. Spectrum by SASW: (a) amplitude and (b) complex phase of NES signals.
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Fig. 7. Spectrum by CWRA: (a) normalized scalogram of channel-X, (b) that of channel-Y, (c) amplitude, and (d) complex phase of the ridges of NES signals.
Fig. 8. Spectrum by PWCA: (a) the scalogram of channel-X, (b) that of channel-Y, (c) amplitude, and (f) complex phase of NES signals.
Fig. 9. Dispersion curves of numerical examples: (a) NES and (b) NED.
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model, but 10% degraded modulus was assigned for the top layer of NED case. Table 1 shows the assumed material properties of the model. The simulation results are shown in Fig. 4, where it was hard to figure out the difference between the two surface waveforms. 3.1.2. Verification results The proposed technique in Eq. (6) was applied in order to evaluate a single representative R-wave velocity. The proposed PWCA found the peak on the squared scalogram, which resulted in determining the TOF. Fig. 5 schematically shows the TOF determination using two conventional methods (peak-to-peak [5] and cross-correlation [6]) in addition to the proposed technique. In special, Fig. 5(d) shows the extracted component that was used for computing the R-wave velocity. As reported in Table 2, all three methods, including the proposed technique, show an acceptable performance with the error bound of about 1% in the case of NES example. The R-wave velocity for NED example decreases, where the results indicate the R-wave velocity of the upper damaged layer (MAT1). If relatively low-frequency R-wave is investigated, the estimated velocity would be closed to the bottom layer (MAT2). The frequency dependency of the NED degradation was deeply analyzed by evaluating the dispersion curve of the surface waveforms. Two other conventional methods were adopted for comparison: spectral analysis of surface waves (SASW) and continuous wavelet ridge analysis (CWRA). SASW converts the signals from the time domain to the frequency domain via Fourier transform [13], as shown in Fig. 6, where the signals were zero padded in order to refine the frequency resolution up to 0.1 kHz. CWRA is one of the time–frequency domain techniques [14] that give the amplitude and phase of the ridges, as indicated in the white points in Fig. 7(a and b). In this study, CWRA was performed using the Gabor wavelet that had the shaping factor of 3.5 and a time increment of 0.1 ms. The proposed PWCA also determined the Table 3 Dynamic properties of concrete specimens. Specimen
B-specimen
D-specimen
Mix-proportion (W:C:S:G)a Age (months) Density r [kg/m3] Young’s modulus E [GPa] Shear modulus G [GPa] Calculated Poisson’s ratiob Calculated Rayleigh wave velocity (m/s)b
0.54:1:2.13:2.96 18 2330 31.8 12.8 0.246 2152
0.38:1:1.55:2.11 32 2310 32.9 13.4 0.232 2205
a Mix proportion is expressed as the weight ratio of water, cement, sand, and gravel. b Poisson’s ratio is calculated as n ¼ E/2G 1 and the Rayleigh wave velocity is calculated with Viktorov’s approximate solution: VR ¼(0.8+ 1.12n)/(1+ n) O(G/r).
amplitude and phase of the extracted wave component indicated as the white lines in Fig. 8(a and b), where the time increment for the real wavelet transform was 0.1 ms. Finally, Fig. 9 shows the computed phase velocity of the R-wave. As shown in Fig. 9, the three techniques yield the same results for R-wave velocities within the frequency band of interest (30–50 kHz that correspond to 3 dB). Large error on the SASW result is found at frequencies of around 80 kHz, and it is thought to be due to spectral errors, such as spectral leakage by truncating the transient wave. This corresponds to the observation in Fig. 6(a), where a side lobe appears at around 80 kHz. Spectral leakage by the rectangular window (without time windowing) diverts some energy of the main lobe into the side. In addition, SASW seems to evaluate Lamb mode at frequencies lower than 20 kHz. The lowfrequency contents of the surface waveform have too small energy to be analyzed by neither CWRA nor PWCA. On the other hand, as shown in Fig. 9(b), the dispersive wave velocity declines when the frequency increases. This means that the R-wave components with shorter wavelengths transfer more slowly than those with longer wavelengths because of surface degradation. Therefore, via the test with the numerical examples, it is concluded that the proposed PWCA can compute the accurate R-wave velocity.
3.2. Reliability with experimental examples 3.2.1. Experimental sample preparation Two parallelepiped specimens that have a dimension of 400 mm 400 mm 150 mm were prepared for experimental validation. Table 3 shows the mix-proportion and age of the concrete specimens. The material properties of the used concrete were experimentally measured by the impact-resonant frequency method (ASTM C215) [16], where a cylindrical specimen of 150 mm diameter and 300 mm height was prepared. The determined dynamic moduli of elasticity and rigidity, as well as the density of the concrete, are reported in Table 3. The concrete samples including the parallelepiped specimens were cured carefully in order for the surface or interior of the specimen not to be damaged. Thus, it was thought that each specimen had a uniform stiffness and did not contain any internal damage. Their dispersion curves can be determined using the Rayleigh–Lamb frequency relation [17]. Fig. 10 shows the calculation results with the measured material properties in Table 3. As shown in Fig. 10, the wave velocities of fundamental Lamb modes converge to the R-wave velocity when the product of frequency and thickness is higher than about 3000 kHz mm. Considering that the concrete members are generally thicker than 150 mm, a frequency band higher than 20 kHz should be used to generate and measure R-wave. In this paper, a 6 mm-diameter plastic projectile fired by an air-shot gun was utilized as an impact source, and it generated transient wave of which
Fig. 10. Calculated dispersion curves: (a) B-specimen and (b) D-specimen.
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ques. Table 4 shows the computation results and their statistical analysis. The cross-correlation technique gave the largest confidence interval, indicating a wide distribution of the R-wave velocity results. This is because the R-waves were contaminated with the noise that followed. The peak-to-peak method and the proposed technique analyzed the front part of the waveforms having little noise, which led to more reliable results. However, the confidence interval by the heuristic peak-to-peak method is wider because the first positive peak in the time domain depends not only on the R-wave velocity, but also on attenuation. The proposed technique using the energy peak automatically reduces the influence of noise and attenuation, which results in giving better reliability. The three techniques for dispersion curve were also applied to the experimental examples. Fig. 13 shows several interim-stage graphs of a typical waveform. As previously described in the introduction, prior to using SASW, a time windowing shown in Fig. 13(a and b) was required to exclude the effects of unwanted reflection waves in the surface waveforms. The time windowing is a cumbersome and tricky part in the Fourier transform of a transient wave because the window center and length need to be manually assigned to respond to the wave’s characteristics. In this study, Hann window was used for centering of the second positive peak in the time domain, and its length was determined case-bycase by looking at the shape of the resultant windowed signals; this was the thumb rule based on various trials. On the other hand, the time–frequency domain analysis, such as CWRA and PWCA, robustly detected a group that has major wave energy. The scalograms shown in Fig. 13(c–f) are the wave-energy distribution based on the used mother wavelet: Gabor wavelet for CWRA and Gaus1 wavelet for PWCA. Fig. 14 shows the computed R-wave phase velocities and the coherence functions. The overall pattern for the dispersion curves looks similar, regardless of the used computing technique, but there is meaningful difference shown in Fig. 14(b). The SASW and CWRA results show that the R-wave velocity for 20–60 kHz decreases over frequency, while the proposed PWCA gave non-dispersive output. Which is correct? As previously described, the concrete specimens for the experiments were carefully cured with no damage.
acceleration had a frequency band of 20–40 kHz with high repeatability. Two miniature accelerometers (PCB 352A60) that has the resonant frequency higher than 95 kHz and the frequency range of 5 Hz to 60 kHz for73 dB were adopted for measuring surface waves on concrete. The analog signal was recorded with a signal conditioner (PCB 480B12) and an analog-to-digital converter (NI USB5133) that had 100 MS/s sampling rates and 8 bit resolution, as shown in Fig. 11. The impact was applied at the middle point of an edge. The accelerometers mounted with petro-wax were located 150 and 250 mm apart from the impact, on the center of the specimens. A total of 10 surface waves per specimen were measured to investigate reliability. Fig. 12 shows the measured surface waveforms. Contrary to the numerical examples in Fig. 4, it was observed that the measured waves have noise and scattered/ reflected waves that follow the main R-wave component.
3.2.2. Experimental validation In the same manner with the numerical examples, the representative R-wave velocity was computed using the three techni-
Fig. 11. Apparatus for measuring surface waveforms.
Table 4 Rayleigh wave velocity of the measured surface waves. Technique
Peak-to-peak Cross-correlation PWCA
11 waveforms on B-specimen
11 waveforms on D-specimen
Mean
S.D.
C.I.
Mean
S.D.
C.I.
2265 2310 2242
13.82 21.08 5.039
7 13.21 7 20.14 7 4.815
2315 2335 2307
17.14 17.12 5.759
7 16.38 7 16.36 7 5.503
The values of the sample mean, the standard deviation (S.D.), and the 99% confidence interval (C.I.) computed based on Student’s t-distribution are in the unit of (m/s).
Fig. 12. Measured surface waveforms: (a) B-specimen and (b) D-specimen.
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Fig. 13. Typical waveforms and their scalograms: (a) intact and windowed signals for channel-X, (b) those for channel-Y, (c) CWRA scalogram having the ridges for channel-X, (d) that for channel-Y, (e) PWCA scalogram having the principal components for channel-X, and (f) that for channel-Y.
Fig. 14. Evaluation results of the experimental samples: (a) coherence function of B-specimen, (b) its dispersion curve, (c) coherence function of D-specimen, and (b) its dispersion curve.
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Therefore, the surface wave having frequency higher than 20 kHz (R-wave) must be non-dispersive. In order to confirm the nondispersion explicitly, an impact-logging method was applied to measure sample’s modulus over thickness. The detailed procedure and results are reported in the next section, but the answer is reported in advance: non-dispersion is correct. Only the proposed PWCA could compute the precise R-wave velocity. The reason for this is described in the discussion section.
3.2.3. Impact-logging test result The interest frequency band of 20–40 kHz is high enough to get a stable R-wave velocity, excluding the fundamental Lamb mode, as previously stated. Hence, the only proof to show nondispersion is by examining whether the elastic property along the thickness of the B-specimen is consistent or not. The impactlogging method resembles the cross-hole test for deep foundations [4], but an impact hammer and a miniature accelerometer was used to determine the P-wave TOF. Fig. 15 shows the
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apparatus for the method. The procedure is as follows: (1) an impact hammer generates an impulse obtaining a force signal and (2) a sensor then measures the vibration on the opposite side of the specimen. In this study, the used impact hammer (B&K8204) had the sensitivity of 24.55 mV/N and a contact diameter of about 1 mm. The miniature accelerometer (PCB353 B15) that had the contact diameter of 10 mm was used as a sensor. Fig. 16(a) shows one of the measured signals at both sides, and 50 measurements were stacked at each depth from 10 to 140 mm. The TOF was computed with the rise time at both channels and the apparent P-wave velocity was calculated with a known specimen width of 401 mm. Fig. 16(b) shows the apparent velocity profile, where the B-specimen has an almost consistent property within73 percentage of the mean. Therefore, via the impact-logging test, it is confirmed that the concrete sample has a constant modulus over thickness. The non-dispersive R-wave should be obtained for the frequency of interest.
4. Discussion
Fig. 15. Impact-logging method for measuring thickness-varying properties.
In the last experimental example, the proposed PWCA computed the correct R-wave velocity. The reason why it gave better performance was due to the robust time windowing on the fore part of the surface waveforms. As shown in Fig. 13(e and f), the extracted components, displayed as white lines, are located on the fore part. Even though CWRA executed a robust time windowing, the determined ridges were based on a relatively wide district and centered in the middle part of the surface waveforms, as shown in Fig. 13(c and d). Moreover, in the case of the first channel-X in Fig. 13(c), two peaks (around 25 kHz) on the scalogram approach each other and then the arrival time of the ridges is delayed. Finally, the delay on the channel-X increased the R-wave velocity of around 25 kHz-frequency contents. This interference also happened on the SASW results. SASW used tapering in the time domain, and then it could not take out the low-frequency contents from the windowed surface waveforms, as shown in Fig. 13(a and b). Therefore, CWRA and PWCA gave rapid R-wave velocity for around 25 kHz-frequency contents showing the fake dispersion. Robust discovery of the fore part of a transient wave, by PWCA, also improves the reliability because the fore part of the surface waveform is less prone to having noise and reflection waves. The white line composed of the PWCA components in Fig. 13(e and f) is smoother than that composed of the CWRA ridges in Fig. 13(c and d). Consequently, the coherence function calculated with the PWCA components is also superior, as it always has the value over 0.97, as shown in Fig. 14. In addition, as previously described, the short confidence intervals (Table 4) are another evidence for the high-quality reliability of the proposed PWCA.
Fig. 16. Results of the impact-logging method: (a) measured force signal at channel-X and the measured acceleration at channel-Y and (b) apparent velocities along thickness.
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5. Conclusion R-wave is very efficient for NDT application to concrete structures, because it requires only one side of the concrete. However, a measured surface waveform on concrete can be easily contaminated by various noise and reflection waves. A technique named PWCA was proposed to compute R-wave velocity. First, the accuracy of the proposed technique was verified with the numerical examples. In addition, with the measured surface waves on concrete, the proposed PWCA’s reliability was tested via statistical investigation for multiple measurements. Better performance of the proposed technique was found due to its robust time windowing of the fore part of the surface waveforms, where relatively less noise and reflections potentially appear. Therefore, the proposed technique is recommended for computing the R-wave velocity in situations where a surface waveform has potential noise, for example, propagation in concrete. References [1] Malhotra VM, Carino NJ. Handbook on nondestructive testing of concrete. 2nd ed.. CRC Press; 2004. [2] ASTM International. Standard test method for pulse velocity through concrete. American Society for Testing and Materials. ASTM C597-02, 2002. [3] ASTM International. Standard test method for measuring the P-wave speed and the thickness of concrete plates using the impact-echo method. American Society for Testing and Materials. ASTM C1383-04, 2004.
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