## Abstract

Accurate inversion is vital for quantitative imaging, including ultrasonic guided wave tomography, where thickness maps of plate-like structures are reconstructed to quantify corrosion damage. The dispersive properties of guided waves are often exploited to enable thickness maps to be produced from wave speed reconstructions. Ray tomography, diffraction tomography and a hybrid algorithm combining their features were investigated to reconstruct wave speed. Test data produced from simple defects of different sizes using a realistic full elastic guided wave model and the equivalent idealized acoustic model were passed to the imaging algorithms, generating wave speed maps, and, from these, thickness maps. For both datasets, ray tomography exhibited poor resolution. Diffraction tomography performed better, but was limited to shallow, small defects. The hybrid algorithm achieved the best results, giving a resolution around 1.5–2 wavelengths from the realistic test data compared to half wavelength from the idealized case. These results were validated with experimental data, and also extended to a realistic corrosion patch confirming the trends demonstrated with simple defects. The resolution loss with realistic data compared with idealized data indicates the acoustic model cannot accurately capture guided wave scattering and an alternative approach is necessary for better resolution reconstructions.

## 1. Introduction

Quantitative imaging, i.e. the reconstruction of maps of properties through a domain from sets of measurements, has applications in a number of areas, including medicine [1,2] and geophysics [3]. One important application of quantitative imaging methods is guided wave tomography. For this, Lamb-type guided waves are excited in a plate, transmitted across a domain, and then measured at several points by an array; these measurements are then used to produce a map of plate thickness across the domain. Determining the remaining wall thickness of vessels and pipes is very important for corrosion detection and quantification in a number of industries, particularly petrochemical. Conventional ultrasonic thickness maps are produced by scanning a probe across all points on the surface, and obtaining the thickness from the arrival time of the wave reflected from the backwall. By contrast, guided wave tomography removes the requirement to have direct access to all points on the surface, and is also faster since the entire surface need not be scanned. In many cases any curvature is sufficiently low that they can be approximated as flat plates, so the standard Lamb wave dispersion relations can be used.

The process of quantitative imaging can be considered an inversion problem, to determine what particular map of properties would have caused a given set of measurements. A significant challenge of quantitative imaging is to identify a suitable forward model to describe the behaviour of the wave propagation. An ideal scenario would be a model which is (a) accurate at describing the physics of interest, (b) relatively fast, (c) highly sensitive to changes in the properties to be reconstructed, (d) insensitive to changes in any unwanted/irrelevant properties, and (e) easily invertible. The first two points (a) and (b) do not need additional explanation; (c) and (d) are important because different models respond in different ways to changes in the properties and hence can be more or less sensitive to these properties or unwanted noise. The last requirement (e) can take a number of forms; some forward models can be directly inverted (e.g. [4,5]), while some may require an iterative, gradient stepping approach (e.g. [6,7]), which would place a strong emphasis on models which contain few local minima and have straightforward methods of gradient calculation. The ultimate speed of the algorithm will be a combination of the speed of the forward model and the complexity of its inversion. While the focus of this paper is on inverting mechanical wavefields, similar research has been ongoing for the inversion of electrical impedance measurements [8] and electromagnetic waves [9].

The dispersive nature of guided waves in plates provides a widely exploited approach to simplify the inversion. Lamb waves at a certain frequency will have particular known phase and group velocities for an infinite plate of particular thickness. Based on this, a commonly used assumption in guided wave tomography inversion is that a guided wave travelling in a plate of varying thickness and dimensions *x* and *y* in-plane and *z* out-of-plane will behave in the same way as an acoustic wave travelling in a medium across the *x* and *y* dimensions with varying velocity, where the velocity at each location is given by the velocity of the Lamb wave at that frequency and thickness. This is a very valuable assumption for greatly simplifying the inversion process, enabling the full three-dimensional elastic problem to be recast as a two-dimensional acoustic problem and has been widely demonstrated in the production of thickness maps [10,11]. However, the effects of this assumption on the quality of the inversion are unclear; no attempts have been made to quantify the errors or to understand the circumstances where the assumption fails.

The ray theory of geometric optics is another widely used assumption and can be used to invert an acoustic wavefield. The waves are assumed to propagate as rays, with the arrival time being a line integral of the slowness (reciprocal of the velocity) along the ray path. Variations exist: the rays can be assumed straight [12,13], or can be bent to account for refraction as they pass through the domain [14,15]. However, the act of ignoring diffraction causes the reconstruction to be severely resolution limited for the majority of applications [16]. Despite this, the model has widespread use in guided wave tomography reconstructions [10,17,18].

Higher resolution attempts have been made using diffraction tomography [19], which uses the Born approximation as the forward assumption. This, however, relies on the object being weakly scattering, so any defect imaged must be small and of low contrast (i.e. very shallow), which is unlikely to be the case for the majority of defects of interest. The Rytov approximation has been considered for this problem [20] but the main challenge is that it requires the phase of the field to be unwrapped, which is difficult for complex scatterers in the presence of noise. HARBUT (the hybrid algorithm for robust breast ultrasound tomography) [21] uses a low-resolution ray tomography background to reduce the effective contrast of any remaining scatterers, which can then be imaged using a Born approximation approach, and has been successfully applied to invert breast ultrasound tomography data [22]. The approach was applied to the acoustic inversion stage of guided wave tomography in [11], enabling the higher resolution associated with diffraction tomography to be achieved across a wider variety of defects. While for purely acoustic problems, it is known that the resolution is limited to the diffraction limit λ/2, it is unclear what the resolution performance is when test data are produced from the realistic fully three-dimensional guided wave model.

This paper aims to comprehensively and quantitatively evaluate the accuracy of the underlying assumptions used in guided wave tomography, and to understand the physical mechanisms causing these to fail. Section 2 outlines the background theory for the guided wave propagation and the imaging algorithms. Initial investigations outlined in §3 reconstruct sets of simple defects across a range of sizes and depths, with the results being validated with some experimental tests. Section 4 investigates imaging with more complex defects to confirm whether the results generalize and §5 discusses the results and their implications.

## 2. Background

### (a) Guided wave theory

Lamb waves [23] are guided elastic waves which travel in infinite plates of constant thickness. As with all guided waves, they can be considered as a superposition of bulk waves (longitudinal and shear) combined using particular boundary conditions to form a particular characteristic mode shape. For Lamb waves, the mode shapes of displacement and stress can be calculated across the thickness of the plate, to fit the ‘sound soft’ (i.e. zero pressure) free boundaries at the top and bottom surfaces. Several solutions can be found to these equations depending on the frequency and thickness, and software packages such as Disperse [24,25] exist to calculate these. Figure 1*a* plots the mode shapes for the fundamental *A*_{0} and *S*_{0} modes, which exist at low frequencies. Figure 1*b* and *c* plots the phase velocity and group velocity dispersion curves, respectively, as a function of the product of plate thickness and frequency.

The Lamb wave formulation assumes that the plate is of uniform thickness, which in general is not the case for guided wave tomography since the focus is on estimating varying thickness. However, provided that the variation in thickness is slowly varying, it can be assumed that each mode behaves as a Lamb wave with the acoustic properties (phase and group velocities) associated with the local thickness. Under this assumption, a two-dimensional acoustic bulk wave model can be defined with equivalent velocities which would approximate the behaviour of the guided waves. An aim of this paper is to identify how rapidly the thickness can vary before the assumption becomes invalid.

As the guided waves are dispersive, the acoustic wave equation is expressed in the frequency domain and can be described by
*p* represents the Fourier transform of a scalar parameter of the wavefield (the out-of-plane displacement is appropriate in this case) and *k*=*ω*/*c* is the local wavenumber in terms of the angular frequency *ω* and the phase velocity *c*. Having made the assumption that the waves follow this model, this equation itself must be inverted via an acoustic inversion technique.

### (b) Imaging algorithms

Acoustic velocity reconstruction algorithms are the primary algorithms used in guided wave tomography, and are thus dependent on the accuracy of the mapping between elastic guided waves and acoustic waves discussed earlier. This paper will consider three of these algorithms, which each comprise two core features: a set of physical assumptions which are used to describe how the waves propagate through the domain and a mathematical inversion method, which is used to calculate the best solution which fits the data. These three algorithms cover the majority of the acoustic wave speed inversion approaches used, and are the focus of this paper as they have been demonstrated to be fast, robust and practical for a wide range of configurations, but it is recognized that alternative approaches exist, such as full wave inversion [26] and the Novikov algorithm [27].

#### (i) Ray tomography

The first algorithm uses the ray assumption, which neglects diffraction. Diffraction is a phenomenon which occurs when a wavefield interacts with an object that is small relative to the wavelength. If a situation exists where the effects of diffraction can be neglected, such as for large scatterers, then the ray theory of geometrical optics will be suitable to describe the wave propagation. By considering a solution of the form *p*=*A*e^{−iωτ(x,y)}, where *τ* represents a time delay and *A* is an arbitrary complex constant, it is possible to rewrite equation (2.1), as
*ω* tends to infinity
*ω*^{2} (note that *k*^{2}=*ω*^{2}/*c*^{2}) have been dropped, and both sides have been divided by *A*e^{−iωτ} to remove the common factor. The equation can be rearranged to become
*ωτ* in the exponential term represents the (unwrapped) phase of the wavefield, so *τ* corresponds to the ‘phase’ arrival time. The difference in arrival time between two adjacent locations for a one dimensional model is *d*/*c*, where *d* is the distance between them. The eikonal equation above is simply a more general expression of this for higher dimensions, taking the differential limit as *d* tends to zero. Based on this physical explanation, the equation can be extended to use other velocity fields. As ray tomography reconstructions are typically based on the arrival times of the wavepackets *τ*_{gp}, the equation is altered to use group velocity *c*_{gp}
*τ*_{gp}(*x*,*y*) on a uniform grid representing the domain. The inversion process of ray tomography then involves determining a suitable group velocity field to fit this model to a set of arrival times extracted from measurements. The initial step is therefore to extract the arrival times from the experimental time traces, and a wide variety of arrival time pickers have been developed to achieve this, with developments mainly made in the field of geophysics. These use ideas such as cross correlation [31,32], the Akaike information criterion [33] and wavelet transforms [34] but the challenges of determining accurate arrival times mean that manual arrival time picking is still common (e.g. [26]). In this paper, in the reconstructions from simulated data, which is very clean, a thresholding approach is used. The arrival time is taken when the Hilbert envelope of the signal exceeds 5% of the signal peak; a similar process was performed on the input signal and the difference between them was taken as the travel time. For the experimental datasets, where noise was present, the arrival times were manually picked.

Having obtained the arrival times, a technique must then be developed which produces a map of wave velocity to match these. In many inversion algorithms, the approach is to locally ‘linearize’ the model of wave propagation in some way. In some cases, the solution can then be calculated by a direct inversion, if the linear model accurately describes the wave behaviour between the starting point (typically a homogeneous field) and the solution. Alternatively, if the linearity only holds locally, then many small steps can be taken, with the gradient recalculated at each step, which is an iterative gradient-type approach. It is noted that alternative approaches to determining a global minimum such as evolutionary methods [35,36] have been applied to imaging inversions but these have not found widespread use.

Both the direct and gradient-based approaches can be applied to ray tomography. The eikonal equation (2.5) can be rewritten as an integral of ‘slowness’ (reciprocal of speed) along the ray path (*x*(*r*), *y*(*r*))
*τ*_{gp} can be seen to be linear with respect to slowness, owing to the linearity of the integral operator. This enables it to be inverted using a direct approach, such as the filtered backprojection method, widely used in practice for reconstructing attenuation maps from the amplitudes of X-ray projections [37], which is an equivalent problem.

For more accurate reconstructions, refraction should be included by bending the ray paths; however, this makes the problem nonlinear, so it can no longer be solved with a single direct step. This paper uses this more accurate model, with the conjugate gradient method exploited to perform the inversion. This is a common approach and full details of the implementation are available elsewhere [38,39]. In summary, a cost function is defined as
*N*×*N* different send–receive pairs; the goal is to fit *τ*_{model} (the arrival times modelled with the eikonal solver) to *τ*_{meas} (the measured arrival times) by minimizing *C*, which is achieved by a gradient-based stepping approach. For this, the gradient of *C* for variations in the modelled slowness field *s*=1/*c*_{gp} must be calculated. A discretized form of *s* is considered, which consists of *M* tiles with independent slowness values, given by *s*_{m}, for *m*=1..*M*, to represent the slowness field. For the change in *C* with the slowness in tile *m*
*i* within the tile *m* as *l*_{i,m}, by inspection of equation (2.6) the gradient can be written as

The resolution limit, defined in this paper as the wavelength of the maximum wavenumber component accurately reconstructed in the image, is dependent on exactly how the arrival times are picked. If a cross-correlation approach is used for this, then the resolution is close to the width of the first Fresnel zone *L* is the distance between the furthest two transducers in the array and λ is the wavelength [16,40]. Nolet & Dahlen [41] provide a physical explanation of the causes of this resolution limit; in summary, diffraction causes ‘wavefront healing’ to occur, which smooths the wavefronts once they have interacted with any features and hence reduces the resolution of any reconstruction from the arrival times. By focusing on arrival time pickers which extract the earliest arrivals rather than the centre of the wavepacket (as is the case with the cross-correlation approach), the result will be less affected by the diffraction-type scattering and it may be possible to improve on this resolution limit to an extent. Using the ray model to describe wave propagation is very convenient for inversion, despite the limited resolution, because there are very few local minima. This is straightforward to understand conceptually with straight rays, since one would expect the term *τ*_{meas}−*τ*_{model} to be monotonic with changes in the contrast and size of the scatterer. Using bent rays to account for refraction means this monotonic nature cannot be guaranteed, but since ray bending is typically fairly low, local minima only appear rarely.

It should be noted that the Rytov approximation [37] employs a similar approach to the eikonal model by expressing the field as a phase; this approach is not considered here since the algorithm requires phase unwrapping which can be challenging given practical issues such as noise and limited spatial sampling.

#### (ii) Diffraction tomography

Diffraction tomography commonly uses the first Born approximation to produce a reconstruction [37,42], although the Rytov approximation discussed above can also be used. Since the Born approximation accounts for diffraction, the resolution limit is significantly improved over the ray tomography approaches. The Helmholtz equation, equation (2.1), can be reformulated as
*p*_{0} is the incident field (i.e. what would be measured if no scatterer were present), *G* is the free space Green's function and *O* is the object function defined as
*k*_{0} and *c*_{0} being the background wavenumber and phase velocity respectively.

The first Born approximation makes the assumption that the scatterer is weakly scattering, i.e. it is small, and of low contrast. A criterion for this assumption being acceptably accurate is that the phase shift through the scatterer (which causes the difference between *p* and *p*_{0}) must be small, less than a quarter of a cycle [37]. Under this, as a first-order assumption, the difference between *p* and *p*_{0} will be small, so the *p* term within the integral can be replaced with *p*_{0} leading to
*p*−*p*_{0} varies linearly with *O*, and as before a direct solution can be obtained. To analyse this more, if the sources and receivers are in the far field such that the paraxial approximation holds [42,43] (although it should be noted that this is not a necessary condition for diffraction tomography algorithms), both the incident field and Green's functions can be expressed as plane waves, so
*p*−*p*_{s}) for source and receiver directions given by the unit vectors *x*′ and *y*′ has been replaced with a single integral over the two-dimensional space ** x**′. By expressing the two-dimensional Fourier transform of

*O*as

*p*

_{s}appears when

*k*

_{0}; this means that the wavefield can only encode a low-pass filtered version of

*O*, making the well-known diffraction limit for resolution λ/2.

#### (iii) HARBUT

The limitation of the Born approximation to small, weakly scattering objects contrasts to the ray tomography approach, which can only reconstruct the large features where diffraction is limited. HARBUT (the Hybrid Algorithm for Robust Breast Ultrasound Tomography) aims to combine the complementary features of these two algorithms, to enable high-resolution reconstructions for a range of scattering sizes and contrasts, and was introduced for breast ultrasound tomography in [21] and applied to guided wave tomography in [11]. The HARBUT algorithm decomposes the object function into two components
*O*_{b} is a smoothly varying approximation of *O* and *O*_{δ} is the remainder. *O*_{b} can be obtained from the ray tomography reconstruction, and equation (2.14) can be rewritten as
*p*_{b} and *G*_{b} are estimates of the incident wavefield and Green's functions, respectively, in the background *O*_{b}; complete details of how they are calculated are in [21]. As before, this forms a linearization, provided *p*_{b} and *G*_{b} remain constant, which enables *O*_{δ} to be calculated directly.

Huthwaite & Simonetti [11] recognized that, in many cases in guided wave tomography, reconstructions could be improved by doing additional iterations. This is achieved by setting *n* being the iteration number, updating *p*_{b} and *G*_{b} accordingly, calculating the new *O*^{(n)}. Instead of being a direct approach, or a gradient stepping approach, this iterative formulation follows the Newton–Raphson method. At each point, the problem is linearized against the background *O*_{b}, and *O*_{δ} is the solution to the problem under that linearization. The linearization is then updated for this new point and the process repeats. By iterating, any nonlinearity present in the model around the solution point are accounted for, which allows a greater range of sizes and defect depths to be accurately reconstructed. The value of using the ray tomography image as a starting point for iterative HARBUT, rather than just using a homogeneous background is to minimize the likelihood of convergence to a local minimum. The theoretical resolution of HARBUT and iterative HARBUT maintain that of diffraction tomography, λ/2.

## 3. Simple defect analysis

This section aims to present a broad picture of the performance and implications of the different assumptions, by reconstructing the thicknesses of a series of different sized defects. A simple axisymmetric defect shape is chosen, allowing each defect to be characterized purely by its outer diameter and its peak depth. To produce the necessary test data for the different defects in this study, simulations must be performed. Given the potential for confusion between these simulations and the forward models which describe the wave propagation within the algorithms themselves, a naming convention will be used throughout this paper. The simulated experimental test data will be referred to as ‘type 1’ and ‘type 2’ data depending on how they are produced, as outlined below. By contrast, ray tomography, diffraction tomography (DT) and HARBUT, which are used to subsequently produce the reconstruction from the data, will be referred to as ‘acoustic imaging algorithms’.

### (a) Configuration

A diagram of the configuration is presented in figure 2*a*. The plate is made from 10 mm thick steel (although it should be noted that the results are presented normalized, so are valid for any plate thickness provided that the waves are excited at the same frequency-thickness point), and an axisymmetric defect is placed at the centre of the plate. A circular array of radius 180 mm is positioned coaxially, with *N*=64 transducers; in theory *N*×*N* time traces should be recorded through the different send–receive pairs to enable all the information to be collected. However, owing to symmetry, just a single source may be used for illumination; the data from this can then be replicated to produce a full matrix of data. Symmetry can also be exploited to generate a set of measurements across all 360^{°} from the source by taking measurements at only half the angles (marked in grey in figure 2*a*) and mirroring the results. A five-cycle Hann-windowed toneburst, at a centre frequency of 50 kHz, was used to excite an *A*_{0} wave at the source. This gives a wavelength of around 37 mm at the centre frequency for the 10 mm plate.

The defect cross section needs to be selected; this can then be rotated to sweep out the axisymmetric shape. The initial focus is on simple defects, so a raised cosine Hann function is used
*W* defining the outer diameter of the defect, *D* defining the depth, *T*_{0} representing the nominal plate thickness and *r* being the radius from the centre. This function has a smoothly varying shape, and the two parameters *D* and *W*, illustrated in figure 2*b* can be used to vary the dimensions of the defect to enable a variety of cases to be tested.

Owing to the asymmetry of the defect with respect to the centreline of the plate, mode conversion will occur as the waves interact with it. The *A*_{0} component is selectively measured by recording the out-of-plane displacements at the centreline of the plate if possible. In experiments, measurements of the out-of-plane displacements at the surface are used and these will contain some *S*_{0} components, but these are extremely small since *S*_{0} has very little out-of-plane displacement and the amount of *S*_{0} is low since there is little significant mode conversion.

### (b) Elastic model

Various solution approaches exist for simulating ultrasonic waves in elastic media. For simulating the configuration outlined above, the finite-element (FE) method was selected because it is flexible enough to simulate a variety of complex shapes; throughout this paper, following the convention above, the test data produced from this model will be referred to as ‘type 1’ data. This is solved in the time domain using explicit finite difference steps; this explicit formulation allows larger models to be simulated as no memory demanding matrix inversion steps need to take place. The majority of the models were solved with the open source Pogo software package [44], an FE solver which runs very quickly on Nvidia graphics cards. Since this software is relatively new, initial models were performed using both Pogo and the Abaqus Explicit package to validate the accuracy of the software, confirming that the results were the same.

The meshing approach used in the FE models in this paper is illustrated in figure 3*a*. A structured uniform grid of hexahedral brick elements is used to model the plate, then to model the thickness reduction caused by corrosion, the out-of-plane dimension of the elements is reduced accordingly. This approach reduces the mesh scattering that occurs with free meshing, although there is likely to be a degree of anisotropy in the wave propagation due to the element compression, but this should be small provided the mesh is sufficiently refined. Typical estimates of the errors associated with this are available in [45]. Also, it is known that around the chosen operating point (0.5 MHz mm with *A*_{0}) the mode shape does not change significantly as thickness reduces, so this meshing approach has the beneficial property that the same number of elements is available to capture the wavefield variation through the thickness.

For the models, first-order linear hexahedral brick elements were used. In Abaqus, these had reduced integration with hourglass control, while the Pogo elements were fully integrated. The elements were sized 1 mm along the two in-plane dimensions, and 10 elements were used through the plate thickness, making 1×1×1 mm^{3} sized cubic elements in the 10 mm thick region away from the defect. Given the wavelength of around 37 mm for the *A*_{0} mode at 50 kHz, this gives 37 elements per wavelength, which is well above the typical 20–30 elements per wavelength considered necessary to accurately capture the behaviour of the elastic waves [45].

The plate was defined to be 560×560 mm^{2} in size. Absorbing boundaries were applied to the edge of the plate to avoid boundary reflections. The Absorbing Layers using Increasing Damping (ALID) technique outlined in [46] was used; the absorbing region extended 80 mm from each boundary (leaving a ‘working region’ of 400×400 mm^{2}) and the mass proportional damping factor was defined as
*l* is the fraction of the distance through the layer, and *K* is a constant. From testing, *K*=2×10^{6} was chosen to minimize the boundary reflections.

### (c) Acoustic model

An important aim of this paper is to distinguish where in the thickness reconstruction process the errors occur; of the simplifying assumptions, these could either be caused by the guided wave to acoustic mapping of §3*a*, or by the acoustic imaging algorithms of §3*b* which are subsequently used to perform the reconstruction. To aid this distinction, an acoustic bulk wave model was run, with velocities chosen to match those of the plate based on the Lamb wave propagation; the test data produced will be referred to as ‘type 2’ data. The velocity map at 50 kHz for a defect 3.2λ wide and 60% deep is shown in figure 3*b*. This model reproduces the same acoustic assumption as is used in the forward models of the imaging algorithms. The images generated from these data can be compared to the images generated from the type 1 data from the full elastic guided wave models. As the velocities are frequency dependent, simulations are performed in the frequency domain, with a different velocity field at each frequency. At each frequency, the output from this will describe a transfer function between the source and receiver; this can be multiplied by an input source (typically, the Fourier transform of an input signal) to determine the received signal. The inverse Fourier transform of all these frequencies can be taken to obtain the output in the time domain. To model the 50 kHz five-cycle Hann-windowed toneburst, 40 frequencies from 2.5 to 100 kHz were simulated; this captures the bandwidth of the toneburst without significant components being removed due to filtering.

As discussed previously, modelling the plate in three dimensions requires an approach to capture the complex shape; the solution was to use the FE method. For the two-dimensional acoustic model, the features are instead represented by wave velocity variations, so there is no need to model complex boundaries and the solution can instead by calculated on a uniform grid; this grid is shown in the inset of figure 3*b*. Therefore, the simpler finite difference method is used. This was written in Fortran 95 and consists of forming a sparse matrix equation by discretizing the wave equation of equation (2.1), and then using the Pardiso (Parallel Direct Solver) version available as part of the Intel MKL (Math Kernel Library) to calculate a solution to the matrix for a particular source.

In the model, perfectly matched layers (PMLs) of the form described in [47] were used to prevent artificial reflections appearing at the boundary of the domain, although since the simulation was undertaken at a single frequency the convolutional formulation described in the paper was unnecessary. The 1120×1120 nodes were arranged in a uniform grid, spaced by 0.5mm; this is large enough to model the complete array and the PMLs at the boundaries.

### (d) Results

Initially, the results from the reconstructions generated from type 2 data are considered. The focus here is purely on the performance of the algorithms themselves, since both the simulated experimental data and the imaging algorithms use the acoustic assumption. Cross sections through the reconstructions from type 2 data are presented in figure 4 for defects of depths 10, 30 and 60% of the nominal plate thickness (1, 3 and 6 mm), with widths of 15, 30, 60 and 120 mm, which correspond to 0.8λ, 1.6λ, 3.2λ and 6.4λ.

Ray tomography is known to be resolution limited, and clearly performs poorly as the width of the defect is reduced, across all defect depths. The typical measure of resolution limit for ray tomography is given as *b*. As the size of the defect is reduced to 3.2 wavelengths, the accuracy of the reconstruction suffers, reflecting the limited resolution of the algorithm. Further reductions in width result in reconstructions with little in common with the original thickness map, owing to the domination of diffraction at such scales.

The DT algorithm is limited by the validity of the Born approximation, which requires that the reconstructed objects must be sufficiently small and low contrast that the phase shift is low. Following this, there are several cases with the shallow, small defects, where the performance of DT matches the true thickness map well. However, figure 4*a*, *b*, *d*, *g* show the DT reconstruction giving a significant deviation; this is a widely known result of the scatterer being too strong for the Born approximation to be valid [37]. The performance of the HARBUT algorithm is very good at almost all defect sizes. The largest deviations occur for the narrowest, deepest defects; for the worst case (figure 4*j*) the discrepancy is around 1.5 mm (15% of the nominal thickness) at the deepest point; for all other cases, the maximum error is less than 1 mm (10%).

Having evaluated the performance of the acoustic algorithms by using test data obtained from the type 2 model, now the performance of the algorithms is evaluated when using the more realistic three-dimensional elastic guided wave forward model to model the experiment, producing type 1 data. Figure 5 presents the cross sections for the same set of defects as before, with depths of 10, 30 and 60% of nominal plate thickness and widths of 0.8λ, 1.6λ, 3.2λ and 6.4λ.

The ray tomography reconstructions for the 6.4λ width defects notably overestimate the defect depth, which is not the case with the same reconstructions from figure 4. This suggests that the defect introduces an additional shift into the arrival time of the wavepacket in the type 1 data which is not captured by the acoustic model used in the inversion, although this has not affected the HARBUT or DT reconstructions, which are based on the phase and amplitude at a single frequency. Similar to the results in figure 4, the ray tomography algorithm is unable to generate accurate thickness maps for the defect 3.2λ wavelengths wide, although it should be noted that the overall shapes are more accurately reconstructed than before, which could give an indication that the signals from the elastic guided wave model contain a lower level of diffraction than from the equivalent acoustic model.

The DT reconstructions are limited by the maximum phase shift in the same way as when type 2 data were used, but when this shift is low, the Born approximation is an accurate representation of the wave propagation, so DT typically performs well. However, it is clear that, unlike the reconstructions from type 2 data, for the 1.6λ and 0.8λ widths, the defect depth is significantly underestimated. The HARBUT reconstructions, as was shown previously in figure 4, match the best throughout, but show a similar tendency to fail when the width is reduced to 1.6λ and below. This suggests that the resolution is limited when using acoustic reconstruction algorithms to produce images from type 1 data, preventing accurate reconstructions of small (in the in-plane dimensions) defects.

In guided wave tomography, the primary aim is to determine the minimum remnant wall thickness; therefore, one metric for quantifying the performance of the algorithm is to take the error in the maximum defect depth, normalized against the plate thickness. However, this parameter makes no assessment of the quality of the shape reconstruction. For a single simple defect such as this, this is relatively unimportant, but when considering a more complex defect which might be formed from a number of superposed simple defects then the accuracy of the reconstructed shape of each is important to ensure that the correct peak is determined. Because of this, an alternative metric is defined, based on the root mean squared (RMS) average
*T*(*x*,*y*) is the thickness of the original map at coordinates *x* and *y*, and *T** is the approximate thickness map produced by the reconstruction algorithm. *w* has been defined as a windowing function, defining the extent and weighting to apply to the error prior to integration. The selected choice is to define the window as
*A* defines the outer radius of the window. *A* is set to double the radius of the defect, but is limited to 180 mm, the radius of the array, for the largest defects. The window has a peak at *r*=0 and varies smoothly down to zero at *r*=*A*. This biases the error metric towards errors at the centre of the defect, where the deepest point of the defect is, while also measuring the shape of the defect away from this point. The window extends beyond the defect itself to measure how accurately the background is captured; this is important if two defects were superposed close together. To perform the calculation in practice, the integral is replaced by a discrete sum of the values across each pixel in the image.

This error metric will be zero in the case of a perfect reconstruction, and otherwise will represent the weighted RMS error as a fraction of the nominal plate thickness. The error metric is plotted in figure 6 for the reconstructions from type 2 data, and figure 7 for the reconstructions from more realistic type 1 data. In both figures, the trends, for HARBUT and ray tomography in particular, seem quite consistent between the three defect depths. The error for DT increases for deeper defects since the higher contrast causes the Born approximation to become violated quicker.

Plotting the results in this way enables clearer comparisons to be drawn between the reconstructions from type 1 data and type 2 data. Interestingly, the local peak in the ray tomography reconstruction around 1.5λ is reduced slightly when reconstructing from type 1 data, compared to type 2 data. This suggests that the diffraction at the defect, which limits the applicability of the ray assumption, is reduced. The acoustic model used to produce type 2 data cannot represent mode conversion at the defect, so all diffracted wavefield components appear in the predicted measurements; by contrast, the diffracted components in the type 1 data will be lower since some energy will be lost to the *S*_{0} or *SH*_{0} modes. Given that the reconstructions of such defects are known to be poor from ray tomography anyway, however (figure 5*g*–*i*), it is unlikely that this improvement could be exploited.

There is a clear shoulder around defects of diameter 1λ which appears in the error curves for reconstructions from type 1 data (figure 7), which is not present for the type 2 data reconstruction (figure 6). This reduction in reconstruction accuracy for type 1 data (from the elastic guided wave simulation of the experiment) relative to type 2 is due to the inaccuracy of the elastic guided wave to acoustic mapping outlined earlier, as type 2 data are generated from an acoustic simulation of the experiment and hence are unaffected by the mapping does not rely on the mapping. This suggests that better reconstructions may be possible from type 1 data if a more accurate representation of the guided wave scattering was developed and incorporated into the inversion processes, rather than using the mapping.

Ultimately, the main concern for users of the method is the maximum defect depth. As explained earlier, in order to capture this for more complex defects, the overall shape reconstruction must be reasonable, and once this is achieved, it is important to study the accuracy of the maximum depth reconstruction. Therefore, in addition to the weighted average RMS error, *e*, defined above, a second parameter is defined as
*m* represents the minimum thickness of the original, *T*, and reconstructed, *T**, thickness maps. If the reconstruction obtains the maximum wall loss to within 10% of the plate thickness and the average reconstruction within 5%, then it is considered to be sufficiently accurate, i.e. if *e*_{d}<0.1 and *e*<0.05. This latter threshold is marked on figures 6 and 7. As an example, from the cross sections from type 2 data, in figure 4*h* (1.6λ wide and 30% deep) the ray tomography deepest value estimation is quite accurate, so *e*_{d}<0.1, but the shape is a poor representation, so *e*>0.05. By contrast, both DT and HARBUT satisfy both criteria in this case. For the type 1 data, in figure 5*d* (3.2λ wide and 60% deep), again the ray tomography maximum depth assessment is within 10% of the total wall thickness, but again the shape is insufficiently well reconstructed to satisfy the shape requirement. DT clearly satisfies neither criterion in this example, but HARBUT satisfies both.

To apply these criteria, simulated experimental data were generated using type 1 and type 2 models, and reconstructions were generated at a matrix of all the different combinations of defect depths (0.5, 1, 2, 3, 4, 5, 6 mm, i.e. 5% to 60%) and defect widths (360, 240, 160, 120, 80, 60, 40, 30, 20, 15, 10, 5 mm, i.e. 9.6λ down to λ/8), and for each of these, *e* and *e*_{d} were calculated. These were interpolated to a 400×400 uniformly sampled grid, to enable the results to be plotted. The two criteria *e*_{d}<0.1 and *e*<0.05 were applied to the three different algorithms, ray tomography, DT and HARBUT, reconstructing from both type 1 and type 2 data, and the results for when both are satisfied are shown in figures 8 and 9, respectively. It should be noted that the errors in reconstruction are likely to increase fairly monotonically with contrast; this is caused by the linear, or near linear, nature of the algorithms. However, measures *e* and *e*_{d} both present error as a fraction of nominal plate thickness, rather than the depth of the defect, which causes the error for the shallower (i.e. lower contrast) defects to appear lower, and hence the plots show that all the algorithms typically perform well under the criteria presented for the shallower defects.

When reconstructing from both type 1 and type 2 data, these plots demonstrate that the HARBUT algorithm performs well across all depths for most defect widths; for the shallower defects the resolution limit becomes apparent. The ray tomography reconstruction is resolution limited, so is only sufficiently accurate for larger defects. Diffraction tomography is limited in two ways, firstly by the fundamental resolution limit of the Born approximation, where it matches the HARBUT algorithm. As discussed earlier, the Born approximation also requires the phase shift through the object to be low; since the phase shift is a product of the size and the contrast (related to defect depth for guided wave tomography) of the object, an accurate reconstruction becomes a trade-off between these two parameters, producing the near-hyperbolic curve A marked in figure 8. To produce this curve, an approximation was made where the raised cosine Hann function from equation (3.1) describing the defect shape was replaced with a simple flat-bottomed hole with a diameter half that of the Hann function and a depth equal to its peak. This significantly simplifies the estimation of the phase shift through the defect, while still providing a reasonably accurate estimate. The maximum phase shift for this homogeneous defect could then be calculated as in [37] as 4*πn*_{δ}*a*/λ, where *n*_{δ}=|*c*_{0}/*c*−1| is the difference in the index of refraction relative to the background and *a* is the radius; curve A marks the line where a phase shift of around *π* would be caused by the defect, confirming that this criterion defines well the limit of the applicability of the Born approximation.

The most notable difference between figures 8 and 9 is that the minimum size of the defects, which can be reconstructed by HARBUT and DT, is reduced when using experimental data obtained from the more realistic elastic guided wave model, reflecting a reduction in the resolution. While the actual limit varies to an extent with defect depth, it is around the λ/2 line marked B in figure 8 for the HARBUT reconstruction from type 2 data, while for the same reconstruction from type 1 data the limit, shown as C, is around 1.8λ. This suggests a loss of resolution in the reconstructions when using a purely acoustic inversion approach to reconstruct from the realistic elastic guided wave type 1 data compared with using the same acoustic inversion methods with the type 2 data obtained from the approximate acoustic forward model.

### (e) Experimental validation

Two experimental validation tests were performed to confirm that the three-dimensional FE model of the plate was producing accurate results. The configuration was as described in §3*a*, to match the FE models, and the two defects were 120 mm (3.2λ) wide and 6 mm (60%) deep, and 60 mm (1.6λ) wide and 3 mm (30%) deep, matching figure 5*d* and *h*. Both defects were machined in the same 1.2×1.2 m^{2} plate, separated by 0.5m, with interference between them unlikely to be significant in the reconstructions, and both were positioned sufficiently far from the plate edge to enable the initial wavepacket to be separated in time from any boundary reflections.

As in the simulations, owing to symmetry, only a single source was needed for each defect. For this, a piezoelectric transducer was used to excite the *A*_{0} wave; this transducer is as described in [19]. This is around 10 mm in diameter (i.e. small relative to the wavelength) and predominantly excites an out-of-plane force on the upper surface. It has been shown to produce very pure *A*_{0} guided waves for a 10 mm thick plate. This transducer was bonded to the plate with epoxy to give good coupling, then excited with the toneburst directly from a Handyscope HS3 (TiePie Engineering, Sneek, The Netherlands).

Measurements were taken via a pair of Polytec OFV-505 laser Doppler vibrometers as described in [18]; these were positioned with their beams at 45^{°} to the plate surface, as illustrated in figure 10, configured to measure the out-of-plane and one component of the in-plane velocities of the plate surface. For the reconstructions, only the out-of-plane components, corresponding to the *A*_{0} mode, were needed, so the in-plane values could be discarded for this study; while the out-of-plane components can be measured with a single laser beam normal to the surface; in this case, the in-plane components were simultaneously acquired for other research. A semicircular reflective strip was attached to the plate to maximize the light reflected back to the laser vibrometers along the scanned path and improve the signal-to-noise ratio. The laser heads were scanned to measure a 180^{°} arc from the source transducer to a point directly opposite, with measurements taken at 65 points, corresponding to an effective 128 transducer array; this path is marked by the reflective strips visible in figure 10. Clearly, the early measurement locations coincide with the source transducer, so these terms are discarded from the measurements.

Figure 11*a* presents a single time trace from the measurements (experimental data are available in the electronic supplementary material). The initial, direct wavepacket is clearly visible, and the later arrivals are the reflections from the boundaries of the plate. The defects and measurement locations are positioned on the plate to avoid any reflections interfering with the signals of interest, so the signals are straightforward to separate. The traces are gated before and after to reduce the noise in the signals, as outlined in [11]. The start of the signal is approximated from the group velocity and geometric distances, neglecting the disturbances caused by the defect, and is assumed to finish 0.11 ms later to capture the five cycles at 50 kHz, extended by 10% to account for some dispersion. A cosine taper window is taken around this, smoothed via the cosine function to zero over a time of 0.04 ms, equivalent to two cycles. An FFT (fast Fourier transform) algorithm is then applied to this signal to enable frequency domain data to be extracted.

Extracting the arrival times from data, as discussed earlier, is a complete research area in itself; in this example, since relatively few arrival times needed to be determined, manual picking was chosen. It is felt that manual picking is likely to give the most accurate results to enable a representative comparison to be made. To achieve this, firstly the majority of the wavepacket distortion due to dispersion was removed by performing a backwards propagation, by multiplying each signal by *d* is the straight-line distance from the source to each receiver. If there were no defects, the arrival times of these signals should then all be zero. The arrival times are then manually picked, as illustrated in figure 11*b*; in this case, they have been chosen to follow an early peak in the signals. Note that the offset to zero is subsequently removed by subtracting a constant; this relates to the constant offset in arrival time which, as mentioned earlier, is present in the transmitted signal. Finally, to account for the backwards propagation step performed initially, the arrival times must be re-propagated by adding *d*/*c*_{gp}. As with the numerical models, full matrices of frequency domain and arrival time data were produced by replicating the signals, exploiting the symmetry of the configuration.

Figure 12 presents the reconstructed images for the experimental reconstructions compared to the equivalent reconstructions from FE data. Both the HARBUT and DT reconstructions match very well between the FE model and the experimental data, although there is notable noise in the background of the experimental images. As observed in [11], in most normal scenarios any noise present in each trace will be incoherent between the different measurements in the array; when using a reconstruction algorithm, the noise in the final image will be reduced to an extent by averaging. However, the act of copying measurements to synthesize a full array of measurements introduces coherence between the noise components, which will not be averaged out to the same extent. Thus, one would expect the effect of noise to be lower in practice than demonstrated here.

Ray tomography shows more significant differences between the simulation and the experimental data, primarily due to the difficulties in determining the arrival times of the signals, as discussed above. However, the resolution limitations of the algorithm due to diffraction are clearly visible in both sets of reconstructions, and the overall behaviour between the simulations and experiments is very similar.

## 4. Complex defect comparison

Previous analysis has focused on simple defects, enabling a general analysis of the performance of the algorithms to be identified. Such defects are not representative of the results in practice, however, since defects are typically complex with many features. One approach could be to consider that the complex defects are a particular combination of the simpler defects, with the resulting image being a linear combination of the images of these simple defects. However, this relies on the linearity of both the wave behaviour and the algorithm itself, which as discussed earlier cannot be guaranteed under general circumstances. Given that the assumptions in the algorithms do allow the wave scattering problem to be explained with a degree of linearity, however, it could be expected that the trends identified for the simple defects would be present also for a complex defect.

To investigate this problem, a realistic complex defect is calculated from a laser scanning of the surface profile of a corrosion defect found at a pipe support. A thickness map based on these measurements is presented in figure 13*a* and is available in the electronic supplementary material. At present, RAM limitations on the GPU restrict the maximum size of the model, limiting the usable model size to around 400×400 mm^{2}. Therefore, it is not possible to model large arrays, so this has been set to 360 mm in diameter. The defect has been scaled from the original by a factor of around 0.7 in the in-plane directions to enable it to fit within this array. To achieve a maximum 50% wall loss (i.e. 5 mm for the 10 mm plate), the defect is similarly scaled by a factor of 0.7 in the out-of-plane direction. This defect was selected because it has a number of interesting features, including a particularly deep point close to the edge; it should therefore form a significant challenge to the reconstruction algorithms. Simulations of this model are performed using the elastic FE model and the acoustic frequency domain equivalent, to produce type 1 and type 2 experimental data, respectively. These datasets are subsequently used to test the acoustic imaging algorithms as before.

Figure 13*b*,*c* and *d* presents the reconstructions for ray tomography, DT and HARBUT, respectively, all when using type 1 data. The ray tomography reconstruction is poor, with the smaller features present having little in common with the original thickness map, although the overall outline of the object is fairly well matched. Diffraction tomography is very poor since the contrast and the size are too large for the Born approximation to be valid. HARBUT gives a reasonable reconstruction, although it is clear that the resolution is limited, preventing the algorithm from being able to pick out the finer details, so importantly the algorithm is unable to determine the deepest point in the thickness map. For comparison, the HARBUT reconstruction from type 2 data is presented in figure 13*e*, then cross sections along the line marked in figure 13*a* for the thickness maps of figure 13*a*,*d* and *e* are plotted in figure 13*f*. It is clear that when using HARBUT with type 2 data, the reconstruction is much higher resolution than with type 1 data, confirming the results seen with the simple defects.

To confirm that the inability to reconstruct the deepest point is down to resolution limitations when using acoustic imaging algorithms with type 1 data, the same defect is scaled up by a factor of four in each of the in-plane dimensions and a HARBUT image is generated from these data. Since this would extend beyond the edge of the 360 mm diameter array, a circular window is applied to limit the extent. The resulting thickness map used is shown in figure 14*a*. The reconstruction is presented in figure 14*b*, and the cross sections are compared in figure 14*c*, showing how HARBUT is now able to better capture the deepest point in the image, confirming that the failure of the algorithm previously was a result of the scale of the defect. This gives a strong indication that the rapid changes in thickness do cause errors in the assumption that the acoustic imaging algorithms can capture all the behaviour of the guided waves as they travel though the plate.

As discussed earlier, the standard resolution limit of HARBUT is λ/2, based on the Born approximation [37]. This resolution limit is defined as the period of the highest frequency component which will exist in the image, and in the case of the Born approximation the limit occurs because the reconstruction is, in effect, filtered to remove any wavenumber components above 2*k*_{0}, where *k*_{0} is the wavenumber of the background medium. The resolution can be determined by comparing the various reconstructions to the original in the Fourier domain. Figure 15 plots the two-dimensional Fourier transforms of figure 13*a*,*d* and *e* to enable the resolution to be compared.

The original, in figure 15*a* clearly has significant features at a wide variety of wavenumbers. Both of the HARBUT reconstructions, using type 1 experimental data (extracted from the full elastic guided wave model) in figure 15*b* and type 2 data (from the approximate acoustic model) in figure 15*c* are notably filtered. The approximate limits in *k*_{x} where the amplitudes of the features drop significantly below those of the original are marked by the dashed white lines. In the case of figure 15*c*, this filtering matches the 2*k*_{0} limit for the Born approximation methods, which is expected since both the forward model and the inversion are performed using an acoustic model, but the limit of figure 15*b* is much lower. The radius where the component amplitudes drop significantly below the original is around 0.6*k*_{0}, which suggests a resolution limit of around 1.5−2λ, matching closely the observations made earlier in the paper for the simple defects.

## 5. Analysis and discussion

The results presented in this paper for a variety of different scenarios have shown that resolution is significantly limited when reconstructing from type 1 experimental data obtained from the realistic elastic guided wave model, when compared to the equivalent reconstruction using type 2 data extracted from the approximate acoustic forward model. To analyse this further, a small, point-like flat-bottomed defect which lies within the Born approximation can be considered. The chosen defect is 5% of the nominal thickness (0.5 mm) deep and of diameter λ/4 (10 mm). As before, type 1 and type 2 experimental data are produced from the full elastic guided wave model and the acoustic approximation, respectively, then the resulting scattered field is calculated by subtracting the incident field. This scattered field, output for different scattering angles as defined in figure 16*a*, is plotted in figure 16*b* and *c*, for amplitude and phase, respectively. In these, there is a clear region where the amplitude and phase match within the limit of 6 dB amplitude and 0.7 radians, from scattering angles of 0–0.64 radians, marked in grey, although the amplitudes in particular are clearly diverging. Outside this region, there is a significant difference between the acoustic and the elastic models. Based on this, there is a region of transmitted measurements of around ±0.64 radians either side of through-transmission where the acoustic model forms a reasonable approximation of the elastic data.

Under the Born approximation, as discussed in §2*b*(ii), the different measurements of the scattered field can be directly mapped to the ‘K-space’, the two-dimensional Fourier transform of the image space plotted in figure 15. For a given illumination direction ** s_{0}** and measurement direction

**, the measured scattered field component maps to the point in K-space at**

*s**θ*is the angle between the vectors

*a*. From this, it is clear that the transmitted components (

*θ*tends to zero) contain information about the low wavenumber components of the object, whereas the reflected components (

*θ*tends to

*π*) contain the highest wavenumber components, around 2

*k*

_{0}. In this example, scattered data from the acoustic and elastic models only match up to an angle of ±0.64 radians either side of through-transmission, so all the wavenumber components within a circle of radius

*b*.

While this analysis has been undertaken under the Born approximation, the results do generalize outside it. HARBUT produces its reconstruction by using the Born approximation with corrections to allow it to work relative to a known background, and it would therefore be anticipated that the HARBUT reconstruction would be similarly limited. This is why the resolution limitation of HARBUT when using data from the elastic forward model matches this theory so well.

The practical implication of the resolution limit when using acoustic imaging algorithms for this problem is that the finer details of the thickness map will be removed. As demonstrated for the complex defect of figure 13, this can prevent the minimum remnant wall thickness from being accurately identified under certain circumstances. There is therefore a need to improve the resolution. One option is to use a higher frequency, to reduce the wavelength, so the resolution will improve due to the shorter wavelength. However, there are a number of practical issues associated with changing the operating point. Exciting a pure mode will become more challenging because of the presence of additional modes at higher frequencies, making transduction significantly more critical. Additionally, the sensitivity of the guided waves to thickness changes will often vary with frequency, and there can be additional issues including attenuation; a full study of these is outlined in [48]. A final point is that a shorter wavelength will require more transducers to enable the field to be sufficiently sampled.

The other potential solution for improved resolution would be to use a reconstruction approach which is more faithful to the full behaviour of the three-dimensional guided waves, rather than relying on the limited accuracy of the guided wave to acoustic mapping. This could be achieved through a ‘full wave inversion’ approach, where the three-dimensional FE model is repeatedly run and updated to match the experimental data. Clearly, this is susceptible to any local minima present in the solution, and is likely to be sensitive to noise, although the issues associated with the former in particular could be minimized by using the lower resolution reconstruction from HARBUT as a starting point. It should also be recognized that while the λ/2 resolution has been achieved when using acoustic models in both simulation of experimental data and the subsequent imaging techniques, there is no guarantee that such a resolution can be achieved when a full elastic inversion approach is used with experimental data from an elastic model.

## 6. Conclusion

This paper has clearly defined the performance of the ray tomography, diffraction tomography and HARBUT algorithms for thickness mapping via guided wave tomography. When using simulated experimental data generated by an acoustic model (type 2 data), it was clear that ray tomography is limited by the presence of diffraction that occurs at small scales, preventing small-scale objects from being reconstructed. With the same data, diffraction tomography is limited by the Born approximation, which prevents it from generating accurate reconstructions when the contrast and size combine to cause a large phase shift through the defect. HARBUT shows a good step forward over both of these, performing as well as either within their respective limits, but also producing good reconstructions in the intermediate region where neither performs well, achieving a resolution of around λ/2 across all tested defect depths. When the simulated experimental data are generated by a more realistic elastic model instead (i.e. type 1 data were used), this paper has shown that the achievable resolution drops to around 1.5–2λ. The accuracy of the data produced from this elastic model was also confirmed with a set of experiments. For the defects across the ranges tested (5–60% wall loss and 0.125–9.6λ diameter), HARBUT was shown to reconstruct the peak wall loss within 10% of nominal plate thickness and have an average shape error of less than 5%, down to the resolution limit of 1.5–2λ for type 1 data, and λ/2 for type 2 data.

A physical investigation of the resolution loss when reconstructing from simulated experimental data from the elastic model (type 1 data) compared with simulated experimental data from the acoustic model (type 2) has explained how the scattered field differs between the two models used, showing that there is a limited range of angles either side of through-transmission where the difference between the two is relatively low. Analysis has shown how this range of angles only allows components up to the 1.5–2λ resolution limit to be extracted when acoustic inversion approaches are used, indicating that this is a fundamental limit which restricts the resolution of this widely used approach in guided wave tomography. A clear opportunity for future resolution improvements is to develop inversion approaches which can more accurately capture the physics of the elastic guided wave scattering occurring in the type 1 data, rather than approximating it with an acoustic model.

## Acknowledgements

The author is grateful to Prof. P. Cawley for the critical reading of the paper and the helpful discussions, and also to J. Isla for the assistance with the experimental measurements.

- Received January 24, 2014.
- Accepted March 3, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.