inhomogeneous media using the conjugate gradient

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1 Rad Scence Vlume 34 Number 6 Pages Nvember-December 1999 Slvng the vlume ntegral equatn n axsymmetrc nhmgeneus meda usng the cnjugate gradent fast Hankel transfrm methd $. Y. Chen and W. C. Chew Center fr Cmputatnal Electrmagnetcs Electrmagnetcs Labratry Department f Electrcal and Cmputer Engneerng Unversty f Illns Urbana W. D. Kennedy Mbl Explratn and Prducng Techncal Center Dallas Taxes Abstract. In ths paper we present a new methd t slve the vlume ntegral equatn gvernng the electrmagnetc wave prpagatn n axsymmetrc nhmgeneus meda. The ntegral equatn s frst frmulated usng a ne-dmensnal Green's functn and then slved usng the cnjugate gradent fast Hankel transfrm methd. The use f the ne-dmensnal Green's functn reduces the number f unknwns sgnfcantly thereby acceleratng the teratve prcedure. Several numercal results are used t shw the effcency and accuracy f ths new apprach as well as ts applcatns n electrmagnetc subsurface sensng. 1. Intrductn Electrmagnetc wave prpagatn n axsymmetrc nhmgeneus meda s a subject f great nterest. Its applcatns can be fund n many areas such as ntegrated ptcs [e.g. $afaa- Jaz and Yp 1980] mcrwave ntegrated crcuts [e.g. Gl and Martnez 1985] and electrmagnetc subsurface sensng frm well bres. [e.g. Andersn and Chang 1982; Chew et al ; $hen and Zhang 1985; Lu and Chew 1993; Lu 1993]. In a cylndrcal crdnate system (p b z) the parameters f the axsymmetrc nhmgeneus meda depend nly n p and z. In general an arbtrary surce wll stll excte a three-dmensnal feld. The case wth axsymmetrc nhmgeneus medum and arbtrary surce s classfed as a 2.5-dmensnal prblem [Chew et al. 1984; Lu and Chew ; Lu 1993]. Hwever the prblem reduces t tw-dmensn when the surce s an axal magnetc dple pntng t the z drectn. In ths paper we wll restrct ur dscussns t the tw-dmensnal Cpyrght 1999 by the Amercan Gephyscal Unn. Paper number 1999RS / 99 / 1999 RS $11.00 prblem and fcus n ts applcatns n electrmagnetc subsurface sensng. Amng varus methds used t slve ths prblem the vlume ntegral equatn methd has been studed extensvely by many researchers [Meyer 1993; Lu and Chew ]. Recently a cnjugate gradent fast Furer-Hankel transfrm (CG-FFHT) methd derved frm the k space methd [Bjarsk 1971] has been prpsed by Lu and Chew [1994]. Ths new methd uses the CG-FFHT requrng nly O(Nlg 2 N) cmplex multplcatns per teratn. In ths paper we fllw the same prcedures. Hw- ever nstead f usng the hmgeneus medum Green's functn a ne-dmensnal Green's functn s used. The use f ths ne-dmensnal Green's functn re- duces the number f unknwns sgnfcantly. Furthermre the ntal guess usng a ne-dmensnal Green's functn s usually clser t the slutn. Cnsequently fewer fast Hankel transfrms (FHT) are needed and faster cnvergence can be acheved Our paper s rganzed as fllws: In sectn 2 we brefly revew the FHT algrthm and ntrduce sme new mdfcatns t the exstng FHT algrthm. In sectn 3 a tw-dmensnal axsymmetrc ntegral equatn s frmulated usng a ne-dmensnal Green's functn. A CG-FHT scheme s then ut- 1339

2 1340 CHEN ET AL.- SOLVING THE VOLUME INTEGRAL EQUATION lned t slve fr the unknwn nduced current densty. Fnally several numercal results are used t demnstrate the applcatns f CG-FHT. 2. A Bref Revew f the FHT Algrthm We defne the Hankel transfrm f rder v f functn f (p) as [Krn and Krn 1961] F(kp) -- pf (p)jv(kpp)dp (1) where Jv(x) s the vth-rder Bessel functn. The nverse transfrm s gven by where f (p) - kpf(kp)j kpp)dkp (2) I(P) - I lm[f(p - e) + f(p + e)] whch unquely determnes the nverse transfrm whenever t s cntnuus. T avd the ndcated numercal ntegratn and evaluatn f Bessel functns the changes f varables are ntrduced t rewrte (1) and (2) n a cnvlutn frm [Jhansen and Srensn 1979; Andersn ; Lu and Chew ] that can be mplemented effcently usng the fast Furer transfrm (FFT) algrthm. T avd dscrete Furer transfrmng the kernel pjp(kpp) whch s a very ll behaved functn Jhansen and $rensen [1979] and Lu and Chew [1994] suggest usng a clsed-frm expressn fr the Furer transfrm f pj kpp). Ths mdfed FHT algrthm reduces the number f samples requred t btan results f the same accuracy. In ur CG-FHT teratve scheme the fllwng peratn s frequently encuntered: & - (p (4) where 7-/s the Hankel transfrm defned n (1) and J½(p z) s the nduced current densty. When the nduced current densty J (p z) s a smth functn f p such as n layer 2 shwn n Fgure 1 the FHT algrthm gves accurate results. Hwever the nduced current densty n layer 3 wll have abrupt changes at the cylndrcal bundares Pl and p2. In ths case the FHT algrthm wll have a large alasng errr near the axs. Ths can be seen mre clearly by apply- ' (3'! 2 T ( (z p")!!!.: : '--... ' R 4 Fgure 1 Schematc plt f a generc tl n an asym- metrc medum. z ng 7-/and then?_ --1 t the current densty J (p z). If the FHT algrthm s accurate we shuld recver J (p z) exactly. The resultng current densty func- tn J&(p z) and the rgnal ne J (p z) are cmpared n Fgure 2. Errr s bserved at the small p. Ths errr wll cause the teratve prcedure t cnverge slwly r even dverge. Hwever a Hannng wndw can be used t remve the hgh-frequency cmpnents befre the nverse dscrete Furer transfrm s appled. The result btaned usng the Han- nng wndw s als shwn n Fgure 2. The mprvement s evdent. 3. Vlume Integral Equatn In ths sectn we frst frmulate the vlume ntegral equatn usng the ne-dmensnal Green's functn and then utlne the CG-FHT scheme. In an axsymmetrc medum the cnductvty a(pz) and delectrc cnstant er(p z) are functns f p and z nly. The magnetc permeablty/ s assumed t be a cnstant/ 0. The electrc feld prduced by a caxal lp antenna can be expressed n terms f the EO cmpnent nly whch satsfes the fllwng ntegral equatn [Chew and Lu 1994]: E½(pz p z ) -- k-.nc -tf_% dz" f_% dp" gb(p z p" z")j½(p" z") (5) where the nduced current densty J½(p"z") s defned by J½(p"z")- Ak2(p"z")E½(p"z"p'z ') (6)

3 CHEN ET AL.' SOLVING THE VOLUME INTEGRAL EQUATION 1341 x I I l' ' Re (J) -. 10[ I... Re(H-I J]) - Re ( H- /-/[J]) wth wndw 0 I -v I I./ '... '-'... ' Index f p X IIIII1'1!! I! - Im(J) ' Im(H -./-IJ]) / 3 t: 6!!1!!. Im (H- J]) wth wndw -I... I.I. I.... """"" '"" -- I. I I II I _1 '[ 0 VVVVVV..."?r './ '.... l l. I I I I I I Index f p Fgure 2. The results fr the mdfed fast Hankel transfrm (FHT) algrthm. The errrs at the small rad are reduced by usng Hannng wndw befre the nverse dscrete Furer transfrm ( ). and the cntrast functn Ak2(p"z ") s gven by (p" ") - (p" ") - (p" "). (7) In the abve E 0 als satsfes the equatn Op" Op + Oz "2 + k2 EO = - z 0( " - ')( " - ') (8) where k 2 - w2 e(er + a/ e) and g s the nedmensnal Green's functn satsfyng the equatn ( ) ( p" z") = -(p- p")( - "). (9) The slutn f (9) n a ne-dmensnal layered meda can be wrtten as [C hen and Chew 1997] gb(p Z pt Zt') p" -- dkp zj(wp)j(wp")r( t' ) (10) where F(z z" kp) s a cmplcated functn dependng nt nly n the kp but als n z and z" as well as n the a and er f each layer. The expressn f F(zz"kp) s gven by Chew et al. [1984]. The ncdent feld can be wrtten n terms f a ne- dmensnal Green's functn as nc :l lgb (p z p' z') Makng use f (10) we can rewrte (5) as 1 Enc(p z) - /kk2(pz ) Jck(p z) / 2 dp" jz (kpp")j(p" z") dz F(zz"kp) 5 aww (Wp) z( ") (p ) (p" ")e (11) (12) where R dentes the nhmgeneus regn. We nte that by usng a ne-dmensnal Green's functn R shuld nly nclude the shaded areas n Fgure 1;

4 1342 CHEN ET AL.- SOLVING THE VOLUME INTEGRAL EQUATION thus the vlume f ntegratn s much smaller than t wuld be usng the hmgeneus Green's functn. The frst and thrd ntegratns n the rght-hand sde f (12) are the nverse Hankel and the Hankel transfrms respectvely. Defnng an peratr as rl- l 1 _?_[--1 c we can rewrte (12) as dz" F (z z" kp ) kz(ztt) 7-/[ ]} (13) œ J (p z) - t?nc (p z). (14) The rght-hand sde f (14) s the ncdent feld excted by a caxal lp antenna n a ne-dmensnal layered medum. The CG algrthm prpsed by Peters and Vlaks [1988] s used t slve (14) tera- tvely. The ntal guess J (p z) s gven by the Brn apprxmatn as J (pz)- Ak2(pz)E nc(pz) (15) and the adjnt peratr œ encuntered n the CG algrthm can be btaned frm (13) as [] dz ttf*(z'ztt'kp) [ ]. (16) After btanng the nduced current densty J (p z) we can cmpute the feld everywhere usng (5). T estmate the cmputatnal cmplexty we use Np and Nz pnts t dscretze the unknwn regn alng the p and z drectns respectvely. The number f unknwns wll be less than NpN because the nhmgenetes exst nly n parts f the layers f a ne-dmensnal Green's functn s used. Further f the layer des nt have any cylndrcal bundares (e.g. layer I n Fgure 1) n dscretzatn needed n that layer. A wndw n the z drectn s used t truncate the ntegratn wth respect t z n (12). Because the cntrbutns frm small nhmgenetes far away frm the vlume f nterest are neglgble n lssy meda layers utsde the wndw are apprxmated by the ne-dmensnal layered meda. Ths assumptn reduces N sgnfcantly. The number f cmplex multplcatns requred n each teratn s O(2NzNp lg 22Np + 2NpN 2). The factr f 2 s due t the zer-paddng alng the p drectn. The number f teratns s als reduced. Fr lw- frequency applcatns n lssy meda such as n the nductn well-lggng the new algrthm usually cnverges wthn tw teratns whle the algrthm usng hmgeneus medum Green's functn requres fve t ten teratns fr the same prblem. The faster cnvergence results fr tw reasns. Frst usng the ne-dmensnal Green's functn sgnfcantly reduces the number f unknwns; therefre fewer teratns are needed. Secnd the better ntal guess f unknwn current densty als cntrbutes t faster cnvergence. Overall the new algrthm s 6-10 tmes faster than the ne usng a hmgeneus medum Green's functn. 4. Numercal Results We frst demnstrate the accuracy f the new algrthm. The setup f ur frst example s the same as the ne used by Lu and Chew [1994]. One transmtter and ne recever are used. The transmtter- recever spacng s m. The transmtter perates at an nductn frequency f 20 KHz; therefre t can be apprxmated by a magnetc dple. The feld at the recever s frst cnverted nt vltage and then nt apparent cnductvty [Mran and Kunz 1962]. In all the examples cnsdered n ths paper the delectrc cnstant er = 1 and the cnductvty rr vares I R! "-" '"' " ---' -'-: " :: :! ;½ ' "'"':"'"'"'":'"'"'"'' :...!]::"'"' " '"'""' ':':' l..--..=-:-..:... :::....:.. :½:a :.... : S ' ':'z:':':':':':':' :(::z:':;"' : : :. : :>.. :.:.:.;: 3.81 Fgure 3. The gemetry f a nne-bed mdel fr Fgure 4 (dmensns n meters and cnductvty n semens per meter). 8.89

5 _ CHEN ET AL.- SOLVING THE VOLUME INTEGRAL EQUATION _ ' <r E $ 0 rr -20 I I I v -12._ => ' c <r CG-FHT-1D - -. CG-FFHT 0 NMM O b......} I I I _E Fgure 4. Results fr Fgure 3. Three methds cnjugate gradent fast Furer-Hankel transfrm methd (CG-FFHT) cnjugate gradent fast Hankel transfrm methd usng ne-dmensnal Green's functn (CG-FHT-1D) and numercal mde matchng (NMM) are used t cmpute the results. Shwn are the (a) real and (b) magnary parts f the apparent cnductvty. The gemetry f the frst example s shwn n Fgure 3. If a hmgeneus medum Green's functn s used we have t dscretze mre layers such as the tw layers wth the cnductvty rr = 0.1 S/m. Fr ths tw-dmensnal prblem the numercal mdematchng methd (NMM) [Chew et al ; Chew 1995] can als be used t calculate the apparent cnductvty. The results btaned by these three methds are shwn n Fgure 4. Very gd agreement has been bserved amng these three sets f results. In the next example a standard nductn tl 6FF40 (trademark f Schlumberger) s used. The tl cmprses three transmtters and three recevers and perates at 20 khz. The gemetry f the tl and the mdel are shwn n Fgure 5. In ths case the advantages f usng a ne-dmensnal Green's functn becme mre evdent. The unknwns wll exst nly n the shaded regns. The number 'f unknwns s much smaller than wuld result frm usng a hmgeneus medum Green's functn. The cmparsn f CG-FHT-1D results wth CG-FFHT and NMM results n Fgure 6 shw very gd agreement. (s/m) I : T R1 R T2 T1 '-- 1 R " ' "< '" "" " * ' "" 0.5 Z Fgure 5. Gemetry f a nne-bed mdel f; Fgure 6 (dmensns n meters and cnductvty n semens per meter). A standard nductn tl 6FF40 s used. The apparent cnductvtes are cmputed usng CG-FHT- 1D CG-FFHT and NMM. The unknwns fr the CG- FHT-1D methd exst 0nly n the shaded regn.

6 1344 CHEN ET AL.' SOLVING THE VOLUME INTEGRAL EQUATION 1.1 I -- CG -FHT-1D II -- CG-FFHT II O NMM ' Fgure 6. The apparent cnductvty results fr Fgure 5 usng 6FF40. The cmparsn amng the three methds shws gd agreement. 0.8 I I I I I I I -- CG-FHT-1 D I 0 NMM :) I -' CG-FHT-1D anmaly 0.6-._.> "' 0.4- ( w+ 4+ _ _+_m_. _4.+ h-+%. \ I _ _ I I I I I I I Fgure 7. The apparent cnductvty results fr Fgure 5 usng 6FF40 wth brehle. The dashed lne and the crss are the respnses f 2-D anmales cmputed by usng CG-FHT-1D and NMM respectvely.

7 ._ CHEN ET AL.' SOLVING THE VOLUME INTEGRAL EQUATION 1345 (s/m) I ; 0.01 ' Z Fgure 8. Schematc plt f a dual prpagatn resstvty (DPR) tl n a seven-bed mdel wth three center beds nvaded (dmensns n meters and cnductvty n semens per meter). Fr the ntended applcatn mdelng a brehle s essental. T test the algrthm n such a case we nclude a brehle n the abve example. The brehle radus s m and the cnductvty f the mud s 10 S/m. The results are shwn n Fgure 7. T shw the accuracy f the vlume ntegral equatn we als gve the respnses frm the tw-dmensnal anmales nly n the same fgure. Gd agreement s bserved. The algrthm can als be used t analyze the respnses f the measurem-ents-whle-drllng (MWD) resstvty tls whch perate at medum frequency (2 MHz). One wdely used MWD resstvty tl s the dual prpagatn resstvty (DPR) (trademark f Baker-Hughes INTEQ) tl. The cnfguratn f the DPR s shwn n Fgure 8. The DPR makes tw measurements: the phase dfference and the ampltude rat between the recevers R1 and R2. The phase dfference and ampltude rat are then cnverted ndependently nt tw resstvtes Rpha and Ramp as thugh the tl were placed n a hmgeneus medum [Cpe et al. 1984; $hen 1991]. The transmtter and recevers f DPR are wund n a very cnductve mandrel wth slatng materals between the mandrel and the cls. The hghly cnductve mandrel may cause dffcultes n the vlume ntegral equatn methd. Hwever as lng as nly the phase dfference and ampltude rat are cncerned the effects caused by the metal mandrel 10 3 I I Rph a CG-FFHT-1 D Rpha NMM Ram p CG-FHT-1 D Ramp NMM ' 10 2 I E v E Fgure 9. Results fr Fgure 8. The respnses f DPR are calculated usng CG-FHT-1D and NMM.

8 1346 CHEN ET AL.: SOLVING THE VOLUME INTEGRAL EQUATION can be neglected [Zhu and Hllker 1991]. Therefre the man&el has been remved n the fllwng calculatns. The gemetry f the example s shwn n Fgure 8. It s a seven-bed mdel wth three center beds nvaded wth an nvasn radus f m [Meyer 1993]. Fgure 9 shws the results btaned by ur methd and NMM. The cmparsn shws the accuracy f ur methd at medum frequency. Fnally we want t address the advantages and dsadvantages f usng a ne-dmensnal Green's functn. The dsadvantage s that we cannt use FFT alng the z drectn. Hwever ths dffculty s greatly allevated by ntrducng a wndw n the z drectn. The wndw mves wth the tl. The layers utsde the wndw wll be treated as a ne-dmensnal medum. The cntrbutns frm these layers have already been ncluded n the ne-dmensnal Green's functn. Cnsequently the number f unknwns wll nt ncrease even fr a large-sze prblem. Fr the layers nsde the wndw n dscretzatn s needed f they d nt have cylndrcal bundares. If a brehle exsts the brehle part wll be treated the same way as ther twdmensnal anmales. Hence the number f the un- knwns fr the ntegral equatn wll ncrease. Ths nly slws dwn the cde by a lttle. Hwever the number f unknwns s stll much smaller than thse requred usng a hmgeneus medum Green's functn. Hence the methd s effcent when a brehle s ncluded n the gemetry. The advantages f usng a ne-dmensnal Green's functn can be summa- rzed as (1) the number f unknwns s much smaller (2) a smaller number f FHT peratns and teratns s needed and (3) t s effcent fr a large-sze prblem. In general the NMM methd s mre effcenthan the vlume ntegral equatn methd when the frmatn cntans many thck layers r nly the feld alng the brehle axs s f nterest. Ths s usually the case fr the frward prblem. Hwever the vlume ntegral equatn apprach s preferable when the layers are thn r fnely dscretzed such as the case n the nverse prblem [Chew and Lu 1994]. Als the feld thrughuthe dman R s cmputed at the same tme wthut ntrducng extra cmputatnal burden. Ths full slutn s mprtant fr sme nversn schemes such as the dstrted Brn teratve methd (DBIM) [Chew and Lu 1994]. 5. Cnclusns We use the vlume ntegral equatn methd t slve fr the nduced current densty n the axsymmetrc nhmgeneus medum. The ntegral equatn s frmulated usng a ne-dmensnal Green's functn and slved usng the CG-FHT methd. By usng the ne-dmensnal Green's functn the number f unknwns s reduced sgnfcantly leadng t fewer fast Hankel transfrm evaluatns and teratns than f the hmgeneus medum Green's functn s used nstead. Several numercal results are used t demnstrate the effcency and accuracy f the new algrthm n ts applcatn t electrmagnetc subsurface sensng. Acknwledgment. The authrs wsh t acknwledge the cmments frm tw annymus referees. References Andersn B. and S. K. Chang Synthetc nductn lgs by fnte element methd Lg Anal Andersn W. L. Numercal ntegratn f related Hankel transfrm f rders 0 and I by adaptve dgtal flterng Gephyscs Andersn W. L. Fast Hankel transfrms usng related and lagged cnvlutns A CM Trans. Math. Sftware Bjarsk N. N. K-space frmulatn f electrmagnetc scatterng prblem Rep. AFAL- TR Ar Frce Avncs Lab. Wrght-Pattersn Ar Frce Base Oh March Chen S. W. C. Chew and W. D. Kennedy Inversn f 6FF40 nductn tl measurement usng the dstrted Brn teratve methd Prc. IEEE/APS Chew W. C. Waves and Felds n Inhmgeneus Meda IEEE Press N.J Chew W. C. and Q. H. Lu Inversn f nductn tl measurements usng the dstrted Brn teratve methd and CG-FFHT IEEE Trans. Gesc. Remte Sens. GE Chew W. C. S. Barne B. Andersn and C. Hennessy Dffractn f axsymmetrc waves n a brehle by bed bundary dscntnutes Gephyscs Chew W. C. Z. Ne Q. H. Lu and B. Andersn An effcent slutn fr the respnse f electrcal well lggng tls n a cmplex envrnment IEEE Trans. Gesc. Remte Sens. GE Cpe D. L. C. Shen and S.C. Huang The thery f 2 MHz resstvty tl and ts applcatn t measurement-whle-drllng Lg Anal Gl F. H. and J.P. Martnez Analyss f delectrc res-

9 CHEN ET AL.: SOLVING THE VOLUME INTEGRAL EQUATION 1347 natrs wth tunng screw and supprtng structure IEEE Trans. Mcrwave Thery Tech. MTT Jhansen H. K. and K. Srensen Fast Hankel transfrm Gephys. Prspe- ct Krn G. A. and T. M. Krn Mathematcal Handbk fr Scentsts and Engneerng McGraw-Hll New Yrk Lu Q. H. Electrmagnetc feld generated by an ff-axs surce n a cylndrcally layered medum wth an arbtrary number f hrzntal dscntnutes Gephyscs Lu Q. H. and W. C. Chew A CG-FFHT methd fr the scatterng slutn f axsymmetrc nhmgeneus meda Mcrwave Opt. Tech. Left Lu Q. H. and W. C. Chew Applcatns f the cnjugate gradent fast Furer Hankel transfrm methd wth an mprved fast Hankel transfrm algrthm Rad Sc Meyer W. H. 2-MHz prpagatn resstvty mdelng n nvaded thn beds Lg Anal Mran J. H. and K. S. Kunz Basc thery f nductn lggng and applcatn t study f tw-cl sndes Gephyscs Peters T. J. and J. L. Vlaks Applcatn f a cnjugate gradent FFT methd t scatterng frm thn planar materal plates IEEE Trans. Antennas Prpag. AP Saœaa-Jaz A. and G. L. Yp Scatterng frm an arbtrarly lcated'ff-axs nhmgenety n a step-ndex ptcal fber IEEE Trans. Mcrwave Thery Tech. MTT Shen L. C. Thery f a cl-type resstvty sensr fr MWD applcatn Lg Anal Shen L. C. and G. J. Zhang Electrmagnetc feld due t a magnetc dple n a medum cntanng bth planar and cylndrcal bundares IEEE Trans. Gesc. Remte Sens. GE Zhu Q. and D. J. Hllker MWD resstvty tl respnse n a layered medum Gephyscs S. Y. Chen and W. C. Chew Department f Electrcal and Cmputer Engneerng Unversty f Illns 1406 west Green Street Urbana IL (e-mal: chen7@uuc.edu w-chew@uuc.edu) W. D. Kennedy Mbl Explratn and Prducng Techncal Center Mdway Rad Dallas TX (e-mal: davd_kennedy@emal.mbl.cm) (Receved March ; revsed July ; accepted July )