## Abstract

Structural health monitoring (SHM) using guided waves is one of the only ways in which damage anywhere in a structure can be detected using a sparse array of permanently attached sensors. To distinguish damage from structural features, some form of comparison with damage-free reference data is essential, and here subtraction is considered. The detectability of damage is determined by the amplitude of residual signals from structural features remaining after the subtraction of reference data. These are non-zero due to changing environmental conditions such as temperature. In this paper, the amplitude of the residual signals is quantified for different guided-wave SHM strategies. Comparisons are made between two methods of reference signal subtraction and between two candidate sensor configurations. These studies allow estimates to be made of the number of sensors required per unit area to reliably detect a prescribed type of damage. It is shown that the number required is prohibitively high, even in the presence of modest temperature fluctuations, hence some form of temperature compensation is absolutely essential for guided-wave SHM systems to be viable. A potential solution is examined and shown to provide an improvement in signal suppression of approximately 30 dB, which corresponds to two orders of magnitude reduction in the number of sensors required.

## 1. Introduction

The recent special issue of *Phil. Trans. R. Soc. A* indicates the increasing significance of structural health monitoring (SHM) across a range of industrial sectors. SHM is a broad subject that includes the detection of a wide variety of structural or component degradation, including machine condition monitoring, generalized corrosion, fastener loosening, etc. The current paper is concerned specifically with the detection of unexpected localized damage that may occur at any location in a structure using guided acoustic waves. The detection of localized damage is a generic and as yet unsolved SHM challenge that has a huge range of potential applications to industry, e.g. impact-induced delaminations in composite aerospace structures, localized corrosion in petrochemical plant and malicious penetrations of shipping containers.

The background to SHM for localized damage detection and the logic for using guided waves will be discussed in the following introductory paragraphs. However, the primary purpose of the paper is to quantify the fundamental sensitivity of the guided-wave technique to damage and to show what the implications of this are for a practical SHM system.

To ensure the reliable operation and assess the condition of a structure, non-destructive testing (NDT) is required. Current techniques to measure the conditions of a structure and detect damage involve rigorous and time-consuming NDT. Typically, the approach is to make multiple point measurements using ultrasound, eddy current or X-rays to cover the whole structure (Hull & John 1988). The slow nature of these techniques hence represents a significant cost.

SHM is a potential alternative by which the condition of a structure is permanently monitored, through embedded or attached arrays of sensors. By monitoring the condition, or change in conditions, of the structure, detection of damage at any location within the structure becomes possible.

A number of strategies for SHM have been investigated (Farrar & Worden 2007), and Boller (2000) indicates how they can be integrated into a system. One strategy is a very dense network of sensors integrated into the structure with the spacing between sensing points similar to or smaller than the scale of the anticipated damage. The sensing requirements for an individual sensor are then very modest, since the damage need only be detected at the sensor position itself, and can be accomplished with conventional NDT techniques. This is the biologically inspired nervous system approach and requires complete integration of the sensing system into the host structure. Although attractive, the power, weight and data transfer requirements of a large-scale integrated conventional NDT array are prohibitive. To make such an approach feasible, significant developments in sensor technology are required. Some progress has been made in this direction through the work of Qing *et al.* (2005), showing how a number of techniques can be applied together and integrated into a structure.

A similarly biologically inspired approach is that of Pang & Bond (2005*a*,*b*) and Trask & Bond (2006) who embed hollow glass fibres within a composite structure. When the structure is damaged, the glass breaks and releases epoxy resin to repair the structure and fluorescent dye to indicate that damage has taken place. Such an approach has significant benefits, in both indicating the location and temporarily repairing the damage. However, it is inherently not suitable for existing structures, and the repair methods are limited to composite structures. This immediately eliminates many of the situations where SHM would be most useful.

An obvious alternative to these dense biologically inspired sensing networks is a sparse network of discrete sensors, with the sensor separation far larger than the scale of the anticipated damage. The major advantage is that it can be potentially retrofitted to existing structures if the sensors can be sufficiently sparsely spaced and are non-intrusive. Continuing advances in wireless data transmission, power scavenging and energy storage mean that the supporting architecture for a distributed sensor network is almost, if not already, a reality. Indeed, distributed networks of sensing nodes are already used for distributed environmental monitoring (www.crossbow.com, Crossbow Technology, Inc., 2006, http://www.xbow.com). However, the challenge for SHM is to demonstrate reliable detection of damage remote from a sensor location. To this end, various approaches have been applied.

Linear global testing techniques have been developed, such as the measurement of natural frequencies (Ewins 1984; Farrar *et al.* 2001) and several optical and thermal techniques (Heller 2001). While each of these approaches has found some applications, they suffer from limitations. Natural frequency measurements have limited sensitivity to small defects and such approaches are sensitive to changes in boundary conditions and geometry. Global large-area optical and thermal NDT techniques have the required sensitivity but are limited to surface-breaking defects. More recently, nonlinear techniques have been investigated. Hillis *et al.* (2006) detect the nonlinear properties of cracks in a structure using a bispectrum analysis to indicate the presence of a defect. The understanding of the underlying physics of this approach and its practical implementation are still at an early stage. The general trend for all techniques involving stress waves is that increased frequency leads to increased sensitivity but at the expense of detection range.

Guided acoustic waves in the tens to hundreds of kilohertz range, which are the subject of this paper, are arguably the only detection mechanism that combines a reasonable sensitivity to damage with significant propagation range. Guided acoustic waves propagate along a plate-like structure and, in certain circumstances, can propagate over several metres. Viktorov (1970) gave a detailed description of Lamb waves, but Worlton (1961) first proposed their use for NDT of plates. There is already an established NDT industry using deployable guided-wave systems for the inspection of pipelines (Lowe *et al.* 1998; Alleyne *et al.* 2001; Rose *et al.* 2002; Long *et al.* 2003) and rails (Wilcox *et al.* 2003). In these guided-wave systems, a significant amount of effort in the design of transducers and signal processing is devoted to obtaining modal and directional purity, thus ensuring that only desired modes are excited and that any excited modes are in non-dispersive regions (Alleyne & Cawley 1992*a*,*b*). In the case of a deployable system, data must be interpreted with minimal *a priori* knowledge of the structure. Typically, an array of sensors is attached to the structure and used to inject a pulse of guided-wave energy. The same array is then used to record the reflected signals, and the position of these signals in the time domain can be easily related to the position of features (e.g. flanges, welds) in the structure. Any signal that cannot be related to a known feature is assumed to be a defect. The problem of signal interpretation means that guided-wave NDT remains fundamentally limited to geometrically simple structures with low feature densities. As the number of features becomes higher, their associated signals merge together obscuring the reflected signals from damage.

However, in an SHM system with permanently attached or embedded sensors, it becomes possible to make highly accurate measurements throughout the life of the structure by using reference (or baseline) signal subtraction (Kagawa *et al.* 1998; Lu & Michaels 2005; Konstantinidis *et al.* 2006). This potentially enables the guided-wave technique to be extended to more complex structures, such as those likely to be encountered in the aerospace or power generation industries, and is compatible with the axioms of SHM as outlined by Worden *et al.* (2007). The success of guided-wave SHM is thus critically dependent on the ability of reference signal subtraction to suppress signals from benign structural features. Ultimately, the sensitivity of an SHM system is governed by the residual signal (termed noise in this paper) left after subtraction.

This paper addresses this problem through the following ways.

Presenting an analytical model to describe the noise after reference signal subtraction caused by the first-order temperature effect.

Applying this model to different wave modalities and sensor geometries, showing how detection limits can be related to both temperature and propagation distance.

Demonstrating the effect of a temperature compensation technique on required sensor spacing to enable the detection of defects.

## 2. The effect of temperature on guided-wave signal subtraction

In order for any subtraction technique to be of practical value, it is important to have an understanding of the factors affecting the variability in the measurements that result in imperfect subtraction of the reference signal, resulting in residual signals which could be mistaken for damage. Lu & Michaels (2005) and Konstantinidis *et al.* (2006) showed that the main source of fluctuations in the received signal was through the effects of temperature, both the changes in the material properties of the structure due to thermal effects including thermal expansion and temperature-induced changes to the transducers and their bonds. It is possible through careful selection of adhesives and transducers to minimize the variability from the bonds; however, the changes in the material properties of the structure cannot be altered. This section details an analytical model that can be used to describe the effects of temperature on the post-subtraction signal.

An initial reference time trace, *g*_{0}(*t*), containing guided-wave signals is recorded by some means from a structure. For the purposes of this analysis, it is assumed that this time trace contains just one signal resulting from the presence of just one benign structural feature and that there is no dispersion present. It is important to note that, although the regions of operation used in this work are largely non-dispersive, guided waves often exhibit dispersive behaviour. If it is necessary to operate in a dispersive region, it is probable that some form of compensation for this will be required as described by Wilcox (2003). In the reference time trace, the signal from the structural feature arrives at time *t*_{0} and corresponds to wave propagation over a distance *d* at velocity *v*. Some time later, the temperature of the structure has changed by δ*T* and another time trace, *g*_{1}(*t*), is recorded. The purpose of this analysis is to estimate what residual signal from this benign structural feature remains when *g*_{0}(*t*) is subtracted from *g*_{1}(*t*). To do this, further information about the shape of the signal and the method of subtraction is necessary. Here it is assumed for simplicity that the signals in both *g*_{0}(*t*) and *g*_{1}(*t*) are time-delayed copies of a Hanning-windowed toneburst, *s*(*t*), initially centred on *t*=0. These signals are widely employed in guided-wave applications (e.g. Lowe *et al.* 1998), as the toneburst ensures that a limited part of the frequency spectra is excited (Alleyne & Cawley 1992*b*). The use of the windowing technique allows the signal to be resolved in space, and a Hanning window suppresses sidelobes from the signal in the frequency domain. The signal is defined as(2.1)where *f* is the centre frequency of the toneburst; *n* is the number of cycles; *u*_{0} is the amplitude of the signal; and *w*(*t*) is the Hanning window. Hence,(2.2)where δ*t* is the change in the arrival time of the signal due to the change in temperature. Note that the change in temperature causes a translation of the received signal in time rather than stretching or compression of the time axis. It is now necessary to relate δ*t* to δ*T*. Starting from *t*=*d*/*v* and partially differentiating with respect to both *d* and *v* gives(2.3)

The relationship between distance and temperature is δ*d*/δ*T*=*αd*, where *α* is the coefficient of thermal expansion of the material and is typically of the order of 10^{−5}°C^{−1} for metals. The relationship between wave velocity (due to changes in stiffness) and temperature can be written as δ*v*/δ*T*=*k*, where *k* is typically of the order of −1 m s^{−1}°C^{−1}. Combining these expressions with equation (2.3) gives(2.4)

Note that the size of *k*/*v* is typically one to two orders of magnitude larger than *α*, so the change in velocity with temperature is the dominant effect rather than the thermal expansion of the structure. Also, the presence of *d* in the numerator indicates that the time shift associated with a change in temperature is proportional to propagation distance, hence the performance of reference signal subtraction decreases as propagation distance increases. Finally, the presence of *v* in the denominator suggests that faster modes are less affected by temperature than slower modes. Equations (2.2) and (2.4) enable both the time traces *g*_{0}(*t*) and *g*_{1}(*t*) to be predicted at two different temperatures. Two different methods of signal subtraction are now considered.

### (a) RF signal subtraction

In the first instance, algebraic subtraction of the RF signal *g*_{0}(*t*) from *g*_{1}(*t*) is considered. The quantity of interest is the peak amplitude of the result. The subtraction *g*_{1}(*t*)−*g*_{0}(*t*) is equal to . If it is assumed that δ*t* is small compared with 1/*f*, then the maximum amplitude of this subtraction occurs where the gradient of *s*(*t*) is highest, which is at *t*=0. So the subtraction becomes(2.5)

Expanding the sine terms, assuming that δ*t* is sufficiently small so that a small-angle approximation can be applied, leads to(2.6)

Substituting equation (2.6) into equation (2.4) to express δ*t* in terms of δ*T* gives(2.7)where *v*_{ph} is the phase velocity of the guided wave and *k*_{ph} is the coefficient relating changes in *v*_{ph} to temperature. The phase velocity is used here since this determines the shift of the individual waves in the wave packet, which causes *u*_{noise}.

### (b) Envelope subtraction

An alternative to subtracting the RF signals, which has been used by the current authors (Konstantinidis *et al.* 2006) and others, is to first calculate the envelope of the signals using Hilbert transforms and then subtract these. For this case, the subtraction of the signal envelopes is equal to , where *w*(*t*) is a Hanning window given by equation (2.1). The difference between these signals is given by(2.8)where *n* is the number of cycles in the toneburst. This can be rearranged to give(2.9)

The maximum is located at 2*πft*/*n*=*π*/2 and, if δ*t* is assumed to be small, this relationship can be expressed as(2.10)

Substituting equation (2.4) into equation (2.10) for δ*t* gives(2.11)where *v*_{gr} is the group velocity of the guided wave and *k*_{gr} is the coefficient relating changes in *v*_{gr} to temperature. Group velocity is used because the *u*_{noise} is due, in this case, to the shift in the envelope of the wave packet.

### (c) Example and comparison

Having established the relationships describing the post-subtraction noise due to temperature effects for two different subtraction methods, an example of the effect of temperature on Lamb waves in a plate-like structure is considered. The structure is a 1 mm thick aluminium plate that has the following reference material properties at a certain temperature: density (*ρ*) 2700 kg m^{−3}; shear velocity (*V*_{T}) 3130 m s^{−1}; and longitudinal velocity (*V*_{L}) 6320 m s^{−1}. The excitation signal is a five-cycle Hanning-windowed toneburst with a centre frequency of 1 MHz. At this frequency, only the fundamental modes exist. Fundamental modes at relatively low frequencies are selected, as at higher frequencies the presence of multiple modes makes the resulting signals extremely complex (Alleyne & Cawley 1992*b*). The following results are scaleable to other plate thicknesses if the frequency-thickness product is maintained, and propagation distances are expressed as multiples of plate thickness. The temperature parameters for aluminium used in the simulation were *α*=23×10^{−6}°C^{−1}, *k*_{T}=−0.752 m s^{−1}°C^{−1} and *k*_{L}=−1.089 m s^{−1}°C^{−1}, where *k*_{T} and *k*_{L} refer to bulk shear and longitudinal waves, respectively. These parameters were used to modify the reference material properties to material properties at a temperature 1°C higher. The reference and modified material properties were then used to generate two sets of dispersion curves from which the modal temperature parameters given in table 1 were extracted for the two modes at the frequency thickness of interest. For more information on the generation of dispersion curves, see Lowe (1995) and Long *et al.* (2003).

The values shown in table 1 were then substituted into equations (2.7) and (2.11) to predict the post-subtraction noise, as shown by the solid lines on the graph in figure 1 for a 1°C temperature change. This shows the amplitude of *u*_{noise}/*u*_{0} versus propagation distance (expressed in multiples of plate thickness) for both subtraction methods and both the fundamental Lamb wave modes. The open circles on the graph were obtained by performing complete time-trace simulations from the dispersion curve using Fourier decomposition of the chosen input signal. The simulation exactly models dispersive propagation through the plate, but it models transduction as point excitation by a normal force and point detection by a normal displacement. The RF waveforms or their Hilbert envelopes at the two different temperatures are subtracted and the maximum magnitude of the resulting signals is extracted. As such, the full simulation method can be regarded as the exact counterpart to the approximate method derived previously in equations (2.7) and (2.11). It can be seen that the approximate method is in very good agreement with the exact method over the distances and temperature change considered.

Several observations can be made from figure 1. First, the *S*_{0} lines are below the equivalent *A*_{0} ones, due to the effect discussed previously where the higher *S*_{0} velocity reduces its sensitivity to temperature. Second, the curves for envelope subtraction are approximately 20 dB lower than those for RF signal subtraction. This is entirely expected since comparison of equations (2.7) and (2.11) suggests that, if the modal parameters are equal (i.e. *k*_{ph}=*k*_{gr} and *v*_{ph}=*v*_{gr}), envelope subtraction gives a reduction in noise by a factor of 2*n* compared with RF signal subtraction. The difference is due to the fact that the envelope is a more slowly varying function of time than the RF signal and is hence less sensitive to time shifts.

In order to confirm that structural temperature effects are dominant in the physical system, it is instructive to compare the model predictions with the experimental measurement. An experiment was carried out using a 3 mm thick aluminium plate, 1 m×1.25 m. Signals were recorded five times a second between a pair of transducers operating in pitch-catch mode attached to the plate 400 mm apart. The transmitted signal was a five-cycle Hanning-windowed toneburst with a centre frequency of 250 kHz (i.e. equivalent to a frequency thickness of 0.75 MHz-mm). The transducer separation and the size of plate were chosen to be sufficiently large to enable the signals due to direct transmission of the fundamental modes to be separated in the time domain from each other and the edge reflections. The first signal recorded was used as the reference signal. The temperature of the plate was then controlled in an oven over a 10°C range. The reference signal was then subtracted from later signals and the resulting *u*_{noise}/*u*_{0} ratio was plotted. This operation was carried out for the two fundamental modes and for both RF and envelope subtraction. Figure 2*a* shows the experimental results, the performance of the approximate method (see equations (2.6) and (2.10)) and the exact simulations. As in figure 1, there is a very good agreement between the approximate method and the exact simulations despite the wider temperature range. More importantly, figure 2 shows an excellent agreement between the simulations and the experiments for an RF subtraction of both the fundamental wave modes.

The agreement between simulation and experiment is somewhat worse when a comparison is made using envelope subtraction. For the *S*_{0} mode, this is to be expected, as the absolute values after subtraction lie in the region of incoherent electrical and acoustic noise present in the experiment. However, this degradation of performance was not predicted for the *A*_{0} mode and, at this point, the explanation is unclear, with investigations ongoing into its cause.

Having shown in figure 2 that the experiment and the model are in good agreement across a range of temperatures, the performance can be compared across a range of temperatures and propagation distances. It is worth noting that equations (2.7) and (2.11) can be generalized to give an overall expression for the noise signal remaining after subtraction, as shown here(2.12)

The value of *β* is dependent on the wave mode that is to be used and the type of subtraction being carried out (envelope or RF),(2.13)

(2.14)

In order to show how well this approximate method works, experimental results for *u*_{noise}/*u*_{0} are plotted against the approximate method in figure 3. In this figure, *u*_{noise}/*u*_{0} is plotted against the product of propagation distance and temperature difference, for a single value of *β*. The value of *β* is calculated for RF subtraction of the *A*_{0} mode and is 0.1689. The agreement between the experiment and the model is excellent, showing that the method is capable of predicting *u*_{noise} over a range of temperatures and propagation distances.

Having shown that the model accurately recreates the behaviour of the physical system, it is now possible to perform calculations to predict the behaviour and capabilities of various sensor arrangements of practical importance for SHM systems.

## 3. Implications for guided-wave SHM

Figure 4*a* shows a portion of a large plate-like structure instrumented with a sparse array of guided-wave sensors arranged in an isometric grid pattern with sensor pitch *p*. The choice of an isometric grid is arbitrary and the analysis could easily be extended to other configurations.

For simplicity in this analysis, the requirement of the system is to detect (but not necessarily locate) a pre-defined level of localized damage anywhere in the structure. It should be stressed that the analysis could be extended to the localization of damage, the underlying principles being the same. Two possible modes of operation are defined in the following, both of which are based on reference signal subtraction.

*Pulse-echo-based detection*. Each sensor operates in pulse-echo mode; damage detected by the appearance of a new reflected signal.*Pitch-catch-based detection*. Each pair of adjacent sensors operates in pitch-catch mode; damage detected by the appearance of a new diffracted signal.

In both cases, damage can be detected only if, after reference signal subtraction, the signal due to damage exceeds the noise associated with benign structural and environmental changes such as temperature. The purpose of the following analysis is to compare the two approaches by determining the worst-case signal-to-noise ratio, *S*.

### (a) Pulse-echo-based detection

Figure 4*b* shows the hexagonal area around a sensor which represents the minimum area of structure within which that sensor must be able to detect damage if the pulse-echo approach is used. Within this area, the worst-case location for damage is at an apex of the hexagon as shown in figure 5*a*, since here the propagation distance to and from the sensor is greatest.

The amplitude of the reflected signal from damage at this position may be written as(3.1)where *u*_{input} is the amplitude of the transmitted signal (measured at unit distance from the sensor); *R*_{damage} is the reflection coefficient of the damage (defined in terms of the scattered wave amplitude at unit distance from the damage); and *r* is the distance from the sensor to the damage. The terms are due to the spreading of the input and scattered waves resulting in an overall 1/*r* dependence. This can be expressed in terms of the sensor pitch, *p*, noting that ,(3.2)

The worst-case noise signal will be due to the largest possible reflector at the same distance from the sensor as the damage. The largest possible reflector likely to be encountered in practice is a straight edge and will result in a reflected signal with amplitude, *u*_{edge}, given by(3.3)where *R*_{edge} is the reflection coefficient of the edge and is assumed to be unity. Note that, for a straight edge, there is no secondary beam spreading so that there is an overall dependence in contrast to the 1/*r* dependence for the damage signal. Again, substituting gives(3.4)

A term describing the noise can be created by setting *u*_{0}=*u*_{edge} in equation (2.12) and substituting for the propagation distance of this signal to give(3.5)

The signal-to-noise ratio for the pulse-echo approach can then be formulated as(3.6)

### (b) Pitch-catch-based damage detection

Figure 4*c* shows the diamond-shaped area between a pair of sensors which represents the minimum area of structure within which that sensor pair must be able to detect damage if the pitch-catch approach is used. In this case, the worst-case situation for damage cannot be specified without some assumption about the angular dependence of the diffracted signal from the damage. If for the purposes of this simple analysis it is assumed to be constant (i.e. independent of the angle and therefore equal to the value used in the pulse-echo case), then the worst-case location for damage is at one of the corners indicated in figure 5*b*, since this again is the point where propagation distance to and from the sensor is greatest. As in the pulse-echo case, the amplitude of the reflected signal from damage at this position may be written as(3.7)

This can be expressed in terms of sensor pitch *p* as(3.8)

In this configuration, there is always a directly transmitted signal that will give rise to a noise term if the temperature changes and, as in the pulse-echo case, there is also the possibility of an edge reflection that could arrive at the same time as the signal from damage. Both these cases are shown in figure 5*b*. In fact, the difference in the amplitude of these two possibilities is small, and here the slightly higher amplitude and therefore slightly worse case of an edge reflection will be used, owing to the post-subtraction noise being greater for the increased propagation distance in this case. This gives rise to *u*_{edge},(3.9)which can be substituted into equation (2.12), and following the same procedure as used for the pulse-echo approach yields the expression for SNR in the pitch-catch approach given by(3.10)which is exactly the same as in the pulse-echo case.

### (c) Comparison of approaches

Equations (3.6) and (3.10) indicate that the signal-to-noise ratio, *S*, for the pulse-echo and pitch-catch approaches is identical and inversely proportional to *p*^{(3/2)}. This rapid fall-off in signal-to-noise ratio with sensor pitch is due to the *p*^{−1} decrease in the damage signal combined with the increase in the noise signal.

An important difference between the pulse-echo and pitch-catch approaches is that the sensitivity of the pulse-echo approach is governed by the amplitude of the directly back-scattered signal from the damage, while that of the pitch-catch approach is governed by the amplitude of the forward-scattered signal over a range of angles. This then makes the choice of configuration dependent on the scattering characteristics of the anticipated damage. Diligent *et al.* (2002) showed that the forward-scattered signal amplitude for a through-thickness circular hole has nulls at certain angles, implying that this type of damage would actually not be detected at all in the pitch-catch configuration if it occurred at certain positions in the structure. The amplitude and angular dependency of the reflection coefficient of the damage is therefore crucial in fully assessing the performance of a pitch-catch guided-wave SHM system. Examples of the types of measurements that are required to characterize defects can be found in the work of Diligent *et al.* (2002, 2003) and Diligent & Lowe (2005).

## 4. Envelope versus RF signal subtraction

It has already been noted with reference to figures 1 and 2 that envelope subtraction gives a significant improvement in sensitivity, typically of the order of 20 dB compared with RF signal subtraction. On this basis, envelope subtraction seems preferable, but there is a more fundamental reason for why RF subtraction should be used, which is described in this section.

If the reference RF signal obtained from a structure in an undamaged state is denoted by *u*_{ref}, then the introduction of damage means that a subsequently recorded RF signal, *u*_{later}, is given by(4.1)where *u*_{damage} represents all the additional signals introduced due to scattering by the damage. This includes not only the directly scattered incident wave but also higher order multiple scattering involving the damage and other structural features. The goal of subtraction is to isolate *u*_{damage} from *u*_{later}. Since RF subtraction is a linear operation, the isolation of *u*_{damage} is achieved as(4.2)

However, because enveloping a signal by any method removes phase information, the linearity is lost and, in general,(4.3)

In fact, the result of the envelope subtraction, , is a function of both *u*_{ref} and *u*_{damage}. This is a crucial point of great practical significance, which is best illustrated by the use of a sensitivity map. The sensitivity, *S*, is a function of spatial location * r* and is defined as the amplitude of the signal that would be measured if an ideal point reflector was introduced into the structure at the location

*. An ideal point reflector in this context is one that has a scattering coefficient of unity for all incident and scattering directions. In a real system, the sensitivity map provides the calibration against which subsequent measurements are normalized and must be either calculated or measured experimentally. For the purposes of this discussion, the sensitivity map for a pair of sensors operating in pitch-catch mode is calculated for both RF and envelope subtraction to illustrate the disadvantage of the latter.*

**r**The locations of the transmitter and the receiver are denoted by position vectors **r**_{T} and **r**_{R}, respectively, and the position vector * r* is used to denote a location somewhere in the monitored region of the structure. For this illustrative example, the mode is assumed to be non-dispersive over the bandwidth used and the input signal,

*u*

_{input}(

*t*), is a windowed toneburst,(4.4)where

*w*(

*t*) is a suitable window function (e.g. Hanning) centred on

*t*=0 and

*f*is the centre frequency of the toneburst. An analytic input signal is used for simplicity so that the Hilbert envelope of this and subsequent signals generated from it are given directly by the modulus. In the absence of any structural features in the vicinity of the transducers, the reference signal,

*u*

_{ref}, in this configuration contains only the directly transmitted signal and is therefore given by(4.5)

To calculate the sensitivity, the signal, *u*_{point}(*t*), generated by an ideal point scatterer at location * r* is required and is given by(4.6)

The total signal recorded with the scatterer present in the structure is equal to *u*_{ref}+*u*_{point}. A difference signal, *u*_{diff}(*t*), may then be calculated using either envelope or RF subtraction. The magnitude of the resulting signal at the point in time corresponding to the arrival of the scattered signal from the point scatterer (i.e. at ) is the sensitivity to damage at that location, based on the directly scattered signal. By evaluating this at different points throughout the monitored region, it is possible to produce the complete sensitivity map,(4.7)

For RF subtraction, this yields(4.8)

The RF sensitivity function has peaks at the transducer locations and elsewhere is a smoothly varying function that is always greater than zero. An example of the sensitivity map for RF subtraction using a pair of transducers separated by a distance of 30 wavelengths is shown in figure 6*a*. For envelope subtraction, the sensitivity function is given by(4.9)

Figure 6*b* shows the equivalent sensitivity map for envelope subtraction, again using a pair of transducers separated by a distance of 30 wavelengths and with a five-cycle Hanning-windowed toneburst input signal. In the envelope subtraction case, a series of null lines are seen cutting through the sensitivity map. These are due to the loss of phase information in the enveloping process resulting in interference effects between the directly transmitted signal and the scattered signal. Note that the envelope sensitivity function depends on both the input and reference signals. An input signal containing more cycles results in a greater number of null lines extending outwards from the centreline between the transducers. Likewise, if additional reflections from structural features are present in the reference signal, these will give rise to more null lines and complicate the sensitivity map further. The nulls in the sensitivity map correspond to blind spots where defects cannot be detected and show that envelope-based subtraction is a fundamentally unsound technique for damage detection. For this reason, the rest of this paper will focus on the use of the RF subtraction technique.

## 5. Required sensor pitch

In order to detect damage in a structure, the signal from any damage must be greater than that left as a result of imperfect subtraction. This is illustrated graphically in figure 7, which defines the overall detection problem. The magnitude of a signal reflected from a worst-case structural feature decreases with propagation distance as a result of beam spreading. However, equation (2.12) shows that, as a result of changes in temperature, the noise resulting from imperfect subtraction increases with propagation distance. The magnitude of any damage signal will decrease with propagation distance. The result is that, beyond a certain propagation distance, the magnitude of the signal from the defect will be smaller than *u*_{noise} resulting from imperfect subtraction. Under these circumstances, the defect will be undetectable. With these points in mind, it is possible to formulate an equation to determine the sensor pitch which will result in the required detection sensitivity.

Equations (3.6) and (3.10) can be used to predict the signal-to-noise ratio (*S*) for a given sensor pitch. Rearranging either of these equations can yield a relationship to give the required sensor pitch of a system,(5.1)which can then be used to formulate a relationship to describe the number of sensors required per square metre of structure, *N*, as *N*∝1/*p*^{2},(5.2)

If a term describing sensor geometry (i.e. pitch-catch or pulse-echo) is also included, then(5.3)where *G* is a factor determined by the sensor geometry.

Equations (5.2) and (5.3) are important, allowing the number of sensors per square metre to be specified. The required sensor density is fundamental to any SHM system and has in the past proved difficult to determine with any confidence. All of the parameters in equation (5.3) can be derived through either simple experiments or FE modelling, from which the required sensor pitch for defect detection in the guided-wave SHM system can be defined.

### (a) Practical example

In order to predict the behaviour of the complete system, it is first necessary to rearrange equation (5.1) to include the constant required for pulse-echo operation given in equation (3.6),(5.4)

Values are now needed for each of the parameters in equation (5.4). *R*_{damage} is dependent on the guided-wave mode, operating frequency, type and size of the defect and the angles of the transmitter and the receiver relative to that defect. By choosing to examine the pulse-echo case, only a single value is required, that of the back-scattered reflection coefficient. In this example, the *S*_{0} guided-wave mode will be used and a through-thickness circular hole will be considered as the defect. This is a relatively simple defect and wave mode combination that has been studied in detail using both FE simulations and experimental measurements as in the study of Diligent *et al.* (2002). They showed that this reflection coefficient can be normalized to hole size to yield a general equation for the back-scattered reflection coefficient, which is given by(5.5)where *d* is the diameter of the hole in metres. For the case of a 6 mm hole, this yields a value for *R*_{damage} of 0.043. *β*_{RF} can be evaluated for the *S*_{0} wave mode using equation (2.13) to give *β*=0.0962.

If it is assumed that a signal-to-noise ratio of 6 dB is required for detection (i.e. the signal is twice the size of the noise), then the chances of false positives being received are greatly reduced. With the detection threshold selected, the required sensor pitch for detection can be evaluated for the different values of δ*T*. This can in turn be used to calculate the area each sensor must cover based on the hexagons illustrated in figure 5*a*. The sensor pitch and the sensor density are shown in table 2. The values suggested in table 2 are far too high to be practically useful, due to the cost and complexity of the resulting system.

From equation (5.2), it can be shown that, to reduce *N* by an order of magnitude, *β*δ*T* must be reduced by a factor of 6 (15 dB). Thus, if the effects of temperature can be compensated for (i.e. reducing), then the requirements of the SHM system in terms of numbers of sensors may be made more economically feasible. In real terms, this means applying some kind of processing to the gathered data to shift the resulting post-subtraction noise down (as shown in figure 8), thereby increasing the spacing between sensors over which damage is detectable.

## 6. Discussion

The previous sections have outlined techniques to define the spacing required between sensors in an SHM system and have shown how the effects of temperature can severely limit the feasibility of guided-wave SHM. With this in mind, this section will look at a method for controlling the effects of temperature.

Lu & Michaels (2005) and Konstantinidis *et al.* (2006) illustrated a method of temperature compensation using multiple reference signals (baselines). Instead of using a single baseline for subtraction purposes, a series of baselines were used, covering the range of operating conditions of the structure. Thus, there will be a set of baselines, among which there should be one taken under conditions similar to those encountered at any time in subsequent operation. When the subtraction is made, the baseline that results in the lowest noise (defined as post-subtraction signal as described in this paper) is used. This approach is known as optimum baseline subtraction (OBS). The details of the implementation of the OBS technique are beyond the scope of this paper (for a full explanation, see Konstantinidis *et al.* (2006)).

An example of the application of OBS can be seen in figure 9, showing *u*_{noise}/*u*_{0} against the product of δ*T* and propagation distance as shown in figure 3. In the example shown, the baseline set consists of 250 samples and the time window used for the OBS process is 1.6×10^{−4} s, which is the beginning of the time-series up to the first edge reflection as shown in figure 2*a*. The data are the same as those used to generate figure 2. The solid lines in figure 9 are obtained using equation (2.12), with the value of *β* adjusted to produce the best fit to the original and OBS data. The fit of equation (2.12) to the OBS data is not as good as it is to the original data, but the data still clearly follow the same trend.

The OBS temperature compensation approach has shifted the post-subtraction noise down by approximately 40 dB. The result is that, in the *S*_{0} case for a δ*T* of 10°C, the number of sensors per square metre reduces from 97 to 0.2, with the separation between each sensor increasing to 2.3 m. The increased spacing means that the costs of implementing a guided-wave SHM have also been reduced by two orders of magnitude, perhaps to a point where the costs of running a permanently attached system are smaller than those associated with conventional NDE approaches.

## 7. Conclusion

This paper has outlined a methodology consisting of a series of simple modelling steps to allow the performance of a guided-wave SHM system to be analysed in terms of its ability to detect specific flaw types. This requires the sensor arrangement to be known and any temperature changes to be homogeneous. Each stage of this approach uses analytical techniques to quantify the behaviour and has been experimentally verified, giving good confidence in the predictions for the two sensor modalities (pitch-catch and pulse-echo) investigated here.

The effect of temperature on the reference signal subtraction approach for guided-wave SHM has been quantified and shown to be a major environmental factor limiting the sensitivity of such systems. A relationship between sensitivity and sensor pitch has been found and shown to be similar for two different operating modalities. It has been noted that reference signal subtraction using signal envelopes gives a significant improvement in sensitivity over RF subtraction but at the expense of introducing blind spots into the structural coverage.

The relationship between sensitivity and sensor pitch has been extended to show that a basic subtraction-based SHM approach requires prohibitive numbers of sensors. The use of the OBS temperature compensation technique lowers the post-subtraction noise, significantly reducing the number of sensors required. The result of this is that guided-wave SHM systems potentially become economically and practically feasible, requiring something of the order of 1 sensor per square metre.

To have confidence in the detection of defects, the nature of the defects themselves requires a greater level of understanding. For this reason, the next stage in the implementation of guided-wave SHM must involve the detailed characterization and understanding of the response from various types and sizes of defects.

## Footnotes

- Received May 24, 2007.
- Accepted August 2, 2007.

- © 2007 The Royal Society