Ultrasonic Measurement Models for Imaging with Phased Arrays

Transcription

1 Ultaoni Meauement Model fo Imaging with Phaed Aay Lete W. Shme J. ab Bady J. Engle ab Alexande Sedov and Xiongbing Li d a Cente fo NDE Iowa State Univeity Ame IA USA b Dept. of Aeopae Eng. Iowa State Univeity Ame IA USA Dept. of Meh. Eng. Lakehead Univeity Thunde Bay ON Canada P7B 5E1 d CAD/CAM Intitute Cental South Univeity Changha Hunan P.R. China Abtat. Ultaoni imaging meauement model (IMM) ae developed that geneate image of flaw by inveion of ultaoni meauement model. Thee IMM ae genealization of the yntheti apetue fouing tehnique (SAFT) and the total fouing method (TFM). A peial ae when the flaw i mall i hown to genealize phyial opti fa field invee atteing (POFFIS) image. The ultaoni IMM povide a ational bai fo geneating and undetanding the ultaoni image podued by delay-and-um imaging method. Keywod: Phaed Aay Syntheti Apetue Fouing Tehnique Total Fouing Method Phyial Opti Fa Field Invee Satteing Phaed Aay Imaging PACS: INTRODUCTION With the gowing ue of ultaoni phaed aay in NDE inpetion imaging ha beome an impotant tool fo identifying and evaluating flaw. The yntheti apetue fouing tehnique (SAFT) and the total fouing method (TFM) fo example ae two of the mot fequently ued imaging method beaue of thei ability to eate high quality image. Both SAFT and TFM ae o-alled delay-and-um method that ae baed on ad-ho poedue to fom an image. Howeve imaging at it mot fundamental level i the olution of an invee poblem o it i impotant to undetand how the ultaoni meauement ytem ontibute to a flaw image though uh an inveion if one wihe to extat quantitative infomation on the flaw fom that image. Ultaoni meauement model have been ued fo ove thity yea to imulate the fowad poblem of ultaoni atteing i.e. they pedit how the ultaoni voltage meaued in an ultaoni tet i elated to the atteing popetie of a flaw the wave geneated by the inteogating tandue() and the eleto-mehanial element peent. Sine all image ae fomed in ome fahion fom thee meaued voltage it i ou view that one hould teat the image fomation poe a an inveion of an ultaoni meauement model. Hee we will demontate that uh an inveion i indeed poible uing meauement model in onjuntion with a tong atteing Kihhoff appoximation. Thi type of inveion we will all an imaging meauement model (IMM). It will be hown that both SAFT and TFM ae in fat appoximate fom of ou IMM appoah and we will identify expliitly what pat of the flaw epone SAFT and TFM ae tying to image. Fo mall flaw ou IMM appoah will be hown to be genealization of a well-known invee method alled POFFIS (phyial opti fa field invee atteing). Thu ou imaging meauement model will povide a foundation fo undetanding and quantitatively evaluating ome of the mot impotant type of imaging method that have been developed and ae uently ued with ultaoni phaed aay ytem.

2 x 1 p 1 x x x flaw ufae p FIGURE 1. An immeion etup fo imaging a flaw with the pule-eho epone of a -D phaed aay (eond dimenion not hown). PULSE-ECHO IMAGING The SAFT imaging method wa oiginally developed fo ue with ingle element tandue. A the tandue wa anned ove a ufae the pule-eho epone of a flaw wee olleted at eah tandue loation. The ame data olletion an be done with a phaed aay a hown in Fig. 1. Uing eipoity elation the meaued voltage fom eah element of the aay an be imulated by a meauement model of the Auld type. Fo an immeion meauement fo example we find [1] 1 1 I x V x ji x vi x ji x vi x n j x ds x. (1) 1 p1s v x S f In Eq. (1) the numbe in paenthee indiate two diffeent tate labeled (1) and (). State (1) i the atual flaw meauement etup whee the flaw i peent while tate () i whee the eeiving element (ame a the ending element fo pule-eho meauement) at a a tanmitte intead of a eeive and the flaw i abent. The eeived voltage in the fequeny domain fo an element whoe entoid i loated at x i V x i the ytem funtion whih x whee i the iula fequeny. The funtion haateize all the eletial and eletomehanial element peent in the meauement ytem fo a given element. Sine the ytem funtion may be diffeent fo diffeent element we have indiated thi by iuding x in the agument of thi funtion. Thi ytem funtion an be haateized fo eah element in a alibation etup. Similaly vx i the diving veloity on the fae of the element whih i aumed to at a a piton oue of ound. The quantitie 1 p 1 ae the denity and ompeional wave peed of the fluid adjaent to the element and S i the aea of an element (aumed hee to be the ame fo all element). The field ufae m m x v x m 1 ij j S f of the flaw whoe outwad unit nomal omponent ae I ae the tee and veloity omponent on the n j x fo tate (1) and (). In tate (1) thee field iude both the inident wave field fom the element and the atteed wave field fom the flaw while in tate () thee field ae due oy to the inident wave. Sine meauement model like Eq. (1) an be obtained fom pimaily eipoity and lineaity aumption they ae appliable in lightly

3 modified fom to almot all flaw meauement etup. Hee fo peifiity we will oy ue Eq. (1) whih i valid fo the ae of immeion inpetion with an aay that geneate bulk wave. When the wave fom an element of the aay inteat with a flaw atteed wave of vaiou type ae geneated. Thee iude wave that ae efleted fom the font ufae of the flaw (whee by font we mean that pat of the flaw that i illuminated dietly by the aay) flaw that eflet intenally eeping wave et. Howeve the wave that ae peulaly efleted fom the font ufae of the flaw play an epeially impotant ole ine they ae often the wave having the laget amplitude and hene ae the mot ignifiant wave that geneate an image. Thu we will aume that it i thee font ufae peulaly efleted wave that ontibute to the meaued voltage in Eq. (1). In thi ae Shme [] ha hown that by ue of the Kihhoff appoximation to model the wave that ae peulaly efleted fom the flaw ufae one an expe the integand in Eq. (1) in a muh moe expliit fom to give x V x R V x x i x xds x () I ; exp 1p 1S Slit whee lit ae the denity and wave peed (of type whee p o fo inident (and atteed) ompeional o hea wave epetively) and S i the flaw ufae dietly illuminated by the aay ; R i the plane wave efletion oeffiient fo nomal inidene of a wave of type (fo both inidene and efletion) at the flaw ufae. The field V x x i the veloity at a point x on the flaw ufae geneated by an element loated at x divided by the diving veloity vx on the fae of the element. Thi nomalized veloity field an be dietly alulated with ultaoni beam model. Finally x x i the (one-way) tavel time fom an element to the point x on the flaw. If we now define a ingula funtion (volumeti) pae a S x of the lit ufae [3] then we an ewite Eq. () a an integal ove all x V x R xv x x i x xdv x. (3) I ; S exp 1p 1S V Equation (3) i a fundamental eult ine it i poible to invet thi elationhip to fom an image of the flaw. Fit we will define ou flaw image x a imply the ingula funtion of the flaw ufae multiplied by the plane wave efletion oeffiient i.e. ; x R S x. With thi definition Eq. (3) i in a fom imila to that that of a 3-D Fouie tanfom of the flaw image. Thi ugget that at leat at high fequenie by pefoming an opeation imila to an invee Fouie tanfom we hould be able to invet ou meauement model elationhip and eove the flaw image. Spae doe not allow u to give all the detail but one an follow the tep outlined in Bleitein Cohen and Stokwell [4] fo a imila ala poblem to obtain y V x x x y 1p 1S exp i d d x y I V (4) whee the integation i ove a olid angle at the image point y to the ending/eeiving element. Thi integation an be tanfomed into one ove the plana ufae of the aay by witing it a

4 y x x V x y S V d 1 p 1 exp i x y ds I ds d. (5) Fo inpetion in a ingle medium the angle to aea atio d / ds o / whee i the ditane fom the point x on the plane of the aay to an image point y and i the angle that the ay fom y to x make with epet to the nomal to the plane of the aay. Fo moe geneal inpetion poblem it i elatively eay to alulate thi atio via ay theoy. Equation (5) i a ontinuou integal fom of ou imaging meauement model ine it aume that both the fequeny and the element loation x ae ontinuou vaiable. But it i eay to tanfom thee integal into the diete um that would be found in patie fom a et of diete element and ampled fequenie to obtain a band and apetue limited image BA given by BA y d x yexp im. ds x y V x m x x y L1 L x M 1x 1 p1s Re m m1 n1 l1 I m V m (6) Hee "Re" indiate "eal pat of" and x1 x ae the paing between the entoid of the aay element in the x1 x dietion epetively o they ae jut the pithe of the aay in thoe dietion. The quantity i the paing of fequeny omponent (in ad/e).the point x 1 x x ae the diete loation of the element entoid m ae the diete ampled fequenie and y i the image point (whih will alo be ampled dietely in patie). The quantity d / ds i the olid angle to aea atio that i alulated fom eah element entoid to an image point along a ay path while x the time it take fo a wave to tavel fom the entoid of an element to the image point. The diviion peent in Eq. (6) epeent a deonvolution of the meaued voltage with ytem funtion and the inident wave field. Thi diviion i eaonable to expet ine one mut emove thee ontibution fom the meaued voltage (ontibution that ae not elated to the flaw) if we wih to obtain an image of flaw haateiti oy. Howeve uh deonvolution i inheently untable in the peene of noie and o one would need to eplae the taight diviion by an equivalent but table poe. In patie thi an be done with a Wiene filte [1] []. One an ompae thi IMM with SAFT. In the fequeny domain a SAFT image SAFT i given by L L y x x y (7) M 1 Re V exp i SAFT m m m 1 n 1 l 1 whih how that both method ae delay and um method but that SAFT doe not ompenate fo ytem effet inident beam effet and othe fequeny and geometial fato peent in the inveion tem of the IMM. When the flaw i mall the beam oetion tem in Eq. (7) an be eplaed by thei value at a fixed point y 0 nea the flaw and the time tem an be expeed in appoximate fom to fit ode a y i

5 x x x x x flaw ufae x x 1 p 1 p FIGURE. An immeion etup fo imaging a flaw with the full matix aptue of all the epone of a -D phaed aay (eond dimenion not hown). x y x y e y y whee e i a unit veto in the dietion of popagation of the 0 0 / inident wave at the flaw. In thi ae Eq. (7) beome y 1p 1S V x I x V x y0 exp i x y0 d exp i ke y y0 dsd. ds (8) But in the mall flaw ae we an identify the ombination of tem appeaing in the integand a A e ; e giving popotional to the pule-eho fa field plane wave atteing amplitude of the flaw y d ik A ; exp i dsd. e e e y y ds (9) 0 Equation (9) i in a fom idential to that of the ala phyial opti fa field invee atteing (POFFIS) theoy of Bojaki a modified by Bleitein [3]. Engle et al. [5] how that thi ame eult an be obtained dietly by an inveion of the Thompon-Gay meauement model whih i a meauement model valid fo the mall flaw ondition. Thu ou pule-eho IMM of Eq. (7) an alo be thought of a a genealization of POFFIS to the ae of lage flaw. It i alo hown in [5] though exat imulation that although the full atteing amplitude of the flaw i ued to geneate the flaw image oy the font ufae peula epone of the flaw whih i the epone ontained in i popely imaged. Othe epone uh a bak ufae efletion and eeping wave appea a image atifat.

6 FULL MATRIX CAPTURE IMAGING Pule-eho epone ae oy a fation of the poible ending and eeiving element ombination poible with an aay. In full matix aptue one ollet the pith-ath epone of all the element pai. Uing thoe epone in a um and delay poe imila to SAFT eult in an imaging method alled the total fouing method (TFM). Geneally TFM i onideed to eult in "bette" image than SAFT and beaue it ue all the poible epone infomation in the aay it i ometime alled the "gold tandad" of imaging. Howeve what flaw popetie ae being imaged with TFM that ae diffeent fom that of SAFT and what ae the eal quantitative impovement of TFM ove SAFT? An imaging meauement model appoah an anwe thee quetion. To fom up a TFM-like image with a meauement model appoah one an fit onide the ae whee one fixed element i ued a a tanmitte and all the aay element ae eeiving. In the eimology liteatue thi i alled a fixed hot onfiguation. An imaging meauement model an be fomed fo thi ae in a vey imila fahion to that of the pule-eho ae o we will oy tate the final eult hee whih i the pith-ath analog of Eq. (5): V x x x x x y x y S y 16 1 p1 1 o I V V d exp i x y x y dsd. ds (10) ; Hee y R e n k k S y i a genealized flaw ufae efletivity whee ; R i the plane wave efletion oeffiient fo an inident wave of type and a efleted wave of type and eknk e n i the oine of the angle that the efleted wave (of type ) make with the outwad nomal of the flaw. The veloity tem V i the veloity amplitude of the inident ound beam geneated by the ending element divided by the diving veloity on the fae of the element and V i the oeponding nomalized veloity if the eeiving element at a a tanmitte. The funtion I x x ending and eeiving element and i the ytem funtion fo a pai of x y x y ae the tavel time fom the ending element and the eeiving element epetively to the image point y. The angle (ee Fig. ) i the angle between the inident and atteed dietion and a in the pule-eho ae the quantity d / ds i the atio of a olid angle to aea along the atteed ay. To fom up a full matix image one an now vay the eeiving element loation x and alulate the integation of the flaw efletivity ove a olid angle a meaued along the inident ay i.e. we ompute d whih give an imaging meauement in the fom y V x x x x x y x y S 16 1 p1 1 o I V V d d exp i x y x y dsdsd. ds ds (11) We an ompae thi imaging meauement model to a TFM image witten alo in ontinuou integal fom a

7 y 1 V exp i ds ds d. x x x y x y (1) TFM Fom Eq. (11) and (1) we an ee that the TFM like SAFT ignoe the beam popagation and ytem effet o it geneate a filteed veion of the angula integated font ufae efletivity of the flaw. In the tudy of Engle et al. [5] it i hown that the impovement in TFM ove SAFT ome fom the angula integation peent whih eliminate atifat like eep wave but that othe atifat till emain in the image with the oy "tue" pat of the image till being aoiated with the font ufae efletivity of the flaw. By making the mall flaw aumption one an alo geneate a POFFIS-tyle imaging meauement model fo the full matix aptue ae. In thi ae we again appoximate the beam field oetion by thei value at a fixed point y 0 nea the flaw and expand the delay time to fix ode a whee x y x y x y0 x y0 1 e y y / e y y / e e ae unit veto along the inident wave dietion at the flaw along ay path fom the ending and eeiving element epetively. In thi ae Eq. (11) beome (13) y V x x S 16 exp 1 p1 1 I x x V x y0 i x y0 1 o d d exp 0. i y y dsdsd V 0 exp i 0 ds ds x y x y (14) whee 1 / / e e (15) A in the pule-eho ae the image an alo be expeed in tem of the pith-ath fa field atteing 1 amplitude A e ; e to give ik 1 y ; 1 o 4 e e 1 A d d exp i y y0 dsdsd. ds ds To ou knowledge Eq. (16) i a new imaging model of the POFFIS-type fo full matix imaging. (16) SUMMARY We have developed imaging meauement model that ae genealization of both SAFT and TFM imaging and alo iude POFFIS-type of image a peial ae. Thee imaging meauement model

8 how that SAFT i attempting to image a efletivity defined expliitly a a efletion oeffiient multiplied by a ingula funtion that define that flaw ufae. In ontat TFM i tying to image an angula integated fom of a imila flaw ufae efletivity. It i thi angula integation that allow TFM to podue "bette" image than SAFT by eduing ome of the atifat and othe "noie" in the imaging poe. Both SAFT and TFM howeve iude ytem and beam effet that ae not flaw elated and whih modify the image they podue. It i alo hown that POFFIS-tyle imaging i loely elated to SAFT and TFM imaging when the inident field do not vay ignifiantly ove the flaw ufae. Thu imaging meauement model fom a ational bai fo jutifying and extending all of thee ommoy ued delay-and-um type of imaging method. ACKNOWLEDGMENTS Thi wok wa uppoted fo L.W. Shme and B. J. Engle by the National Siene Foundation Induty/Univeity Coopeative Reeah Cente pogam at the Cente fo NDE Iowa State Univeity. A. Sedov wihe to aknowledge uppot by the Natual Siene and Engineeing Reeah Counil of Canada. X. Li would like to expe hi gatitude to the National Natual Siene Foundation of China ( and ) fo uppot. REFERENCES 1. L. W. Shme and S.-J. Song Ultaoni Nondetutive Evaluation Sytem Model and Meauement New Yok N.Y.: Spinge L. W. Shme Fundamental of Ultaoni Nondetutive Evaluation A Modeling Appoah New Yok N.Y.: Plenum Pe N. Bleitein Mathematial Method fo Wave Phenomena New Yok N.Y.: Aademi Pe N. Bleitein J.K. Cohen and J.W. Stokwell J. Mathemati of Multidimenional Seimi Imaging Migation and Inveion New Yok N.Y.: Spinge B. J. Engle L. W. Shme and A. Sedov "Quantitative Ultaoni Phaed Aay Imaging" (thi poeeding).