Investigation of the 1-D inverse Born technique

C.A. Chaloner, BScL.J. Bond, BSc, PhD, CEng, MIEE, CPhys, MlnstP

Indexing terms: Nondestructive testing, Ultrasonics, Image processing

Abstract: An ultrasonic digital signal processingtechnique which can be used to size isolated inho-mogeneities in an otherwise homogeneousmedium is examined. The technique is known asBorn inversion and is theoretically valid for appli-cation to weak scatterers. However, accurateradius predictions have also been obtained experi-mentally for strong scatterers (voids). The tech-nique was examined using synthetically generatedultrasonic signals from weak and strong scatterers,as well as experimental data from voids in highstrength metals, to ascertain the application limitsof the algorithm and to investigate why the tech-nique is insensitive to different classes of scatterer.It was found that the part of the frequency spec-trum used in the inversion routine has a verysimilar profile for weak and strong scatterers.

Accurate sizing using Born inversion is onlyobtained if the bandwidth of the interrogatingprobe is adequate. However, the probe bandwidthrequirements depend on the size of the flaw beingexamined, which is not known a priori. A method-ology is presented which determines whether thematch of flaw size to probe bandwidth is suitable.This has proved critical in obtaining accurate sizeestimates from the inversion.

1 Introduction

Signal processing techniques in general and inversionroutines in particular have for many years been widelyused for information retrieval in the fields of radar, sonarand seismology [1]. The problems of ultrasonic defectlocation and sizing are, essentially, similar to those in theabove fields, and over the last decade there have beenincreasing demands for the development of quantitativenondestructive testing (NDT) techniques which give fulldefect characterisation. Such characterisations arerequired for combination with stress and materials datafor use in what are called 'damage tolerance', 'retirementfor cause' and 'remaining life' analyses. There has there-fore been a period in which NDT has been recognised to

Paper 5151A (E4, S6), first received 19th December 1985 and in revisedform 22nd July 1986Mrs. Chaloner is with Royal Ordnance PLC, Westcott, Aylesbury,Buckinghamshire HP 18 ONZ, United KingdomDr. Bond is with the Department of Mechanical Engineering, Uni-versity College London, Torrington Place, London WC1E 7JE, UnitedKingdom

be of growing importance and a fundamental part of thedesign and production process. Furthermore, it is nowbeing recognised that there is a need to design for test-ability, in particular when new materials are used, such aspowder metals, ceramics and composites, and when novelforms of design are involved.

In conventional ultrasonic imaging processes contrastand resolution become poor as the wavelength of the illu-mination approaches the dimensions of the object details.Defect detection and sizing using ultrasonic nondestruc-tive means becomes increasingly more difficult underthese conditions. This situation is encountered when thedefect to be characterised is small, the material attenu-ation reaches a high value at low frequency or the soundvelocity in the material is very high. Alternatives weretherefore investigated for quantitative assessment ofdefects under such conditions, for example in high-strength metal alloys and composites for use in com-ponents requiring a high degree of structural integrity.

The method considered in this paper is known asBorn inversion and has received much attention in theUSA. The theory is derived for weak scatterers, but ithas been successfully applied to strongly scatteringdefects in high-strength metals [2, 3]. This particularmethod was chosen as it involves a direct inversion of thebackscattered signal, obtained in a single pulse-echo mea-surement, to find a characteristic function for the defectsize. The method is sensitive to low-frequency informa-tion making it potentially useful for materials with ahighly frequency-dependent attenuation response. Thiswork is an independent study and explanation of a signalprocessing technique which remains the subject of somedebate [4].

The contents of this paper are as follows. Section 2outlines the generalised inverse problem and gives thetheoretical and practical background to the Born inver-sion technique. A new methodology for data classi-fication, practical difficulties with the inversion schemeand some results obtained for spheroidal voids in high-strength metals are given in Section 3. The anomalies inthe technique are examined in Section 4 using exact theo-retical scattering data with a view to ascertaining theapplicability of the method. The final Section contains asummary and short discussion on the results presentedherein and conclusions.

2 Theory

2.1 InversionThe inverse problem, viewed with particular reference todefect or target characterisation, is that of gaining infor-mation on the features of an unknown or concealed

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987 257

body, which can be made to cause a disturbance, bymaking observations of its effects. In practice, this oftenmeans that energy in some form impinges on the objectof interest, and the change effected on the incident energyis used to infer something about the object. More for-mally, most inverse problems can be reduced to solvingan integral equation of the type

(fa) •fJo

K(v, t)p(v) dv 0 ^ T < oo

for p(y) from measurements of g{x\ where K is the kernelrelating the governing parameters T and v; i.e. a Fred-holm integral equation of the first kind. An equivalentmatrix formulation can also be used.

In most examples in which an inverse problem is to besolved the related forward problem must be fully under-stood. Further, it often arises that the forward problemmust be solved in conjunction with the inverse problem,and these must be uncoupled in some way, usually by anapproximation of the forward problem.

In situations involving experimental measurements theproblem of the signal-to-noise ratio must also be con-sidered. Many inversion algorithms are ill-posed andhence inherently unstable when a finite data set (allreceiving and transmitting equipment is bandlimited), orerrors in the data, are present. Often, practical inversionis only possible if the measured data and possible solu-tions are constrained so that the problem is over-determined.

Inversion can be divided into two classes; direct andindirect. The direct method involves mathematical oper-ations in which the field data are the inputs, and theinterpretation of those data is the output. Indirect inver-sion usually means finding the best fit between field dataand a previously assumed model.

Perhaps the greatest effort in inversion has been in thefield of seismology where methods in use include forwardproblem approximations [5], trial and error searchingtechniques [6], iterative methods [7] and direct datatransforms [8]. Complementary work has also beencarried out on the significance of the solution obtained[9]. However, the inversion of ultrasonic data has onlybecome topical in recent years as more general modelsfor the forward problem involving elastic waves havebecome available.

Finally, it should be noted that very few processes arepurely inversion techniques; they usually involve iter-ation, 'guesswork', and prior knowledge. Often the severerestrictions placed on the possible value of a problem sol-ution mean that generalised inversion techniques are notapplicable.

2.2 Born inversion22.1 Theory: A brief description of the theory and prac-tical implementation of Born inversion is given as it isnot presented in an easily assimilated form elsewhere.Detailed theoretical treatments are given in References 10and 11.

The general scattering situation is depicted in Fig. 1.As | r | -*• oo, for longitudinal incident ultrasound the fieldscattered by an inhomogeneity, in an otherwise homoge-neous isotropic medium, can be represented by

BL^Texp(ip\r\)\r\ (1)

flaw centroid. a and /? are the wave numbers for longitu-dinal and transverse waves, respectively. AL^L representsthe amplitude of the longitudinal-to-longitudinal wavescattering, and BL_T represents the amplitude of the

Fig. 1 Scattering situation from volumetric defect

longitudinal-to-trans verse scattering. The form of AL_Land BL_T is dependent on the scatterer, its orientationwith respect to the transmitting and receiving transducersand the wavenumbers.

The solution to the forward scattering problem is

U? (2)

where u( is the total scattered displacement field in the ithdirection and is the sum of the incidence field uf and thefield scattered in that direction by the inhomogeneity uf.The scattering solution was formulated in terms of anintegral equation which facilitated the matching of fieldsacross any generally shaped void/host boundary. It wasshown that

«J = uf + Spco2 I g(r, r')u{r') dVx

SCjklm f g'(r, r')u\r') dVl (3)JR

Us, the scattered displacement field, is dependent on themagnitude of the position vector | r | measured from the

where dp and SCjklm are the changes in density andelastic stiffness constant between host and inclusion,;, k, Iand m are direction indices, g(r, r') is the Green's func-tion for the response at r due to a point source at r', u isthe displacement field at r', and the integral is over Vx inR, i.e. within the flaw volume, g' is a spatial derivative ofg and u' is a derivative of u. Eqn. 3 is evaluated, using thestandard expression for the Green's function for an infin-ite volume, to obtain a scattering expression in the formof eqn. 1. Thus, to solve the forward problem, completeinformation on the fields inside the scatterer is required.This information usually cannot be obtained, and aforward problem approximation must be used for ageneral shaped flaw. In the Born approximation theunknown fields inside the flaw are replaced by the knownincident fields, and scattering amplitude expressions

258 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987

become

4n

dpcosd-

5k + 2dfi cos2 6k + 2\i

and

-.T = ^ j - ^ sin 20 - -^ sin

x S(aet - aes) (4a)

,. - pes) (4b)

where A and \i are the Lame parameters of the host,dp = P — Pf, Sk = k — Ay and dp. = fi — nf, the subscript/ refers to properties of the flaw. S is called the 'shapefactor' and is dependent on the wave numbers e(, whichis the unit vector in the direction of the incident wave,and es, the unit vector in the scattered wave's measure-ment direction. Expressions similar to 4a and 4b can beobtained for incident transverse waves.

Considering only the backscattered direction simplifiesthe expressions (et = — es, 0 = 180°) since BL^T = 0. Eqn.4a becomes

AL_L = a2 const S(ae,- + aet)

The shape factor is given in Reference 10:

(5)

S(pet - qes) -Illlimits definingflaw volume

exp (i(pei - qes) • r) d3r (6)

where p and q are wavenumbers and r is measured fromthe flaw centroid.

Since the limits of the integrand are the unknowns tobe evaluated, eqn. 6 is expressed as

rcrS(pe{ - qes) = y(r) exp (i(pet - qes) • r) d3r (7)

all space

where y(r), known as the characteristic function of theflaw, is defined as 1 inside the flaw and zero elsewhere.

Eqn. 5 now becomes

rrrAL^L = a2 const y(r) exp (2iae,- • #•) d3r (8)

JJJall space

which is the solution to the direct problem for the back-scattered direction.

Taking the inverse Fourier transform of eqn. 8 givesthe characteristic function

y(r) = const — ^ exp (-2iae,- • r) d3(ae{) (9)

all ate;

If di(ae) is expanded, and terms of order (da)2 andgreater are neglected, this can be reduced to the form

y{r) = const a2 doc — ^

all a all ei

x exp(-2iae i , - r) d2e{ (10)

For the case of spherical symmetry, the 1-dimensionalBorn inversion is dependent on the magnitude of thewave vector only (i.e. a) and eqn. 10 becomes

y(r) = const2ar

da (11)

since y(r) is real and AL^L has only a real componentwhen the time domain origin coincides with the flaw cen-troid [10]. Eqn. 11 is the inversion algorithm, AL^L as afunction of frequency being the only measured parameter.The integrand is evaluated over all a for discrete values ofr.

If infinite bandwidth were available, i.e. a -* oo, thenthe Born characteristic function would be a step function,the discontinuity occurring at the flaw/host boundary. Inthe bandlimited case the function is smoothed and theradius location has to be estimated.

22 ImplementationTo obtain AL^L, the backscattered time-domain signalfrom a flaw is recorded, as is a reference signal, usuallyfrom the plane back surface of the sample containing theflaw (see Fig. 2). The latter is assumed to be representa-tive of the spectrum of the complete interrogating system.

contact immersion

transducer

watercouplant

reference \Ksignal path | |

1

1

1

cz

f law signalpath

>

Fig. 2A Schematic diagram of experiment set up in contact andimmersion modes

monitoringoscilloscope

ultrasonictransmitter

andreceiver

driveamplifier

analogue output

digit iserand

line store

triggeroutput

digitaldelayunit

triggercapturewindow

digital output

controlinput

computersystem

alia

Fig. 2B Equipment for implementation of Born inversion

Both signals are digitised at 100 MHz, 8-bit resolution,stored in a computer and their spectra calculated using afast Fourier transform algorithm. These are then decon-volved in the frequency domain using a desensitisingWiener filter [12].

The distance travelled by the wave through the samplecould not be maintained the same for both flaw and ref-erence although these were made as close as practicableand both measurements were made in the probe far field.No compensation was incorporated for differencescaused by dispersion.

The deconvolved signal is then timeshifted so that theflaw centre coincides with the time origin. Methods ofachieving this are detailed in Section 3.1. The real part ofthe backscattered signal can now be used in eqn. 11 to

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987 259

obtain the flaw characteristic function and, from this, anestimate of the flaw radius.

3 Practical constraints and refinements to Borninversion

In this Section, details of the difficulties encountered inthe practical implementation of the inversion and therefinements made to the basic algorithm are given.

3.1 Transducer bandwidth and timeshiftingObtaining an adequate backscattered signal from theflaw is the most difficult step practically because of thetransducer bandwidth requirement and the low signal-to-noise ratio of a wave scattered from a small flaw. Thespectrum should ideally cover the range cca = 0.5 to 2.5,where V is the flaw radius; thus a 400 fim diameter flawin titanium alloy would require a spectral bandwidth of2.4 to 12 MHz. This range has been calculated by usingideal Born data in the inversion algorithm. The intersec-tion of the Wiener filter level (which was set at about10% of the maximum of the reference magnitude fre-quency spectrum) with the reference spectrum was usedto estimate the available bandwidth.

The majority of experiments were performed using aspecially constructed lead metaniobate transducer [13] asthe bandwidth requirements are at the limits of com-mercially available probes. This problem is accentuatedsince, in general, the flaw size is not known a priori.Various other criteria therefore have to be used to deter-mine the suitability of a particular transducer response,and to test the quality of any data obtained; these arenow discussed below.

Initially, the transducer bandwidth, estimated from thefilter intersection levels, is used to indicate the range offlaw sizes that can be correctly treated by the inverseBorn algorithm; if the final result lies out of this range itcan be discarded. The peak/trough pattern of the decon-volved frequency spectrum can also be used to give aninitial estimate of flaw size. Fig. 3 is a plot of back-

0 000 10 20 30 40 50 6 0 7 0 8 0 9 0 10 0

txa

Fig. 3 Backscattered magnitude spectrum for a void in titanium alloyplotted with respect to dimensionless quantity eta

scattered amplitude against eta for an ideal strong scat-terer; the peak separation is approximately \aa.Unfortunately, this estimation cannot be made if theprobe bandwidth only covers one peak.

Flaw radius predictions from the characteristic func-tion are sensitive to the timeshift chosen to locate theflaw centroid; various methods of locating the zero-of-

260

time were therefore investigated using ideal Born weak-scatterer data, ideal strong-scatterer data andexperimental data. The four methods considered were:

(a) Area function: From the Born approximation for aweakly scattering flaw it can be shown that the cross-sectional area of the flaw perpendicular to the incidentwave direction is related to the back-scattered signal[14], the time-domain signal being proportional to thesecond derivative of the area function. For a weakly scat-tering spherical flaw and normally incident ultrasoundthe area function has a parabolic shape with a maximumat the flaw centroid. This can therefore be located byintegration of the time-domain signal.

(b) Maximum flatness: Using ideal Born data it isfound that if the Born predicted radius is calculated for alarge number of timeshifts, a pattern is produced wherethe correct timeshift is located at the minimum of theportion indicated in Fig. 4. This pattern is evident for

27794

23956

201 18

162 79

124 41

E 86 02

3 47-64

» 9 26

-29 13

-6751-0-15 -0 12 -0 09-0-06 -003 000 003 006 009 012 015

T- shift , M s

Fig. 4 Evaluation of Born radius for various timeshifts using idealBorn dataArrow indicates portion containing correct timeshift

ideal strong-scatterer and experimental data. The Borncharacteristic functions for the timeshifts in this regionare examined to determine when the function is maxi-mally flat, and this point is chosen as the timeshift [15].

(c) Minimisation of imaginary part (MOIP): In the low-frequency scattering regime, the scattered spectrum ispurely real when the coordinate system is centred on theflaw [16]. Under this condition the Born approximationpredicts that the imaginary part of the spectrum is zeroat all frequencies. In practice, neither of the above aretrue owing to noise, however, the integral of the imagin-ary part of the spectrum plotted against frequency tendsto a minimum at the correct timeshift. Ideal weak- andideal strong-scatterer data showed a marked differencebetween the two scattering types, the correct timeshiftbeing the second minimum in Fig. 5b.

(d) Low frequency examination: The amplitude of anelastic wave scattered by a local inhomogeneity has theform

A(co) = An(on + i An(o"

n even n odd

at low frequencies [16], where An are constants and a> isthe angular frequency of the ultrasound. In the Rayleighscattering regime A2co2 is the dominant term if the coor-dinate origin coincides with the flaw centre, i.e.

and this can be used to find the required timeshift.

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987

With ideal data this method was found to be more effi-cient than using the area function formulation, for

0.00-0.61 -0.30 0.00

T-shift.psa

0.30 0.61

-0.48 -0.24 0.00 0.24 0.489.19

Fig. 5 MOIP resultsArea under imaginary part curve for(a) Ideal Born data(b) Ideal strong scatterer data

example crude initial estimates of the origin yield anaccurate value of the required timeshift after one iter-ation. However, experimental data were, in general,found to be too noisy in this region (cca < 0.5), and thismethod was not pursued further.

The investigation findings were as follows:(i) The area function formulation was the most sensi-

tive to very low-frequency noise, if very low frequencieswere not truncated from approximately 0.5aa andreplaced by a polynomial extrapolation to zero, anom-alous timeshifts occurred.

(ii) The maximum flatness method was the least sensi-tive to low-frequency noise, indeed extrapolation of thelow frequency data often made the pattern less welldefined.

(iii) MOIP results were not forced to the 'correct'second minimum, and consequently some instabilitybetween the two minima resulted.

(iv) The shape of the MOIP against timeshift plot wasfound to be degraded if inadequate bandwidths or noisydata were used.

(v) Anomalous predictions of the area function routineoften coincided with the MOIP first minimum.

Most importantly, however, it was found that for gooddata, i.e. frequency spectra covering an adequate band-width and containing no anomalous signals, the first

three methods considered predicted timeshifts usuallywithin one or, at most, two resolution points (frequencysampling, 100 MHz: resolution 10 ns). This was thereforeused to classify the data as suitable or as unsuitable forinversion. From these results it was found that 'adequatebandwidth' was in fact 0.5 < aa < 2.0 as a lack of higherfrequency components did not significantly affect time-shift predictions.

32 Radius estimationData that have passed the above classification processare inverted using eqn. 11 to obtain a characteristic func-tion. Radius estimations from this function are used asanother data classifier.

Three methods of radius estimation were considered:(a) (area under characteristic function)/(peak of charac-

teristic function)(b) distance corresponding to the point that is 50% of

the peak value (see Fig. 6)(c) point of steepest descent.

It was found that, even using the ideal bandlimited Borndata method, (c) gave underestimates of around 15%.Methods (a) and (b) agreed to within 10% and often towithin 5% for experimental and ideal data. Radius pre-dictions by these two methods which did not agree towithin 10% were rejected as not conforming to theexpected characteristic function shape determined usingideal data.

o.o50 100 150 200

distance, (jm

250 300 350 400

Fig. 6 Characteristic function for experimental data backscatteredfrom 200 fim radius void in Ti-alloy illustrating 50% point

Table 1 shows timeshifts and radius predictions by thetwo methods for ideal and experimental data from spher-oidal voids. The results demonstrate that the Born inver-sion will give incorrect radius predictions if unsuitabletransducer bandwidths are used, however, if bandwidthsuitability can first be classified then the results agreewith the expected values usually to well within 10%.

3.3 ResultsTable 2 summarises results obtained by using Born inver-sion on spheroidal voids manufactured in diffusionbonded blocks [22] of high-strength metals.

4 Ideal strong scattering data

In the preceding Section it has been shown that Borninversion, which is based on a weak-scatterer theory, canaccurately size voids in metals that are strong scatterers.The assumption of the Born technique, that the fieldsinside the scatterer are the same as the fields outside, isobviously not true for a void. Thus, the question arises:why does Born inversion work for voids?

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987 261

Table 1: Born data classification using timeshift and radius predictions

File andinput datainformation

TimeshiftAr Area fnA7" MOIP, seesAr Max. flatness

Radiusarea/peak,//m

Radius50% peak,//m

Accept/reject

Nominalradius,fjm

Ideal Borndata (weakscatterer)Ideal datafor void inTi-alloyafter Yingand TruellExperimental datafrom void inTi-alloyprobe incontactExperimental datafrom void inTi-alloyimmersionExperimental datafrom void inmaragingsteel contactExperimental datafrom void inTi-alloyimmersion

1.85 x10"6

1.85 x10"6

1.85 x10-6

8.0x10"'8.0 x 10-'8.0 x 10"'

-1 .0x10- '-9.0x10-8-9.0 x 10-8

-7.0 x lO-8

-8.0 x lO-8

-7.0 x lO-8

1.4x10"'1.3x10"'1.2x10"'

-3.4x10" '-3.2 x 10"'-3 .0x10" '

197197197

212212212

202195193

206195206

282296294

434431423

188188188

203203203

198193193

197188197

255255255

362338338

accept

accept

accept

accept

reject

reject

200

200

200

200

400

1000

To answer this a set of programs, which generate exactscattering data from spherical voids and inclusions inelastic media, were written [17] using the simple scat-tering theory of Ying and Truell [18]. These data werethen used to interrogate the Born technique.

Fig. 7 shows a comparison of the real part of the time-shifted spectrum (AL_L) for experimental data, Ying and

Truell theoretical data and Born theoretical data calcu-lated for a 0.2 mm radius void in Ti-alloy (Ti-6A1-4V)and the corresponding radius predictions. From theseresults spherical voids in Ti-6A1-4V can be accuratelysized using the Born inversion procedure provided suffi-cient bandwidth is available. The mechanism by whichthis arises is now examined.

Table 2: Experimental results using the 1-D Born inversion technique

Material andflaw data

Ti-6AI-4VOblatespheroidalvoid bydiffusionbonding

Ti-6AI-4VCylindricalvoids

Ti-6AI-4VNominallyellipsoidalvoid bydiffusionbonding

Maragingsteelnominallysphericalvoids bydiffusionbonding

Sampleface

ABCDABCDEF

TTSS

ABCDABAD

Nominalradius,/vm

200200200200200200200200165165

250500

1000

225 ± 40225 ± 40440 ±40440 ±40

200200200200300300400400

Measuredradius//m

186188191204200219193209167158

245528611

189163402316

238221231217266253418374

% difference

- 7- 6-4.5+2

0+9.5-3.5+4.5+ 1- 4

- 2+5.5-39

within error-27.5within error-28

+ 19+ 10.5+ 15.5+8.5-11-16+4.5-6.5

Availablebandwidthaa

0.16-2.20.16-2.20.16-2.20.16-2.20.16-2.30.16-2.30.16-2.30.16-2.30.16-2.00.16-2.0

0.26-2.30.4-2.31-9.3

0.14-2.10.12-1.90.27^40.23-3.6

0.44-20.23-20.23-20.23-20.43-2.30.43-2.30.44-2.30.44-2.3

Comments

ContactContactContactContactImmersionImmersionImmersionImmersionContactContact

ImmersionImmersionImmersion

ImmersionImmersionImmersionImmersion

ContactContactContactContactContactContactContactContact

262 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987

4.1 Mode analysisThe partial wave expansion approach of the Ying andTruell theory does, in fact, make physical interpretationof results quite intractable in comparison with other

o.o 1.9 3.8 5.7 7.6 9.5 V ^ 13.3 15.2 17.1 x19.0

frequency, MHz x x

xx .xxx

gating the flaw surface. Experiments have shown this tobe the case [21]. Examination of the scattered wave at

0.20t imers

-0.02L

Fig. 8B Time-domain signal from bandlimited data (0 < cut < 4)

Fig. 7 Real part of backscattered timeshifted spectrum from a 200 \imradius void in Ti-alloy

experimental data radius prediction 190 jimideal scattering data (Reference 18) radius prediction 212 /an

x x x Born weak scattering data (Reference 10) radius prediction 197 /on

theories such as the geometrical ray theory [14]. How-ever, examination of each of the terms (m = 0, 1, 2 ...) inthe series summation of amplitude response has yieldedsome information on contributions from various vibra-tional modes. Gaunard and Uberall [20] found, onexamination of each term, that the spectrum from a fluid-filled cavity could be identified as a series of resonance'spikes' superimposed on the smooth background of thespectrum for a void. Simple comparison of the summa-tion terms of a weak scatterer and a cavity did not revealan obvious relationship, although it has been shown [23]that the scattering amplitude from an elastic inclusionconsists of a nonresonant background term, due to longi-tudinal scattering from a soft (cavity) sphere, with a seriesof superimposed resonances. In both cases, however,modes 0, 1 and 2 are the most important for the fre-quencies of interest.

4.2 Time-domain analysisAnalysis of the backscattered time-domain signal from aBorn weak scatterer shows reflections from the front andback of the inclusion (Fig. 8A). If the spectrum is band-limited these are not fully resolved (Fig. 8B). Similar plotsfor voids (Fig. 9) show a secondary signal which can beexplained by postulating creeping waves circumnavi-

-0.07 L

Fig. 8A Time-domain Born ideal weak backscattered signal frombroadband (0 < <xa < 10) data

90° to the incident direction (Fig. 10) also shows a reflec-ted and a creeping wave. In comparison with Fig. 9A, thecreeping wave has a larger amplitude relative to thereflected wave and the separation between them isreduced. This is consistent with creeping waves beingtangentially launched and reradiated.

By examining the separation of the peaks in the back-scattered magnitude spectrum for a void, the velocity ofthe creeping waves can be shown to be approximatelyequal to the longitudinal wave velocity. If C,

Fig. 9A Ideal time-domain backscattered signal for void in Ti-alloyfrom broadband (0 <<xa < 10) data

200 250 300time, ns

Fig. 9B Time-domain signal from bandlimited data {0 < <xa <4)

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 3, MARCH 1987 263

(longitudinal) ~ Cc (creeping), then A/, the peak separa-tion in Hz, would be

C,(n + 2)a

a = flaw radius

Aaa =n + 2

= 1.22

For this case, the average peak separation is measured tobe 1.14.

Thus it is expected that the Born inversion routine willwork reasonably well for voids in elastic media whichsupport a creeping wave of detectable magnitude. Thequalification 'reasonably well' is due to the fact that, for a

400

Fig. 10 Ideal time-domain signal for void in Ti-alloy at 90° to incidentdirection from broadband (0 < eta < 10) data

weak scatterer the path difference between the front andback face signals is proportional to 4a, whereas for avoid, the signal path difference is proportional to (2 + n)aassuming Ct~ Cc. In fact, Cc is approximately 90% ofC,, calculated from

„ 100 x n x (peak separation of oca) . _Cc = — — - ; —-— % of C,

2n — 2(peak separation of aa)

This further accentuates the difference between weakscatterers and voids. Indeed, if strong- and weak-scattering ideal data are treated in an exactly equivalentmanner in the inversion algorithm (i.e. same bandwidth,filter, window and extrapolation points) the strong scat-terer does give larger radius predictions (see Table 1)although not as significantly different as expected frompath and velocity differences (> 30%).

It is the correspondence of the lower frequency wavecomponents, which are less important in the resolution ofthe front face and creeping wave, that makes the algo-rithm work due to what is effectively a low-frequencyweighting function in the inversion equation.

There remains the problem of detecting the creepingwave, which even in the ideal case has an amplitude ofabout 15% of the specularly reflected wave in the back-scattered direction. For small voids, low signal-to-noiseratios or voids with rough interfaces it is likely that thecreeping wave signal will have deteriorated or will not bedetected. To ascertain the effect on inversion results theamplitude of the creeping wave was degraded gradually.The inversion is not seriously affected; for example,reducing the secondary impulse amplitude by over 80%causes a change of less than 8% in the Born radius pre-diction.

264

However, the high-frequency content of the signal(> 3aa) is greatly reduced, and the shape of the area func-tion peak and MOIP output became less well defined.The timeshift predictions, and therefore the inversionresult, become inaccurate if the creeping wave signal isset to zero as this degrades both the low- and high-frequency spectral content. Thus, the algorithm dependson the relative scattering positions of low- and higher-frequency wave components, attenuation of the higher-frequency content being less serious. (High frequenciesare scattered from the defect surface whereas low fre-quencies appear to scatter from the flaw centroid).

5 Summary and discussion of results

The 1-D Born inversion routine has been tested usingideal weak-scatterer, ideal strong-scatterer and experi-mental data from spheroidal voids manufactured by adiffusion bonding process in high-strength metals. It wasfound that, provided sufficient bandwidth was available,the Born routine could size the above three data typesusually to well within 10% of the true value. The suffi-ciency condition was found to be approximately0.5 < aa < 2.5 which agrees with other workers. Theinversion is more seriously affected by lack of low-frequency information, indeed, an upper limit as low as2aa has proved adequate.

The Born technique appears to work on voids that arestrong scatterers due to the presence of creeping waveswhich circumnavigate the void/host boundary and arelaunched and reradiated tangentially. These have a veloc-ity near that of the longitudinal mode in the host and canbe greatly degraded before the inversion becomes inaccu-rate. This agrees with work that suggests that the Borntechnique is insensitive to noise. The algorithm in effectlocates the relative positions of high- and low-frequencyscattering.

An advantage of the inversion is that it does notrequire measurement of the absolute amplitude of thebackscattered signal, although this can be utilised as aguide to flaw size and composition. The inversion can beadversely affected by misuse of filtering levels, Hanning-window and low-frequency extrapolations. The optimalapplication of these signal processing routines is found byconditioning them on ideal Born bandlimited data.

The greatest problem with the inversion procedure isobtaining a probe bandwidth/flaw match if the approx-imate flaw size is not known a priori. A methodology forprobe bandwidth suitability classification has been devel-oped involving various methods of timeshifting andradius estimation.

The use of ideal strong-scattering data has illustratedthe application limitations of the Born inversion to voids,namely:

(a) elastic hosts which support a detectable creepingwave

(b) void/host media boundaries which are relativelysmooth, i.e. no 'edges' from which further sets of creepingwaves may be launched

(c) small voids ( < 1 cm in most media). Creepingwaves travelling round large voids would be severelyattenuated due to continual tangential reradiation intothe host over a longer circumferential path. Further, thebandwidth requirements for accurate sizing of largervoids may become unattainable.

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6 Conclusions

The application of indirect inversion methods to ultra-sonic defect characterisation is, as yet, limited in com-parison to related fields such as seismology.

Direct inversion algorithms are usually restricted toparticular cases where the forward problem can be fullydescribed. It has been demonstrated that the applicabilityof the 1-D Born algorithm can be successfully extendedfrom weak scatterers to voids due to a mechanism(creeping waves) not described in the original algorithmtheory. Provided sufficient bandwidth is available spher-oidal voids can usually be sized to within 10% of thecorrect radius.

A methodology for data classification has been devel-oped as it was found that flaw size/probe bandwidth mis-match was the most critical factor adversely affectingBorn radius predictions.

The limitations and advantages of the technique havebeen illustrated.

7 Acknowledgments

This work was performed with the support of the RoyalOrdnance PLC and the Procurement Executive, MOD.

8 References

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