Ultrasonic Time of Flight Estimation Using Wavelet Transforms
S. Laddada1,2, S. Lemlikchi1,F. Guendouzi1
1Centre de Développement des Technologies Avancées, Baba-Hassen, 16303 Algérie
E-Mail: sladdada@cdta.dz
M. O. Si-Chaib2, H. Djelouah3
2Laboratoire de Mécanique des Solides et Systèmes, UMB.
Boumerdès, Algérie
3Laboratoire de Physique des Matériaux, USTHB, Algérie
Abstract—The accurate estimation of the time-of-flight (TOF) of the ultrasonic echoes is important in non-destructive testing (NDT). The main interest of the TOF estimation is the flaw detection, localization and coating thickness evaluation. Usually, the backscattered echoes from a thin coating or a flaw located close to the interface overlap in the time domain, so TOF estimation becomes more complicated and requires advanced signal processing methods. In this paper, the wavelet transform was investigated in order to evaluate the TOF. The two applied methods are the continuous wavelet transform based on the scale-averaged power (SAP) and the discrete stationary wavelet transform (DSWT). These methods were applied, firstly, on simulated Gaussian echoes for two practical cases: without and with partially overlapping echoes. Several numerical tests have been carried out to select a suitable mother wavelet for each method and for each case. The performance was evaluated through the mean square error (MSE) between the estimated and the reference value of the TOF. The numerical tests showed that both methods give a low error for non-overlapping echoes, while a reasonable error was obtained in case of partially overlapping echoes. An experimental validation was performed on a real signal taken from a thermally coated sample in order to evaluate the coating thickness. An overall agreement was observed between simulation and experiment.
Keywords—Time of Flight estimation; ultrasonic echo; SAP;
DSWT
I. INTRODUCTION
The time of flight (TOF) estimation isconventionallyperformed by gating and peakdetection. The separation of the differentechoes and the estimation of the corresponding TOF is a challengingtask, especially in the case
of overlappingechoes. Since the
ultrasonicsignalscontainseveral non-stationary or transientcharacteristics, whichoftencontainuseful information of the signal, the WaveletTransform (WT) is a convenienttool for processingsuchsignals. It has a strongcapability in time and frequencydomainanalysis. It describes a signal of interest by using the correlationwith translation and dilatation of a functioncalledmotherwavelet [1, 2]. The WT includes the ContinuousWaveletTransform (CWT) and the DiscreteWaveletTransform (DWT).
The ContinuousWaveletTransform (CWT) maybedefined as a mappingdependent on the specification of an
auxiliaryfunctioncalled the analyzingwavelet [2]. It computes the innerproduct of the signal with the translated and dilated versions of the analyzingwavelet. Manypreviousworks have shownthat the continuouswavelettransform (CWT) offerspromisingtools for the estimation of TOF. In the field of tissue characterization, Georgiouet al. [4] examined the ultrasound fluctuations in the wavelet power over the range of scalesusing the scale-averagedwavelet power (SAP) of the CWT. In the structural damage evaluation, Su et al.[5] used the SAP spectrum to determine the exact arrival time of damage-scatteredwave in a captured signal. Hosseinabadiet al.
[6] used the envelope of scale-averagedwavelet power (SAP) to determine the TOF of the guidedultrasonicwave (GUW) signals.
The discretewavelettransform (DWT) is an implementation of the wavelettransformusing a discrete set of waveletscales and translations. The methodoffers a lowcomputationalcomplexity due to the implementation of the waveletfilterbanks. However, itsuffersfrom a serious drawback in thatitis not a time-invariant transform and is a criticallysampledtransform. Owing to this drawback, the DSWT wasdesigned as an extension method to the DWT. The time-invariance propertyisparticularly important in keeping the waveshape [3]. The DSWT has been alsoused in manyworks to estimate the TOF of ultrasonicechoes. Zhong and Oyadiji [7] studied and applied DSWT algorithm to the field of crack detection of beamslike structures to locate the presence of cracks. The authorsfoundthat the DSWT detail coefficient provides a better crack indication compared to the DWT detail coefficient especially in the case of small damages. The envelope of the DSWT detail coefficients wasexploited to estimate the TOF of ultrasonicechoesusing the analyticrepresentation [8]. The analyticrepresentation of a real-valuedfunctionis an analytic signal comprising the original function and its Hilbert transform (HT). The envelope of the analytic signal isobtained by the use of the Hilbert transform [9]. Chen et al. [10] proposed HT approach to identify the arrival time of overlappingultrasonicechoes in a time-of-flight diffraction flawdetection. Ingarocaet al. [11]
used the DSWT and the Hilbert transform for locating and measuring the size of cracks in beams. In the pharmaceuticalindustry, differentcoatingthicknesses of drugtabletscoatedwithKollicoatwereestimatedusingthisenvelop e [8].
In thispaper, twomethodsbased on WT are used to extract the TOF. The continuouswavelettransformbased on the scale-
averaged power (SAP) and the
discretestationarywavelettransform (DSWT).
Thesemethodswereapplied, firstly, on
simulatedGaussianechoes for twopractical cases: without and with partiallyoverlappingechoes. Severalnumerical tests have been carried out for eachmethod, and for each case, to select a suitablemotherwavelet. The performance wasevaluatedthrough the mean square error (MSE) between the estimated and the
reference value of the TOF. The
resultsshowedthatbothmethodsgivelowerror for non- overlappingechoes and a reasonableerrorwereobtained in the case of partiallyoverlappingechoes. An experimental validation wasperformed on a real signal takenfromthermallycoatedsample in order to evaluate the coatingthickness.
II. WAVELET TRANSFORM METHODS A. SAP method
The ContinuousWaveletTransform (CWT) isdefined by the innerproduct of the function and the basis wavelet. The expression of the ContinuousWaveletTransform (CWT) for one-dimensional real signal isdefined as [12]:
CWT( , τ) = 1
√s y(t)ψ ∗ t − τ
s dt (1)
where is the mother wavelet,∗denotes conjugate, ( )is the signal to betransformed, s and are the scale and translation parameters, respectively.
For the discreterealization of the CWT, we deal withsampled versions of the real signal. The signal issampledat a sampling rate and recorded in the vectory (n), described by N sampling points, ( n = 1, 2, …, N ). The CWT of the discrete signal y (n) is defined as the convolution of y (n)with a scaled and translated version ofψ (n)[4].
( , ) = ( ) ∗ ( − ) (2) The envelope of CWT basedscaleaveragedwavelet power (SAP) isdefined as [4] [6]:
( ) = 1
, (3)
= 1, . . . ,
where denote the scale and M is the largestscaleduring CWT.
B. DSWT method
The discrete stationary wavelet transform DSWT is the undecimated version of the discrete wavelet transform [2].
The DSWT procedure is illustrated in Fig.1.The stepj + 1convolves the approximation coefficients at level j with upsampled versions of the appropriate original filters
toproducethe approximation and detail coefficientsat levelj + 1[13, 14].
Fig.1. One-dimensional DSWT [14]
For getting better performances for the estimation of the TOFs, the analytic signal is used in making the envelope of DSWT detail coefficients. An analytic signal is a complex signal, where the real part is the original signal and the imaginary part is its Hilbert transform. For an arbitrary signal
( ), the analytic signal ( )isdefined as
( ) = ( ) + ( ) (4)
wherei is the imaginary unit, and y(t) is its Hilbert transform. The analytic signalz(t)can also be expressed as
( ) = | ( )| ( ) (5)
where |z(t)|is the envelope of the analytic signal and ( )isits phase [15].
In order to extract the envelope of DSWT, the detail coefficient at level j (DSWT (j; t)) is made equal to ( ). Once the two strongest peaks of |z (t)| are found, the TOF could be estimated [8].
III. Results and discussion
The ultrasonic signals are simulated using the Gaussian echo model [16]:
( ) = ( ) cos (2 ( − ) + ∅) + ( )(6) where is the real signal and is a WGN sequence.
, , , , and ∅arethe amplitude, bandwidth factor, arrival time, center frequency and phase, respectively. The studiedsignals are the sum of twoGaussianechoes (6) simulated for twopractical cases: without and with partiallyoverlappingechoes. In each case,the first echoparametersareβ = 1, α = 200(MHz) ,f = 15 MHz, and ∅ = 0 radand the second echoparametersareβ = 0.8, α = 180 (MHz) , f = 14 MHz, and ∅ = 0 rad.The
signalsweresimulated with a sampling frequency of500 MHz and WGN of 60 dB signal-to-noise ratio (SNR).
To createthe two situations, only the arrival time of the second echowasvaried. The arrival time and TOF of the twoechoes in each case are reported in Table I.
Table I. Simulated signals Arrival time(μs)
Without overlapping Partially overlapping
First echo 0.9 0.9
Second echo 1.17 1.02
TOF 0.27 0.12
Fig. 2 shows the simulatedsignals. Fig. 2(a) shows the case of non-overlappingechoeswith a TOF equal to 0.27 µs. Fig.
2(b) shows the case of partiallyoverlappingechoeswith a TOF equal to 0.12 µs.
Table II shows the main TOF estimation resultsobtainedusing the SAP method. This table contains the analyzingwavelet, the actual and estimated TOF, the MSE and the scale. The waveletsincluded in the studybelong to the followingfamilies: Daubechies (db1, db2 and db3), Symlet (sym1 until sym10), and Coiflet (coif1, coif3 and coif5). All otherfamilies are eliminatedbecausetheyprovidedhigherrors. In the case of non-overlappingechoeswhere the actual TOF=0.27 µs, weobtained a good estimation of the TOF usingdifferentmotherwavelets: db1 until db3, sym2 until sym10 and coif1, coif3 and coif5 for rangescalesfrom 7 until 15. The obtained MSE waslower than 0.36 %. Concerning the second case, partiallyoverlappingechoeswhere the actual TOF isequal to 0.12 µs, the best resultwasobtainedusing db1 wavelet for the scale 12 with MSE of 8.6 %. The estimated TOF using sym1 wavelet for scales 12 and 15 wasequal to 0.11 µs with MSE of 9.95 %.
Fig. 3 depicts the SAP envelope of the two cases; the blue line represents the SAP and the red line is its envelope. For the case of non-overlapping echoes, Fig. 3(a) showsthe SAP and
Fig. 2.Simulated signals. (a) Withoutoverlapping echoes with TOF=0.27µs.(b) Partially overlapping echoes with TOF=0.12µs itsenvelopeobtainedusing db1 wavelet for the scale 12.The two envelope peaks are located at 0.90 µs and 1.17 µs respectively. This corresponds to a TOF of 0.26935 µs. Fig.
3(b) shows the SAP and its envelope obtained using db1 wavelet for the scale 12 in the case of partially overlapping echoes. The two peaks in the envelope plot correspond to arrival times of 0.90 µs and 1.01µs respectively. This corresponds to a TOF of 0.11 µs.
Table III shows the main TOF estimation results obtained by the use DSWT method. The analyzing wavelet, the actual and estimated TOF, the MSE and the scale are reported in Table III. The wavelets included in the study belong to the following families: Daubechies (db1, db2 and db4), Symlet (sym1, sym2, sym3, sym5 and sym9). All other families are eliminated because they provided high errors. As can be seen, the estimated TOF in the first case is relatively close to the actual value 0.27 μs. The obtained MSE is inferior at 0.36 % using different mother wavelets: (db1, db2 and db4) and (sym1, sym2, sym3, sym5, sym9) for the level 4. The best estimate of the TOF in the second case was equal to 0.11μs by the use of sym1 wavelet for the level 4.
Table II. SAP Method Results Wavelet Estimated TOF
(µs) Err (%) Scale
Actual TOF = 0,2 µs
db1, db2, db3 0.26935 0.2389 7, 8,9,11,12
sym2, sym3,sym4 0.26935 0.2389 7,8,9,11
sym5, sym6,sym7,
sym8 0.26935 0.2389 10,12
sym9, sym10 0.269354 0.23895 13,14
coif1 , coif3,coif5 0.270970 0.35842 9,11,12,13,14 Actual TOF= 0.12 µs
db1 0.10967 8.6083 12
sym1 0.10806 9.9462 12, 15
Fig. 3. SAP envelope. (a) Withoutoverlapping echoes using db1 wavelet for thescale 12. (b) partially overlapping echoes using db1 wavelet for the scale 12The corresponding MSE is equal to 9.94 %. The estimated TOFusing sym2 waveletfor the level 5 was equalto 0.11μs.The correspondingMSE increased to 10 %.
Fig. 4 shows the DSWT envelope for the two cases. Fig.
4(a) depicts the DSWT envelope of non- overlappingechoesusing db1 wavelet for the level 4. The twostrongestpeaks are locatedat 0.89 µs and 1.16 µs respectively. This corresponds to a TOF of 0.27 µs.Fig. 4(b)
depicts the DSWT envelope of
partiallyoverlappingechoesusing sym1 wavelet for the level 4.
The twopeaks in the envelope plot correspond to arrival times of 0.89 µs and 0.99 µs respectively. This corresponds to a TOF of 0.11 µs.
For the case of partiallyoverlappingechoes the SAP methodis more accuratethan the DSWT method in terms of the MSE. It has foundthat the MSE using the SAP methodislowerthanthatobtainedusing the DSWT method.
Table III. DSWT Method Results
Wavelet Estimated TOF (µs) Err (%) Level Actual TOF = 0,2 µs
db1, db2, db 4 0.269354 0.238948 4
sym1, sym2, sym3, sym5,
sym9 0.270967 0.358422 4
Actual TOF= 0.12 µs
sym1 0.108064 9.946666 4
sym2 0.108000 10 5
Fig. 4.DSWT envelope. (a) Without overlapping echoes using db1 wavelet for the level 4. (b) partially overlapping echoes using sym1 wavelet for the level 4
Experimental test
In order to highlight the effectiveness of the simulatedresults,experimentalmeasurementwascarried out on aknownthickness thermal sprayedcoating. The sprayedcoatingwith the thickness of 350 µmisdeposited onto stainlesssteelsubstrate. The backscatteredechoes are recorded on digital oscilloscope with 0.5 GHz samplingfrequency. The correspondingreflectedsignalsexhibitpartiallyoverlappingecho es as shown inFig. 5(a). The SAP and DSWT methods have been applied and testedwithseveralmotherwavelets. The estimatedthicknessusing the SAP methodwas354 µm witherror of 1.12 %. This resultisobtainedusing db1 wavelet for the scale 12 as shown in Fig. 5 (b). Concerning the DSWT method, the best resultwasobtainedusing sym1 waveletatlevel 4 as shown in Fig. 5(c). The estimatedthickness and MSE were 361.95 µm, 3.41% respectively.
It has been foundthat the estimatedthicknessusing the SAP methodis more accuratethanthatobtainedusing the DSWT method.
Fig. 5. (a) Partially overlapping echoes. (b)SAP approach envelope using db1wavelet for the scale 12. (c) DSWT envelope using sym1wavelet for the level 4
IV. CONCLUSION
This paper presented a wavelet transform based methods to evaluate the TOF of ultrasonic signals. These methods have been investigated by means of tests on both simulated and experimental signal taken from measured coating thickness.
The numerical simulations provided the possibility to select the appropriate mother wavelets as well as the appropriate level of decomposition to analyze partially overlapping echoes during experimental test for measuring coating thicknesses.
Simulation results have shown that in case of non-overlapping echoes, the DSWT and SAP method have given good results with a MSE less than of 0.36 %. However, in case of partially overlapping echoes, an agreement was observed between simulation and experiments. The good result has fallen on those obtained on simulation part. These results showed good benefits of the used wavelet methods for time of flight estimation of ultrasonic signals.
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