KSCE Journal of Civil Engineering (2013) 17(2):357-363 DOI 10.1007/s12205-013-1636-7
Geotechnical Engineering
www.springer.com/12205
Phase Velocity Evaluation of Two-Layered Gypsums by Using Wavelet Transform Jong-Sub Lee*, Hyon-Sohk Ohm**, Sungsoo Yoon***, and In-Mo Lee**** Received July 11, 2011/Revised February 3, 2012/Accepted May 21, 2012
···································································································································································································································
Abstract A non-destructive method to evaluate the phase velocity in two-layered gypsums using the wavelet transform of surface waves is proposed. Model tests were performed using gypsum composed of two layers. A piezoelectric actuator with frequencies ranging from 150 Hz to 5 kHz was selected as a harmonic source. The surface waves were measured with two accelerometers. Wavelet transform analyses were performed to obtain the dispersion curves from the measured responses. The experimental results show that the near-field effects can be neglected if the distance between the two receivers is greater than three times the wavelength. A simple inversion method using a weighted factor based on the normal distribution is proposed. The inversion shows that the predicted phase velocity agrees reasonably well with the measured phase velocity when the wavelength influence factor is 0.2. The propagation depth of surface waves is from 0.41 to 0.67 times the wavelength. The range of wavelength varying with the phase velocity in a dispersion curve matches well with that estimated using the inversion method. This study shows that the simple wavelet transform can effectively produce the phase velocity of the two-layered gypsums. Keywords: phase velocity, wavelet transform, surface waves, non-destructive methods ···································································································································································································································
1. Introduction For the subsurface characterization, many non-destructive methods based on the elastic waves such as ultrasonic waves and surface waves have been suggested. The ultrasonic reflection method was used for the detection of the discontinuity in the laboratory (Lee et al., 2009). The directivity of the ultrasonic transducer and the coupling layer between the transducer and the medium may affect the resolution of the images. Note that the ultrasonic reflection method can be performed in a limited space by using rotational scanner. A surface wave method is a non-destructive method for nearsurface site characterization based on the following properties of Rayleigh waves: (1) Rayleigh waves generated by a point source are attenuated more slowly than body waves, thus the nearsurface becomes dominated by Rayleigh waves. (2) Rayleigh waves in a heterogeneous medium are dispersive: the velocity is dependent on the frequency and the wavelength. For the analysis of the surface wave method, the spectral analysis method (Spectral-Analysis-of-Surface-Waves, SASW) was suggested by Nazarian and Stokoe (1984). Note that the measured surface waves include a significant number of modes if the soil profile is
heterogeneous. The numerical codes were developed for the selection of the fundamental mode in a heterogeneous soil (Gucunski and Woods, 1992; Tokimatsu et al., 1992). Note that the data analyses were based on the phase information (spectral analyses). In addition, multiple receivers (array-based) with advanced signal processing techniques such as frequency-wave number analyses were adopted to effectively receive the meaningful mode (Park et al., 1999; Zywicki, 1999). The distortion of surface waves due to the near field effects was considered using an array-based technique (Yoon, 2005). The material damping ratio was evaluated for the surface waves in an array-based technique (Rix et al., 2000, 2001; Lai et al., 2002). The Harmonic-Wavelet-Analysis-of-Waves (HWAW) method uses the harmonic wavelet transform to evaluate the phase velocity. The HWAW method can minimize the effects of noise because it mainly uses the signal portion of the maximum signal/ noise ratio and this method can determine the dispersion curve of the whole depth from one experimental setup. In addition, the HWAW method applies a single array inversion, which uses the theoretical dispersion curve calculated from one experimental setup to determine the phase velocity profile in a multi-layered site (Park and Kim, 2001).
***Member, Associate Professor, School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 136-701, Korea (E-mail: jongsub @korea.ac.kr) **Graduate Student, School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 136-701, Korea (E-mail: hyonsohk @gmail.com) ***Research Fellow, Samsung Engineering & Construction, Seoul 136-701, Korea (E-mail:
[email protected]) ****Member, Professor, School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 136-701, Korea (Corresponding Author, Email:
[email protected]) − 357 −
Jong-Sub Lee, Hyon-Sohk Ohm, Sungsoo Yoon, and In-Mo Lee
Two-layered gypsums have been modeled and the phase velocity of gypsums has been evaluated using the wavelet transform of surface waves. Two-layered gypsums were composed of two gypsum layers with different phase velocities. This paper includes a review of wavelet transform techniques, experimental studies of gypsum plates composed of two layers and analyses of measured surface waves using the wavelet transform. Dispersion curves were obtained by the wavelet transform analyses and a simple inversion method using a weighted factor based on the normal distribution is suggested.
2. Wavelet Transforms The wavelet transform is fundamentally a cross-correlation between a given original signal and the mother wavelet. A Gabor wavelet, used as a mother wavelet, provides the best resolution in both the time and frequency domains (Inoue et al., 1996). The dispersions of the group and phase velocities are obtained using the wavelet transform of surface waves. 2.1 Scaling and Translation of Mother Wavelet The mother wavelet (in the wavelet transform) is as follows: 1 a
t–b a
ψa, b ( t ) = ------ ψ ⎛⎝ ---------⎞⎠
(1)
where ψ (t) is the mother wavelet, a is the scaling parameter and b is the translation parameter. The wavelet transform of a function f (t) is:
transform are:
+∞
Wf ( a, b ) =
Fig. 1. Group and Phase Velocities of the Wavelet Transform at the Selected Frequency: (a) Source and Receivers, (b) Magnitude of the Wavelet Transform at the Selected Frequency, (c) Phase Information of Receiver 1 and Receiver 2, (d) Unwrapped Phase Information of Receiver 1 and Receiver 2
∫ f ( t ) ψ ( t ) dt * a, b
(2)
–∞
where f (t) is the original signal and ψ *(t) is the complex conjugate of the mother wavelet. The wavelet transform is expressed as a cross-correlation between the original signal and the mother wavelet. Because the shape of the mother wavelet varies with the scaling parameter, the wide range of frequencies can be analyzed. As the mother wavelet moves with the translation parameter, the similarity between the original signal and the mother wavelet can be captured. In this study, the Gabor wavelet is adopted as the mother wavelet because it provides the best time frequency resolution as confirmed by the uncertainty principle (Inoue et al., 1996). The Gabor wavelet and its Fourier transform are expressed as: ( ω0 / γ ) 2 1 ω - t exp( i ω0 t ) ψg ( t ) = ------- ----------0 exp – ---------------4 γ 2 π
(3)
( γ / ω0 ) 2π γ - ( ω – ω0 )2 Ψg ( ω ) = ---------- ----- exp – ---------------4 2 π ω0
(4)
d Vphase = ------------------tph2 – tph1
(6)
tph1 = tgr1
(7)
θ2 – θ1 tph2 = tgr2 – --------------
(8)
ω
Note that the slope of lines shown in Fig. 1(d) is ω . By substituting Eqs. (7) and (8) into Eq. (6), Eq. (6) can be rewritten as: d Vphase = -----------------------------------------------θ2 – θ1⎞ ( tgr2 – tgr1 ) – ⎛⎝ -------------ω ⎠
where γ is π 2/ln2 and ω 0 is 2π. 2.2 Evaluation of Phase Velocity using Wavelet Transform The group velocity and the phase velocity from the wavelet
(5)
where tgr1 and tgr2 are the group delay time of receiver 1 and receiver 2, respectively, tph1 and tph2 are the phase delay time of receiver 1 and receiver 2, respectively and d is the distance between receiver 1 and receiver 2 as shown in Fig. 1. Fig. 1 shows the unwrapped phase information of receiver 1 and receiver 2. Based on the geometry relationship, tph1 and tph2 can be expressed as:
2
2
d Vgroup = ------------------tgr2 – tgr1
(9)
where θ1 and θ2 are the phases at tgr1 and tgr2, respectively.
− 358 −
KSCE Journal of Civil Engineering
Phase Velocity Evaluation of Two-Layered Gypsums by Using Wavelet Transform
3. Experimental Setup 3.1 Gypsum Models Three models were prepared by using three types of gypsum. The properties of the three gypsum types are summarized in Table 1. Each model was prepared with different sizes and properties of gypsum as shown in Fig. 2 and Table 2. The phase velocities of all two-layered gypsums were lower in the top layer than in the bottom layer. All of the gypsum models thus have a soft layer on the top and a stiff layer at the bottom. The gypsum materials were chosen so that the difference in phase velocity is significant. The mixing ratio was 0.26 for Gypsum 1, 0.45 for Gypsum 2, and 0.75 for Gypsum 3. The specimens were mixed by hand until the gypsum started to solidify and the specimens were quickly poured into the acrylic mold. For the gypsum models of Case 1 and Case 2, the specimens were poured vertically into the acrylic mold, whereas for the gypsum model of Case 3 the specimen was poured horizontally while the acrylic mold was lying horizontally. The uniformity of the entire specimen was confirmed by using the transmission method of the ultrasonic waves by 0.5 MHz transducers. 3.2 Measurement Systems Measurement systems included a waveform generator, power amplifier, source, receiver, amplifier and oscilloscope. Square waves of 1 Hz frequency were generated by a waveform generator (Agilent, 33220A), with an amplitude of 10 V amplified by a power amplifier (Piezo Systems, EPA-104), and were used to
operate a piezoelectric actuator (PCB, 712A02). The operating frequency was about 150 Hz to 5 kHz. The vibration of the actuator generated the elastic waves, which propagated through the gypsum plates. The propagated elastic waves were detected by two receiver accelerometers (PCB, 353B15). The signals measured by the accelerometers were amplified by a signal conditioner (PCB, 482A16) and were digitized and recorded by the oscilloscope (NI, PXI-5112). The 10,000 signals were averaged to obtain high quality data.
4. Test Results and Analyses 4.1 Test Procedures and Results The distance from the source to receiver 1 and the distance from receiver 1 to receiver 2 were set to be equal to minimize the near-field effects and the aliasing between the two receivers (Rix et al., 2000; Rix et al., 2001). The distances between the two receivers were selected to be 10 cm, 20 cm, 30 cm, 40 cm and 50 cm to determine the most appropriate distance between the two receivers. Typical results are plotted in Figs. 3 and 4 for receivers 1 and 2, respectively. The original time signals measured by the two receivers, which are represented in Figs. 3(a) and 4(a), were analyzed by the wavelet transforms to obtain the amplitude of the wavelet transform and the unwrapped phase information at the selected frequency. The results of the wavelet transform are plotted in Figs. 3(b) and 4(b). The group delay times, tgr1 and tgr2,
Table 1. Properties of Gypsum VP* (m/sec) VS* (m/sec) VR** (m/sec) Gypsum 1 3400 2000 1834 Gypsum 2 2500 1600 1444 Gypsum 3 2000 1500 1268 * VP and VS are measured by the transmission method using 0.5MHz transducers (Panametrics NDT, V318). ** VR is a theoretical value based on VP and VS.
Fig. 2. Specification of Gypsum Models Table 2. Specification of Gypsum Models
Case 1 Case 2 Case 3
Bottom layer m1 Gypsum 1 Gypsum 1 Gypsum 2
Vol. 17, No. 2 / March 2013
Top layer m2 Gypsum 2 Gypsum 3 Gypsum 3
Height of bottom layer h1 (mm) 150 300 700
Height of top layer h2 (mm) 50 100 300
Fig. 3. Signal Information of Receiver 1: (a) Original Time Signal, (b) Wavelet Transform, (c) Amplitude of the Wavelet Transform at the Selected Frequency, (d) Unwrapped Phase Information at the Selected Frequency − 359 −
Jong-Sub Lee, Hyon-Sohk Ohm, Sungsoo Yoon, and In-Mo Lee
Fig. 4. Signal Information of Receiver 2: (a) Original Time Signal, (b) Wavelet Transform, (c) Amplitude of the Wavelet Transform at the Selected Frequency, (d) Unwrapped Phase Information at the Selected Frequency
were selected at the maximum amplitude of the wavelet transform at the selected frequency, as shown in Fig. 3(c) and 4(c), respectively. The phases θ 1 and θ 2 at the given group delay times were selected from the unwrapped phase information, as shown in Fig. 3(d) and 4(d), respectively. The phase velocity was calculated by substituting the selected values into Eq. (9). 4.2 Dispersion Curves The dispersion curves, which relate the phase velocity to the wavelength determined at different frequencies, are plotted in Fig. 5. Fig. 5 includes all of the dispersion curves obtained at the different distances. The phase velocity of Rayleigh waves VR is related to the phase velocity of compression VP and shear waves VS as follows (Achenbach, 1975; Stokoe and Santamarina, 2000; Santamarina et al., 2001). 2
2
2
VR VR R ⎞ ⎛2 – V ------ – 4 1 – ------ 1 – ------=0 2 2 ⎝ V2⎠ VP VS S
Fig. 5. Dispersion Curves at Different Receiver Distances: (a) Case 1, (b) Case 2, (c) Case 3 Table 3. Maximum Wavelength at the Minimum Receiver Distance
Case 1 Case 2 Case 3
Minimum receiver distance dmin (cm) 30 40 50
Maximum wavelength λmax (cm) 12 25 63
dmin/λmax 2.5 1.6 0.8
(10)
minimum distance should be greater than about three times the wavelength.
The minimum distance was evaluated based on the theoretical value of the phase velocity of Rayleigh waves which was calculated by substituting the compressive and shear wave velocities of gypsum into Eq. (10). The wavelengths at which the phase velocity increases were selected and were summarized in Table 3. Table 3 shows that the maximum wavelength at the minimum distance affects the calculated phase velocity. The
4.3 Inversion Inversion analyses can be performed by constructing a theoretical dispersion curve (Thomson, 1950; Haskell, 1953; Kausel and Roesset, 1981) from a trial profile and minimizing the error between the theoretical dispersion curve and the experimental dispersion curve. In this study, a simple inversion method using the weighted factor based on the normal distribution is proposed.
− 360 −
KSCE Journal of Civil Engineering
Phase Velocity Evaluation of Two-Layered Gypsums by Using Wavelet Transform
The weighted factor is expressed as: 2
1 w ( λ ) = ------------------- e 2π αλc
( λ – λc ) ------------------2 2 2α λc
(11)
where w(λ) is the weighted factor that varies with wavelength, λc is the central wavelength and α is the wavelength influence factor. The range and the centre of the weighted factor widen and deepen with increasing wavelength, representing the characteristics of surface waves. The predicted phase velocity is calculated by the multiplication of the weighted factor and the assumed phase velocity. The error between the predicted and the measured phase velocities is calculated by the least square method: E = Vmeas − Vpred
(12)
The inversion was conducted with a receiver distance of 50 cm to avoid the near field effects. The least square L with the wavelength influence factor aaa and the inversion range for each case is plotted in Fig. 6. The error between the predicted and measured phase velocities was quantified using the least square method by changing both the wavelength influence factor and the influence range. From each pair of wavelength influence factors and influence ranges, the minimum least square value was selected. Fig. 6 shows that the optimal wavelength influence factor is 0.2 for all cases. Furthermore, the optimal influence range can be selected as 0.4 m in Case 1, 0.7 m in Case 2 and 1.5 m in Case 3. The inversion results are presented in Figs. 7, 8 and 9 for Cases 1, 2 and 3, respectively. The calculation of the phase velocity consists of following steps: (1) the phase velocity profiles with
where Vmeas is the measured phase velocity, and Vpred is the predicted phase velocity. The least square method is expressed as: 1 --2 2
L = (∑(E) )
(13)
Fig. 6. Least Square L for Each Case: (a) Case 1, (b) Case 2, (c) Case 3 Vol. 17, No. 2 / March 2013
Fig. 7. Inversion Results for Case 1: (a) Assumed Phase Velocity with the Wavelength, (b) Weighted Factor with Wavelength, (c) Predicted and Measured Phase Velocities
− 361 −
Jong-Sub Lee, Hyon-Sohk Ohm, Sungsoo Yoon, and In-Mo Lee
Fig. 8. Inversion Results for Case 2: (a) Assumed Phase Velocity with the Wavelength, (b) Weighted Factor with Wavelength, (c) Predicted and Measured Phase Velocities
respect to the wavelength were assumed; (2) by multiplying assumed phase velocity profiles and the weighted factors, predicted phase velocities are calculated; (3) the predicted phase velocities are fitted using polynomial equations to calculate the error between the predicted and measured phase velocities; (4) steps (1) ~ (3) are iterated until the assumed phase velocity profile yields the minimum L value; (5) the phase velocity profile with the minimum L is selected as the final phase velocity profile. The location of the gypsum layer can be found using the assumed phase velocity versus wavelength. The upper layer thickness was compared with the wavelength based on Fig. 7, 8 and 9, and the results are summarized in Table 4. Table 4 shows that the propagation depth of the surface waves is about 0.41 to 0.67 times the wavelength, which agrees with 0.5 times the wavelength suggested by Nazarian and Stokoe (1984). The range
Fig. 9. Inversion Results for Case 3: (a) Assumed Phase Velocity with the Wavelength, (b) Weighted Factor with Wavelength, (c) Predicted and Measured Phase Velocities Table 4. Layer Thickness versus Wavelength
Case 1 Case 2 Case 3
Upper layer Thickness D (m) 0.05 0.10 0.30
Wavelength λ (m) 0.080 ~ 0.100 0.210 ~ 0.245 0.450 ~ 0.525
D/λ 0.50 ~ 0.63 0.41 ~ 0.48 0.57 ~ 0.67
of wavelength variation with the phase velocity (transitional zone) in the dispersion curve and the thickness of the upper gypsum layer (separate zone) are summarized in Table 5. Table 5 shows that the centre of the wavelength range varying with the phase velocity in the dispersion curves matches well with the
− 362 −
KSCE Journal of Civil Engineering
Phase Velocity Evaluation of Two-Layered Gypsums by Using Wavelet Transform
Table 5. Range of Wavelength Varying with Phase Velocity in the Dispersion Curve
Case 1 Case 2 Case 3
Transitional zone (m) 0.05 ~ 0.14 0.13 ~ 0.32 0.37 ~ 0.63
Separate zone (m) 0.08 ~ 0.10 0.20 ~ 0.24 0.50 ~ 0.60
Center of transitional zone (m) 0.095 0.225 0.500
location of the gypsum layer estimated by the inversion technique.
5. Conclusions A non-destructive method to evaluate the phase velocity in the two-layered gypsums using the wavelet transform of surface waves is proposed in this study. For the generation and detection of surface waves, one source and two receivers are used. The experimental results show that the near-field effects can be neglected if the distance between the two receivers is chosen to be three times the wavelength. A simple inversion method using a weighted factor based on the normal distribution is also suggested. The minimum error between measured and predicted phase velocities is calculated using the least square method by changing the wavelength influence factor and influence range. The inversion results show that the predicted phase velocities agree well with the measured phase velocities when the wavelength influence factor is 0.2. Note that the optimal influence ranges are from 0.4 m to 1.5 m. This study shows that the propagation depth of surface waves is from 0.42 to 0.63 times the wavelength. The range of wavelength variation with phase velocity in the dispersion curves matches well with that estimated by the inversion technique. As a future study, different types of anomalies such as faults, fractures, and cavities need to be examined. This study demonstrates that a simple test setup and inversion may yield a reasonable phase velocity and thus the wavelet transform of the surface waves can be effectively used for the evaluation of the phase velocity in the field tests.
Acknowledgements This work was supported by the Korea Institute of Construction & Transportation Technology Evaluation and Planning (KICTEP) (Program No.: 09 Technology Innovation-E05).
References Achenbach, J. D. (1975). Wave propagation in elastic solids, North-
Vol. 17, No. 2 / March 2013
Holland Publishing Company, New York. Gucunski, N. and Woods, R. D. (1992). “Numerical simulation of the SASW test.” Soil Dynamics and Earthquake Engineering, Vol. 11, No. 4, pp. 213-227. Haskell, N. A. (1953). “The dispersion of surface waves on multilayered media.” Bulletin of the Seismological Society of America, Vol. 43, No. 1, pp. 17-34. Inoue, H., Kishimoto, K., and Shibuya, T. (1996). “Experimental wavelet analysis of flexural waves in beams.” Experimental Mechanics, Vol. 36, No. 3, pp. 212-217. Kausel, E. and Roesset, J. M. (1981). “Stiffness matrices for layered soils.” Bulletin of the Seismological Society of America, Vol. 71, No. 6, pp. 1743-1761. Lai, C. G., Rix, G. J., Foti, S., and Roma, V. (2002). “Simultaneous measurement and inversion of surface wave dispersion and attenuation curves.” Soil Dynamics and Earthquake Engineering, Vol. 22, Nos. 9-12, pp. 923-930. Lee, I. M., Truong, Q. H., Kim, D. H., and Lee, J. S. (2009). “Discontinuity detection ahead of a tunnel face utilizing ultrasonic reflection: Laboratory scale application.” Tunnelling and Underground Space Technology, Vol. 24, No. 2, pp. 155-163. Nazarian, S. and Stokoe, K. H. (1984). “In situ shear wave velocities from spectral analysis of surface waves.” Proceedings of the 8th World Conference on Earthquake Engineering, Vol. 3, pp. 31-38. Park, H. C. and Kim, D. S. (2001). “Evaluation of the dispersive phase and group velocities using harmonic wavelet transform.” NDT&E International, Vol. 34, No. 7, pp. 457-467. Park, C., Miller, R., and Xia, J. (1999). “Multichannel analysis of surface waves.” Geophysics, Vol. 64, No. 3, pp. 800-808. Rix, G. J., Lai, C. G., and Foti, S. (2001). “Simultaneous measurements of surface wave dispersion and attenuation curves.” Geotechnical Testing Journal, Vol. 24, No. 4, pp. 350-358. Rix, G. J., Lai, C. G., and Spang, A. W. (2000). “In situ measurement of damping ratio using surface waves.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, No. 5, pp. 472-480. Santamarina, J. C., Klein, K. A., and Fam, M. A. (2001). Soils and waves, John Wiley & Sons, New York. Stokoe, K. H. and Santamarina, J. C. (2000). “Seismic-wave-based testing in geotechnical engineering.” International Conference on Geotechnical and Geological Engineering, pp. 1490-1536. Thomson, W. T. (1950). “Transmission of elastic waves through a stratified solid medium.” Journal of Applied Physics, Vol. 21, No. 2, pp. 89-93. Tokimatsu, K., Tamura, S. and Kojima, H., 1992, “Effects of multiple modes on rayleigh wave dispersion characteristics.” Journal of Geotechnical Engineering, Vol. 118, No. 10, pp. 1529-1543. Yoon, S. S. (2005). Array-based measurements of surface wave dispersion and attenuation using frequency-wavenumber analysis, PhD Dissertation, Georgia Institute of Technology. Zywicki, D. J. (1999). Advanced signal processing methods applied to engineering analysis of seismic surface waves, PhD Dissertation, Georgia Institute of Technology.
− 363 −
