Application of the factorisation method to limited aperture ultrasonic phased array data

This paper puts forward a methodology for applying the frequency domain Factorisation Method to time domain experimental data arising from ultrasonic phased array inspections in a limited aperture setting. Application to both synthetic and experimental data is undertaken and a multi-frequency approach is explored to address the difficulty encountered in empirically choosing the optimum frequency at which to operate. Additionally, a truncated singular value decomposition (TSVD) approach is implemented in the case where the flaw is embedded in a highly scattering medium, to regularise the scattering matrix and minimise the contribution of microstructural noise to the final image. It is shown that when the Factorisation Method is applied to multi-frequency scattering matrices, it can better characterise crack-like scatterers than in the case where the data arises from a single frequency. Finally, a volumetric defect and a lack-of-fusion crack are both successfully reconstructed from experimental data, where the resulting images exhibit only 3\% and 10\% errors respectively in their measurement.


Introduction
Ultrasonic nondestructive testing uses high frequencymechanical wavest oi nspect components of safety critical structures, ensuring that theyo perate reliably without compromising their integrity.I ti sr outinely used within the non-destructive testing (NDT)i ndustry due to the relatively inexpensive and portable equipment it requires and its potential for automation and real-time results. The production and implementation of ultrasonic phased array transducers (which are capable of simultaneously transmitting and receiving ultrasound signals across multiple array elements)has surged in the last ten years [1]. These multi-element transducers allowfor greater coverage (and potentially faster inspection times)t han that afforded by single probe inspections, and provide the possibility of performing inspections with ultrasonic beams at various angles and focal lengths, giving rise to aricher set of data. When each of the N elements are fired sequentially,t he N 2 time traces arising from each transmit-receive pair of elements (N being the number of elements, usually between 32 and 256)c an be processed and stored in a3 D matrix (N × N × T ,w here T is the number of sample points in the time domain), usually termed the Full Matrix Capture (FMC) [2]. The current industry benchmark for interpreting the FMC is the Total Focussing Method (TFM) [ 2]; ad elay and sum imaging technique based in the time domain where the area of inspection is discretised into agrid and the signals from every transmit-receive pair are subsequently focussed at each pixel and su mmed. In its most basic form, the TFM can struggle with the detection and characterisation of flaws embedded in highly heterogeneous media. However, efforts have been made to improve the algorithm so that it can handle such environments. Modifications include the implementation of frequencyfiltering [3], the incorporation of the directional dependence of the ultrasonic velocity (caused by anisotropy) [4], and the consideration of multiple wave modes [5].
An alternative approach to analysing the FMC could be to operate in the frequencydomain. Assuming that the location of the flawi sk nown ap riori,ad iscrete Fourier transform can be taken overthe relevant time interval, allowing examination of the frequencyspectrum of the wave scattered by the flaw. This information can be presented in the form of scattering matrices, and analysis of these matrices has become more prevalent in the non-destructive testing literature in recent years [6,7,8,9]. One method which could be used to exploit this frequencydomain information is the Factorisation Method [10,11,12,13]. The Factorisation Method is the continuous analogue of the MUSIC algorithm [14,15] and belongs to ac lass of non-iterative methods known as sampling methods, which deal with the inverse problem of shape identifi-cation. Other such methods include the Linear Sampling Method [16,17], the Probe Method [17,18] and the Singular Sources Method [17,19]. These sampling methods are so named since theywork on the basis of determining whether sampled points within an imaging domain meet some criteria which determines whether theyf all within the support of the flawdomain D.
The NDT community have yet to fully explore the potential of the Factorisation Method for improvedflaw characterisation and this paper endeavours to put forward a framework for applying it to time domain experimental data arising from limited aperture phased array inspections. Some interesting work has already been carried out in [13,20], where sampling methods were used to image cracks in acoustic waveguides, which of course has important implications for the NDT of pipelines. However this work showed only results from simulated data with Gaussian noise and is limited to the inspection of plate-like structures. One important contribution of the work shown in this paper is that it presents af ramework for interrogating time domain ultrasonic phased array data arising from the inspection of welds, by the Factorisation Method. Application of sampling methods to time domain data has been studied before in [20,21,22,23,24] howevert he authors believe that this paper presents application of the Factorisation Method to experimentally collected time domain ultrasonic phased array data in alimited aperture setting for the first time. The case where the host medium is inhomogeneous (resulting in poor signal to noise ratio)i sfi rst considered via synthetically generated data. The phased array inspection of aweld with ahighly scattering material microstructure (taken from experimental electron backscatter diffraction (EBSD)measurements)is modelled within afi nite element package. This allows us to study noisy signals which closer resemble the data arising from experiment than those created when the simulation is run with ahomogeneous host medium and then retrospectively perturbed by random noise. Note that the implementation of the factorisation methodology used in this paper assumes ahomogeneous host medium and receives no information on the scattering host microstructure and so anyinverse crimes are avoided. The reconstructions of both volumetric and crack-likescatterers embedded in this heterogeneous environment are presented. Crack-specific adaptations to the Factorisation method and the linear sampling method have been developed in [13,25,26,27]. In the study by Boukari et al. [27], an expression for the far-field pattern of as mooth non intersecting open arc is presented and employed within the indicator function used to reconstruct the scatterer.However,in [25], an open arc scatterer with Dirichlet boundary conditions is reconstructed using the far-field pattern for apoint source. In this paper,o nt he grounds of simplicity,w ew ill take the second approach. To begin, ab rief overviewo ft he method is given. The truncated singular value decomposition (TSVD)isused to regularise the scattering matrices that arise from the FMC data in the cases where the flaw is embedded in ahighly scattering host medium. It is well known that the largest scatterers can be associated with the Figure 1. Scattering problem geometry where D is av olumetric scatterer with boundary Γ, u i is the incident plane wave and u s is the resulting scattered field. largest eigenvalues of the scattering matrix [28] and so, by using the TSVD to set the smallest eigenvalues to zero, interference from microstructural heterogenetites (which can be thought of as noise)can be reduced, enhancing the signal to noise ratio of the resulting image.
Additionally,amulti-frequencyapproach, as previously explored in [29], is adopted. In taking at ime windowed Fourier transform of collected time domain data, arange of scattering matrices spanning multiple frequencies is made available. Choosing the center frequencyofthe transducer does not necessarily give rise to the optimal reconstruction of the flawa nd an empirical strategy to choose the most appropriate frequencyr equires ap riori knowledge of the defect'sc haracteristics. To avoid this, am ulti-frequency approach is proposed, where the scattering matrices are summed overt he range of frequencies which span the bandwidth of the transducer.A st his approach allows increased exploitation of the available data, improvedcharacterisation is subsequently facilitated.

The Factorisation Method
The forward scattering problem states that there is an incident plane wave, u i (x, θ) = e ikx·θ ,x ∈ R,t ravelling in direction θ ∈ S 2 ,w here S 2 = {x ∈ R 3 : |x| = 1} is the unit sphere in R 3 .O ne ncountering ad efect, in this case the region D with boundary Γ,t he wave scatters, giving rise to the scattered field u s (see Figure 1).T he sum of the incident and scattered fields results in the total field u, which satisfies the Helmholtz equation where k is the wavenumber.A lthough we are primarily interested in the elastodynamic case for the purposes of NDT,byconsidering only longitudinal waves(mode conversion does occur howeverthe method we use to extract the scattering matrices is based on first times of arrivaland so is dominated by the longitudinal waves-see Section 3) then it is sufficient to study the Helmholtz equation. The scattered field u s satisfies the So mmerfeld radiation condition uniformly in all directionsx = x/|x|,e nsuring that the wave is radiating outwards and decays sufficiently fast so that there are no sources at infinity. The scattered field u s also solves the exterior Dirichlet problem where f = −u i and v satisfies the So mmerfeld radiation condition givenin (2). The Factorisation Method [10,11,12] attempts to solve the inverse problem of determining the shape of D from the scattered field. The methodology exploits the relationship between the data-to-pattern operator G and the shape of the scatterer.T ob egin with, let u sph be the fundamental, radiating solution to the Helmholtz equation in R 3 (a spherical wave generated at apoint source z and measured at point x,inahomogeneous host medium)given by As the distance between x and z gets large (far-field), the spherical wave begins to resemble aplane wave at point x. This can be approximated by where u ∞ is the far-field pattern when we have an incident wave arising from apoint source. The Herglotz wave function describes the superposition of plane waves with density g ∈ L 2 (S 2 ). The far-field pattern arising from an incident plane wave applied to some function g is the far-field pattern of the Herglotz wave function with density g.B ydenoting the far-field pattern of the scattered field (obtained from our measured data), by u ∞ s ,wecan define the far-field operator F by Note that F is anormal operator and compact in L 2 (S 2 ). Deriving the following factorisation of the operator F [10, Theorem 1.15], is the basis for the Factorisation Method. Here P * : H −1/2 (Γ) → H 1/2 (Γ)isthe L 2 adjoint of the single layer boundary operator P : and effectively converts the incoming wave to an outgoing wave on the defect boundary.The operator G * : L 2 (S 2 ) → H −1/2 (Γ)isthe L 2 adjoint of G : H 1/2 (Γ) → L 2 (S 2 ), the data-to-pattern operator,defined by Critically,t he range R(G)o ft he operator G has ad irect relationship to the shape of the domain D.F or z ∈ R 3 , φ z ∈ L 2 (S 2 )isdefined by It follows that if z ∈ D,then, from equations (5) and (11), φ z = u ∞ and so, from equation (10), The converse is also true according to Theorem 1.12 in [10]. To gain an exact characterisation of R(G)i n terms of the known operator F ,w ec an relate G to F by equation (8).T oproceed, some further technical assumptions are required. It is assumed that F ,f or the Dirichlet boundary conditions (Equation 1),isnormal, the operator I + ikF/8π 2 is unitary and k 2 is not aDirichlet eigenvalue of −Δ in D (these conditions are justified in [10]). It then holds that the range of (F * F ) 1/4 coincides with that of G. Hence, the sampling point z ∈ R 3 lies in D if and only if for some g ∈ L 2 (S 2 ). By Picard'sc riterion, equation (12) is solvable if and only if the condition φ z ∈R((F * F ) 1/4 )issatisfied (this is shown to hold by equations (5)-(10)).Itthen follows that, where {λ j ,ψ j } forms an eigensystem of the normal operator F such that the eigenvectors define ac omplete orthonormal system in L 2 (S 2 )a nd the Fourier coefficients decay to zero faster than the eigenvalues. Using the spectral theory of an ormal operator,[ 10, equation 1.74] it is observed and so equation (13) does indeed hold. From equations (12) and (14),the solution g is givenby and the following result is obtained In practice, we use the N × N scattering matrix in place of our operator F (where N is the number of array elements). Assuming F is normal (and thus diagonalizable), it holds that there exist N linearly independent eigenvectors. Thus, when using this discrete, limited aperture, we truncate equation (17) to where ε>0. By plotting w(z)f or all sampling points, z, it is possible to recoverthe shape and size of the defect.

The F # Operator
It wasshown above that asampling point z lies within the domain D of the scatterer if and only if there exists asolution in L 2 (S 2 )t oequation (12).However,this criterion only holds if the far-field operator F is normal, which is not always the case when limited angles of inspection or heterogeneous host materials are present. To circumvent this, the positive,s elf-adjoint operator F # is introduced [10,30] where and It is helpful to note here that for agiven self-adjoint operator J ,if where E λ is the spectral family of the operator J [13].
As F is compact in L 2 (S 2 )i tfollows that F * is compact in L 2 (S 2 )a nd thus so is F # [31]. It can be subsequently shown that as ample point z belongs to the domain D if and only if the integral equation has as olution in L 2 (S 2 ) [ 10] (here φ z is as defined in equation (11)). It follows that, by plotting where {λ # j ,ψ # j } j∈N forms an eigensystem of the selfadjoint operator F # such that the eigenvectors define a complete orthonormal system in L 2 (S 2 ), an image of the scatterer can be reconstructed.

Truncated SVD of the Scattering Matrix
From equation (24),w ec an see that W (z)i sl arge when φ z is orthogonal to the eigenvectors of F # ,w hich occurs when the sampling point lies within the spatial domain occupied by the flaw. The other occasion when W (z)c ould be large is when λ # j is large for some j = 1,...,N,e ven when z ∈ D and so (φ z ,ψ # j ) L 2 (S 2 ) = 0. To minimise the contribution of these cases, an artificial nullspace is created. This is achievedbytaking the singular value decomposition (SVD)o ft he N × N scattering matrix, F * ,a nd approximating it using the m largest singular values via thus creating an ullspace with dimension N − m.I nthe case of subwavelength non-isotropic scatterers, the largest eigenvalue is associated with the spherically sy mmetric part of the scattering amplitude and there are three eigenvalues associated with the directional part [32,33]. Where the scatterer is larger than the wavelength, there exist many singular values associated with it. And so, in the work beloww em aket he constraint that m ≥ 4. Aside from this lower bound, we typically assume that the singular values which are greater than 10% of the largest singular value correspond to scattering by the defect and those belowt his threshold correspond to noise and scattering by the microstructure [15,34]. However, by studying the distributions of the singular values it can be observed that this threshold may not always be optimal and may require some tuning subject to the system parameters.

Application to NDT
The results in this paper arise from application of the Factorisation Method to data collected (orm odelled)i nt he time domain. To interrogate the data using the Factorisation Method, we require af requencyd omain representation of the scattered signals overat ime interval corresponding to the wave's interaction with the flaw. To ensure that the flawscattering dominates in the frequencydomain and that other experimental artifacts (such as the back wall of the sample)d on'to bscure the flaw'ss cattering signature, the time domain FMC data must be processed. Firstly the location of the flawisrequired apriori (itmust be remembered that the Factorisation Method is being applied here as ap ost-imaging tool for flawc haracterisation). In this paper,the defect is located using an image generated by the standard TFM [2] I where I (x, z)i st he image intensity at the pixel with coordinates (x, z), A s,r denotes the A-scan (that is, the time Then, the distance of the defect from the array coupled with the estimated wave speed givesrise to atime interval pertaining to scattering by the defect. The Fourier transform is applied to the FMC data in this interval. Then, the amplitude at aspecified frequency is plotted for each transmit-receive pair to generate the scattering matrix. series data)w hen awaveisemitted at location x s and receivedatlocation x r and c is the estimated constant wave speed throughout the host medium. When the distance of the defect from the array is known, it can be coupled with the estimated wave speed c to give rise to atime pertaining to scattering by the defect. Some interval is taken around this value and adiscrete Fourier transform is applied. From the resulting spectral data ascattering matrix can be generated at achosen frequencybyplotting the amplitude of the power spectrum at that frequencyf or every transmitrece ivepair.The scattering matrix is then assigned as the operator F and the Factorisation Method can be applied accordingly.T his process is depicted in Figure 2a nd can be summarised in four keysteps: 1. Apply the TFM algorithm (see equation (26))t ot he rawFMC data to findthe location of the defect. 4. The amplitude of Y s,r at as pecificf requency f is assigned as the element F s,r of the scattering matrix F at that frequency. Note that the size of the time interval depends on the size of the defect and its proximity to other scatterers (we would ideally not incorporate data arising from other scatterers within the interval). It is clear that, for every FMC dataset there exists as et of scattering matrices, each one at ad i ff erent frequency. Choosing af requencya tw hich to operate is not straightforward: the best reconstructions rarely arise from the scattering matrix generated at the center frequencyofthe transducer.Identifying which single frequencyr esults in the optimal reconstruction of the flawrequires apriori knowledge of the defect dimensions. Hence, amulti-frequencyapproach (where scattering matrices spanning at least the −6dBbandwidth of the transducer are summed at regular intervals)i si ntroduced, exploiting more of the information made available by the bandwidth of the transducer whilst removing the subjective aspect of identifying the frequencywhich affords the best flawreconstruction.

Simulated and Experimental Data Sets
In this paper,t he Factorisation Method is applied to data arising from twosources. Firstly,the scattering of an ultrasonic wave by aflaw is simulated using atime domain finite element method in the software package PZFlex [35]. In this paper,t hree FMC datasets generated using this method are examined (the parameters are listed in Tables  Ia nd II). Firstly the scattering by a5mm crack with 40 • orientation (relative to the horizontal axis)embedded in a homogeneous medium wassimulated. The same flawwas then placed in aheterogeneous environment where the locally anisotropic microstructure of an austenitic steel weld (derivedf rom experimental electron backscatter diffraction measurements [36])w as embedded in the simulation (see Figure 3).I nb oth instances the domain wasm eshed with elements of dimension λ/15, where lambda is the wavelength. The 1.5 MHz sinusoidal excitation used thus gave rise to elements approximately 200 µm square, which is sufficient to accurately model the wave propagation. In the heterogeneous case, the weld structure consisted of grains where contigious crystallites with similar orientations were grouped together to form locally anisotropic regions. The correlation length [37] wasestimated as λ/8 and the RMS longitudinal velocity through this heterogeneous medium wasestimated as 5758 m/s with astandard deviation of 146 m/s (calculated using the times corresponding to the backwall echo in the A-scans where transmission and reception took place on the same element). The location of the flawa nd this estimated average wave speed were then used to isolate the time interval pertaining to the flawa nd the relevant scattering matrices were thus obtained (see Section 3and Figure 2).Note that by isolating the time interval using the estimated longitudinal velocity,shear wave scattering (which should occur at alater time)isneglected, as is secondary scattering which occurs after the wave has reflected off the back wall and interacts  with the defect on its return journey. The third simulated dataset examines an alternative scenario where a2 .5 mm side-drilled hole is embedded in the parent stainless steel material to the left of the weld microstructure, at adepth of 30 mm from the array (this is similar to the experimental sample as detailed in Table III). As chematic is shown in Figure 4, with parameters recorded in Table II. The RMS longitudinal velocity through this sample medium wasestimated as 5801 m/s with as tandard deviation of 363 m/s (relatively large as approximately N/2 elements lay over the homogeneous parent material and the others overt he anisotropic weld structure).  Secondly,t he Factorisation Method wasa pplied to experimental data. The first test sample considered in this paper is as teel block containing an ultrasonically noisy manual metal arc (MMA)w eld. The defect of interest is a3mm diameter side drilled hole lying to the left of the weld, 30 mm from the front face of the 50 mm thick sample, as shown in Figure 5. The inspection wascarried out by a2.25 MHz linear array (Vermon, France)asspecified in Table III combined with the Zetec DYNARAY ® (Zetec, Canada)a rray controller.T he RMS longitudinal velocity through this heterogeneous medium wase stimated as 5580m/s with as tandard deviation of 192 m/s. Note that these values differ from the comparable simulation described in Table II as the weld geometry used in the simulation does not come from this particular sample and provides only an estimate of the effects of multiple scattering in the ultrasonically noisy MMA weld present in the experiment. TFM images were constructed using this experimentally derivedp hase velocity to identify the location of the defect before scattering matrices were generated as discussed in Section 3.
The second experimental test sample considered was manufactured from welded austenitic steel plates with implanted defects. The defect of interest is a7.8 mm lack-offusion crack between the weld and steel plate, lying at a   50 • angle, relative to the horizontal axis, in close proximity to the back surface of the sample (see Figure 6).The inspection wascarried out by a5MHz linear array (Vermon, France)asspecified in Table IV, combined with the Zetec DYNARAY ® (Zetec, Canada)array controller.The RMS longitudinal velocity through this heterogeneous medium wase stimated as 5820 m/s. As before, relevant time intervals were isolated using TFM imaging and scattering matrices were thus generated.

Application to FMC Data Generated by the Finite Element Method
In [10] it is indicated that in the case of limited aperture data, where the far-field pattern u ∞ (x, θ)isknown only for x, θ ∈ U , U ⊂ S 2 (that is, where there is alimited angle of inspection), the far-field operator F is not normal. As the FMC data used in this paper arises from ultrasonic inspections by alinear phased array,the Factorisation Method is applied to the F # operator here (see Section 2.1). Figure 7depicts reconstructions of a5mm crack orientated at 40 • to the horizontal axis, embedded in ah omogeneous medium, using FMC data generated by the time domain finite element method (see Section 3.1, Table I). Note that the sampling domain is a2 0mm 2 region centered on the flaw( see Figures 3a nd 4) and this remains constant for all reconstructions shown in this paper.A s discussed in Section 3, only as ubset of the FMC dataset (arising from the central 44 elements)has been considered in order to exclude scattering by the backwall. This offers an angular aperture of only 83 • ,l ess than one quarter of the full aperture (360 • )atwhich the Factorisation Method performs optimally.NoTSVD has been taken here as the flawi se mbedded in ac ompletely homogeneous medium and the null space of the scattering matrix already exists. The image is plotted overa6dB dynamic range (a standard threshold for measuring defects larger than one wavelength)where the outermost contour is aligned to this limit. Figure 7a shows the known geometry and size of the defect. Image (b) depicts the reconstruction generated by applying the Factorisation Method to the single frequency( 1.5 MHz)s cattering matrix. Although the resulting image is oversized (11.7 mm in length)a nd includes twol ower amplitude artefacts, the method has identified the defect as atilted ellipse-likescatterer.Asdiscussed in Section 3, it is possible that improvedr econstructions of the flawm ay be achieveda td i ff erent frequencies. However,without apriori knowledge of the flaw'sdimensions, the optimal frequencycannot be deduced. Thus the multifrequencya pproach wasa dopted to generate image (c) where scattering matrices were generated overt he range 0.75 MHz-2.25 MHz, at intervals equal to the sampling frequency f s ,and then summed. Again, the result is atilted ellipse although this time the additional artifacts have disappeared and an improvedcrack length estimate of 9.6 mm is achieved.
To further assess the suitability of the Factorisation Method for application in NDT,i th as been applied to data arising from the FEM simulation of an ultrasonic wave scattered by a5mm crack of 40 • orientation (relative to the horizontal axis)e mbedded in ah eterogeneous medium (see Section 3.1 for details and Figure 7a for the defect size and geometry). Again, to exclude the signals where the flawscattering is conflated with that of the back wall, only the data arising from the central 44 array elements is considered (ana ngular aperture of only 83 • ). In Figure 8, results from application of the Factorisation Method to variants of the scattering matrix are plotted. Image (a) shows the reconstruction arising from the sin- image (d) is obtained which better represents the nature of the tilted crack defect. However, at the −6dBt hreshold, the diameter of the reconstructed flawis8mm and its orientation is 69 • ,both presenting significant errors when compared to the known 5mmlength and 50 • orientation. Afinal simulated dataset where a2.5 mm diameter disc wasembedded in the parent material to the left of the weld wasalso interrogated. Here data arising from the central 64 elements of the 128 element linear array wasused to generate the scattering matrices for inspection by the Factorisation Method, affording an angular aperture of 112 • .The exact defect geometry is shown in Figure 9(a).Image (b) shows the results from interrogation of the scattering matrix arising at 2.25 MHz whilst image (c) arises from the multi-frequencys cattering matrix generated overt he frequencyr ange 1.125 MHz-3.375 MHz. In this case, taking the TSVD of the scattering matrix inhibited the method's ability to characterise the flaw. This can be attributed to the fact that the flawisembedded in ahomogeneous medium and since its scattering has been successfully isolated from that of the weld by limiting the aperture of inspection, taking the TSVD only serves to remove data relevant to the flaw. Additionally,t he question of howm anys ingular values should be considered can be avoided (although we have as uggested at hresholding technique in Section 2.2, howt oo ptimally truncate the SVD remains an open question). The diameters of the disc (measured along the longest dimension)a re 3.4 mm and 2.8 mm respectively. By using the multi-frequencys cattering matrix, not only is the flawsize estimate improvedbut alower aspect ratio ellipse is yielded and thus the disc nature of the defect is better defined.
To comment on the potential of the Factorisation Method as at ool for improvedfl aw characterisation, the standard time domain imaging algorithm, the Total Focussing Method (TFM) [ 2], has been applied to the datasets which gave rise to the reconstructions shown in Figures 8a nd 9, and the results are plotted in Figure 10. On initial examination, it can be observed that the image of the crack in 10a better captures the 40 • orientation of the defect. However, the −6dBt hreshold givesr ise to a crack length estimation of 9.8 mm, which compares poorly to the measurements obtained by each version of the Factorisation Method presented in Figure 8. Additionally,we observetwo distinct peaks, aside effect which is inherent to the TFM imaging algorithm. This effect can also be observed (toalesser extent)inplot 10 (b),which shows the reconstruction of the side-drilled hole embedded to the left of the austenitic weld. The reconstructed defect measures 6.7 mm along the horizontal, which compares poorly to the 2.8 mm measurement obtained using the multi-frequency factorisation method (recall the flawh as ad iameter of 2.5 mm). Through further comparison of Figures 8, 9and 10, it is also observed that the Factorisation Method better distinguishes between the crack defect and disc defect when the multi-frequencydata are interrogated (images 8b Depth (m) Figure 9. Reconstructions of a2.5 mm diameter disc as shown in image (a) from FMC data arising from finite element simulation (see Table III)b yt he Factorisation Method applied to (b) the scattering matrix arising at 2.25 MHz and (c) the multi-frequency scattering matrix. and 8d and 9c). This improvedflaw identification complements previous work on flawclassification [38,39] where the primary objective is to distinguish between crack defects and volumetric scatterers. Figure 11 depicts reconstructions of the 3mmd isc from the experimental data as detailed in Table III. Once again, the aperture had to be cropped to exclude interference of the flaws cattering by that of the back wall and so the scattering matrix used arises from the central 64 elements placed directly above the flaw(again affording an angular aperture of 112 • ). Image (a) shows the known defect size and geometry,image (b) arises from application of the factorisation method to the single frequencyscattering matrix  Tables Iand II respectively) Figure 12 depicts reconstructions of the 7.8 mm crack from the experimental data as detailed in Table IV. Once again, the aperture had to be cropped to exclude interference of the flaws cattering by that of the back wall and so the scattering matrix used arises from the 42 elements placed directly above the flaw. This givesr ise to an angular aperture of 122 • .N ote that due to the close proximity of the flawt ot he backwall, it wasn ot possible to entirely separate the scattering by the flawf rom that of the backwall. Image (a) shows the known defect geometry and size. Image (b) arises from the single frequency scattering matrix at 5MHz and image (c) arises from the multi-frequencys cattering matrix generated by summing overthe range 2.5 MHz-7.5 MHz at intervals equal to the sampling frequency, f s .Measuring the defects along their longest dimension givesr ise to defect measurements of 9.3 mm and 8.6 mm respectively,exhibiting relative errors of 19% in the case where the scattering matrix at asingle  Figure 11. The geometry of the 3mmdiameter disc is plotted in (a).Reconstructions from FMC data arising from the phased array inspection of asteel block (see Table II)bythe Factorisation Method applied to (b) the scattering matrix arising at 2.25 MHz and (b) the multi-frequencyscattering matrix.

Application to Experimental FMC Data
frequencyisexamined and an improved10% error where multiple frequencies are considered. Furthermore, by using the multi-frequencys cattering matrix, the defect can be better identified as crack likethan from the image arising from the data at as ingle frequency: the ratio of the crack length to crack width (which should be large)isapproximately 3.8 in the single frequencycase, increasing to 5.2 in the multi-frequencycase.

Conclusions
This paper has put forward aframework for using the Factorisation Method as atool for flawcharacterisation in the ultrasonic NDT industry.Abrief derivation wasprovided  Figure 12. Reconstructions of the 7.8 mm crack orientated at 50 • with respect to the x-axis shown in (a) from FMC data arising from the phased array inspection of welded austenitic plates (see Table IV)bythe Factorisation Method applied to (b) the scattering matrix arising at 5MHz and (c) the multi-frequencyscattering matrix.
to introduce the algorithm and subsequent application to synthetic time domain data as modelled in the software package PZFlexw as carried out. Every case considered in this paper arose from limited aperture inspections with angles of inspection ranging between 83 • and 122 • (compared to the ideal 360 • full aperture cases at which this method optimally performs). Amethod for isolating flaw scattering in the time domain before converting it to frequencydomain scattering matrices waspresented. Due to the time domain nature of the data, information wast hen made available for arange of frequencies at which scattering matrices could be generated and subsequently interrogated by the Factorisation Method. Initial implementation of the algorithm wasc arried out at the center frequency of the transducer array before amulti-frequencyapproach wasadopted. In the case where the flawwas embedded in highly scattering medium it wasshown that by using only asubset of the singular values, interference by noise could be minimised. Additionally,the method outperformed the standard imaging algorithm (the total focussing method, TFM)i ns izing and differentiating between twod i ff erent types of flaws: av olumetric side-drilled hole and an angled crack-likescatterer.The algorithm wasapplied to two experimental data sets, one in which aside-drilled hole lay next to aweld and the other where alack-of-fusion crack lay on the boundary of the weld. In both cases, taking the TSVD inhibited the Factorisation Method'sp erformance (this can be attributed to the exclusion of valuable data). However, the multi-frequencyapproach yielded good flaw size estimates with errors of 0.1 mm for the 3mmdiameter side drilled hole and 0.7 mm for the 7.8 mm crack. And so, there remains much work to be done in order for the Factorisation Method to be adopted by end-users. However, this paper proposes af ramework in which time domain, limited aperture data can be brought into the Factorisation Method domain. One natural direction to extend this work would be to consider the elastodynamic equations (rather than the Helmholtz equation). Also, it would be of interest to investigate the use of the Factorisation Method in the time-frequencydomain [40].

Funding Statement
This work wasf unded through the UK Research Centre in NDE Targetted Progra mme by the Engineering and Physical Sciences Research Council (grant number EP/I019731/1)and latterly the iNEED project (grant number EP/P005268/1). Experimental samples were provided by Amec Foster Wheeler and Rolls Royce and software and support were supplied by Thornton-Tomasetti.