Ultrasonics 55 (2015) 15–25

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Ultrasonics

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Wave separation: Application for arrival time detection in ultrasonicsignals

http://dx.doi.org/10.1016/j.ultras.2014.08.0190041-624X/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +1 818 331 3161.E-mail addresses: avanesians@email.arizona.edu (P. Avanesians), moe.momayez

@arizona.edu (M. Momayez).1 Co-author. Tel.: +1 520 621 6580.

Patrick Avanesians ⇑, Moe Momayez 1

Department of Mining and Geological Engineering, University of Arizona, United States

a r t i c l e i n f o

Article history:Received 19 August 2013Received in revised form 24 June 2014Accepted 14 August 2014Available online 26 August 2014

Keywords:Wave separationNonlinearDecompositionTime-frequency analysisArrival time

a b s t r a c t

A method to detect and accurately measure the arrival time of wave packets in ultrasonic signals using anonlinear decomposition technique is presented. We specifically address the problem of extractingevents that are not well separated in the time, space and frequency domains. Analysis of complex ultra-sonic signals, even those composed of poorly separated echoes, provided exceptional estimates of thedesired time of arrival, from the media under investigation.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The ultrasonic pitch-catch technique is one of the common testsfor inspecting the quality of materials. Two piezoelectric transduc-ers separated by a certain distance are placed on a free surface ofthe test object, where one transducer is used to transmit an ultra-sonic pulse inside the material while the other sensor capturesreflections due to differences in the acoustic impedance betweentwo media. One goal of conducting ultrasonic tests is to measurethe thickness of the material under investigation or to locatedefects/objects inside the material. Typically, the equationd = (m � t)/2, where m is the ultrasound speed and t is the arrivaltime is used to determine the distance from the transducer tothe target. Thus, distance is related directly to arrival time if veloc-ity is known.

Fig. 1 shows two signals recorded on the same test object. Trac-ing the signal in time, where the wave typically travels through thematerial, the recorded amplitude is zero or close to zero. The directwave (one that travels near the surface of the material when thetransmitting and receiving transducers are place close to eachother) is usually the first one to reach the receiver as shown bythe change in the amplitude in the recorded signal (in the case ofFig. 1 the first arrival time is around 45 ls). However, in ultrasonic

tests, this arrival time is occasionally used to get an estimate of thewave velocity since the distance between the transducers isknown. The typical ultrasonic signal is composed of multiplemodes and reflections with different frequencies. In testing con-crete, since the reflections from defects inside the slab are rarelywell separated in time, it becomes highly problematic to identifythe arrival time from the desired targets. The detection of the arri-val times from multiple scatterers inside the medium has been thesubject of intense research over many decades. Different methodshave been proposed to decompose the recorded ultrasonic wavesand measure the associated arrival times. The following is aninventory of the main techniques that have become popular amongthe nondestructive testing practitioners:

1. Application of the traditional quadrature-demodulationscheme to suitably extract the envelope of the main echoand locate its onset [1].

2. Correlation between input and output for high signal-to-noise ratio waveforms and L2-norm with low noise level[2].

3. Application of the square-root unscented Kalman filter(SRUKF) to identify the shape parameters of an ultrasonicecho envelope [3].

4. Application of Synthetic Aperture Focusing Technique(SAFT) where a constructive interference procedureenhances the lateral spatial resolution and signal-to-noiseratio. The arrival time is estimated from enhanced imagesof the interior of the medium [4].

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8

Time (us)

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(a)

(b)

Fig. 1. (a) Well separated ultrasonic signal; (b) mixed ultrasonic signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

4000

4500

5000Model Space

Time [ seconds ]

1000

1500

2000

2500

3000

3500

Freq

uenc

y [ H

z ]

Time-Frequency Distribution

Time [seconds]

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uenc

y [H

z]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 2. Model space and the associated Wigner–Ville transform.

16 P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25

5. Separation of non-linear and non-stationary ultrasonic sig-nals by means of the empirical mode decompositionmethod to obtain the intrinsic mode functions used forsignal reconstruction. The arrival time is then obtainedfrom the reconstructed signal [5].

6. Combination of both the improved self-interference driv-ing technique and the optional optimization signal pro-cessing algorithms for measuring the arrival time [6].

In some ultrasonic signals, picking the first reflection (arrivaltime) from the desired target is an easy task since the directwave and the main reflection are well separated in time (with lit-tle or no interference from other sources of ultrasonic scattering),and therefore can be discriminated visually. Fig. 1(a) shows a sig-nal in which the direct wave and the main reflection are rela-tively well separated and the arrival time was correctlydetermined by visual inspection. Fig. 1(b) on the other hand,depicts a commonly encountered situation in practice wheremultiple waves are superimposed, turning the detection of themain reflection into a challenging problem. In these cases, signaldecomposition has shown promise and is used in this study toseparate the signal into different modes in order to extract thedesired reflected wave pack. More specifically, the synchro-squeezed transform (SST), a new signal decomposition method,is shown to provide useful results.

Conventional time–frequency representations, such as theshort-time Fourier transform (STFT), the Wigner–Ville transform(WVT), the Wavelet Transform (WT), and the Empirical ModeDecomposition (EMD) method, perform poorly when individualcomponents of the recorded signal are not sufficiently separatedin the time–frequency domain. The synchro-squeezed transformbuilds on the philosophy of the EMD method using a more robusttheoretical foundation and employs a different approach for theconstruction of a signal’s constituent components. In fact, syn-chro-squeezing can be overlaid with many commonly used time–frequency methods [7–10]. SST is a combination of the wavelettransform and reassignment technique [11].

2. Theoretical background

The simplest form of a time–frequency representation is aspectrogram created using a short-time Fourier transform (STFT).This transform employs a short-time window that limits the eval-uation of the Fourier transform to some specified region along thetime index. It is well-known that STFT spectrograms suffer fromthe time–frequency resolution trade-off, that is, a shorter windowprovides better resolution in the time domain, and the largerbandwidth of the window’s spectrum affords a poorer resolutionin the frequency domain. The Wigner–Ville transform (WVT)takes advantage of the powerful concept of match filtering, effec-tively employing the time-reversed copy of the signal as the win-dow to carry out a short-time window Fourier transform. A

Fig. 3. (a) Concrete slab with a 15� bottom surface; (b) rectangular grid on concrete slab; (c) ultrasonic transmitter and receiver; (d) ultrasonic transducers while conductingthe experiment; (e) typical recorded signal.

P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25 17

number of properties make the WVT valuable for decomposingcomplex signals. For example, the analysis is performed usingonly the transform’s ability to be entirely localized in the caseof frequency modulated signals. The trade-off in using WVT isthe manifestation of prominent interference (oscillations for themost part) between any two interacting components of the signal[12]. The effect is illustrated in Fig. 2 below. Smoothing some-what enhances the readability of the transformed signal throughthe application of the pseudo-Wigner–Ville distribution which isessentially a windowed version of the WVT. Here, the cross-termsare attenuated at the cost of spreading signal energy along thefrequency axis, adversely affecting the time–frequency localiza-tion of the signal content.

To optimize the tradeoff between attenuating oscillations andfocusing the signal energy in the time–frequency domain, the coef-ficients (t, w) of the WV distribution are reassigned to new pointsð~t; ~wÞ which are the center of gravity of the signal energy

distribution around (t, w). The reassignment method was firstintroduced by Kodera et al. [13] to overcome the spectrogram’sdrawbacks such as biased estimation of the signals instantaneousfrequency, group delay, and the Gabor–Heisenberg inequalitywhich controls the locality in either the time or frequency domain,but never both [14].

The concept of reassignment can also be applied to improve thespectral resolution obtained from wavelet transforms. In this con-text, the process has been dubbed synchro-squeezing, and was firstintroduced to analyze speech signals [15]. Subsequent studies [8–11,16] showed that synchro-squeezing performance is superiorand could be considered as a viable alternative to the EMD method.

In this paper, we show that the synchro-squeezing methodcould also be used to decompose ultrasonic signals into their con-stituent wave packets, providing the means to detect and deter-mine the arrival time of the desired echoed pulses from insidethe medium under investigation. This is especially valuable since

0 50 100 150 200 250 300 350 400-10

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Time index for maximum amplitude

121.6

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10Truncated and Padded signal

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Fig. 4. (a) Original signal with lower and upper time interval and time index formaximum amplitude; (b) truncated and padded signal.

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Time - Frequency - Amplitude

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Central Frequency = 0.8

(a)

(b)

Fig. 5. (a) TFA distribution; (b) SST distribution at central frequency of 0.8.

18 P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25

the shape of the mother wavelet is tuned to closely match theultrasonic pulse waveform used in the inspection process.

The following is a brief description of the synchro-squeezingtransform (SST). If s(t) is the desired signal, its continuous wavelettransform (CWT), Ws(a, b) is given by:

Wsða; bÞ ¼1ffiffiffiap

Z 1

�1sðtÞw� t � b

a

� �dt ð1Þ

where w⁄ is the complex conjugate of the mother wavelet (a contin-uous function in both the time and frequency domains), and b is thetime shift applied to the mother wavelet which is scaled by a. Wecan think of the CWT as the cross-correlation of the signal s(t) withthe scaled and time shifted wavelet wððt � bÞ=aÞ=

ffiffiffiap

. This cross-cor-relation provides a measure of similarity between the signal and thescaled shifted wavelet. The wavelet coefficients Ws(a, b), however,frequently spread out along the scale dimension a, producing ablurred image in the time-scale domain.

The next step in the process involves the transformation of thepoints in the time-scale domain into the time–frequency domainfor all Ws(a, b) – 0. Different expressions may be used to handlethe transformation. We adopted the following relationship [16]:

xsða; bÞ ¼1

2p�ddb� arg½Wsða; bÞ� ð2Þ

where arg (�) denotes the unwrapped phase of the complex coeffi-cients and the 1/2p factor is introduced to convert from circularto normal frequencies. Eq. (2) is the expression for instantaneousfrequency (derivative of the unwrapped phase of the continuouswavelet transform). When the coefficients xs(a, b) are determined,we select the desired frequencies xk and form the frequency inter-val [wi–Dw/2, wi + Dw/2] where Dw = wi–wi�1. The synchro-squeezing transform Ts(wi, b) is computed only at the centers wi

of the frequency range as:

Tsðwi; bÞ ¼1

Cw

Xaj:jwðaj ;bÞ�wi j6Dw=2

Wsðaj; bÞ � a�3=2j � Daj ð3Þ

where Daj are the distances between two adjacent scales aj, andCwB 1

2

R10 wðeÞ de

e , with wðeÞ being the Fourier transform of the motherwavelet. Eq. (3) indicates that synchro-squeezing takes place alongthe frequency/scale axis, producing a more accurate and detailedtime–frequency representation.

The synchro-squeezed transform is invertible and the originaltime domain signal can be reconstructed (with no loss of informa-tion) using [11]:

sðtÞ ¼ ReX

i

Tsðwi; bÞ ¼X

i

Tsðwi; bÞ�����

����� cos argX

i

Tsðwi; bÞ !

ð4Þ

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Fig. 6. Input signal and its TFA distribution.

Synchrosqueezed Wavelet Transform

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Fig. 7. Selection of frequency bands.

P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25 19

The choice of the mother wavelet is based on many factors, andshould reflect the type of features present in the signal. A com-monly used function is the Morlet wavelet, consisting of a planewave modulated by a Gaussian function. The Morlet wavelet isnon-orthogonal, complex and offers a good tradeoff betweendetecting oscillations and peaks or discontinuities making it espe-cially suited for analyzing ultrasonic signals. We used the following

form of the wavelet because, compared to other mother wavelets,it represents most accurately the actual ultrasonic pulse thatprobed the interior of the concrete material in this study:

wðtÞ ¼ Bf 0ei2pf 0t � e�

12ð2pf 0Þ

2� �

e�t22 ð5Þ

where Bf 0is a normalization factor equal to

ffiffiffiffiffiffiffiffiffiffiffiffiffi2=pf 0

pand f0 is the so-

called central frequency, which determines the relationshipbetween the time and the scale (or frequency) resolutions of thewavelet transform.

3. Proposed technique

We describe a method to unscramble closely spaced eventscommonly encountered in ultrasonic testing of materials. This con-dition occurs when the direct wave and the reflection from a scat-terer superimpose and create an interference in the recorded signalbecause the events are not well separated in time, space, and fre-quency domains, or in other words, the length of the ultrasonicpulse is larger than the time of flight between the echoes. We spe-cifically tackle the case where the detection and estimation of thearrival time of the first reflection is needed. Our approach usesboth the synchro-squeezed transform and the measurement ofinstantaneous frequencies to decompose a signal into its compo-nents. The arrival time is then determined for the desiredreflection.

First, the arrival time associated with the highest peak is deter-mined from the time domain signal. Let us call this tmax. Since boththe input and output signals have a Gaussian shaped envelope, weknow that the actual arrival time should be before tmax. Next, a cer-tain range is selected to truncate the signal and make the decom-position procedure go faster. The lower limit is chosen at somepoint before the onset of the direct wave arrival, while the upperlimit is a point after tmax where the change in the shape of theenvelope of the signal is discernable, or tmax + tmax/3 in extremelymixed waveforms. The signal is then truncated and padded withzeroes to the same length as the original signal.

To decompose the signal and measure the arrival time, the fol-lowing steps are performed. Calculate the instantaneous frequen-cies and plot the time–frequency–amplitude (TFA) distribution.Note that the instantaneous frequency is the derivative of theunwrapped instantaneous phase, which is obtained by performinga Hilbert transform on the truncated s(t) signal. The amplitude val-ues in the TFA distribution are taken from the original time domainsignal and color coded for better readability. Next a synchro-squeezed transform is carried out on the truncated signal usingan appropriate central frequency. The choice of the central fre-quency is crucial and will affect the outcome of the analysis. Therule we propose for the selection of the central frequency is basedon visual interpretation of both the SST and time–frequency–amplitude distributions. The instantaneous frequency plot is com-pared with the SST and the optimum central frequency value isfound from the closest match between the two distributions.

The best match between the TFA and SST distributions isobtained by changing the value for the central frequency in theSST and comparing the resultant maximum amplitudes (values inthe third dimension) in the TFA and SST plots. The time and fre-quency values corresponding to the maximum amplitudes shouldbe the same or very close for the optimal central frequency. Inalmost all cases, the central frequency is found to be between 0.5and 1.

We continue the process by defining the frequency bandsneeded for signal decomposition. Since the first reflection normallyhas the strongest echo, we look for the frequency range in both theTFA and SST plots where the highest amplitudes (absolute value)are located.

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D

Fig. 8. Decomposed signal and associated waveforms from defined frequency bands.

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Time-Frequency-Amplitude

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-9.6

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-0.00497

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5.75

7.67

9.59

Sudden Change inFrequency Value

1

4

32

Fig. 9. Sudden changes in the time–frequency–amplitude distribution.

20 P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25

Once the frequency bands are selected, we extract the wave-form from each band by performing an inverse synchro-squeezedtransform of the SST coefficients. The reconstructed time-domainsignal with the highest energy is the waveform associated withthe main reflection from the discontinuity. In some cases, it is

possible that the main frequency band is selected incorrectly andthe resultant waveform does not correspond to the actual reflec-tion. To mitigate those special circumstances, the two methodolo-gies described below can be used to validate the results.

Method 1: In the time–frequency–amplitude plot, abruptchanges in frequency are indicative of different wave typesdetected by the sensor. The last peak before the cluster of highamplitude values corresponds to the time index of the mainreflected wave.Method 2: In the time–frequency–amplitude plot, select the fre-quency range where the maximum amplitude values arelocated (the cluster of red and blue dots). Use this frequencyrange to reconstruct a time-domain waveform from the originaltime-domain signal. In other words, amplitudes outside of thisrange are removed from the original signal s(t).

4. Experimental setup

The proposed decomposition method was tested by analyzingsignals recorded on three concrete slabs, each with a surface areaof 1 � 1 m2 and a sub-horizontal inclined bottom surface of 5�,10�, and 15� respectively. Fig. 3(a) shows the slab with a 15�inclined bottom surface. The concrete used in this work had a max-

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106.3

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8Decomposed Signal : Fmin=120000 Fmax=140000

Time (us)

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itude

C

107

Fig. 10. Comparison of frequency changes with reconstructed waveform forvalidating Method 1.

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250

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y (K

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Time-Frequency-Amplitude plot

-8

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0

2

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6

8

90KHz

180KHz

Fig. 11. Selection of the frequency band in the TFA distribution for validatingMethod 2.

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107

Fig. 12. Signal in time domain after removing amplitudes value outside certainfrequency range.

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Fig. 13. Validation of the main reflection wave packet arrival time using Method 2for signal S15.

P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25 21

0 50 100 150 200 250 300 350 400-20

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Time index formaximum amplitude

94.68

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Tuncated and Padded Signal

Fig. 14. (a) Original signal with lower and upper time interval; (b) truncated andpadded signal.

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Central Frequency = 0.6

Fig. 15. (a) TFA distribution; (b) SST distribution at central frequency of 0.6.

Synchrosqueezed Wavelet Transform

Freq

uenc

y K(

Hz)

Time (us)40 60 80 100 120 140

100

120

140

160

100KHz

150KHz

130KHz

Fig. 16. Selection of frequency bands.

22 P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25

imum grain size of 20 mm and compressive strength of 32 MPa. Itwas made by a local company with the same properties as the nor-mal concrete used in constructions to simulate a ‘‘real world’’ con-crete structure. The incline surface at the bottom of the slabs wasbuilt using a layer of wood with a thickness of 5 mm. The slabssimulate wide inclined cracks in large concrete structures such asdams where water can seep through and expand the cracks result-ing in significant damage.

Data were collected from rectangular grids mapped in the mid-dle of slabs with 10 mm spacing between the points as shown inFig. 3(b). The size of the grid was different for each slab and variedfrom 21 to 27 cm in both x and y directions, corresponding to adepth of 100–300 mm, as shown in Fig. 3(a).

The tests were conducted in pitch–catch mode using two ultra-sonic transducers, one as a transmitter and the other as a receiver.Fig. 3(c) shows ultrasonic transducers used b this experiment andFig. 3(d) demonstrates ultrasonic transducers conducting experi-ment. The transducers had a nominal frequency of 140 kHz andan actual bandwidth of 400 kHz. A pair of flat acrylic plates wereattached to the surface of the transducers as a protection measureagainst scratch and damage. The thickness of the plates are 5 cm,and together, they introduce an additional delay of 26 ls in thesignal.

A two-channel data acquisition and function generator wasused to input a 140 kHz square pulse and to capture the reflectedwaveforms. The data were collected at a high sampling frequencyof 25 MHz using 12-bit resolution. For each waveform, 10,000 sam-ples with a peak-to-peak amplitude of 12 V were collected. The sig-nal captured from the test medium was passed through a signal

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68

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-1

-0.5

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Time (us)

Ampl

itude

Fig. 17. Decomposed signal and associated waveforms from defined frequencybands.

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Time (us)

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68

Fig. 18. Comparison of frequency change with reconstructed waveforms forvalidating Methods 1.

140

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130KHz

P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25 23

conditioner with the appropriate low-pass filter before feeding it tothe data acquisition system. Fig. 3(f) shows a screenshot of thesoftware used in this work displaying a typical waveform obtainedfrom the test slab and recorded by the data acquisition system.

Calibration tests on five cylindrical concrete samples providedan average velocity of 4100 m/s. This value is used to calculatethe depth of the inclined bottom surface.

20 40 60 80 100 120 140

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100

120

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-15

-10

-5100KHz

Fig. 19. Selection of the frequency band from TFA distribution for validatingMethod 2.

5. Data analysis

For the sake of brevity, the analysis results for two signals arepresented in this section. The first signal is from the 15-degree slab(see Fig. 3 above) at a known depth of 16.69 cm. This gives anactual arrival time of 81.41 ls, using the velocity of 4100 m/s. Asexplained in the experimental setup section, there is a delay of26 ls due to the thickness of the guides, and therefore the totaltime we should get after signal decomposition is 107.41 ls(81.41 + 26). We name this signal S15.The second signal, S05, istaken on the 5-degree slab at a known depth of 8.5 cm. The arrivaltime for this signal is 67.47 ls (41.47 + 26).The analysis steps areexplained in detail for the S15 signal only. For signal S05, we onlypresent the results.

The first step is the selection of tmax which is 121.6 ls for S15.We take as the lower limit, the time index of 26 ls and as upperlimit, the time index 162 ls. Fig. 4(a) shows the signal and the

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Reconstructed Signal : Fmin=10000 Fmax=130000

68

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Time (us)

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itude

68

Fig. 20. Validation of the main reflection wave packet arrival time using Methods 2for signal S05.

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15

Time (us)

Ampl

itude

93.93

0 50 100 150 200 250 300 350 400-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (us)

Ampl

itude

102.8

Fig. 21. Checking accuracy of arrival time by subtracting tmax of two consecutivesignals.

24 P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25

selected time interval. The signal is then truncated and padded tozeroes as shown in Fig. 4(b). Next, the instantaneous frequencies,and the synchro-squeezed transform (at different central frequen-cies) are computed, and the time–frequency–amplitude (TFA) andthe SST distributions are plotted. The optimal value of 0.8 for thecentral frequency is found by comparing the maximum values inthe TFA and SST distributions, and the best match is selected.Fig. 5(a) shows the TFA distribution and Fig. 5(b) presents theSST at the optimum central frequency for signal S15.

Let is take a closer look at the SST plot and select the frequencyband that encompass the largest SST coefficients and call it themain reflection. Also, the selected frequency band should includethe main input signal frequencies (80–140 kHz in our case).Fig. 6 shows the input signal and its associated time–frequency–amplitude plot.

Using the SST distribution shown in Fig. 7, we identify the mainfrequency range from 120 kHz to 140 kHz, and we define the fol-lowing frequency bands for signal decomposition: 50–90 kHz,90–120 kHz, 120–140 kHz, and 140–180 kHz. Once the waveformsare reconstructed in the time-domain for each of the defined fre-quency band, the waveform with the largest energy (sum of theamplitudes squared) is used to determine the arrival time. Fig. 8

presents the reconstructed waveforms for each of the defined dif-ferent frequency bands.

From the previous analysis, we conclude that the time corre-sponding to the wave reaching the sensor 107 ls with a frequencybandwidth from 120 kHz to 140 kHz. The last step involved indetermining the time of arrival is to verify that the reconstructedwaveform carries the correct information as discussed in the pro-posed method section.

5.1. Method 1

We consult the time–frequency–amplitude distribution, shownin Fig. 9, and locate the frequency peak (peak #4) before the clusterof high amplitude values in the signal. The time index correspond-ing to the peak #4 equals to 106.3 ls which is very close to theactual arrival time of 107.41 ls. Fig. 10 shows the comparison ofthe sudden change in the time–frequency–plot with the decom-posed signal (Fig. 8(c)) in order to confirm that the correct reflec-tion wave packet is selected.

5.2. Method 2

Let is take a look at the time–frequency–amplitude distributionand define the frequency range encompassing the highest ampli-tude values. In the case of signal S15 illustrated in Fig. 11, the lower

P. Avanesians, M. Momayez / Ultrasonics 55 (2015) 15–25 25

and upper limit of the frequency range are 90 kHz and 180 kHz.Then, amplitude values outside this range are removed from thesignal s(t), and the desired arrival time is selected from the recon-structed waveform as shown in Fig. 12.

A comparison of the reconstructed waveform in the band 120–140 kHz (Fig. 4C) and the reconstructed waveform shown inFig. 13, validates the results of the analysis and provides the correcttime of arrival.

The second signal from the 5-degree slab has an arrival time41.47 ls corresponding to a depth of 8.5 cm. The actual total timeis 67.47 ls (includes the delay from the acrylic guides). As in theprevious example, we start with the time–frequency–amplitudedistribution plot and compare this with the SST distribution inorder to find the optimal central frequency. This step is followedby the selection of frequency bands, extraction of the SST coeffi-cients and reconstruction of the waveform for each frequencyband. The waveform with the highest energy content which alsocontains the main input frequency range is used to determinethe time of arrival. Validation is performed by comparing thechange in the TFA distribution with the reconstructed signal fromthe main frequency band (Method 1), and by removing the sampleswith amplitudes outside the defined frequency range from the ori-ginal signal s(t) (Method 2). The reconstructed waveforms are plot-ted and compared to obtain the arrival time. Figs. 14–20 illustratethe preceding steps for S05.

6. Discussion

Over 500 signals were analyzed in this study and arrival timeswere successfully determined. The results were verified withknown depths of the bottom discontinuity in the concrete slabs.In the process, we observed that the choice of the central frequencyaffects the outcome of the analysis to some extent, especially withrespect to the creation of the synchro-squeezed transform plot. Asmentioned above, at present, the selection of the central frequencyrelies on a visual inspection and comparison of both the SST andTFA distributions. The best match between the instantaneous fre-quency and SST defines the optimal value for the central frequency.Care should also be exercised in the selection of the main fre-quency interval and the corresponding frequency bands used forthe extraction of the SST coefficients and the subsequent recon-struction of waveforms in the time-domain. The suggestedapproach where the size of the frequency bands is taken to beequal to the width of the main frequency interval (obtained fromthe SST distribution) is a good initial guess and provided accurateestimates of the time of arrival for all analyzed signals. To obtainthe arrival time for the first reflection, the original signal s(t), thetime-instantaneous frequency–amplitude and the decomposedwaveforms using the synchro-squeezed transform are analyzedsimultaneously. The accuracy of the measured values can furtherbe investigated by comparing two closely-spaced signals. Forexample, if time indexes corresponding to the maximum ampli-tude for the first and second time-domain signals are t1max andt2max , and the actual arrival times are ta1 and ta2 , then differencebetween t1max and t2max must be equal to the difference betweenta1 and ta2 . From Fig. 21 below, we read off the following values:

ta1 ¼ 39:73 ls; ta2 ¼ 48:63 ls; t1max ¼ 93:93 ls; t2max ¼ 102:8 ls

We then form the differences for the maximum amplitude timeindexes and the measured arrival times:

Dta1a2 ¼ jta2 � ta1 j ¼ j48:63� 39:73j !yieldsDta1a2 ¼ 8:9 ls

Dt1max2max ¼ jt2max � t1max j ¼ j102:8� 93:93j !yieldsDt1max2max ¼ 8:87 ls

Dta1a2 ffi Dt1max2max

7. Conclusions

The synchro-squeezed transform is an effective tool for decom-posing many types of noisy, time-varying frequency, and nonlinearsignals. In this paper, we presented a new technique for determin-ing the arrival time of ultrasonic signals used in nondestructivetesting of geomaterials. The method uses the synchro-squeezedtransform to decompose the signal into constituent waveformsreconstructed from suitably defined frequency bands. We haveshown that the time of arrival of the primary reflection (from a dis-continuity inside the material) can be accurately measured fromsignals where the events are poorly separated in time and fre-quency. This technique could also be used to detect the arrival timeof well separated time, frequency and space domain signals. It wassuccessfully applied in the processing of lower frequency elasticwaves from earthquake events and seismic exploration.

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