Efficient temperature compensation strategies for guided wave ...
Ultrasonics 50 (2010) 517–528
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Efficient temperature compensation strategies for guided wave structural health monitoring Anthony J. Croxford a,*, Jochen Moll b, Paul D. Wilcox a, Jennifer E. Michaels c a
University of Bristol, Mechanical Engineering, Queens Building, University Walk, Bristol, Avon BS8 1TR, United Kingdom University of Siegen, 9-11 Paul Bonatz Str., Siegen, S7076, Germany c Van Leer Electrical Engineering Building, 777 Atlantic Drive NW, Atlanta, GA, 30332-0250, USA b
a r t i c l e
i n f o
Article history: Received 14 July 2009 Received in revised form 26 August 2009 Accepted 5 November 2009 Available online 26 November 2009 Keywords: Structural health monitoring Guided waves Temperature compensation
a b s t r a c t The application of temperature compensation strategies is important when using a guided wave structural health monitoring system. It has been shown by different authors that the influence of changing environmental and operational conditions, especially temperature, limits performance. This paper quantitatively describes two different methods to compensate for the temperature effect, namely optimal baseline selection (OBS) and baseline signal stretch (BSS). The effect of temperature separation between baseline time-traces in OBS and the parameters used in the BSS method are investigated. A combined strategy that uses both OBS and BSS is considered. Theoretical results are compared, using data from two independent long-term experiments, which use predominantly A0 mode and S0 mode data respectively. These confirm that the performance of OBS and BSS quantitatively agrees with predictions and also demonstrate that the combination of OBS and BSS is a robust practical solution to temperature compensation. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The detection of damage inside a structure is a generic problem for nondestructive testing (NDT) and has been extensively studied. Historically damage has been detected by temporarily placing sensors on the surface of a structure, performing an inspection of some kind, and then removing the sensors. This process is repeated if subsequent inspection is required. Structural health monitoring (SHM) represents a change to this basic approach. Sensors are permanently attached to the structure, allowing for highly accurate repeat measurements. This repeatability enables the recording of baseline measurements to track changes in the time-traces that can potentially be related to structural damage. The use of such baselines means that structures with complex geometries and responses can be monitored and a high degree of automation is possible. A further advantage of this approach is that no disassembly of the structure to be inspected is necessary, which can result in significant cost savings. Examples of SHM applications include damage detection in aircraft structures [1], bridges [2], offshore wind energy plants [3], pipes [4] and rails [5]. The current limiting factor of this SHM strategy is the difficulty in differentiating changes due to damage and those caused by changing environmental and operational conditions (EOC). Meth* Corresponding author. Tel.: +44 117 3315909; fax: +44 117 9294423. E-mail address: [email protected] (A.J. Croxford). 0041-624X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2009.11.002
ods that have been developed in the last decades (for an overview see, e.g., [6,7]) perform well in laboratory conditions but often fail in real-world situations where changing EOC are present. The underlying influences of temperature, humidity, changing boundary conditions, etc. can be sufficient to mask any changes due to damage to a degree that it might not be detected. It has been shown by Worden et al. [8] and Sohn [9] that damage-sensitive indicators are sensitive to EOC as well. Thus, it is important to focus on EOC in order to make damage assessment robust for in situ applications. In particular, expensive false alarms need to be avoided without decreasing damage sensitivity to enhance confidence in SHM technology. Guided waves have shown great potential in SHM applications to detect damage in plate-like structures. They can travel over long distances and thus cover large areas with only a limited number of sensors. The fact that the entire thickness is interrogated makes it possible to detect damage inside the structure (e.g., cracks [10] and delaminations [11]) as well as on the surface (e.g., corrosion [12]). Of the EOC affecting guided waves, temperature has been shown to be one of the dominant effects [13]. In addition to altering the condition of the structure, temperature can also affect the transducers and their bonding. For small temperature variations of a few degrees, the effect of temperature on transducer performance has been shown to be significantly less than the effect of temperature on wave propagation within the structure [13]. Transducer bonding has also been shown to be remarkably consistent throughout
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fatigue tests lasting many weeks and tens of thousands of fatigue cycles [14]. For large temperature variations, changes in transducer and bonding properties may become more significant. However, this variability is one that can potentially be minimized via the choice of transducer types, materials and manufacturer. This is in contrast to the unavoidable effect of even small temperature changes on wave propagation within a structure, which is the subject of this paper. Strategies for compensating the effect of temperature on the structure have been developed in recent years. In particular Lu and Michaels [15] and Konstantinidis et al. [16] introduced a methodology, often referred to as optimal baseline selection (OBS), that uses multiple baseline measurements recorded over a range of temperatures. This approach has been the basis for further developments (see [13,16–24]), and a detailed discussion of this method will be presented in the next sections. The purpose of this paper is to quantitatively analyze different methods of temperature compensation and show that their performance can be estimated from fairly simple models. The outline is as follows. Firstly, a simple mathematical description of the guided wave SHM process is presented and used to analyze the temperature compensation methodologies. Then results from two independent experiments are presented. From these the experimental performance of the different temperature compensation strategies is obtained and compared with earlier predictions. The analysis presented enables a quantitative assessment to be made of the efficacy of temperature compensation for specific guided wave SHM scenarios.
2. Temperature compensation A typical guided wave SHM system comprises a number of transducers permanently attached to the surface of the structure. A suitable excitation signal is sent to one of the transducers and the time-domain responses (time-traces) from this and other transducers are recorded. This process is repeated using different transducers as the transmitter. Time-traces that are recorded on a different transducer to the transmitting one are referred to as pitch-catch mode, while those recorded on the transmitting transducer are referred to as pulse-echo mode. In this paper, a single pitch-catch time-trace from one pair of transducers is considered. A typical pitch-catch time-trace contains various overlapping wavepackets corresponding to different wave paths between the transducers. The first wavepacket (assuming a single propagating mode is present) corresponds to the direct arrival, and subsequent wavepackets are due to reflections and scattering from structural features. A significant class of SHM applications that employ active guided wave monitoring is based on the algebraic difference between the current time-trace and a baseline time-trace recorded when the structure was undamaged. The signal remaining after the subtraction of a baseline time-trace is referred to as a residual time-trace. In the best-case scenario, this subtraction allows the wavepackets due to direct transmission and scattering from benign structural features to be eliminated and any remaining wavepackets in the residual time-trace to be attributed to scattering from damage. The implication is that even complex structures can potentially be monitored. However, in the presence of changing temperature, the effectiveness of the subtraction process is reduced and the residual time-trace contains coherent noise that is primarily due to small shifts in the arrival times of wavepackets from structural features. The cause of these time-shifts is both thermal expansion and changes in wave velocities with temperature. It has been shown by different authors [15,16,24] that in the presence of changing temperature, simple subtraction of one base-
line time-trace from the current time-trace is not sufficient to discriminate between changes due to damage and those due to temperature variations. In order to mitigate temperature effects, the concepts of optimal baseline selection (OBS) and baseline signal stretch (BSS) have been proposed and demonstrated [15,20]. In the following section, these techniques are described and analyzed mathematically. 2.1. Preliminary analysis A baseline time-trace recorded from a guided wave SHM system at an initial temperature T0, is assumed to consist of N superimposed wavepackets corresponding to scattered time-traces from structural features. Mathematically this can be written as:
uðt; T 0 Þ ¼
N X
Aj sj ðt t j Þ
ð1Þ
j¼1
where t is time and Aj, sj and tj are respectively the amplitude, waveform and arrival time of the jth wavepacket. The arrival time of a wavepacket is tj = dj/vgr, where vgr is the group velocity and dj is the propagation distance. Consider how this time-trace changes in appearance if there is a change to a different temperature T = T0 + dT. In general, such a temperature shift may alter the shape, amplitude and arrival time of each of the individual wavepackets, with the shape and amplitude changing because of the possible temperature dependence of dispersion, reflection coefficients and attenuation mechanisms. However, to a first approximation, the dominant effect is a change in arrival time of individual echoes. For dispersive waves temperature actually causes two different time-shift effects, one due to a change in group velocity and one due to a change in phase velocity. The first of these affects the envelope of a wavepacket and the second affects the waves within the wavepacket. These time-shifts are generally not equal, resulting in a change in phase of the waves within a wavepacket. For the reasons described in [24], it is preferable to apply baseline subtraction to RF (i.e., un-envelope-detected) time-traces. In this scenario, the amplitude of the residual after subtracting two nominally identical wavepackets is predominantly governed by the time-shift of waves within the wavepacket, which arises from the change in phase velocity with temperature. The temperature-dependent time-shift, dt, of waves within a wavepacket is described by the fundamental relationship [24] as:
dt j ¼
dj
mph
a
kph
mph
dT
ð2Þ
where dj is the propagation distance, vph the phase velocity, a the coefficient of thermal expansion, and kph the change in phase velocity with temperature. In practice, the thermal expansion term, a, is typically one-to-two orders of magnitude smaller than the kph/vph term, and the above expression can be approximated as:
dt j ¼
kph
m2ph
dj dT
ð3Þ
The effect of the change in group velocity due to temperature on the level of residual time-trace is second order compared to the change in phase velocity. For this reason, it is physically justified and mathematically convenient to assume that the effect of a change in temperature is to simply shift the arrival time of an entire wavepacket by the amount dt given by (3). Thus, after a temperature change, the time-trace of (1) becomes:
uðt; T 0 þ dTÞ ¼
N X j¼1
Aj sj ½t t j bðdTÞ;
ð4Þ
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where b(dT) is the fractional change in arrival time of the wavepackets. From (3):
b¼1
mgr kph dT; m2ph
ð5Þ
where vgr is the group velocity. As previously shown [17], the peak amplitude of the post-subtraction residual associated with each wavepacket can be estimated as:
unoise ¼ Aj x0 jdt j j; j
ð6Þ
where x0 is the center frequency of the wavepacket. Thus, if the temperature change is known, the time-shifts of the wavepackets in the time-trace can be calculated and used to estimate the postsubtraction noise. Having developed a simple model of the changes to a time-trace associated with a temperature shift, the techniques to deal with this change are now analyzed. 2.2. Optimal baseline selection (OBS) OBS uses M baseline time-traces recorded at different temperatures Tm = T0 + dTm referred to as the baseline dataset. The mth time-trace from the baseline dataset can be expressed as:
um ðt; T m Þ ¼
N X
m m Am j sj ½t t j bðdT m Þ:
ð7Þ
j¼1
In this equation, b(dTm) is the fractional shift in arrival times of wavepackets in each time-trace with respect to their values at an arbitrary fixed temperature. To find the best match between a time-trace in the baseline dataset and the current time-trace, recorded at a temperature T, a criterion needs to be defined that characterizes their similarity. Possible criteria include the mean square deviation [15],
mms ¼ arg min m
Z
t2
½uðt; TÞ um ðt; T m Þ2 dt ;
ð8Þ
t1
and the maximum residual amplitude [25],
mmr ¼ arg minfmax juðt; TÞ um ðt; T m Þjg;
ð9Þ
m
where mms and mmr are the indices of the baseline time-trace that best matches the current time-trace according to the criterion used. Note that selecting the best-matched baseline time-trace achieves superior results to simply selecting the baseline time-trace with the temperature that best matches the current time-trace. This is because of the difficulty in accurately measuring the mean temperature of the entire structure. If the nominal temperature of the current time-trace and the best-matched baseline time-trace are compared, noticeable differences are observed [20]. If the time-traces in the baseline dataset are recorded at a range of discrete temperatures with uniform spacing DT, then the noise associated with OBS can be estimated. This estimate assumes that the maximum deviation in the true temperature between the current time-trace and the best-matched baseline time-trace, dT, is DT/2, from which the noise can be estimated using (6). This will be demonstrated experimentally later in this paper. It is worth noting at this point that the OBS does not suppress the signal due to damage. This is due to the changes as a result of damage being much smaller than those induced by temperature change as shown by Lu and Michaels [15]. 2.3. Baseline signal stretch (BSS) Another method for temperature compensation is BSS [15], which in its simplest form requires only one baseline time-trace
to compensate for the effect of temperature. Unlike simple subtraction and OBS, this technique modifies a single baseline time-trace to match the current time-trace. In BSS the time-axis of the baseline time-trace, u(t; T0), is stretched (where the term stretch refers ^ to yield a to both dilation and compression) by a stretch factor b ^ This can be expressed as: ^ ðt; T 0 ; bÞ. new time-trace u
^ ¼ uðt=b; ^ T0Þ ¼ ^ ðt; T 0 ; bÞ u
N X
^ tj Þ Aj sj ðt=b
ð10Þ
j¼1
^ is chosen to match b, the arrival times of the different waveIf b packets match those of the baseline time-trace in (4). However, the ^ comindividual wavepackets, sj, are also stretched by the factor b pared to the baseline time-trace. Furthermore, pure stretching of the time-axis also alters the frequency content of the time-trace. Because experimental time-traces are sampled at discrete points in time, the necessary stretching effectively requires that the time-traces be resampled at a slightly different sampling frequency. The practical implementation of this is most easily achieved in the frequency-domain as described below. 2.3.1. Implementation of BSS In this section, square brackets [ ] denote a discretely sampled function. Consider a continuous time-trace, u(t), sampled with a time step of Dt1, resulting in a discretely sampled time-trace u1[n] = u(nDt1), where n is an integer. In order to perform the time-stretch, u1[n] is first padded with zeros so that it contains a total of m1 points. This time-trace is transformed to the frequency-domain using a Fast Fourier Transform (FFT), resulting in a discretely sampled spectrum, U1[n], containing m1 bins, with a frequency spacing, Df = 1/(m1Dt1), between bins. The spectrum is then either truncated or zero-padded to create a new spectrum, U2[n], that contains m2 bins (the truncation or padding taking place at the appropriate portion of the spectrum so that the conjugate symmetry of the whole spectrum is preserved according to the particular implementation of the FFT algorithm). The modified spectrum is then subjected to an inverse FFT, resulting in a new discretely sampled representation of the original continuous time-trace, u2[n] = u(nDt2), containing m2 points sampled with a time step Dt2 = 1/(m2Df) = (m1/m2)Dt1. Note that strictly, this operation has not performed the time-stretch, it has simply resampled the original continuous time-trace, u(t), with a different time step Dt2. However, if a new continuous time-trace is defined as u2 ðtÞ ¼ ^ where b ^ ¼ m2 =m1 , then it can be seen that u2[n] is the disuðt=bÞ crete representation of u2(t) sampled with the original time step, Dt1. Thus the procedure has produced a discretely sampled stretch ^ equal to m2/m1. of the original time-trace with a stretch factor, b, This implementation thus restricts the possible stretch factors to those corresponding to integral values of m1 and m2. Note that the origin of the stretch is the start of the time-trace u[n], i.e. at n = 0. In the context of temperature compensation, this point in time should therefore correspond to the temporal center of the transmitted time-trace. If this center point is not at n = 0, then a translation (i.e. a time-shift) should be applied to the time-trace prior to stretching [21]. Typically an optimization procedure is applied to obtain the va^ that needs to be applied to a time-trace to correct for temlue of b perature using possible criteria similar to those used in the OBS procedure, such as the mean square deviation,
^ms ¼ arg min b ^ b
Z
t2
^ 2 dt ; ^ ðt; T 0 ; bÞ ½uðt; TÞ u
ð11Þ
t1
and the maximum residual amplitude [42],
n o ^ : ^mr ¼ arg min max juðt; TÞ u ^ ðt; T 0 ; bÞj b ^ b
ð12Þ
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^ can be implemented in an iterNote that the optimization of b ative algorithm. However, this needs to be performed with caution, since metrics such as maximum residual amplitude have multiple minima. Practically, it has been found that the mean square devia^ the maxtion provides a more robust metric for coarse tuning of b; ^ imum residual metric is best restricted to fine tuning of b. It is also worth noting that if a sufficiently long time-trace is recorded, the latter part will contain waves in a diffuse field regime (i.e., the coda). If this is the case, Lu and Michaels [15] showed that ^ can be evaluated directly by the the optimum stretch factor, b, short time cross-correlation of the current and baseline timetraces, yielding a time-dependent local time-shift from which the overall stretch factor can be computed. 2.3.2. Granularization noise Since m1 and m2 are integers, the possible stretch factors that can be applied to a baseline time-trace are themselves discrete and spaced at 1/m1. This introduces what is termed granularization noise into the stretching process. For a wavepacket that nominally arrives at time tj, the stretching procedure can only adjust its arrival time in steps of tj/m1. Thus, when stretching is applied to compensate for a change in temperature, there is potentially an error in arrival time matching of up to tj/(2m1). The granularization noise, ugn, associated with a particular time-trace can therefore be estimated from (6) as:
ugn ¼
Aj x 0 t j : 2m1
ð13Þ
This noise can clearly be reduced by increasing m1 (i.e. the amount of padding applied to the original time-trace), but this requires more computational resources. It should be stressed that the granularization noise does not depend on the sampling frequency of the original time-trace. Provided the Nyquist criterion is satisfied, there is no benefit in using a higher sampling frequency; the critical parameter is the number of points, m1, after zero padding. 2.3.3. Frequency noise In addition to changing the arrival time, stretching also distorts the shape of each wavepacket. Although the arrival times of the mid-points of wavepackets in the baseline and current time-traces may be aligned by stretching, the shape of one will be distorted relative to the other. This distortion gives rise to the second source of noise, which is termed frequency noise. The frequency noise is arrival time independent as it is a function of the distortion of the individual wavepackets, and represents the general limit for the stretching method unless steps are taken to compensate for this effect. The more the time-axis is stretched, the greater the influence of frequency noise. It is theoretically possible to correct for this frequency noise as described below, although practical success with this approach has been limited. If it is assumed that the individual wavepackets are all the same shape; i.e., sj(t) = s(t), then the baseline and current time-traces, (1) and (4) respectively, can be written in the frequency-domain as:
Uðx; T 0 Þ ¼
" N X j¼1
Uðx; T 0 þ dTÞ ¼
# Aj expðixtj Þ SðxÞ " N X
ð14Þ #
Aj expðixt j bðdTÞÞ SðxÞ
ð15Þ
j¼1
where variables with capital letters represent the Fourier transforms of their lower case counterparts. In these expressions, the first term in square brackets corresponds to the Fourier transform of a train of impulses representing the arrival times of the wavepackets, and the second term, S(x), describes the shape of the wavepackets in the frequency-domain. In the time-domain, these two
elements are convolved, but the frequency-domain representation enables them to be written as a product. ^ ¼ b, then the If the baseline time-trace is stretched by a factor b equivalent frequency-domain expression for the stretched counterpart of (14) is
Uðx; T 0 ; bÞ ¼
" N X
# Aj expðixtj bðdTÞÞ bSðbxÞ
ð16Þ
j¼1
It can be seen that the arrival time portion of this expression matches that of the current time-trace in (15), but that the wavepacket term is modified. In order to recover U(x, T0 + dT) precisely, it is theoretically possible to multiply the above expression by a suitable factor:
Uðx; T 0 ; bÞ
SðxÞ ¼ Uðx; T 0 þ dTÞ: bSðbxÞ
ð17Þ
The practical difficulties with this approach are that (a) all the constituent wavepackets must have the same shape, (b) S(x) must be known, and (c) the division by S(bx) may introduce noise at the extremes of the time-trace bandwidth unless care is taken to avoid these effects by filtering. For these reasons, the compensation for frequency noise has not been adopted here. Instead, discussion is limited to estimating the amplitude of frequency noise in the absence of compensation. The upper bound of frequency noise, ufn, can be expressed as:
ufn ¼ Aj max jsðtÞ sðt=bÞj:
ð18Þ
It is not possible to proceed further without specifying the shape of s(t). Consider a Gaussian windowed complex toneburst centered at t = 0. This complex toneburst is of interest because its envelope is readily computed and gives a phase-independent result. Such a toneburst is defined as:
SðtÞ ¼ expðix0 tÞ exp
t 2 ; 2 2r
ð19Þ
where x0 is the toneburst frequency and r is a parameter specifying the width of the toneburst. The difference between this wavepacket and one stretched by a factor b is:
SðtÞ sðt=bÞ ¼ expðix0 tÞ exp exp
: 2r2 b2
t 2 expðix0 t=bÞ 2r 2
t2
ð20Þ
If |1b| 1, then the magnitude of this difference, r(t), can be expressed as:
rðtÞ ¼ jsðtÞ sðt=bÞj ¼ x0 jtð1 bÞj exp
t2 : 2r2
ð21Þ
The position of the maximum (i.e., the location of the peak frequency noise associated with subtraction of the two Gaussian windowed wavepackets) can readily be found by differentiation with respect to t and occurs at t = ±r. The frequency noise is therefore:
ufn ¼ Aj rðt ¼ rÞ ¼ Aj x0 rj1 bj expð1=2Þ:
ð22Þ
The frequency noise associated with other windows, such as Hanning, can also be computed numerically from (18) although closed form solutions, such as that obtained for a Gaussian window, may not be possible. Numerical calculations for the frequency noise associated with Hanning-windowed tonebursts with 5– 50 cycles indicate that the frequency noise is always approximately 1.3 times (i.e., 2.3 dB) higher than for a Gaussian windowed toneburst with the same number of cycles (if the number of cycles in the latter are defined by the number falling within the 40 dB points of the Gaussian function). It was stated earlier that (22) is
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and then using a standard Fourier decomposition technique to propagate each frequency component of the input toneburst over the appropriate distances. This time-trace serves as the baseline. The material properties and propagation distances were altered to simulate a 10 °C temperature change and a new timetrace was generated. BSS was then performed to minimize the maximum residual between the two time-traces. The resulting residual time-trace envelopes are shown on dB scales in Fig. 1b and c for two different values of m1 in the BSS algorithm. Also shown in these figures are dashed lines indicating the estimated values of granularization and frequency noise computed using (13) and (22). It can be observed in the case of a low value for m1 shown in Fig. 1b that the amplitude of the residual increases with time. This behavior, which is consistent with (13), suggests that the dominant effect is granularization noise and that the level of frequency noise is low. It can be seen that the amplitude of the five peaks in the residual time-trace agree reasonably well with the theoretical estimate of granularization noise. Increasing m1 as shown in Fig. 1c reduces the influence of granularization noise so that frequency noise dominates, and this noise level represents the general limit for BSS for this temperature change. The peaks in the residual time-trace are now of fairly constant amplitude and are in good agreement with the estimated line for frequency noise. Increasing m1 further will continue to reduce the granularization noise, but as this effect is now smaller than the frequency noise there is little further improvement.
valid only if |1 b| is small, and adequate results are obtained for b values in the range of 0.95–1.05. It should be noted that this range is ample to cover any temperature swing likely to be encountered in real structures and does not in practice provide any limitation. From (5), the largest value of b likely to be encountered in a given application can be estimated, and (22) (or (18) for arbitrary shaped windows) can be used to estimate the associated frequency noise associated with BSS. This estimate, in combination with the granularization noise model, can be used to characterize the performance of the BSS technique. 2.3.4. Simulations of BSS performance Expressions (13), (18), and (22) provide estimates of the granularization and frequency noise associated with the BSS technique based on some simple approximations. In this section, a complete time-trace simulation is performed and the BSS process is applied to provide a validation of the simplified noise models. The simulation is performed using the propagation properties of the S0 mode in a 3 mm thick aluminum plate. A 10 cycle Hanningwindowed tone burst with a center frequency of 250 kHz is used as the transmitted signal. This center frequency is selected since the S0 mode is largely non-dispersive and it is consistent with typical frequencies used in practice. The simulated time-trace in Fig. 1a consists of five equally spaced echoes of the same amplitude that correspond to propagation distances of 0.25 m, 0.75 m, 1.25 m, 1.75 m and 2.25 m. The time-trace was simulated by first generating the appropriate dispersion curve for the S0 mode in aluminum
a
b
1 0.8
0 Original Residual Gran. noise Freq. noise
−10
0.6 Ampitude (dB)
Ampitude
0.4 0.2 0 −0.2
−20
−30
−40
−0.4 −0.6
−50
−0.8 −1 0
400
−60 0
100
200 300 Time (µs)
400
d
0 Original Residual Gran. noise Freq. noise
−10
Ampitude (dB)
200 300 Time (µs)
Post Subtraction Noise (dB)
c
100
−20
−30
−40
−20 −30 −40 −50 0
10
−50 50,000 −60 0
12
8
m m11 100
200 300 Time (µs)
400
4 100,000
0
2
6 Temperature Difference
Fig. 1. Simulated data illustrating the use of BSS in the presence of a 10 °C temperature change between the current and baseline time-traces. (a) Original RF baseline timetrace containing five wavepackets corresponding to S0 wave propagation over distances of 0.25 m, 0.75 m, 1.25 m, 1.75 m and 2.25 m at a frequency of 250 kHz in a 3 mm thick aluminum plate. (b) Residual time-trace amplitude using BSS with m1 = 25 102. (c) Residual time-trace amplitude using BSS with m1 = 25x104. (d) Surface showing the effect of changing m1 and the temperature difference on maximum residual time-trace amplitude.
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This change from a granularization to frequency noise dominated regime can be used to set the temperature compensation parameters for experimental studies. The surface of Fig. 1d shows the effects of m1 and the temperature difference on post-subtraction noise using the same test signals as shown in Fig. 1a. As previously observed, it is clear that as m1 increases, the postsubtraction noise initially decreases very rapidly. However this drop off slows and reaches a point where further increases in m1 give little reduction in post-subtraction noise for considerable additional processing time. The identification of this point will be used later to select the optimum value of m1 to be used experimentally. The surface also makes it clear that as the temperature difference is increased, the frequency noise increases. Even at these relatively modest temperature changes (
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Efficient temperature compensation strategies for guided wave structural health monitoring Anthony J. Croxford a,*, Jochen Moll b, Paul D. Wilcox a, Jennifer E. Michaels c a
University of Bristol, Mechanical Engineering, Queens Building, University Walk, Bristol, Avon BS8 1TR, United Kingdom University of Siegen, 9-11 Paul Bonatz Str., Siegen, S7076, Germany c Van Leer Electrical Engineering Building, 777 Atlantic Drive NW, Atlanta, GA, 30332-0250, USA b
a r t i c l e
i n f o
Article history: Received 14 July 2009 Received in revised form 26 August 2009 Accepted 5 November 2009 Available online 26 November 2009 Keywords: Structural health monitoring Guided waves Temperature compensation
a b s t r a c t The application of temperature compensation strategies is important when using a guided wave structural health monitoring system. It has been shown by different authors that the influence of changing environmental and operational conditions, especially temperature, limits performance. This paper quantitatively describes two different methods to compensate for the temperature effect, namely optimal baseline selection (OBS) and baseline signal stretch (BSS). The effect of temperature separation between baseline time-traces in OBS and the parameters used in the BSS method are investigated. A combined strategy that uses both OBS and BSS is considered. Theoretical results are compared, using data from two independent long-term experiments, which use predominantly A0 mode and S0 mode data respectively. These confirm that the performance of OBS and BSS quantitatively agrees with predictions and also demonstrate that the combination of OBS and BSS is a robust practical solution to temperature compensation. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The detection of damage inside a structure is a generic problem for nondestructive testing (NDT) and has been extensively studied. Historically damage has been detected by temporarily placing sensors on the surface of a structure, performing an inspection of some kind, and then removing the sensors. This process is repeated if subsequent inspection is required. Structural health monitoring (SHM) represents a change to this basic approach. Sensors are permanently attached to the structure, allowing for highly accurate repeat measurements. This repeatability enables the recording of baseline measurements to track changes in the time-traces that can potentially be related to structural damage. The use of such baselines means that structures with complex geometries and responses can be monitored and a high degree of automation is possible. A further advantage of this approach is that no disassembly of the structure to be inspected is necessary, which can result in significant cost savings. Examples of SHM applications include damage detection in aircraft structures [1], bridges [2], offshore wind energy plants [3], pipes [4] and rails [5]. The current limiting factor of this SHM strategy is the difficulty in differentiating changes due to damage and those caused by changing environmental and operational conditions (EOC). Meth* Corresponding author. Tel.: +44 117 3315909; fax: +44 117 9294423. E-mail address: [email protected] (A.J. Croxford). 0041-624X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2009.11.002
ods that have been developed in the last decades (for an overview see, e.g., [6,7]) perform well in laboratory conditions but often fail in real-world situations where changing EOC are present. The underlying influences of temperature, humidity, changing boundary conditions, etc. can be sufficient to mask any changes due to damage to a degree that it might not be detected. It has been shown by Worden et al. [8] and Sohn [9] that damage-sensitive indicators are sensitive to EOC as well. Thus, it is important to focus on EOC in order to make damage assessment robust for in situ applications. In particular, expensive false alarms need to be avoided without decreasing damage sensitivity to enhance confidence in SHM technology. Guided waves have shown great potential in SHM applications to detect damage in plate-like structures. They can travel over long distances and thus cover large areas with only a limited number of sensors. The fact that the entire thickness is interrogated makes it possible to detect damage inside the structure (e.g., cracks [10] and delaminations [11]) as well as on the surface (e.g., corrosion [12]). Of the EOC affecting guided waves, temperature has been shown to be one of the dominant effects [13]. In addition to altering the condition of the structure, temperature can also affect the transducers and their bonding. For small temperature variations of a few degrees, the effect of temperature on transducer performance has been shown to be significantly less than the effect of temperature on wave propagation within the structure [13]. Transducer bonding has also been shown to be remarkably consistent throughout
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fatigue tests lasting many weeks and tens of thousands of fatigue cycles [14]. For large temperature variations, changes in transducer and bonding properties may become more significant. However, this variability is one that can potentially be minimized via the choice of transducer types, materials and manufacturer. This is in contrast to the unavoidable effect of even small temperature changes on wave propagation within a structure, which is the subject of this paper. Strategies for compensating the effect of temperature on the structure have been developed in recent years. In particular Lu and Michaels [15] and Konstantinidis et al. [16] introduced a methodology, often referred to as optimal baseline selection (OBS), that uses multiple baseline measurements recorded over a range of temperatures. This approach has been the basis for further developments (see [13,16–24]), and a detailed discussion of this method will be presented in the next sections. The purpose of this paper is to quantitatively analyze different methods of temperature compensation and show that their performance can be estimated from fairly simple models. The outline is as follows. Firstly, a simple mathematical description of the guided wave SHM process is presented and used to analyze the temperature compensation methodologies. Then results from two independent experiments are presented. From these the experimental performance of the different temperature compensation strategies is obtained and compared with earlier predictions. The analysis presented enables a quantitative assessment to be made of the efficacy of temperature compensation for specific guided wave SHM scenarios.
2. Temperature compensation A typical guided wave SHM system comprises a number of transducers permanently attached to the surface of the structure. A suitable excitation signal is sent to one of the transducers and the time-domain responses (time-traces) from this and other transducers are recorded. This process is repeated using different transducers as the transmitter. Time-traces that are recorded on a different transducer to the transmitting one are referred to as pitch-catch mode, while those recorded on the transmitting transducer are referred to as pulse-echo mode. In this paper, a single pitch-catch time-trace from one pair of transducers is considered. A typical pitch-catch time-trace contains various overlapping wavepackets corresponding to different wave paths between the transducers. The first wavepacket (assuming a single propagating mode is present) corresponds to the direct arrival, and subsequent wavepackets are due to reflections and scattering from structural features. A significant class of SHM applications that employ active guided wave monitoring is based on the algebraic difference between the current time-trace and a baseline time-trace recorded when the structure was undamaged. The signal remaining after the subtraction of a baseline time-trace is referred to as a residual time-trace. In the best-case scenario, this subtraction allows the wavepackets due to direct transmission and scattering from benign structural features to be eliminated and any remaining wavepackets in the residual time-trace to be attributed to scattering from damage. The implication is that even complex structures can potentially be monitored. However, in the presence of changing temperature, the effectiveness of the subtraction process is reduced and the residual time-trace contains coherent noise that is primarily due to small shifts in the arrival times of wavepackets from structural features. The cause of these time-shifts is both thermal expansion and changes in wave velocities with temperature. It has been shown by different authors [15,16,24] that in the presence of changing temperature, simple subtraction of one base-
line time-trace from the current time-trace is not sufficient to discriminate between changes due to damage and those due to temperature variations. In order to mitigate temperature effects, the concepts of optimal baseline selection (OBS) and baseline signal stretch (BSS) have been proposed and demonstrated [15,20]. In the following section, these techniques are described and analyzed mathematically. 2.1. Preliminary analysis A baseline time-trace recorded from a guided wave SHM system at an initial temperature T0, is assumed to consist of N superimposed wavepackets corresponding to scattered time-traces from structural features. Mathematically this can be written as:
uðt; T 0 Þ ¼
N X
Aj sj ðt t j Þ
ð1Þ
j¼1
where t is time and Aj, sj and tj are respectively the amplitude, waveform and arrival time of the jth wavepacket. The arrival time of a wavepacket is tj = dj/vgr, where vgr is the group velocity and dj is the propagation distance. Consider how this time-trace changes in appearance if there is a change to a different temperature T = T0 + dT. In general, such a temperature shift may alter the shape, amplitude and arrival time of each of the individual wavepackets, with the shape and amplitude changing because of the possible temperature dependence of dispersion, reflection coefficients and attenuation mechanisms. However, to a first approximation, the dominant effect is a change in arrival time of individual echoes. For dispersive waves temperature actually causes two different time-shift effects, one due to a change in group velocity and one due to a change in phase velocity. The first of these affects the envelope of a wavepacket and the second affects the waves within the wavepacket. These time-shifts are generally not equal, resulting in a change in phase of the waves within a wavepacket. For the reasons described in [24], it is preferable to apply baseline subtraction to RF (i.e., un-envelope-detected) time-traces. In this scenario, the amplitude of the residual after subtracting two nominally identical wavepackets is predominantly governed by the time-shift of waves within the wavepacket, which arises from the change in phase velocity with temperature. The temperature-dependent time-shift, dt, of waves within a wavepacket is described by the fundamental relationship [24] as:
dt j ¼
dj
mph
a
kph
mph
dT
ð2Þ
where dj is the propagation distance, vph the phase velocity, a the coefficient of thermal expansion, and kph the change in phase velocity with temperature. In practice, the thermal expansion term, a, is typically one-to-two orders of magnitude smaller than the kph/vph term, and the above expression can be approximated as:
dt j ¼
kph
m2ph
dj dT
ð3Þ
The effect of the change in group velocity due to temperature on the level of residual time-trace is second order compared to the change in phase velocity. For this reason, it is physically justified and mathematically convenient to assume that the effect of a change in temperature is to simply shift the arrival time of an entire wavepacket by the amount dt given by (3). Thus, after a temperature change, the time-trace of (1) becomes:
uðt; T 0 þ dTÞ ¼
N X j¼1
Aj sj ½t t j bðdTÞ;
ð4Þ
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where b(dT) is the fractional change in arrival time of the wavepackets. From (3):
b¼1
mgr kph dT; m2ph
ð5Þ
where vgr is the group velocity. As previously shown [17], the peak amplitude of the post-subtraction residual associated with each wavepacket can be estimated as:
unoise ¼ Aj x0 jdt j j; j
ð6Þ
where x0 is the center frequency of the wavepacket. Thus, if the temperature change is known, the time-shifts of the wavepackets in the time-trace can be calculated and used to estimate the postsubtraction noise. Having developed a simple model of the changes to a time-trace associated with a temperature shift, the techniques to deal with this change are now analyzed. 2.2. Optimal baseline selection (OBS) OBS uses M baseline time-traces recorded at different temperatures Tm = T0 + dTm referred to as the baseline dataset. The mth time-trace from the baseline dataset can be expressed as:
um ðt; T m Þ ¼
N X
m m Am j sj ½t t j bðdT m Þ:
ð7Þ
j¼1
In this equation, b(dTm) is the fractional shift in arrival times of wavepackets in each time-trace with respect to their values at an arbitrary fixed temperature. To find the best match between a time-trace in the baseline dataset and the current time-trace, recorded at a temperature T, a criterion needs to be defined that characterizes their similarity. Possible criteria include the mean square deviation [15],
mms ¼ arg min m
Z
t2
½uðt; TÞ um ðt; T m Þ2 dt ;
ð8Þ
t1
and the maximum residual amplitude [25],
mmr ¼ arg minfmax juðt; TÞ um ðt; T m Þjg;
ð9Þ
m
where mms and mmr are the indices of the baseline time-trace that best matches the current time-trace according to the criterion used. Note that selecting the best-matched baseline time-trace achieves superior results to simply selecting the baseline time-trace with the temperature that best matches the current time-trace. This is because of the difficulty in accurately measuring the mean temperature of the entire structure. If the nominal temperature of the current time-trace and the best-matched baseline time-trace are compared, noticeable differences are observed [20]. If the time-traces in the baseline dataset are recorded at a range of discrete temperatures with uniform spacing DT, then the noise associated with OBS can be estimated. This estimate assumes that the maximum deviation in the true temperature between the current time-trace and the best-matched baseline time-trace, dT, is DT/2, from which the noise can be estimated using (6). This will be demonstrated experimentally later in this paper. It is worth noting at this point that the OBS does not suppress the signal due to damage. This is due to the changes as a result of damage being much smaller than those induced by temperature change as shown by Lu and Michaels [15]. 2.3. Baseline signal stretch (BSS) Another method for temperature compensation is BSS [15], which in its simplest form requires only one baseline time-trace
to compensate for the effect of temperature. Unlike simple subtraction and OBS, this technique modifies a single baseline time-trace to match the current time-trace. In BSS the time-axis of the baseline time-trace, u(t; T0), is stretched (where the term stretch refers ^ to yield a to both dilation and compression) by a stretch factor b ^ This can be expressed as: ^ ðt; T 0 ; bÞ. new time-trace u
^ ¼ uðt=b; ^ T0Þ ¼ ^ ðt; T 0 ; bÞ u
N X
^ tj Þ Aj sj ðt=b
ð10Þ
j¼1
^ is chosen to match b, the arrival times of the different waveIf b packets match those of the baseline time-trace in (4). However, the ^ comindividual wavepackets, sj, are also stretched by the factor b pared to the baseline time-trace. Furthermore, pure stretching of the time-axis also alters the frequency content of the time-trace. Because experimental time-traces are sampled at discrete points in time, the necessary stretching effectively requires that the time-traces be resampled at a slightly different sampling frequency. The practical implementation of this is most easily achieved in the frequency-domain as described below. 2.3.1. Implementation of BSS In this section, square brackets [ ] denote a discretely sampled function. Consider a continuous time-trace, u(t), sampled with a time step of Dt1, resulting in a discretely sampled time-trace u1[n] = u(nDt1), where n is an integer. In order to perform the time-stretch, u1[n] is first padded with zeros so that it contains a total of m1 points. This time-trace is transformed to the frequency-domain using a Fast Fourier Transform (FFT), resulting in a discretely sampled spectrum, U1[n], containing m1 bins, with a frequency spacing, Df = 1/(m1Dt1), between bins. The spectrum is then either truncated or zero-padded to create a new spectrum, U2[n], that contains m2 bins (the truncation or padding taking place at the appropriate portion of the spectrum so that the conjugate symmetry of the whole spectrum is preserved according to the particular implementation of the FFT algorithm). The modified spectrum is then subjected to an inverse FFT, resulting in a new discretely sampled representation of the original continuous time-trace, u2[n] = u(nDt2), containing m2 points sampled with a time step Dt2 = 1/(m2Df) = (m1/m2)Dt1. Note that strictly, this operation has not performed the time-stretch, it has simply resampled the original continuous time-trace, u(t), with a different time step Dt2. However, if a new continuous time-trace is defined as u2 ðtÞ ¼ ^ where b ^ ¼ m2 =m1 , then it can be seen that u2[n] is the disuðt=bÞ crete representation of u2(t) sampled with the original time step, Dt1. Thus the procedure has produced a discretely sampled stretch ^ equal to m2/m1. of the original time-trace with a stretch factor, b, This implementation thus restricts the possible stretch factors to those corresponding to integral values of m1 and m2. Note that the origin of the stretch is the start of the time-trace u[n], i.e. at n = 0. In the context of temperature compensation, this point in time should therefore correspond to the temporal center of the transmitted time-trace. If this center point is not at n = 0, then a translation (i.e. a time-shift) should be applied to the time-trace prior to stretching [21]. Typically an optimization procedure is applied to obtain the va^ that needs to be applied to a time-trace to correct for temlue of b perature using possible criteria similar to those used in the OBS procedure, such as the mean square deviation,
^ms ¼ arg min b ^ b
Z
t2
^ 2 dt ; ^ ðt; T 0 ; bÞ ½uðt; TÞ u
ð11Þ
t1
and the maximum residual amplitude [42],
n o ^ : ^mr ¼ arg min max juðt; TÞ u ^ ðt; T 0 ; bÞj b ^ b
ð12Þ
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^ can be implemented in an iterNote that the optimization of b ative algorithm. However, this needs to be performed with caution, since metrics such as maximum residual amplitude have multiple minima. Practically, it has been found that the mean square devia^ the maxtion provides a more robust metric for coarse tuning of b; ^ imum residual metric is best restricted to fine tuning of b. It is also worth noting that if a sufficiently long time-trace is recorded, the latter part will contain waves in a diffuse field regime (i.e., the coda). If this is the case, Lu and Michaels [15] showed that ^ can be evaluated directly by the the optimum stretch factor, b, short time cross-correlation of the current and baseline timetraces, yielding a time-dependent local time-shift from which the overall stretch factor can be computed. 2.3.2. Granularization noise Since m1 and m2 are integers, the possible stretch factors that can be applied to a baseline time-trace are themselves discrete and spaced at 1/m1. This introduces what is termed granularization noise into the stretching process. For a wavepacket that nominally arrives at time tj, the stretching procedure can only adjust its arrival time in steps of tj/m1. Thus, when stretching is applied to compensate for a change in temperature, there is potentially an error in arrival time matching of up to tj/(2m1). The granularization noise, ugn, associated with a particular time-trace can therefore be estimated from (6) as:
ugn ¼
Aj x 0 t j : 2m1
ð13Þ
This noise can clearly be reduced by increasing m1 (i.e. the amount of padding applied to the original time-trace), but this requires more computational resources. It should be stressed that the granularization noise does not depend on the sampling frequency of the original time-trace. Provided the Nyquist criterion is satisfied, there is no benefit in using a higher sampling frequency; the critical parameter is the number of points, m1, after zero padding. 2.3.3. Frequency noise In addition to changing the arrival time, stretching also distorts the shape of each wavepacket. Although the arrival times of the mid-points of wavepackets in the baseline and current time-traces may be aligned by stretching, the shape of one will be distorted relative to the other. This distortion gives rise to the second source of noise, which is termed frequency noise. The frequency noise is arrival time independent as it is a function of the distortion of the individual wavepackets, and represents the general limit for the stretching method unless steps are taken to compensate for this effect. The more the time-axis is stretched, the greater the influence of frequency noise. It is theoretically possible to correct for this frequency noise as described below, although practical success with this approach has been limited. If it is assumed that the individual wavepackets are all the same shape; i.e., sj(t) = s(t), then the baseline and current time-traces, (1) and (4) respectively, can be written in the frequency-domain as:
Uðx; T 0 Þ ¼
" N X j¼1
Uðx; T 0 þ dTÞ ¼
# Aj expðixtj Þ SðxÞ " N X
ð14Þ #
Aj expðixt j bðdTÞÞ SðxÞ
ð15Þ
j¼1
where variables with capital letters represent the Fourier transforms of their lower case counterparts. In these expressions, the first term in square brackets corresponds to the Fourier transform of a train of impulses representing the arrival times of the wavepackets, and the second term, S(x), describes the shape of the wavepackets in the frequency-domain. In the time-domain, these two
elements are convolved, but the frequency-domain representation enables them to be written as a product. ^ ¼ b, then the If the baseline time-trace is stretched by a factor b equivalent frequency-domain expression for the stretched counterpart of (14) is
Uðx; T 0 ; bÞ ¼
" N X
# Aj expðixtj bðdTÞÞ bSðbxÞ
ð16Þ
j¼1
It can be seen that the arrival time portion of this expression matches that of the current time-trace in (15), but that the wavepacket term is modified. In order to recover U(x, T0 + dT) precisely, it is theoretically possible to multiply the above expression by a suitable factor:
Uðx; T 0 ; bÞ
SðxÞ ¼ Uðx; T 0 þ dTÞ: bSðbxÞ
ð17Þ
The practical difficulties with this approach are that (a) all the constituent wavepackets must have the same shape, (b) S(x) must be known, and (c) the division by S(bx) may introduce noise at the extremes of the time-trace bandwidth unless care is taken to avoid these effects by filtering. For these reasons, the compensation for frequency noise has not been adopted here. Instead, discussion is limited to estimating the amplitude of frequency noise in the absence of compensation. The upper bound of frequency noise, ufn, can be expressed as:
ufn ¼ Aj max jsðtÞ sðt=bÞj:
ð18Þ
It is not possible to proceed further without specifying the shape of s(t). Consider a Gaussian windowed complex toneburst centered at t = 0. This complex toneburst is of interest because its envelope is readily computed and gives a phase-independent result. Such a toneburst is defined as:
SðtÞ ¼ expðix0 tÞ exp
t 2 ; 2 2r
ð19Þ
where x0 is the toneburst frequency and r is a parameter specifying the width of the toneburst. The difference between this wavepacket and one stretched by a factor b is:
SðtÞ sðt=bÞ ¼ expðix0 tÞ exp exp
: 2r2 b2
t 2 expðix0 t=bÞ 2r 2
t2
ð20Þ
If |1b| 1, then the magnitude of this difference, r(t), can be expressed as:
rðtÞ ¼ jsðtÞ sðt=bÞj ¼ x0 jtð1 bÞj exp
t2 : 2r2
ð21Þ
The position of the maximum (i.e., the location of the peak frequency noise associated with subtraction of the two Gaussian windowed wavepackets) can readily be found by differentiation with respect to t and occurs at t = ±r. The frequency noise is therefore:
ufn ¼ Aj rðt ¼ rÞ ¼ Aj x0 rj1 bj expð1=2Þ:
ð22Þ
The frequency noise associated with other windows, such as Hanning, can also be computed numerically from (18) although closed form solutions, such as that obtained for a Gaussian window, may not be possible. Numerical calculations for the frequency noise associated with Hanning-windowed tonebursts with 5– 50 cycles indicate that the frequency noise is always approximately 1.3 times (i.e., 2.3 dB) higher than for a Gaussian windowed toneburst with the same number of cycles (if the number of cycles in the latter are defined by the number falling within the 40 dB points of the Gaussian function). It was stated earlier that (22) is
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and then using a standard Fourier decomposition technique to propagate each frequency component of the input toneburst over the appropriate distances. This time-trace serves as the baseline. The material properties and propagation distances were altered to simulate a 10 °C temperature change and a new timetrace was generated. BSS was then performed to minimize the maximum residual between the two time-traces. The resulting residual time-trace envelopes are shown on dB scales in Fig. 1b and c for two different values of m1 in the BSS algorithm. Also shown in these figures are dashed lines indicating the estimated values of granularization and frequency noise computed using (13) and (22). It can be observed in the case of a low value for m1 shown in Fig. 1b that the amplitude of the residual increases with time. This behavior, which is consistent with (13), suggests that the dominant effect is granularization noise and that the level of frequency noise is low. It can be seen that the amplitude of the five peaks in the residual time-trace agree reasonably well with the theoretical estimate of granularization noise. Increasing m1 as shown in Fig. 1c reduces the influence of granularization noise so that frequency noise dominates, and this noise level represents the general limit for BSS for this temperature change. The peaks in the residual time-trace are now of fairly constant amplitude and are in good agreement with the estimated line for frequency noise. Increasing m1 further will continue to reduce the granularization noise, but as this effect is now smaller than the frequency noise there is little further improvement.
valid only if |1 b| is small, and adequate results are obtained for b values in the range of 0.95–1.05. It should be noted that this range is ample to cover any temperature swing likely to be encountered in real structures and does not in practice provide any limitation. From (5), the largest value of b likely to be encountered in a given application can be estimated, and (22) (or (18) for arbitrary shaped windows) can be used to estimate the associated frequency noise associated with BSS. This estimate, in combination with the granularization noise model, can be used to characterize the performance of the BSS technique. 2.3.4. Simulations of BSS performance Expressions (13), (18), and (22) provide estimates of the granularization and frequency noise associated with the BSS technique based on some simple approximations. In this section, a complete time-trace simulation is performed and the BSS process is applied to provide a validation of the simplified noise models. The simulation is performed using the propagation properties of the S0 mode in a 3 mm thick aluminum plate. A 10 cycle Hanningwindowed tone burst with a center frequency of 250 kHz is used as the transmitted signal. This center frequency is selected since the S0 mode is largely non-dispersive and it is consistent with typical frequencies used in practice. The simulated time-trace in Fig. 1a consists of five equally spaced echoes of the same amplitude that correspond to propagation distances of 0.25 m, 0.75 m, 1.25 m, 1.75 m and 2.25 m. The time-trace was simulated by first generating the appropriate dispersion curve for the S0 mode in aluminum
a
b
1 0.8
0 Original Residual Gran. noise Freq. noise
−10
0.6 Ampitude (dB)
Ampitude
0.4 0.2 0 −0.2
−20
−30
−40
−0.4 −0.6
−50
−0.8 −1 0
400
−60 0
100
200 300 Time (µs)
400
d
0 Original Residual Gran. noise Freq. noise
−10
Ampitude (dB)
200 300 Time (µs)
Post Subtraction Noise (dB)
c
100
−20
−30
−40
−20 −30 −40 −50 0
10
−50 50,000 −60 0
12
8
m m11 100
200 300 Time (µs)
400
4 100,000
0
2
6 Temperature Difference
Fig. 1. Simulated data illustrating the use of BSS in the presence of a 10 °C temperature change between the current and baseline time-traces. (a) Original RF baseline timetrace containing five wavepackets corresponding to S0 wave propagation over distances of 0.25 m, 0.75 m, 1.25 m, 1.75 m and 2.25 m at a frequency of 250 kHz in a 3 mm thick aluminum plate. (b) Residual time-trace amplitude using BSS with m1 = 25 102. (c) Residual time-trace amplitude using BSS with m1 = 25x104. (d) Surface showing the effect of changing m1 and the temperature difference on maximum residual time-trace amplitude.
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This change from a granularization to frequency noise dominated regime can be used to set the temperature compensation parameters for experimental studies. The surface of Fig. 1d shows the effects of m1 and the temperature difference on post-subtraction noise using the same test signals as shown in Fig. 1a. As previously observed, it is clear that as m1 increases, the postsubtraction noise initially decreases very rapidly. However this drop off slows and reaches a point where further increases in m1 give little reduction in post-subtraction noise for considerable additional processing time. The identification of this point will be used later to select the optimum value of m1 to be used experimentally. The surface also makes it clear that as the temperature difference is increased, the frequency noise increases. Even at these relatively modest temperature changes (
