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Methodologies for quantifying changes in diffuse ultrasonic signals with applications to structural health monitoring Jennifer E. Michaels*, Yinghui Lu and Thomas E. Michaels School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332-0250
ABSTRACT Changes in diffuse ultrasonic signals recorded from permanently mounted sensors can be correlated to initiation and growth of structural damage, offering hope that sparse sensor arrays can be utilized for monitoring large areas. It is well-known that benign environmental changes also have significant effects on diffuse ultrasonic signals that are of comparable magnitude to the effects of damage. Several methodologies are investigated for quantifying differences in diffuse ultrasonic signals by computing parameters that can be used to discriminate damage from environmental changes. The methodologies considered are waveform differencing, spectrogram differencing, change in local temporal coherence, and temperature compensated differencing. For all four methods, a set of baseline waveforms are first recorded from the undamaged specimen at a range of temperatures, and subsequently recorded waveforms are compared to those of the baseline set. Experimental data from aluminum plate specimens with artificial defects are analyzed. Results show that the local coherence method is the most effective for discriminating damage from temperature changes whereas waveform differencing is the least effective. Both the spectrogram differencing method and the temperature compensated differencing method offer intermediate performance. As expected, the efficacy of all four methods improves as the number of waveforms in the baseline set increases. Keywords: Ultrasound, diffuse waves, signal comparison, spectrogram, damage detection, structural health monitoring
1. INTRODUCTION Active ultrasonic sensors permanently attached to a structure can be used to interrogate the structure’s health by transmitting an elastic wave with one sensor and receiving it with either the same transducer (pulse echo mode) or with another transducer (through transmission or pitch-catch mode). For many simple structures, such as plates and shells, the structure acts as a wave guide and the received signals can be directly related to flaws, either as reflections or blocked transmissions1,2. For structures that either do not support guided waves or for which there are many internal reflections, the received signals are complex and not readily interpreted. However, it is well known that they are very sensitive to both structural changes and environmental effects, potentially throughout a large interrogation volume. The problem is selectivity -- being able to selectively identify structural changes such as flaws while ignoring changes due to benign causes such as temperature variations and changes in surface conditions. These complex ultrasonic signals, referred to as diffuse ultrasonic waves, are typically analyzed in the frequency domain utilizing time-frequency methods such as the short-time Fourier transform (STFT)3. Such parameters as frequency-dependent decay constants and various moments have been correlated to damage4. Recently, researchers have also been considering the details of the time domain signals, which is only possible for attached sensors for which the transducers themselves, the precise transducer locations, and the coupling conditions are fixed5,6. Small changes in either transducer location or coupling can have as large of an effect on the received signals as significant damage. *
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Health Monitoring and Smart Nondestructive Evaluation of Structural and Biological Systems IV, edited by Tribikram Kundu, Proc. of SPIE Vol. 5768 (SPIE, Bellingham, WA, 2005) 0277-786X/05/$15 • doi: 10.1117/12.598959
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Considered here are four different methods for quantifying changes in lengthy, complex time domain signals and evaluation of their efficacy in discriminating defects from temperature changes. The four methods considered are waveform differencing, spectrogram differencing, changes in temporal coherence, and temperature compensated differencing. These methods are applied to ultrasonic data recorded from aluminum plate specimens where the geometry is such that the large numbers of reflections generate an essentially diffuse field. Before these methods are applied, a set of baseline signals is first recorded from the undamaged structure at different temperatures spanning the range of interest. Subsequently recorded signals are then compared to these baseline signals to detect when damage has occurred.
2. EXPERIMENTAL PROCEDURE The specimens utilized for this work are 6061 aluminum plates, 50.8 mm x 152.4 mm x 4.76 mm (2” x 6” x 3/16”), as previously reported6 and shown in Figure 1. Even though these plates are of suitable thickness for propagation of guided waves, the small lateral dimensions result in large numbers of mode conversions and reflections which result in a diffuse wave field. Two piezoelectric transducers were fabricated and bonded to the top surface of each plate, with each transducer constructed of a 12.5 mm diameter, 2.25 MHz PZT disk. A Panametrics 5072PR pulser receiver was used as both transmitter and receive amplifier. The spike mode excitation and broad bandwidth receiver enabled signals to be recorded in a frequency range of approximately 50 kHz to 5 MHz. Each recorded signal was obtained by averaging 50 signals that were digitized at 12.5 MHz and 8 bits. The recorded time window for both specimens was 1000 µsec, which corresponds to a longitudinal wave making about 40 transits of the length of the specimen (about 6.35 meters). Since the transducers were hand-made, they exhibit considerable transducer-to-transducer variations; these variations, combined with small differences in transducer placement and coupling, result in quite different signals from the nominally identical undamaged specimens.
Figure 1. Photograph of aluminum specimen with bonded piezoelectric transducers and attached thermocouple.
Signals were recorded from both specimens at temperatures ranging from approximately 5°C to 40°C prior to introduction of damage. For the first plate, damage was introduced by machining a through-thickness edge notch of increasing length to simulate a growing crack. The notch was enlarged from 0.64 mm (0.025 in.) to 6.35 mm (0.250 in.) in ten steps of 0.64 mm (0.025 in.). For the second plate, damage was simulated by a through hole of increasing diameter, starting at a diameter of 1.98 mm (5/64 in.) and incrementing in ten steps to a final diameter of 6.35 mm (0.25 in.). For both specimens, ultrasonic signals were recorded at each stage of damage at various temperatures; Figure 2 shows typical received signals from each undamaged specimen. For the first specimen, a total of 65 waveforms were recorded prior to damage, and an additional 397 were recorded at various notch lengths and at a range of temperatures. For the second specimen, 98 waveforms were recorded from the undamaged specimen, and an additional 64 waveforms were recorded at different temperatures as the through hole was enlarged.
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400 500 600 Time (microseconds)
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Figure 2. Typical recorded waveforms from the undamaged specimens: (a) specimen #1 at 24°C, and (b) specimen #2 at 23.3°C .
3. ANALYSIS METHODS All of the analysis methods considered here compare two diffuse ultrasonic signals, x(t) and y(t), both of length T, where x(t) is the signal of interest and y(t) is a baseline signal. The sampled versions of these signals are x(n) and y(n), where n is the sample at time n/fs where fs is the sampling frequency. The discrete time signals are of length N = fs T.
3.1
Time Domain Differencing
This first comparison method is simply subtracting two signals and accumulating the square of the error. However, in order to obtain a result that is independent of the amplitude of the original signals, the signals are appropriately scaled. The first signal, x(n), is scaled to unity energy, and the second signal, y(n), is scaled to minimize the error between the two signals. 1
ax
(1)
N 1
¦ x ( n) 2
n 0 N 1
¦ a x(n) y (n) x
ay
n 0
(2)
N 1
¦y
2
( n)
n 0 N 1
E
¦ [a x(n) a y (n)]2 x
(3)
y
n 0
The error E is a measure of the difference between the two signals.
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This differencing method is very sensitive to any changes between the two signals, and thus would not be expected to be a good discriminator between damage and environmental changes. However, since a set of baseline signals exists from the undamaged structure, waveform differencing can be used to calculate the error between each baseline signal and the current signal of interest. The baseline signal corresponding to the smallest error is selected for subsequent comparisons utilizing the other three methods; this selected signal is the one that was recorded at the temperature closest to that of the current signal.
3.2
Spectrogram Differencing
The spectrogram is a time-frequency representation of a time domain signal computed as the absolute value of the short time Fourier transform (STFT). Time-frequency representations are appropriate for analyzing non-stationary signals for which the spectral content is changing as a function of time7, and thus should be considered as a tool for comparing two diffuse ultrasonic signals. Let X(m,k) and Y(m,k) be the spectrograms of x(n) and y(n), respectively, calculated using a sliding window of length M at an interval L. The discrete indeces m and k are related to continuous time t and frequency f as follows: k
fs
f
(4)
M
mL
t
(5)
fs
A measure of the difference between the two signals can thus be obtained by subtracting their spectrograms and calculating the sum of the squared error over time and frequency windows of interest. Prior to subtraction each spectrogram is normalized to unity energy within the time window m1 to m2 and the frequency window k1 to k2. 1
ax
m2
(6)
k2
¦¦X
2
( m , k )
m m1 k k1
1
ax
m2
k2
¦¦
(7)
Y ( m , k ) 2
m m1 k k1 m2
E
k2
¦ ¦ [a X (m , k ) a Y (m , k )]
2
x
y
(8)
m m1 k k1
The error E is a measure of the difference between the two spectrograms. For the data presented here, the entire time and frequency windows were used (1000 µsec and 6.25 MHz).
3.3
Change in Local Temporal Coherence
The local temporal coherence between two signals is based upon a local cross correlation whereby a sliding window of length T is simultaneously moved along both of the signals. At each window position, the coherence, or normalized cross correlation, is computed between the two windowed signals. The result is a time-dependent map of coherence. Using continuous time notation for clarity, the local cross correlation between two signals is:
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t
RxyT (W , t )
1 T
T 2
³ t
x ( s ) w( s t ) y ( s W ) w( s W t ) ds
(9)
T 2
where w(t) is a windowing function, typically rectangular. The local temporal coherence (LTC) is obtained by normalizing the local cross correlation by the peak autocorrelation of each signal:
RxyT (W , t )
T xy
J (W , t )
(10)
RxxT (0, t ) RTyy (0, t )
If two signals are identical in shape, the peak value of the LTC at every time t is unity even if their amplitudes are different. The peak of the absolute value of the LTC as a function of time is a measure of how the shapes of two signals are changing with time, irrespective of amplitude differences and small time delays. This peak coherence function is:
C xy (t )
max J xyT (W , t )
(11)
W
The difference P between the maximum and average peak coherence, referred to as the coherence change, is defined as:
P
(12)
max C xy (t ) C xy
This single parameter is a measure of the overall difference in shape between the two signals. The local coherence method is described in more detail in [8].
3.4
Waveform Differencing with Temperature Compensation
The first order effect of a temperature change on a diffuse ultrasonic signal is to either stretch or compress it in time9; e.g., x(t) = y(t í Įt) where Į is positive for stretching and negative for compressing. If these two signals are simply normalized and subtracted as per Equation (3), the resulting error could be substantial. However, if the second signal y(t) is first stretched or compressed appropriately, then the resulting error is significantly reduced, better reflecting that the only difference between the signals is due to a benign temperature change. The differencing procedure is as follows: (1) determine Į from the time-dependent time shift obtained from the short time cross correlation10, (2) stretch or compress y(t) by Į, and (3) calculate the error E as per Equation (3). The resulting error E is now a temperature compensated value. This method is used in conjunction with time domain differencing to first select a baseline signal that is closest to the signal of interest as determined by the minimum value of E, and then to stretch or compress this baseline to compensate for the temperature difference; refer to [10] for a detailed description.
4. RESULTS A baseline set of waveforms was selected for each undamaged specimen at temperatures ranging from approximately 8°C to 38°C. Subsequently recorded waveforms were then compared to these baseline signals. Simple time domain differencing was used to select the baseline that most closely matched each signal of interest, and then each of the four methods was used to calculate a parameter to quantify the difference between the signal and the selected baseline. Each parameter is evaluated as to its ability to detect structural damage in the presence of temperature changes. For each specimen, two baseline sets were considered: one with three waveforms recorded at low, medium and high temperatures, and one with seven waveforms recorded at temperatures approximately evenly spaced and ranging from
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about 5°C to 40°C. For each specimen and each baseline set, the following procedure was performed for each of the four analysis methods: 1.
A threshold level was determined to yield a specified percentage of false alarms when applied to the data from the undamaged specimen.
2.
This threshold level was applied to the data from the damaged specimen, and an actual detection percentage was determined.
3.
For signals recorded from the damaged specimen that were misclassified as “no damage”, the one corresponding to the largest defect was found; the size of this defect is the largest missed flaw.
The above steps were performed for desired false alarm percentages ranging from 0 to 10%. Results are shown in Figure 3 for the first specimen and in Figure 4 for the second specimen as plots of actual percent detection vs. actual percent false alarms. Detection Results, Notched Specimen, 3 Baselines
(a)
Actual Detections (%)
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Waveform Differencing Spectrogram Differencing Coherence Change Compensated Differencing
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0
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4 6 Actual False Alarms (%)
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Detection Results, Notched Specimen, 7 Baselines
(b)
Actual Detections (%)
100 95 90 85
Waveform Differencing Spectrogram Differencing Coherence Change Compensated Differencing
80 75
0
2
4 6 Actual False Alarms (%)
8
10
Figure 3. Detection results for the first specimen (edge notch damage) as determined using (a) a three-waveform baseline set, and (b) a seven waveform baseline set. Note that the vertical scales are different for the two plots.
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Detection Results, Specimen w/Hole, 3 Baselines
(a)
Actual Detections (%)
100 80 60 40 Waveform Differencing Spectrogram Differencing Coherence Change Compensated Differencing
20 0
0
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4 6 Actual False Alarms (%)
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10
Detection Results, Specimen w/Hole, 7 Baselines
(b)
Actual Detections (%)
100 95 90 85
Waveform Differencing Spectrogram Differencing Coherence Change Compensated Differencing
80 75
0
2
4 6 Actual False Alarms (%)
8
10
Figure 4. Detection results for the second specimen (through hole damage) as determined using (a) a three-waveform baseline set, and (b) a seven waveform baseline set. Note that the vertical scales are different for the two plots.
Each series of data points in each plot is derived from a range of threshold values with the largest threshold corresponding to the leftmost point. This threshold is the maximum value from the undamaged specimen, which results in a false alarm rate of 0%. As the threshold value is decreased, the percentages of both false alarms and detections increase. The maximum false alarm value considered here is 10%, which corresponds to the largest percentage of detections. If the parameter under consideration is perfectly effective at detecting damage, then the detection value would be 100% for a false alarm rate of 0%; this is the case in Figure 4 for the coherence change method. From these figures it can be seen that the most effective parameter is the change in peak coherence, which equals or outperforms the other parameters for all cases. Table 1 summarizes coherence change results for both specimens and both baseline sets for a desired false alarm rate of 3%. For each case the threshold value, actual false alarm percentage, actual detection percentage, and largest missed flaw are tabulated. The actual false alarm percentage is somewhat different from 3% due to the discrete number of data points.
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Table 1. Summary of detection results for coherence change method.
Notched Specimen, 3 Baselines
Notched Specimen, 7 Baselines
Through Hole Specimen, 3 Baselines
Through Hole Specimen, 7 Baselines
Detection Threshold
0.0637
0.0249
0.0410
0.0247
Actual % False Alarms
3.1 %
3.1 %
3.1 %
3.1 %
Actual % Detections
94.0 %
98.7 %
100 %
100 %
Largest Flaw Missed
1.27 mm
0.64 mm
0 mm
0 mm
Result
5. DISCUSSION OF RESULTS The common element for all four analysis methods is the mechanism for selecting the baseline for comparison. For all methods, the error between the signal of interest and each baseline signal is calculated by waveform differencing, and the baseline signal with the minimum error is selected. Since the baseline waveforms approximately span the temperature range of interest, it is expected that more baseline signals would improve the detection statistics. This expectation is validated for both specimens as can be seen by the dramatic improvement in the detection percentage when comparing results for three vs. seven baseline waveforms. A further expectation, which is suggested but not completely verified here, is that the performance of all four methods will improve and converge as the number of baseline waveforms increases, assuming that their recorded temperatures are evenly distributed across the actual temperature range of interest. Not surprisingly, the waveform differencing method performs the worst for all cases. It is the only method that does not take into account either the non-stationary nature of the signals or their predicted behavior with temperature changes. However, even this method performs fairly well with enough baseline signals. The coherence change method performs the best, achieving reasonable false alarm and detection percentages with only three baseline waveforms and nearperfect performance with seven baseline waveforms. The spectrogram differencing and temperature-compensated differencing methods perform similarly -- significantly better than the waveform differencing method but not quite as well as the coherence change method. The largest flaw missed is also a measure of performance. For the first specimen with a seven-waveform baseline set and at a 3% false alarm rate, the coherence change method was able to detect all signals corresponding to damage except for the very smallest notch size (0.64 mm, 0.025 in.). Even with the three waveform baseline set, all notches larger than 1.27 mm (0.05 in.) were detected. The smallest introduced damage for the second specimen was a 1.98 mm diameter hole, and the coherence change method was able to perfectly detect this size hole (and all larger ones) using both three and seven waveform baseline sets.
6. SUMMARY AND CONCLUSIONS The work presented here evaluates four different analysis methods for comparing two diffuse ultrasonic signals with the objective of detecting structural damage in the presence of temperature changes. All four methods are based upon the concept of having a set of baseline signals recorded at various temperatures which span the expected range of operation. The waveform differencing method, although essential for selecting the best baseline waveform to use for further comparison, is generally not an effective method unless there are a large number of baseline signals recorded at temperatures evenly spaced within the range of interest. The spectrogram differencing method offers reasonable performance due to its ability to represent the non-stationary nature of the signals without being sensitive to the small phase changes caused by temperature variations. In contrast, the method of temperature compensated differencing explicitly measures and compensates for the time-dependent time delays caused by temperature changes. These two methods offer similar performance based upon very different mechanisms. The most effective method is measurement
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of the coherence change between two signals, which offers excellent performance with even a small number of baseline signals. Like the spectrogram method, it is not sensitive to small phase changes between signals, and like the temperature compensated differencing method, it explicitly takes advantage of the fact that the first order effect of temperature variations is to change the phase but not the shape of the ultrasonic signal. Temperature changes are only one type of environmental variation that can strongly affect diffuse ultrasonic signals. Other changes such as variations in surface conditions are much more difficult to discriminate from damage. The challenge remains to develop robust analysis methods for diffuse ultrasonic signals that can reliably discriminate all benign environmental changes from structural damage.
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation under contract number ECS-0401213.
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