Effectiveness of in situ damage localization ... - Semantic Scholar
Effectiveness of in situ damage localization methods using sparse ultrasonic sensor arrays Jennifer E. Michaels School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA 30332-0250 ABSTRACT Sparse ultrasonic arrays spatially distributed over a large area of a structure have been proposed and tested in the laboratory for in situ detection and localization of damage. Detection algorithms are typically based upon comparison to a baseline, where differences not explained by benign environmental effects are interpreted as damage. Most localization methods are either based upon an arrival time analysis of differenced signals or spatial distribution of a damage index. Triangulation and delay-and-sum type methods fall into the first category and, under ideal conditions, can accurately locate discrete damage such as a single crack. Methods in the second category do not rely on precise timing of scattered signals, but are limited in their ability to precisely locate discrete damage using a small number of sensors. This paper evaluates the effectiveness of both types of methods for locating a single site of discrete damage, and considers the degradation in performance resulting from errors in both wave speed and transducer locations. Keywords: Ultrasonics, Structural Health Monitoring, Damage Localization, Damage Index, Sparse Array Imaging
1. INTRODUCTION Guided ultrasonic waves are known to be sensitive to damage in structures supporting their propagation, and their use is well-known for nondestructive inspection [1,2,3]. Guided waves are also under consideration for structural health monitoring because they are one of the few sensing methods that can interrogate large areas with a small number of sensors while still remaining sensitive to damage [4]. Various damage detection algorithms have been proposed, where the challenge is to extract features which are sensitive to damage but not to benign environmental changes [5,6]. Detection algorithms can be based upon signals from either a single sensor acting as transmitter and receiver, or a pair of sensors operating in pitch-catch mode, but for damage localization over a large area, multiple sensors (or sensor pairs) are typically employed. One strategy is to use a spatially compact array of sensors, where both linear [7,8] and circular [9] array geometries have been considered. These arrays are analogous to those used for both medical imaging and nondestructive evaluation with the array being synthetically focused at different angles and distances, typically via delay-and-sum beamforming. Considered here is a spatially distributed, or sparse, array of piezoelectric disc-shaped sensors, where signals are recorded from all possible sensor pairs. This array geometry offers several advantages over a compact array. First, each individual sensor has a very small “footprint,” which simplifies sensor mounting requirements. Second, any defect will be insonified from a variety of directions, and thus is less likely to be oriented unfavorably for detection. And third, rather than relying completely on backscattered energy, signals scattered in the forward direction where the flaw is on or near the direct path of propagation are also possible. A number of methods have been proposed for localizing damage using signals from such a sparse array. All of these methods are differential; that is, they rely upon changes in received signals compared to a baseline, where the baseline is recorded from the undamaged structure. The first of these differential methods is delay-and-sum beamforming, which has been extensively reported in the literature [10,11]. The second method is also of the delay-and-sum type, but is based upon cross correlations between differenced signals. It is referred to as the time-difference-of-arrival method [12,13]. The third method is based upon the expected arrival of energy relative to energy arriving earlier in time; it is referred to as the energy arrival method [14]. The last is the RAPID method, which spatially distributes and sums signal difference coefficients from multiple transducer pairs [15,16].
[email protected]; phone 1-404-894-2994; www.quest.gatech.edu Health Monitoring of Structural and Biological Systems 2008, edited by Tribikram Kundu, Proc. of SPIE Vol. 6935, 693510, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.775788
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Although these methods have been reported upon individually, an overall comparison has not been performed. Doing so is further complicated by the lack of quantitative metrics for characterizing damage localization images. The objective of this paper is to first propose a set of localization metrics, and then to use these metrics to compare the four methods using experimental data from artificial damage in multiple plates.
2. DAMAGE LOCALIZATION METRICS Considered here are images constructed for the purpose of localizing damage using signals from a spatially distributed sensor array. The goal is for damage to appear as a large amplitude, isolated region such as is shown in Figure 1(a). In reality, the image may appear more like that of Figure 1(b) with multiple areas of significant amplitude, or like that of Figure 1(c) where there is a single large region. Several metrics are proposed here for evaluating the efficacy of imaging methods when there is a single known site of discrete damage; multiple damage sites are not considered. 2.1 Geometric Metrics
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The first step in the image analysis process is to segment the image into multiple regions based upon an amplitude threshold, here taken to be 70% of the image peak value. For example, Figure 2 shows the segmented images derived from the images of Figure 1, where the detected regions are shown in white. The first metric is the number of such regions, Nreg, which would be 1, 4 and 1 for the three images of Figure 2.
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Fig. 1. Examples of damage localization images. (a) A single, isolated region, (b) multiple large amplitude regions, and (c) a single large region.
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The second and third metrics are taken from the region that is closest to the actual flaw location; this region is referred to as the flaw region. The distance between the true flaw location and the location of the peak amplitude point in the flaw region is the flaw location error, ǻxerr. The area of this region is the localization area, Aloc. 2.2 Noise Metrics An important aspect of how well a flaw is localized in an image is the signal-to-noise ratio of the localized flaw compared to the background. It is not always straightforward to quantify the signal-to-noise ratio of such an image because the response from the flaw can be “smeared out” over a significant spatial area. Clearly some of this spatial smearing should be considered as a part of the flaw rather than noise, whereas overly extensive smearing should be considered as noise. An attempt is made to take this local smearing into account by excluding all pixels from the noise computations that are within a fixed distance of the flaw region boundary, taken here to be 25 mm (which is about one wavelength). Both the peak amplitude and the average amplitude of the remaining pixels are calculated, and the signalto-noise ratios are determined relative to the peak amplitude of the flaw region,
A
.
SNR peak
flaw , 10 log 10 Apeak Abackground peak
SNR mean
10 log 10
flaw peak
background Amean
(1) (2)
It is assumed that image values are related to energy, hence the factor of 10 in the dB calculations. 2.3 Summary of Metrics To summarize, the following five metrics are computed to evaluate the damage localization efficacy of an image: x
Number of Regions, Nreg
x
Flaw Location Error, ǻxerr
x
Flaw Localization Area, Aloc
x
Signal-to-Peak-Noise Ratio, SNRpeak
x
Signal-to-Mean-Noise Ratio, SNRmean
Small values are desirable for the first three parameters, and large ones for the last two.
3. DAMAGE LOCALIZATION METHODS The four damage localization methods employed here are briefly reviewed. They are: (1) time-of-arrival method, (2) time-difference-of-arrival method, (3) energy arrival method, and (4) RAPID method. All four methods are based upon the signals between pairs of N transducers, where xij(t) is the received signal of interest from transmitter i and receiver j, and yij(t) is the corresponding baseline signal from the undamaged structure. The signal dij(t) = xij(t) – yij(t) is referred to as the differenced signal. The location of the ith transducer is (xi,yi), and the group velocity cg is assumed to be known. For all methods, raw signals are bandpass filtered by convolution with a 225 kHz, 3-cycle Hanningwindowed sinusoid. Additional preprocessing via exponential windowing is performed for some of the algorithms as w' (t )
w(t )e (t t0 ) / td u (t t0 ) .
(3)
Here t0 is the arrival time of the direct echo between two sensors, td is a decay constant, and u(t) is the step function. The purpose of exponential windowing is to suppress unwanted signals which occur later in time, such as from multiple scattering between damage and boundaries. 3.1 Time-of-Arrival Imaging Method The time-of-arrival (TOA) method is delay-and-sum beamforming applied to the differenced signals at every point (x,y) in the image plane [11]. For the results reported here, exponential windowing is performed on the bandpass filtered signals using a decay constant of td = 75 ȝs, and these signals are envelope-detected prior to beamforming. If eij(t) is the envelope-detected signal, the image amplitude at each pixel is
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2
ª N 1 N º «¦ ¦ eij (tij ( x, y ))» , ¬ i 1 j i 1 ¼
ETOA ( x, y )
(4)
where tij(x,y) is the time for the wave to propagate from the transmitter to the imaging point (x,y), and then from (x,y) to the receiver. 3.2 Time-Difference-of-Arrival Imaging Method The time-difference-of-arrival (TDOA) method is delay-and-sum beamforming applied to cross correlations of pairs of signals, where sensors are considered in groups of three with one transmitter and two receivers [12,13]. The difference in the times of arrival between signals received at sensors i and j from transmitter n is
( xi x) 2 ( yi y ) 2 ( x j x) 2 ( y j y ) 2
'tij ( x, y )
cg
.
(5)
If xni,nj(t) is the cross correlation between the differenced signals from transducer pairs n i and n j (transducer n is the transmitter, and transducers i and j are receivers), there should be a peak at ǻtij(x,y) if there is a scatterer at (x,y). Thus, the image amplitude at each pixel is constructed as N N 1
N
¦¦ ¦
ETDOA ( x, y )
xni ,nj ('tij ( x, y )) .
n 1 i 1 j i 1 izn jzn
(6)
Signals are exponentially windowed using a decay constant of td = 75ȝs before computing the cross correlations, and the cross correlations are envelope detected prior to imaging. 3.3 Energy Arrival Method The energy arrival method, originally proposed in [14], is based upon the same calculated arrival times as the TOA method, but the image is computed differently. Let tij(x,y) be the wave travel time from sensor i to point (x,y) to sensor j, and let ǻT be a time window starting at tij(x,y), which is set to 10ȝs for the work reported here. Two signal energies can be computed from the differenced signals as follows: tij ( x , y )
E
cum ij
( x, y )
³d
2 ij
win ij
(t )dt and E
( x, y )
0
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³d
(t )dt .
(7)
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E EA ( x, y )
N 1
N
Eijwin ( x, y ) Eijcum ( x, y )
¦¦E i 1 j i 1
win ij
( x, y ) Eijcum ( x, y )
.
(8)
Additionally, the pixel value is set to zero if EEA is negative. Pixels have a large amplitude if many sensor pairs both have echoes at the computed times and do not have echoes at earlier times. The intent of this method is to effectively localize damage by selectively picking out the first scattered arrivals even if signals are complex because of multiply scattered echoes. 3.4 RAPID Method The RAPID method (acronym for Reconstruction Algorithm for the Probabilistic Inspection of Damage) was first introduced for monitoring of aerospace components [15] and has also been applied to piping [16]. The implementation used here is briefly described. The first step is to compute a signal difference coefficient between the signal xij(t) and the baseline yij(t), which is the drop in the correlation coefficient over a specified time window,
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t 0 'T
³ >x
ij
SDCij
1
@>
@
(t ) P yij (t ) P dt
.
t0 t 0 'T
³ >x
ij
(t ) P
@
2
t 0 'T
³ >y
dt
t0
ij
@
(9)
(t ) P dt 2
t0
The time t0 is that of the direct arrival for each transducer pair, ȝ is the mean of the corresponding signal, and ǻT is a time window. If the signals are identical, the SDC is zero; if the signals are completely out of phase, the SDC achieves its maximum value of 2. Images are generated by spatially distributing each signal difference coefficient on the image plane in an elliptical pattern where the two foci of the ellipse are the two transducer locations. A parameter ȕ controls the size of the ellipse, and the amplitude linearly tapers from its maximum value along the line connecting the ellipse foci to zero on the periphery of the ellipse. The spatial distribution function is defined as
sij ( x, y )
E Rij ( x, y ) for E ! Rij ( x, y ), 1 E
sij ( x, y )
0 otherwise,
(10)
where
Rij ( x, y )
( xi x) 2 ( yi y ) 2 ( x j x) 2 ( y j y ) 2 ( xi x j ) ( yi y j ) 2
.
(11)
2
The image amplitude at each pixel is
E RAPID ( x, y )
N 1
N
¦ ¦ SDC i 1 j i 1
ij
sij ( x, y ) .
(12)
This algorithm is conceptually very simple, as can be seen by Eq. (12), with the ultrasonic response, SDCij, being spatially distributed via a purely geometrical term, sij(x,y). Note that ǻT = 10 ȝs and ȕ = 1.1 for the work reported here.
4. EXPERIMENTS Data were recorded and analyzed from four plate experiments, where the plates were instrumented with a sparse array of sensors and damage was artificially introduced. For all cases, each sensor was a longitudinally polarized, 2.25 MHz PZT disk, 12.5 mm in diameter, housed in either a stainless steel or brass enclosure, and with very little damping. Details of the four plates and the transducer locations are summarized in Table 1 (plate origin is the lower left corner). Table 1. Summary of plates and transducer locations.
Plate No.
Size
Thickness
Transducer Locations (mm)
1
610 mm x 610 mm
0.79 mm
#1: (170.0, 355.0) #2: (355.0, 430.0)
#3: (430.0, 240.0) #4: (240.0, 165.0)
2
610 mm x 610 mm
3.18 mm
#1: (215.9, 375.9) #2: (398.8, 419.1)
#3: (424.2, 205.7) #4: (180.3, 221.0)
3
610 mm x 610 mm
4.76 mm
#1: (184.2, 336.6) #2: (379.4, 412.8)
#3: (454.0, 225.4) #4: (263.5, 401.6)
4
610 mm x 610 mm
4.76 mm
#1: (149.2, 190.5) #2: (315.9, 114.3)
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#3: (469.9, 139.7) #4: (482.6, 444.5)
#5: (292.1, 495.3) #6: (139.7, 419.1)
Signals were recorded between all transducer pairs, with each transducer acting in turn as transmitter and the remaining ones as receivers. A conventional ultrasonic pulser-receiver was used for spike mode transducer excitation and waveform amplification, and a multiplexer switched between transducers on both transmit and receive. Waveforms were digitized at a sampling rate of 25 MHz, and each recorded waveform was the average of 50 signals. Multiple signal-baseline pairs were considered for each plate, where artificial damage was introduced in between recording of the two signals. For all instances considered here, the environmental conditions under which signals and baselines were recorded were well-matched to ensure evaluation of the imaging methods under the best possible conditions. Prior to analyzing the data, the group velocity, cg, was estimated from the times of the direct arrivals. A total of nine images were generated for each imaging method, and the damage conditions for each are summarized in Table 2. As indicated in the comment column, damage was in a variety of locations relative to the convex polygon bounding the sensor array, referred to as the transducer bounding polygon. Table 2. Summary of damage conditions for images generated with each method.
Damage Condition
Plate
Damage Description
Location (mm)
Comment
1
1
6.4 mm diameter through hole
(257.0, 265.0)
Damage inside of transducer bounding polygon
2
1
6.4 mm diameter through hole
(273.0, 390.0)
Damage on edge of transducer bounding polygon
3
2
6 mm diameter through hole (baseline is 1 mm diameter hole)
(355.6, 329.4)
Damage inside of transducer bounding polygon
4
2
6 mm diameter through hole
(279.4, 152.4)
Damage outside of transducer bounding polygon
5
3
6.4 mm diameter through hole
(296.9, 335.0)
Damage inside of transducer bounding polygon
6
3
3.2 mm diameter through hole
(330.2, 180.2)
Damage on edge of transducer bounding polygon
7
3
6.4 mm diameter through hole
(420.7, 377.8)
Damage outside of transducer bounding polygon
8
4
6 mm diameter through hole
(250.8, 381.0)
Damage inside of transducer bounding polygon
9
4
9.5 mm notch starting from through hole at 9 o’clock
(243.1, 381.0)
Damage inside of transducer bounding polygon
5. RESULTS Each of the four methods described in Section 3 was applied to the nine damage conditions summarized in Table 2 for a total of 36 images. For each image there was a single damage condition, and the desired result is an isolated, high intensity region at the damage location. Qualitatively, the best results were obtained from Plate #4 with six transducers (damage conditions #8 and #9), and the worst results were when the damage was outside of the transducer bounding polygon (damage conditions #4 and #7). Representative images are shown in Figures 3 and 4, where Figure 3 is for damage condition #7 (four transducers, damage outside of the transducer bounding polygon) and Figure 4 is for damage condition #8 (six transducers, damage inside of the transducer bounding polygon). All images are shown on a dB scale, unlike those of Figure 1. Note that a threshold of 70% was used to segment the images into regions, which corresponds to a 3.1 dB drop from the peak. Since pixels values resulting from the four imaging methods are all essentially energybased (i.e., constructed by squaring signals), it is appropriate to use the same threshold level for each one.
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Fig. 3. Images from damage condition #7. (a) Time-of-Arrival method, (b) Time-Difference-of-Arrival method, (c) Energy Arrival method, and (d) RAPID method. Circles denote transducers, and the plus denotes the true damage location.
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Fig. 4. Images from damage condition #8. (a) Time-of-Arrival method, (b) Time-Difference-of-Arrival method, (c) Energy Arrival method, and (d) RAPID method. Circles denote transducers, and the plus denotes the true damage location.
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The five damage metrics of Section 2 were calculated for all of the images. Results are shown in Figures 5, 6 and 7 for the geometric parameters (number of regions, flaw location error and flaw localization area), and in Figures 8 and 9 for the noise parameters (signal-to-peak-noise ratio and signal-to-mean-noise ratio). In these figures, the nine bars for each imaging method correspond to the nine damage conditions of Table 2. The median and mean values of all five metrics for each imaging method are summarized in Table 3.
Number of Regions Over Threshold
5
4
3
2
1
0
TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 5. Number of regions exceeding a 70% threshold for each damage condition and imaging method.
Flaw Localization Error (mm)
140 120 100 80 60 40 20 0
TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 6. Flaw location error for each damage condition and imaging method.
4
x 10
Localization Area (mm2)
3
2
1
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TOA Method
TDOA Method
Energy Arrival Method
Fig. 7. Flaw localization area for each damage condition and imaging method.
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RAPID Method
Signal-to-Peak-Noise Ratio (dB)
20
15
10
5
0 TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 8. Signal-to-peak-noise ratio for each damage condition and imaging method.
Signal-to-Mean-Noise Ratio (dB)
35 30 25 20 15 10 5 0
TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 9. Signal-to-mean-noise ratio for each damage condition and imaging method.
Table 3. Median (left) and mean (right) values of the flaw localization metrics for each imaging method.
Flaw Localization Metric Imaging Method
No. of Regions
Flaw Location Error (mm)
Localization Area (mm2)
Signal-to-PeakNoise Ratio (dB)
Signal-to-MeanNoise Ratio (dB)
Time-of-Arrival
1 / 1.0
4.2 / 4.9
1552 / 1797
2.5 / 2.6
11.5 / 10.9
Time-Differenceof-Arrival
1 / 1.3
4.7 / 6.3
4784 / 8213
2.0 / 2.0
4.5 / 4.4
Energy Arrival
1 / 2.0
9.2 / 8.9
3504 / 3399
1.6 / 1.7
14.9 / 15.5
RAPID
1 / 1.1
68.9 / 71.9
8128 / 10965
4.2 / 7.3
19.5 / 20.4
The effects of errors in both group velocity, Cg, and transducer positions were evaluated for the images shown in Figure 4, which were generated from damage condition #8. For Cg, the nominal value is 5.1524 mm/ȝs, which was determined experimentally. It was systematically varied by +/- 10%, as shown in Figure 10(a), and the change in peak amplitude as a function of Cg is plotted in Figure 10(b). Four damage localization metrics (Xerr, Aloc, SNRpk and SNRmn) are also shown as a function of Cg in Figures 10(c), 10(d), 10(e) and 10(f). Transducer positions were systematically varied by linearly interpolating all six x and y coordinates between their nominal values and new (erroneous) values; both sets of coordinates and the paths of each transducer are illustrated in Figure 11(a). The change in peak amplitude is plotted in Figure 11(b), and the four damage localization metrics are shown in Figures 11(c), 11(d), 11(e) and 11(f).
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Fig. 10. (a) Group velocities used for imaging; that of sequence no. 5 is nominally correct. Effect on (b) peak amplitude, (c) flaw location error, (d) localization area, (e) signal-to-peak-noise ratio, and (f) signal-to-mean-noise ratio. 600
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Fig. 11. (a) Actual (circles) and erroneous (squares) transducer locations; as the transducer fractional position varies from 0 to 1, transducer locations for imaging move from the circles to the squares. Effect on (b) peak amplitude, (c) flaw location error, (d) localization area, (e) signal-to-peak-noise ratio, and (f) signal-to-mean-noise ratio.
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6. DISCUSSION Damage localization ability is first assessed by comparing performance under near-ideal conditions. The five performance metrics for the four imaging methods are shown in Figures 5-9 and Table 3, and are discussed as follows: Number of Regions. All of the methods except for the Energy Arrival method perform reasonably well in terms of numbers of regions. The only anomalous situation with the TDOA method was reporting of four regions for damage condition #4; for this condition the damage was outside of the transducer bounding polygon. The RAPID method also reported more than one region for this damage condition, which is not surprising because it is inherently unable to localize damage outside of the area spanned by all of the elliptical distribution functions. Flaw Location Error. The TOA, TDOA and Energy Arrival methods were all able to consistently locate the flaw within about 10 mm. The RAPID method, except for damage condition #8, was relatively ineffective at determining the flaw location. Flaw Localization Area. The TOA and Energy Arrival methods had the smallest flaw localization areas. Except for damage condition #7, where the flaw was outside of the transducer bounding polygon, the TDOA method had reasonable localization ability. The RAPID method was the worst performer. The apparently good performance on damage condition #4 is misleading; the damage, which is well outside of the bounding polygon, was not localized at all. Signal-to-Peak-Noise Ratio. The RAPID method has the best signal-to-peak-noise ratio performance, which is not surprising because of how it works. RAPID images with only a few transducers tend to be dominated by one large region, and are very clean away from it. The Energy Arrival method performs the worst in terms of this metric, with one instance of a negative signal-to-peak-noise ratio, indicating that there was a region with a higher amplitude than the flaw region. The TOA method performs slightly better than the TDOA method, but neither have a very large signal-to-peaknoise ratio, with only 2 to 3 dB separating the flaw from imaging artifacts. Signal-to-Mean-Noise Ratio. The RAPID and Energy Arrival methods are the best performers in this category. Both methods produce images that are identically zero over significant portions of the image area, which results in a high signal-to-mean-noise ratio. The TOA method performs significantly better than the TDOA method, which is readily seen in the higher amplitude background level of the TDOA images in Figures 3(b) and 4(b). Based upon imaging under near-ideal conditions, the TOA method offers the best combination of localization ability and signal-to-noise ratio. The other three methods all have clear weaknesses that adversely affect their performance. It is also instructive to consider the sensitivity of the various methods to errors in imaging parameters. Considered here are errors in both group velocity and transducer locations as shown in Figures 10 and 11. The performance of each method under these non-ideal conditions is discussed as follows: Time-of-Arrival Method. The TOA method is generally well-behaved when the group velocity is varied. All five metrics demonstrate a clear maximum (or minimum) as the group velocity passes through its true value; the flaw location error is the most sensitive. The performance of this method is similarly well-behaved when transducer positions are varied; all metrics degrade in a regular manner. Time-Difference-of-Arrival-Method. The TDOA method demonstrates almost no sensitivity to the changes in group velocity, and it performs similarly to the TOA method in regards to transducer position error. Energy Arrival Method. The Energy Arrival method is also well-behaved regarding group velocity changes, showing more sensitivity than the TDOA method but not as much as the TOA method. It is also very tolerant of transducer position errors, performing similarly to both the TOA and TDOA methods. RAPID Method. At first look, it would appear that the RAPID method should not be at all sensitive to group velocity since arrival times are not part of the algorithm. However, as implemented here, the time windows for computing the signal difference coefficients depend on the calculated times of the direct arrivals. Thus, this method exhibits somewhat unsystematic behavior as errors are introduced into both the group velocity and the transducer positions; this behavior results from time windows moving to different portions of the signals, which can include noise before the first arrival. The TOA, TDOA and Energy Arrival methods can all tolerate both types of variations while maintaining reasonable, although somewhat degraded, performance. The TOA method exhibits the most sensitivity to group velocity errors, and the TDOA method the least; performance of these three methods is similar for errors in transducer locations.
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7. SUMMARY AND CONCLUSIONS The work shown here compares the efficacy of four different damage localization methods based upon signals recorded from spatially distributed transducer arrays. The time-of-arrival method is generally the most effective with both good localization ability and reasonable signal-to-noise ratio. It is also reasonably well-behaved when there are errors in group velocity and transducer locations. The time-difference-of-arrival method, although not quite as effective as the time-of-arrival method, is the most robust to errors in group velocity. The energy arrival and RAPID methods, which both have inherently good signal-to-mean-noise ratios, do not otherwise perform as well. The RAPID method was developed based upon more transducers than were considered here (8 to 16 rather than 4 or 6), and it is not surprising that its performance degrades significantly with fewer transducers. All four methods have their strengths and weaknesses, and combining images from multiple methods using various fusion strategies should be considered.
ACKNOWLEDGEMENTS The support of the National Science Foundation under contract number ECS-0401213 is gratefully acknowledged.
REFERENCES [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16]
D. N. Alleyne and P. Cawley, “The interaction of Lamb waves with defects,” IEEE Transactions on Ultrasonics, Ferrolectrics, and Frequency Control, 39(3), 381-397 (1992). T. Ghosh, T. Kundu and P. Karpur, “Efficient use of Lamb modes for detecting defects in large plates,” Ultrasonics, 36, 791-801 (1998). J. L. Rose, “A baseline and vision of ultrasonic guided wave inspection potential,” Journal of Pressure Vessel Technology, 124, 273-282 (2002). P. Cawley and F. Simonetti, “Structural health monitoring using guided waves – potential and challenges,” Proceedings of the 5th International Conference on SHM, DEStech Publications, 503-510 (2005). A. J. Croxford, Y. Lu, P. D. Wilcox, J. E. Michaels and B. W. Drinkwater, “A comparison of temperature compensation methods for guided wave structural health monitoring,” Review of Progress in QNDE, 27, Presented July 2007, Golden, Colorado, Proceedings in press. Y. Lu and J. E. Michaels, “Consideration of surface variations on ultrasonic structural health monitoring,” Proceedings of the 6th International Workshop on SHM, DEStech Publications, 1275-1282 (2007). V. Giurgiutiu and J. Bao, “Embedded-ultrasonics structural radar for in situ structural health monitoring of thin-wall structures,” Structural Health Monitoring – An International Journal, 3(2), 121-140 (2004). A. S. Purekar, D. J. Pines, S. Sundararaman and D. E. Adams, “Directional piezoelectric phased array filters for detecting damage in isotropic plates,” Smart Materials and Structures, 13, 838-850 (2004). P. Wilcox, “Omni-directional guided wave transducer arrays for the rapid inspection of large areas of plate structures,” IEEE Transactions on Ultrasonics, Ferrolectrics, and Frequency Control, 50(6), 699–709, (2003). C. H. Wang, J. T. Rose and F.-K. Chang, “A synthetic time-reversal imaging method for structural health monitoring,” Smart Materials and Structures, 13, 415-423 (2004). J. E. Michaels and T. E. Michaels, “Guided wave signal processing and image fusion for in situ damage localization in plates,” Wave Motion, 44, 482-492 (2007). J. E. Michaels, A. J. Croxford and P. D. Wilcox, “Imaging algorithms for locating damage via in situ ultrasonic sensors,” Proceedings of the 2008 IEEE Sensors Applications Symposium, 63-67 (2008). A. J. Croxford, P. D. Wilcox and B. W. Drinkwater, “Guided wave SHM with a distributed sensor network,” Proceedings of the SPIE, 6935, 693514:1-9 (2008). J. E. Michaels and T. E. Michaels, “Damage localization in inhomogeneous plates using a sparse array of ultrasonic transducers,” Review of Progress in QNDE, 26A, 846-853 (2007). X. Zhao, H. Gao, G. Zhang, B. Ayhan, F. Yan, C. Kwan and J. L. Rose, “Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network: I. Defect detection, localization and growth monitoring,” Smart Materials and Structures, 16, 1208-1217 (2007). J. K. Van Velsor, H. Gao and J. L. Rose, “Guided-wave tomographic imaging of defects in pipe using a probabilistic reconstruction algorithm,” Insight, 49(9), 532-537 (2007).
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1. INTRODUCTION Guided ultrasonic waves are known to be sensitive to damage in structures supporting their propagation, and their use is well-known for nondestructive inspection [1,2,3]. Guided waves are also under consideration for structural health monitoring because they are one of the few sensing methods that can interrogate large areas with a small number of sensors while still remaining sensitive to damage [4]. Various damage detection algorithms have been proposed, where the challenge is to extract features which are sensitive to damage but not to benign environmental changes [5,6]. Detection algorithms can be based upon signals from either a single sensor acting as transmitter and receiver, or a pair of sensors operating in pitch-catch mode, but for damage localization over a large area, multiple sensors (or sensor pairs) are typically employed. One strategy is to use a spatially compact array of sensors, where both linear [7,8] and circular [9] array geometries have been considered. These arrays are analogous to those used for both medical imaging and nondestructive evaluation with the array being synthetically focused at different angles and distances, typically via delay-and-sum beamforming. Considered here is a spatially distributed, or sparse, array of piezoelectric disc-shaped sensors, where signals are recorded from all possible sensor pairs. This array geometry offers several advantages over a compact array. First, each individual sensor has a very small “footprint,” which simplifies sensor mounting requirements. Second, any defect will be insonified from a variety of directions, and thus is less likely to be oriented unfavorably for detection. And third, rather than relying completely on backscattered energy, signals scattered in the forward direction where the flaw is on or near the direct path of propagation are also possible. A number of methods have been proposed for localizing damage using signals from such a sparse array. All of these methods are differential; that is, they rely upon changes in received signals compared to a baseline, where the baseline is recorded from the undamaged structure. The first of these differential methods is delay-and-sum beamforming, which has been extensively reported in the literature [10,11]. The second method is also of the delay-and-sum type, but is based upon cross correlations between differenced signals. It is referred to as the time-difference-of-arrival method [12,13]. The third method is based upon the expected arrival of energy relative to energy arriving earlier in time; it is referred to as the energy arrival method [14]. The last is the RAPID method, which spatially distributes and sums signal difference coefficients from multiple transducer pairs [15,16].
[email protected]; phone 1-404-894-2994; www.quest.gatech.edu Health Monitoring of Structural and Biological Systems 2008, edited by Tribikram Kundu, Proc. of SPIE Vol. 6935, 693510, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.775788
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Although these methods have been reported upon individually, an overall comparison has not been performed. Doing so is further complicated by the lack of quantitative metrics for characterizing damage localization images. The objective of this paper is to first propose a set of localization metrics, and then to use these metrics to compare the four methods using experimental data from artificial damage in multiple plates.
2. DAMAGE LOCALIZATION METRICS Considered here are images constructed for the purpose of localizing damage using signals from a spatially distributed sensor array. The goal is for damage to appear as a large amplitude, isolated region such as is shown in Figure 1(a). In reality, the image may appear more like that of Figure 1(b) with multiple areas of significant amplitude, or like that of Figure 1(c) where there is a single large region. Several metrics are proposed here for evaluating the efficacy of imaging methods when there is a single known site of discrete damage; multiple damage sites are not considered. 2.1 Geometric Metrics
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The first step in the image analysis process is to segment the image into multiple regions based upon an amplitude threshold, here taken to be 70% of the image peak value. For example, Figure 2 shows the segmented images derived from the images of Figure 1, where the detected regions are shown in white. The first metric is the number of such regions, Nreg, which would be 1, 4 and 1 for the three images of Figure 2.
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Fig. 1. Examples of damage localization images. (a) A single, isolated region, (b) multiple large amplitude regions, and (c) a single large region.
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The second and third metrics are taken from the region that is closest to the actual flaw location; this region is referred to as the flaw region. The distance between the true flaw location and the location of the peak amplitude point in the flaw region is the flaw location error, ǻxerr. The area of this region is the localization area, Aloc. 2.2 Noise Metrics An important aspect of how well a flaw is localized in an image is the signal-to-noise ratio of the localized flaw compared to the background. It is not always straightforward to quantify the signal-to-noise ratio of such an image because the response from the flaw can be “smeared out” over a significant spatial area. Clearly some of this spatial smearing should be considered as a part of the flaw rather than noise, whereas overly extensive smearing should be considered as noise. An attempt is made to take this local smearing into account by excluding all pixels from the noise computations that are within a fixed distance of the flaw region boundary, taken here to be 25 mm (which is about one wavelength). Both the peak amplitude and the average amplitude of the remaining pixels are calculated, and the signalto-noise ratios are determined relative to the peak amplitude of the flaw region,
A
.
SNR peak
flaw , 10 log 10 Apeak Abackground peak
SNR mean
10 log 10
flaw peak
background Amean
(1) (2)
It is assumed that image values are related to energy, hence the factor of 10 in the dB calculations. 2.3 Summary of Metrics To summarize, the following five metrics are computed to evaluate the damage localization efficacy of an image: x
Number of Regions, Nreg
x
Flaw Location Error, ǻxerr
x
Flaw Localization Area, Aloc
x
Signal-to-Peak-Noise Ratio, SNRpeak
x
Signal-to-Mean-Noise Ratio, SNRmean
Small values are desirable for the first three parameters, and large ones for the last two.
3. DAMAGE LOCALIZATION METHODS The four damage localization methods employed here are briefly reviewed. They are: (1) time-of-arrival method, (2) time-difference-of-arrival method, (3) energy arrival method, and (4) RAPID method. All four methods are based upon the signals between pairs of N transducers, where xij(t) is the received signal of interest from transmitter i and receiver j, and yij(t) is the corresponding baseline signal from the undamaged structure. The signal dij(t) = xij(t) – yij(t) is referred to as the differenced signal. The location of the ith transducer is (xi,yi), and the group velocity cg is assumed to be known. For all methods, raw signals are bandpass filtered by convolution with a 225 kHz, 3-cycle Hanningwindowed sinusoid. Additional preprocessing via exponential windowing is performed for some of the algorithms as w' (t )
w(t )e (t t0 ) / td u (t t0 ) .
(3)
Here t0 is the arrival time of the direct echo between two sensors, td is a decay constant, and u(t) is the step function. The purpose of exponential windowing is to suppress unwanted signals which occur later in time, such as from multiple scattering between damage and boundaries. 3.1 Time-of-Arrival Imaging Method The time-of-arrival (TOA) method is delay-and-sum beamforming applied to the differenced signals at every point (x,y) in the image plane [11]. For the results reported here, exponential windowing is performed on the bandpass filtered signals using a decay constant of td = 75 ȝs, and these signals are envelope-detected prior to beamforming. If eij(t) is the envelope-detected signal, the image amplitude at each pixel is
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2
ª N 1 N º «¦ ¦ eij (tij ( x, y ))» , ¬ i 1 j i 1 ¼
ETOA ( x, y )
(4)
where tij(x,y) is the time for the wave to propagate from the transmitter to the imaging point (x,y), and then from (x,y) to the receiver. 3.2 Time-Difference-of-Arrival Imaging Method The time-difference-of-arrival (TDOA) method is delay-and-sum beamforming applied to cross correlations of pairs of signals, where sensors are considered in groups of three with one transmitter and two receivers [12,13]. The difference in the times of arrival between signals received at sensors i and j from transmitter n is
( xi x) 2 ( yi y ) 2 ( x j x) 2 ( y j y ) 2
'tij ( x, y )
cg
.
(5)
If xni,nj(t) is the cross correlation between the differenced signals from transducer pairs n i and n j (transducer n is the transmitter, and transducers i and j are receivers), there should be a peak at ǻtij(x,y) if there is a scatterer at (x,y). Thus, the image amplitude at each pixel is constructed as N N 1
N
¦¦ ¦
ETDOA ( x, y )
xni ,nj ('tij ( x, y )) .
n 1 i 1 j i 1 izn jzn
(6)
Signals are exponentially windowed using a decay constant of td = 75ȝs before computing the cross correlations, and the cross correlations are envelope detected prior to imaging. 3.3 Energy Arrival Method The energy arrival method, originally proposed in [14], is based upon the same calculated arrival times as the TOA method, but the image is computed differently. Let tij(x,y) be the wave travel time from sensor i to point (x,y) to sensor j, and let ǻT be a time window starting at tij(x,y), which is set to 10ȝs for the work reported here. Two signal energies can be computed from the differenced signals as follows: tij ( x , y )
E
cum ij
( x, y )
³d
2 ij
win ij
(t )dt and E
( x, y )
0
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³d
(t )dt .
(7)
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E EA ( x, y )
N 1
N
Eijwin ( x, y ) Eijcum ( x, y )
¦¦E i 1 j i 1
win ij
( x, y ) Eijcum ( x, y )
.
(8)
Additionally, the pixel value is set to zero if EEA is negative. Pixels have a large amplitude if many sensor pairs both have echoes at the computed times and do not have echoes at earlier times. The intent of this method is to effectively localize damage by selectively picking out the first scattered arrivals even if signals are complex because of multiply scattered echoes. 3.4 RAPID Method The RAPID method (acronym for Reconstruction Algorithm for the Probabilistic Inspection of Damage) was first introduced for monitoring of aerospace components [15] and has also been applied to piping [16]. The implementation used here is briefly described. The first step is to compute a signal difference coefficient between the signal xij(t) and the baseline yij(t), which is the drop in the correlation coefficient over a specified time window,
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t 0 'T
³ >x
ij
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1
@>
@
(t ) P yij (t ) P dt
.
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³ >x
ij
(t ) P
@
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³ >y
dt
t0
ij
@
(9)
(t ) P dt 2
t0
The time t0 is that of the direct arrival for each transducer pair, ȝ is the mean of the corresponding signal, and ǻT is a time window. If the signals are identical, the SDC is zero; if the signals are completely out of phase, the SDC achieves its maximum value of 2. Images are generated by spatially distributing each signal difference coefficient on the image plane in an elliptical pattern where the two foci of the ellipse are the two transducer locations. A parameter ȕ controls the size of the ellipse, and the amplitude linearly tapers from its maximum value along the line connecting the ellipse foci to zero on the periphery of the ellipse. The spatial distribution function is defined as
sij ( x, y )
E Rij ( x, y ) for E ! Rij ( x, y ), 1 E
sij ( x, y )
0 otherwise,
(10)
where
Rij ( x, y )
( xi x) 2 ( yi y ) 2 ( x j x) 2 ( y j y ) 2 ( xi x j ) ( yi y j ) 2
.
(11)
2
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E RAPID ( x, y )
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¦ ¦ SDC i 1 j i 1
ij
sij ( x, y ) .
(12)
This algorithm is conceptually very simple, as can be seen by Eq. (12), with the ultrasonic response, SDCij, being spatially distributed via a purely geometrical term, sij(x,y). Note that ǻT = 10 ȝs and ȕ = 1.1 for the work reported here.
4. EXPERIMENTS Data were recorded and analyzed from four plate experiments, where the plates were instrumented with a sparse array of sensors and damage was artificially introduced. For all cases, each sensor was a longitudinally polarized, 2.25 MHz PZT disk, 12.5 mm in diameter, housed in either a stainless steel or brass enclosure, and with very little damping. Details of the four plates and the transducer locations are summarized in Table 1 (plate origin is the lower left corner). Table 1. Summary of plates and transducer locations.
Plate No.
Size
Thickness
Transducer Locations (mm)
1
610 mm x 610 mm
0.79 mm
#1: (170.0, 355.0) #2: (355.0, 430.0)
#3: (430.0, 240.0) #4: (240.0, 165.0)
2
610 mm x 610 mm
3.18 mm
#1: (215.9, 375.9) #2: (398.8, 419.1)
#3: (424.2, 205.7) #4: (180.3, 221.0)
3
610 mm x 610 mm
4.76 mm
#1: (184.2, 336.6) #2: (379.4, 412.8)
#3: (454.0, 225.4) #4: (263.5, 401.6)
4
610 mm x 610 mm
4.76 mm
#1: (149.2, 190.5) #2: (315.9, 114.3)
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#3: (469.9, 139.7) #4: (482.6, 444.5)
#5: (292.1, 495.3) #6: (139.7, 419.1)
Signals were recorded between all transducer pairs, with each transducer acting in turn as transmitter and the remaining ones as receivers. A conventional ultrasonic pulser-receiver was used for spike mode transducer excitation and waveform amplification, and a multiplexer switched between transducers on both transmit and receive. Waveforms were digitized at a sampling rate of 25 MHz, and each recorded waveform was the average of 50 signals. Multiple signal-baseline pairs were considered for each plate, where artificial damage was introduced in between recording of the two signals. For all instances considered here, the environmental conditions under which signals and baselines were recorded were well-matched to ensure evaluation of the imaging methods under the best possible conditions. Prior to analyzing the data, the group velocity, cg, was estimated from the times of the direct arrivals. A total of nine images were generated for each imaging method, and the damage conditions for each are summarized in Table 2. As indicated in the comment column, damage was in a variety of locations relative to the convex polygon bounding the sensor array, referred to as the transducer bounding polygon. Table 2. Summary of damage conditions for images generated with each method.
Damage Condition
Plate
Damage Description
Location (mm)
Comment
1
1
6.4 mm diameter through hole
(257.0, 265.0)
Damage inside of transducer bounding polygon
2
1
6.4 mm diameter through hole
(273.0, 390.0)
Damage on edge of transducer bounding polygon
3
2
6 mm diameter through hole (baseline is 1 mm diameter hole)
(355.6, 329.4)
Damage inside of transducer bounding polygon
4
2
6 mm diameter through hole
(279.4, 152.4)
Damage outside of transducer bounding polygon
5
3
6.4 mm diameter through hole
(296.9, 335.0)
Damage inside of transducer bounding polygon
6
3
3.2 mm diameter through hole
(330.2, 180.2)
Damage on edge of transducer bounding polygon
7
3
6.4 mm diameter through hole
(420.7, 377.8)
Damage outside of transducer bounding polygon
8
4
6 mm diameter through hole
(250.8, 381.0)
Damage inside of transducer bounding polygon
9
4
9.5 mm notch starting from through hole at 9 o’clock
(243.1, 381.0)
Damage inside of transducer bounding polygon
5. RESULTS Each of the four methods described in Section 3 was applied to the nine damage conditions summarized in Table 2 for a total of 36 images. For each image there was a single damage condition, and the desired result is an isolated, high intensity region at the damage location. Qualitatively, the best results were obtained from Plate #4 with six transducers (damage conditions #8 and #9), and the worst results were when the damage was outside of the transducer bounding polygon (damage conditions #4 and #7). Representative images are shown in Figures 3 and 4, where Figure 3 is for damage condition #7 (four transducers, damage outside of the transducer bounding polygon) and Figure 4 is for damage condition #8 (six transducers, damage inside of the transducer bounding polygon). All images are shown on a dB scale, unlike those of Figure 1. Note that a threshold of 70% was used to segment the images into regions, which corresponds to a 3.1 dB drop from the peak. Since pixels values resulting from the four imaging methods are all essentially energybased (i.e., constructed by squaring signals), it is appropriate to use the same threshold level for each one.
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The five damage metrics of Section 2 were calculated for all of the images. Results are shown in Figures 5, 6 and 7 for the geometric parameters (number of regions, flaw location error and flaw localization area), and in Figures 8 and 9 for the noise parameters (signal-to-peak-noise ratio and signal-to-mean-noise ratio). In these figures, the nine bars for each imaging method correspond to the nine damage conditions of Table 2. The median and mean values of all five metrics for each imaging method are summarized in Table 3.
Number of Regions Over Threshold
5
4
3
2
1
0
TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 5. Number of regions exceeding a 70% threshold for each damage condition and imaging method.
Flaw Localization Error (mm)
140 120 100 80 60 40 20 0
TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 6. Flaw location error for each damage condition and imaging method.
4
x 10
Localization Area (mm2)
3
2
1
0
TOA Method
TDOA Method
Energy Arrival Method
Fig. 7. Flaw localization area for each damage condition and imaging method.
Proc. of SPIE Vol. 6935 693510-8
RAPID Method
Signal-to-Peak-Noise Ratio (dB)
20
15
10
5
0 TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 8. Signal-to-peak-noise ratio for each damage condition and imaging method.
Signal-to-Mean-Noise Ratio (dB)
35 30 25 20 15 10 5 0
TOA Method
TDOA Method
Energy Arrival Method
RAPID Method
Fig. 9. Signal-to-mean-noise ratio for each damage condition and imaging method.
Table 3. Median (left) and mean (right) values of the flaw localization metrics for each imaging method.
Flaw Localization Metric Imaging Method
No. of Regions
Flaw Location Error (mm)
Localization Area (mm2)
Signal-to-PeakNoise Ratio (dB)
Signal-to-MeanNoise Ratio (dB)
Time-of-Arrival
1 / 1.0
4.2 / 4.9
1552 / 1797
2.5 / 2.6
11.5 / 10.9
Time-Differenceof-Arrival
1 / 1.3
4.7 / 6.3
4784 / 8213
2.0 / 2.0
4.5 / 4.4
Energy Arrival
1 / 2.0
9.2 / 8.9
3504 / 3399
1.6 / 1.7
14.9 / 15.5
RAPID
1 / 1.1
68.9 / 71.9
8128 / 10965
4.2 / 7.3
19.5 / 20.4
The effects of errors in both group velocity, Cg, and transducer positions were evaluated for the images shown in Figure 4, which were generated from damage condition #8. For Cg, the nominal value is 5.1524 mm/ȝs, which was determined experimentally. It was systematically varied by +/- 10%, as shown in Figure 10(a), and the change in peak amplitude as a function of Cg is plotted in Figure 10(b). Four damage localization metrics (Xerr, Aloc, SNRpk and SNRmn) are also shown as a function of Cg in Figures 10(c), 10(d), 10(e) and 10(f). Transducer positions were systematically varied by linearly interpolating all six x and y coordinates between their nominal values and new (erroneous) values; both sets of coordinates and the paths of each transducer are illustrated in Figure 11(a). The change in peak amplitude is plotted in Figure 11(b), and the four damage localization metrics are shown in Figures 11(c), 11(d), 11(e) and 11(f).
Proc. of SPIE Vol. 6935 693510-9
(b)
5.2 5 4.8
0
2
4 6 Sequence Number
8
-1 -2 TOA TDOA EAR RAPID
-3
-5 4.6
10
4.8
90 TOA TDOA EAR RAPID
80 70 60 50 40 30 20 10
5 5.2 5.4 Group Velocity (mm/Ps)
0 4.6
5.6
4.8
5 5.2 5.4 Group Velocity (mm/Ps)
5.6
4
TOA TDOA EAR RAPID
1.5
1
0.5
0 4.6
4.8
5 5.2 5.4 Group Velocity (mm/Ps)
6 5 4 3 2 1 0 4.6
5.6
(f)
TOA TDOA EAR RAPID
Signal-to-Mean-Noise Ratio (dB)
2
Signal-to-Peak-Noise Ratio (dB)
(e)
2
Localization Area (mm )
0
-4
x 10
(d)
(c) Flaw Location Error (mm)
5.4
4.6
2 1
Amplitude Change (dB)
Group Velocity (mm/Ps)
(a)5.6
4.8
5 5.2 5.4 Group Velocity (mm/Ps)
30 TOA TDOA EAR RAPID
25 20 15 10 5 0 4.6
5.6
4.8
5 5.2 5.4 Group Velocity (mm/Ps)
5.6
Fig. 10. (a) Group velocities used for imaging; that of sequence no. 5 is nominally correct. Effect on (b) peak amplitude, (c) flaw location error, (d) localization area, (e) signal-to-peak-noise ratio, and (f) signal-to-mean-noise ratio. 600
(b) Amplitude Change (dB)
500
mm
400 300 200 100 0
6
(c) 50
TOA TDOA EAR RAPID
5 4
Flaw Location Error (mm)
(a)
3 2 1 0 -1
TOA TDOA EAR RAPID
40 30 20 10
-2 0
200
400
-3
600
0
mm
0.2 0.4 0.6 0.8 Transducer Fractional Position
0
1
0
0.2 0.4 0.6 0.8 Transducer Fractional Position
1
4
(e) Signal-to-Peak-Noise Ratio (dB)
2
Localization Area (mm )
2
1.5
TOA TDOA EAR RAPID
1
0.5
0
0
0.2 0.4 0.6 0.8 Transducer Fractional Position
1
6
(f)
TOA TDOA EAR RAPID
5 4
Signal-to-Mean-Noise Ratio (dB)
x 10
(d)
3 2 1 0
0
0.2 0.4 0.6 0.8 Transducer Fractional Position
1
30 TOA TDOA EAR RAPID
25 20 15 10 5 0
0
0.2 0.4 0.6 0.8 Transducer Fractional Position
Fig. 11. (a) Actual (circles) and erroneous (squares) transducer locations; as the transducer fractional position varies from 0 to 1, transducer locations for imaging move from the circles to the squares. Effect on (b) peak amplitude, (c) flaw location error, (d) localization area, (e) signal-to-peak-noise ratio, and (f) signal-to-mean-noise ratio.
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6. DISCUSSION Damage localization ability is first assessed by comparing performance under near-ideal conditions. The five performance metrics for the four imaging methods are shown in Figures 5-9 and Table 3, and are discussed as follows: Number of Regions. All of the methods except for the Energy Arrival method perform reasonably well in terms of numbers of regions. The only anomalous situation with the TDOA method was reporting of four regions for damage condition #4; for this condition the damage was outside of the transducer bounding polygon. The RAPID method also reported more than one region for this damage condition, which is not surprising because it is inherently unable to localize damage outside of the area spanned by all of the elliptical distribution functions. Flaw Location Error. The TOA, TDOA and Energy Arrival methods were all able to consistently locate the flaw within about 10 mm. The RAPID method, except for damage condition #8, was relatively ineffective at determining the flaw location. Flaw Localization Area. The TOA and Energy Arrival methods had the smallest flaw localization areas. Except for damage condition #7, where the flaw was outside of the transducer bounding polygon, the TDOA method had reasonable localization ability. The RAPID method was the worst performer. The apparently good performance on damage condition #4 is misleading; the damage, which is well outside of the bounding polygon, was not localized at all. Signal-to-Peak-Noise Ratio. The RAPID method has the best signal-to-peak-noise ratio performance, which is not surprising because of how it works. RAPID images with only a few transducers tend to be dominated by one large region, and are very clean away from it. The Energy Arrival method performs the worst in terms of this metric, with one instance of a negative signal-to-peak-noise ratio, indicating that there was a region with a higher amplitude than the flaw region. The TOA method performs slightly better than the TDOA method, but neither have a very large signal-to-peaknoise ratio, with only 2 to 3 dB separating the flaw from imaging artifacts. Signal-to-Mean-Noise Ratio. The RAPID and Energy Arrival methods are the best performers in this category. Both methods produce images that are identically zero over significant portions of the image area, which results in a high signal-to-mean-noise ratio. The TOA method performs significantly better than the TDOA method, which is readily seen in the higher amplitude background level of the TDOA images in Figures 3(b) and 4(b). Based upon imaging under near-ideal conditions, the TOA method offers the best combination of localization ability and signal-to-noise ratio. The other three methods all have clear weaknesses that adversely affect their performance. It is also instructive to consider the sensitivity of the various methods to errors in imaging parameters. Considered here are errors in both group velocity and transducer locations as shown in Figures 10 and 11. The performance of each method under these non-ideal conditions is discussed as follows: Time-of-Arrival Method. The TOA method is generally well-behaved when the group velocity is varied. All five metrics demonstrate a clear maximum (or minimum) as the group velocity passes through its true value; the flaw location error is the most sensitive. The performance of this method is similarly well-behaved when transducer positions are varied; all metrics degrade in a regular manner. Time-Difference-of-Arrival-Method. The TDOA method demonstrates almost no sensitivity to the changes in group velocity, and it performs similarly to the TOA method in regards to transducer position error. Energy Arrival Method. The Energy Arrival method is also well-behaved regarding group velocity changes, showing more sensitivity than the TDOA method but not as much as the TOA method. It is also very tolerant of transducer position errors, performing similarly to both the TOA and TDOA methods. RAPID Method. At first look, it would appear that the RAPID method should not be at all sensitive to group velocity since arrival times are not part of the algorithm. However, as implemented here, the time windows for computing the signal difference coefficients depend on the calculated times of the direct arrivals. Thus, this method exhibits somewhat unsystematic behavior as errors are introduced into both the group velocity and the transducer positions; this behavior results from time windows moving to different portions of the signals, which can include noise before the first arrival. The TOA, TDOA and Energy Arrival methods can all tolerate both types of variations while maintaining reasonable, although somewhat degraded, performance. The TOA method exhibits the most sensitivity to group velocity errors, and the TDOA method the least; performance of these three methods is similar for errors in transducer locations.
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7. SUMMARY AND CONCLUSIONS The work shown here compares the efficacy of four different damage localization methods based upon signals recorded from spatially distributed transducer arrays. The time-of-arrival method is generally the most effective with both good localization ability and reasonable signal-to-noise ratio. It is also reasonably well-behaved when there are errors in group velocity and transducer locations. The time-difference-of-arrival method, although not quite as effective as the time-of-arrival method, is the most robust to errors in group velocity. The energy arrival and RAPID methods, which both have inherently good signal-to-mean-noise ratios, do not otherwise perform as well. The RAPID method was developed based upon more transducers than were considered here (8 to 16 rather than 4 or 6), and it is not surprising that its performance degrades significantly with fewer transducers. All four methods have their strengths and weaknesses, and combining images from multiple methods using various fusion strategies should be considered.
ACKNOWLEDGEMENTS The support of the National Science Foundation under contract number ECS-0401213 is gratefully acknowledged.
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