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HT24 A SOURCE-DETECTOR METHODOLOGY FOR THE CONSTRUCTION AND SOLUTION OF THE ONE-DIMENSIONAL INVERSE TRANSPORT EQUATION

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Ivere Proble geerg: Theory d Prcce 3 rd Ierol Coferece o Ivere Proble geerg Jue 3-8, 999, Por udlow, A, USA HT4 A SOUC-DTCTO MTHODOOGY FO TH COSTUCTIO AD SOUTIO OF TH O-DIMSIOA IVS TASPOT QUATIO Adr T.
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Ivere Proble geerg: Theory d Prcce 3 rd Ierol Coferece o Ivere Proble geerg Jue 3-8, 999, Por udlow, A, USA HT4 A SOUC-DTCTO MTHODOOGY FO TH COSTUCTIO AD SOUTIO OF TH O-DIMSIOA IVS TASPOT QUATIO Adr T. Ku, Aôo J. Slv eo,3 d lo C. obery 3 Boedcl g. Progr, COPP/UFJ, CP 685, CP , o de Jero, J, Brzl, Iuo Polécco, UJ, CP 978, CP 86-97, ov Frburgo, J, Brzl, 3 ucler g. Progr, COPP/UFJ, CP 6859, CP , o de Jero, J, Brzl, Keyword: Bolz equo, Prcpg ed, Source-deecor ehodology, dve he rfer, Merl propere d ource eo. ABSTACT A ource-deecor ehodology preeed for he coruco of vere rpor equo h oce olved provde ee for rdve propere d/or erlly drbued ource prcpg ed. Fro he proper cobo of ource d deecor pr, ye of o-ler equo ebled, g lo codero experel d o he ex rdo fro he edu. Te ce reul re lo preeed. ITODUCTIO The vere ly of rdve rfer prcpg ed h everl relev pplco egeerg, edce, geophyc, rophyc d oher reerch re. Uov (978) eed he exco coeffce d erool prcle cocero ophere. Brov e l. (986) propoed ehodology for he deero of cocero of oo prcle fle. McCorc (979), McCorc d Sche (98), Ho d Öz (988), Schez e l. (99), Subr d Megüç (99), colu e l. (994), d Slv eo d Öz (993, 995), u o e few, olved vere proble for gle cerg lbedo, opcl hce d/or oropc cerg phe fuco eo. Y e l. (99) eed he loco d regh of boluece rdo ource. Fuhy e l. (99) eed he borpo d cerg coeffce d he yery fcor of cerg lvg pl leve. Dffere ype of rdo uch euro, g-ry d phoo hve bee ued for obec defco dury (o-derucve eg), d edce (dgo d herpy). I y of he echque ued, cerg egleced, yeldg relvely ple recoruco proble. Th he ce Copuerzed Toogrphy d Sgle Phoo o Copuerzed Toogrphy (SPCT). he cerg h o be e o ccou (McCorc, 993, Megüç d Du, 994, obery d Olver, 995), uch er Ifrred Opcl Toogrphy (IOT), he recoruco odel becoe uch ore coplex, o-ler, eve requrg he copuo of he rdo feld. Th prculr oogrphc proble plced he e coex rdve he rfer prcpg ed d euro rpor ucler recor, beg he reled phycl pheoe (borpo, eo d cerg) odeled by he lerzed Bolz equo. Slv eo d obery (998, 998) hve bee worg o ource-deecor ehodology for he eo of rdve propere d erlly drbued ource prcpg ed. I h wor he ehodology preeed, well e ce reul for exco d cerg coeffce eo oe-deol hoogeeou ed. MATHMATICA FOMUATIO OF TH DICT POBM A ple-prllel, gry, oropclly cerg lb of hce, wh rpre boudre ubeced o exerl colled rdo ource h y be pooed dffere loco roud he edu, how Fg.. The hecl forulo of h edy-e oedeol rdve rfer proble gve by:, φ ( x, µ ),, µ + σ(x) φ (x, µ ) σ(x, µ, µ ) φ (x, µ )dµ S(x, µ ) x , µ () φ ( b, µ ) Aδ f b, ( µ ) (b) b Copyrgh 999 by ASM where φ, (x, µ ) he rdo ey, x he pl coorde, µ he dreco coe of he rdo be wh he pove x x, σ (x) he ol exco coeffce (borpo + cerg), σ (x, µ, µ ) he cerg coeffce, S(x,µ) erlly drbued ource, A he plude of he regh of he exerl colled rdo b, ource d f ( µ ) repree depedece wh he polr gle. The dce d b repree, repecvely, he urfce whch he ource loced ( or ) d he urfce for whch he boudry codo beg wre (b or b ). The dex repree he loco of he ource,,,,i. Fgure Poble loco for he exerl colled rdo ource. he he geoery, boudry codo, erl propere d he regh of he ource re ow, he rdo ey drbuo, φ, (x, µ ) c be clculed. Proble () he clled he drec proble. O he oher hd, whe y of h foro, or cobo of he, uow, bu experel euree of he red d/or refleced ex rdo re vlble, eo of he uow y be poble. Th ow he vere proble. MATHMATICA FOMUATIO OF TH IVS POBM The vere proble codered here volve he eo of he ource S(x, µ) d exco d cerg coeffce, ug he ource deecor ehodology. Fro he drec proble, gve by q. (), clled here ource proble, do proble coruced, d clled deecor proble. Covolvg he ource proble wh he do fuco, h co of he oluo of he deecor proble, dog egro by pr, d brgg he deecor proble elf o he reulg equo, ye of o-ler equo obed. Th ye clled here he vere rpor equo (IT). Fro he oluo of he IT, g o ccou he experel d o he ex rdo, he uow que re eed. Thee ep wll ow be decrbed ore del. The Source Proble. Th proble co o he drec proble whoe forulo gve by q.(), ug ee for he uow que. Thee ee re obed log he erve procedure doped for he oluo of he vere proble. The Deecor Proble. For ech loco where deecor pooed, do proble foruled. Th forulo obed fro he ource proble by reverg he dreco of rdo rfer,.e., by replcg µ by -µ, φ µ, * (x, µ ),, + σ (x) φ* (x, µ ) σ (x, µ, µ ) φ*, φ * (b, µ ) δ g b, b ( µ ) (x, µ )dµ q *(x, µ ) x , µ () (b) where σ, σ d q* re referece fuco for he uow σ, σ d S repecvely. I dury he referece vlue would be hoe of he erl uder vego perfec ufcurg codo. Ay oly o he erl propere or geoery could he be eed ug he vere echque here decrbed. I edce he referece vlue would be hoe of helhy org or ue. e expec he h he vere proble oluo wll how y devo fro hee referece vlue. The dex repree he urfce whch he eor loced d,,, I, repree he poo of he deecor roud h urfce. By pog he cocdece of he loco of he deecor,,,, I, wh hoe for he ource,,,, I, b, fuco g ( µ ) repree he euree h would be obed by he deecor for he regh of he ource loced h poo, g b, b, ( ) ηaf ( µ ), µ (3) where η he effcecy of he deecor. The Auxlry Proble. everg g he dreco of rdo rfer,.e., replcg µ by -µ, d defg uxlry fuofco * φ (x, µ ) φ (x, µ ) (4) we ob fro he deecor proble he followg uxlry proble φ µ φ,, (,, (x, µ ) σ (x, µ, µ ) φ x, µ ) + σ (x) φ ( b, µ ) δ ηf b, b ( µ ) (x, µ )dµ q (x, µ ) x , µ (5) (5b) Copyrgh 999 by ASM The forulo of he uxlry proble he e h for he ource proble, wh he excepo h he forer ue he referece vlue for he uow que whle he ler ue ee obed log he oluo of he vere proble. The Ivere Trpor quo. I he fr ep o he Ivere Trpor quo (IT) coruco we ulply q. () by he do fuco φ *, (x, µ ), d egre over he pl d gulr do, x[,], µ[-,], repecvely. φ µ, (x, µ ) φ*, (x, µ ) + σ (x) φ (x, µ ) φ*, (x, µ ) dxdµ, φ* (x, µ ) σ (x, µ, µ ) φ (x, µ )dµ dxd µ, [ (x, µ ) φ* (x, µ )] S dxdµ (6) I fc q.(6) repree ye of M(I o +I )x(i o +I ) oler equo, g o ccou ll poble cobo of ource d deecor loco. Iegrg q.(6) by pr, d pluggg eq.() o he reulg equo, we ge [ σ ( x, µ ) + σ (x)] φ (x, µ ) φ*, (x, µ )dxdµ [ σ x, µ, µ ) σ (x, µ, µ )] (, φ (x, µ ) φ* (x, µ )dµ dxd µ, [ S(x, µ ) φ* (x, µ ) q*(x, µ ) φ (x, µ )], [ * (x, µ ) φ (x, µ )] dµ µ φ dxdµ (7) Defg he que σ (x) σ (x) σ (x) σ (x, µ. µ ) σ(x, µ, µ ) σ (x, µ, µ ) (8) (8b) d ug he defo of he uxlry fuco gve by q. (4), q. (7) wre σ (x) φ (x, µ ) φ, (x, µ )dxdµ, σ ( x, µ, µ ) φ (x, µ ) φ (x, µ )dµ dxd µ, [ S(x, µ ) φ (x, µ ) q (x, µ ) φ (x, µ )],, [ (x, µ ) φ (x, µ )] dµ µ φ dxdµ (9) The ye of o-ler equo repreeed by q. (9) here clled he Ivere Trpor quo (IT). Solvg he IT, ee for he uow que re obed. A couple of coe re order. A eoed before, he ource proble d he uxlry proble re very lr. Therefore, he e ehod c be ued for he oluo of boh proble. For he copuol pleeo, h e h he e copuer progr (cully ubroue) c be ued for her oluo. Th why we wre he IT er of he uxlry fuco, ed of ug he do fuco. O he rgh de of q. (9) he experel d coe o plce. Tble urze how he boudry codo d rdo ex euree re e o ccou. b o u d r y Tble - do ey φ(x,µ) d uxlry fuco φ o (x,µ) he boudre x x φ o (x,µ) brce µ gve Af o,o (µ) [ηaf o,o (µ) ] µ eured [clculed] ource loco x x µ gve [] µ eured [clculed] µ gve [] µ eured [clculed] µ eured [clculed] µ gve Af, (µ) [ηa f, (µ)] TH SOUTIO OF TH IVS TASPOT QUATIO To eep h preeo ple poble, we wll coder he vere proble of eg oly he exco d he cerg coeffce. e fr e expo of he uow que gve by q. (8-b), T { σ } ψ(x) σ ψ (x) σ ( x) J T { σ } ψ(x)p ( µ, µ l σ ( x, µ, µ ) l l l + ) J l ( µ l l l () + σ ψ x)p ( µ, ) (b) where ψ(x) he be ued for he pl depedece repreeo, whle for he gulr depedece we ued expo egedre polyol. The dce d l repree he pl d gulr dcrezo, repecvely. The oluo of he IT, becoe ow he proble of d{ }. deerg he coeffce { } σ σ l 3 Copyrgh 999 by ASM Pluggg q. (-b) o q. (9), we ob for he proble wh o erl ource, σ ψ (x) φ (x, µ ) φ o, (x, µ )dµ dx { J, 4 ( l + ) σl ψ (x) φ (x, µ ) l Pl ( µ, µ ) φ (x, µ )dµ d}µ,, [ (x, µ ) φ (x, µ )] dµ µ φ () Sye () h (J+)x uow d M(I +I )x(i + I ) equo. For I I I I K, we ge M4K equo. If we w he ye o be overdeered,.e. M , we c expec o recover { σ } d { σ l } o ore h M/ 4 K /(J+)x elee ech,.e. 4K /J+. For oropc cerg, J, we ge K. Coderg ψ (x), x x x +,,, 3,, -, wh x o d x, d oropc cerg, q. () furher plfed, yeldg dx, σ (x + x ) φ (x +, µ ) φ (x +, µ ) dµ,, σ o ( x+ x) φ ( x+ x+ d d 4, µ ) φ (, µ ) µ µ,, µ [ φ (x, µ ) φ (x, µ )] dµ () Sye (), well ye(), c be wre ore copc for where h,,,..., M (3), { σ σ } T (4), l re coeffce obed egrg ech er o he lef de of q. () d h re obed by ug he proper cobo of he reul of he ource d uxlry proble d he experel d ccordg o Tble o he rgh hd de of he e equo. A ye (3) y be ll-codoed, wh poble proble of exece d uquee, co by le, or co by bloc, lgorh y be uble for oluo. The relvely ller covergece re of uch ehod whe copred wh oher ehod, copeed by robuee o he ree of ll-codoed proble. Ug he Breg dce wh he eropy fuco (Breg, 967) D(, ) l + (5) where vecor wh referece vlue for he uow, we wre he grg coderg oe le Breg proeco (, ) D(, ) λ (, h ) (6) egulrzo occur hrough he proeco o covex e (ech le o ye (3)). I he preece of oe qured redue or dded o q.(6) (lfvg, 989) yeldg Thoov le oler regulrzo fucol (gl e l., 996). h Thoov fucol ddol regulrzo cluded by ll pecrl rlo. rg he uler-grge equo, d olvg for (, ) exp( λ, ) (7) d pluggg o q (3), oe ob he grge ulpler λ fro he oluo of he reulg oler equo wh ewo ehod, λ + λ,, exp [ λ ] exp where he ero couer. Fro q. (7), + [ ] exp,, [ λ ] h, (8) + λ (9) [ ] exp λ, () Pluggg q.(8) o q.(9) d he ug q.(), we wre he co by le lgorh ow MAT (Mulplcve Algebrc ecoruco Techque) (Ceor d e, 98, e d obery, 99): for ul xu ero uber for,,,m (oe le e) for,,,, h + exp,, () 4 Copyrgh 999 by ASM where he ero couer. h he deero of he vecor of uow coeffce,, we ob ee for he uow gve by q. (8) d (). A he coeffce, q.(3) deped o he oluo of he ource proble, h deped o he eed vlue of he exco d cerg coeffce h we w o deere, we hve erve procedure. The Soluo Algorh. The erve procedure urzed follow:. chooe l gue o ;. olve he uxlry proble, q. (5-b), o ob φ, (x, µ ) ; 3. olve he ource proble, q. (-b), o ob φ, (x, µ ) ; 4. eble he IT ug q. () d (3) well Tble; 5. olve ye (3) ug, for exple, he co by le lgorh gve by q. (); 6. clcule σ (x) d σ (x,µ,µ) ug q. (8); 7. ere f oppg crer eblhed pror fed, e.g. + - ε. Oherwe, go bc o ep 3. SUTS To deore he febly of he oluo of he vere proble wh he ource-deeco pproch decrbed he prevou eco, we wll pree oe prelry reul obed for hoogeeou, gry (wh o pecrl depedece), oropc cerg edu. Furherore, he ey of he erlly drbued ource codered eglgble copro o he ey of he rdo cog fro he exerl ource. I Fg. re preeed he reul fro everl ulo coderg hypohecl experel d whou euree error. The crcle repree he exc vlue of he uow propere, (σ ; σ ) (.;.5); (.6;.8); (.7;.8); (.;.9); (.8;.) d (.;.) d he croe repree he l guee eployed. Thee were de equl o he referece vlue, (σ ; σ ) ( σ ; σ ) (.;.5); (.4;.6); (.6;.8); (.7;.8); (.8;.) d (.;.). The u re c - for boh σ d σ. Covergece o he exc vlue ll ulo how Fg. were obed wh o ore h ffee ero. All copuol pleeo h bee doe wh MATAB, d ech ulo h e pproxely fory ue of CPU e o IBM copble perol copuer wh MHz Peu proceor. A σ σ,ll egful e ce hve o be e or bove he dhed le Fg.. Fgure Sulo reul for e ce whou euree error. o exc vlue + l vlue referece vlue uercl expere hve how h covergece o he exc vlue cheved for y vlue of he l guee, excep whe hey re relvely fr d greer h he exc vlue. A rel pplco oe doe o ow wheher he referece vlue re fr fro he exc vlue or o, refee w roduced he lgorh preeed he prevou eco. A oo covergece cheved, y ero *, he referece propere re replced by he eed propere (σ ; σ ) (σ * ; σ * ), d he erve procedure rered fro he begg. If he eed 5 Copyrgh 999 by ASM vlue he prevou ero cycle (σ * ; σ * ) re deed oluo of he vere proble, he ew cycle of ero op he very fr ero, oherwe wll go o ul covergece g cheved. The replcee of he referece vlue doe oce ore d ew cycle of ero ed. Th procedure repeed y e ecery ul o ore vro oberved o he eed propere. Ju o gve exple of h uo, Tble re preeed he ee for he cerg d exco coeffce ech ero for ce wh (σ ; σ ) (.;.5) d (σ ; σ ) (.6;.8). Afer repeg he erve procedure u oe e covergece o he exc vlue h bee cheved. A ofe he ce, le experel d vlble h he uber of uow o be eed, herefore uquee of he oluo c o be ured. I uch uo oe u ry dffere l guee f ore h oe oluo o be foud. e hve o eep d, hough, h our obecve o fd devo fro he referece vlue (flw erl or ole bologcl ue), d h ee y oluo h deve fro he referece y be good eough. I Fg. 3 re preeed he reul for e ce wh he exc vlue (σ ; σ ) (.5;.9) wh he l guee d referece vlue (σ ; σ ) (σ ; σ ) (.;.9) coderg uled experel d wh %, 5% d % error wh repec o he xu eured rdo ey. For ech ru how Fg.3, dffere e of uled experel d w ued. ch e of experel d w obed by ddg rdo error o he exc clculed vlue of he ex rdo. Tble ed vlue for σ d σ ech ero. σ.6 σ.8 σ.39 σ.4468 σ. σ.5 ero # σ (c ) σ (c ) Fgure 3 Sulo reul for e ce wh euree error. exc vlue * * % error o----o 5% error + + % error 6 Copyrgh 999 by ASM I ll ulo preeed here we hve codered poble ource d deecor loco, beg 5 o ech de of he lb. For he oluo of he drec proble we hve ued he dcree orde ehod. COCUSIOS AD FUTU OK The reul obed o fr re very ecourgg. The ehod ee o perfor well regrdg boh ccurcy d copuol perforce (reoble eory d CPU e requree). A he oe we re worg owrd he ulo delg wh erlly drbued ource, heerogeeou d oropclly well oropclly cerg ed. I uch ce he copuo of he coeffce of he ler ye gve by q.(3) becoe ore volved, d he uber of uow o be eed cle up qucly. e re lo rg he odelg of he proble for wodeol rego. Alhough he vere proble hgher deo (-D d 3-D) becoe ore volved copuolly, we expec he ehod preeed here o perfor eve beer, becue ore experel d y be e o ccou. Mhecl d uercl ly of everl pec uch exece, uquee d bly u be perfored order o ge he full poel of he ehod. FCS Brov, F. G., verev, G.., Khpov,. S., d Shhudov,. G. (986), Soluo of Ivere Proble of dve Trpor by Soo Prcle of Coplex Shpe, J., gg. Phy., Vol.49, o.6, pp Breg,. M. (967), The elxo Mehod of Fdg he Coo Po of Covex Se d Applco o he Soluo of Proble Covex Progrg, USS Copuol d Mhecl Phyc Jourl, Vol.7, pp.-7. Ceor, Y. d e, A. (98), A Ierve ow-aco Mehod for Iervl Covex Progrg, Jourl of Opzo Theory d Applco, Vol. 34, o.3, pp lfvg, T. (989), A lgorh for Mxu ropy Ige ecoruco fro oy D, Mh. Copu. Modellg, Vol., o.6, pp gl, H.., He, M. d eubuer, A. (996), egulrzo of Ivere Proble, Kluwer Acdec Publher, eherld. Fuhy,., Fuhy-Krov,. d eowy, A. M. (99), o of Opcl Preer vg Tue by Solvg he Ivere Proble of he Mulflux dve Trfer, Appl. Opc, Vol.3, o., pp Ho, C-H d Öz, M.. (988), Ivere do Proble Ihoogeeou Med, J. Qu. Specroc. d. Trfer, Vol.4, o.5, pp McCorc,. J. (979), Trpor Scerg Coeffce fro efleco d Tro Meuree, J. Mh. Phy., Vol., o.7, pp McCorc,. J. (993), Ivere Phoo Trpor Mehod for Boedcl Applco, Proc. Ivere Proble geerg Coferece: Theory d Prcce, Flord, USA, pp McCorc,. J. d Schez,. (98), Ivere Proble Trpor Clculo for Aoropc Scerg Coeffce, J. Mh. Phy., Vol., o., pp Megüç, M. P. d Du, P. (994), Scerg Toogrphy d I Applco o Soog Dffuo Fle, Jourl of He Trfer, Vol.6, pp colu, V. P., yud, M. d Scdur, J. F. (994), Specrl dve Propere Idefco of Fber Iulg Merl, I. J. He M Trfer, Vol.37, Suppl., pp e, M.. d obery,. C. (99), Mxu ropy Algorh for Ige ecoruco fro Proeco, Ivere Proble, Vol.8, pp obery,. C. d Olver, C... (995), Trpor Model for Toogrphy: Theorecl d Prccl Codero, Proc. h Brzl Meeg o ecor Phyc d Therl Hydrulc, Águ de dó, Brzl, pp Schez,., McCorc,. J. d Y, H. C. (99), Ierve Ivere dve Trfer Mehod o e Opcl Thce d Surfce Albedo, Trp. Theory S. Phy., Vol.9, pp Slv eo, A. J. d Öz, M.. (993), A Ivere Proble of g Therl Coducvy, Opcl Thce, d Sgle Scerg Albedo of Se-Trpre Medu, Proc. Ivere Proble geerg Coferece: Theory d Prcce, Flord, USA, pp Slv eo, A. J. d Öz, M.. (995), A Ivere Proble of Suleou o of do Phe Fuco, Albedo d Opcl Thce, J. Qu. Specroc. d. Trfer, Vol.53, o. 4, pp Slv eo, A. J. d obery,. C.(998), The Source- Deecor Approch o he Soluo of Ivere Proble Prcpg Med, Proc. 5 h - Brzl Cogre Mechcl geerg, Forlez, Brzl, Vol. III, pp ( poruguee). Slv eo, A. J. d obery,. C. (998), Scerg Toogrphy Modellg Ivere Proble, Proc. 4 h Brzl ol Foru o Scece d Techology Helh, Curb, Brzl, pp ( poruguee). Subr, S. d Megüç, M. P. (99), Soluo of he Ivere do Proble for Ihoogeeou d Aoropclly Scerg Med Ug Moe Crlo Techque, I. J. He M Trfer, Vol.34, o, pp Uov,. A. (978), The Ivere Proble of Mulple- Scerg Theory d he Ierpreo of Meuree of Scered do he Cloud yer of Veu, Coc. e., Vol.5, o.5, pp Y, H. C., Schez,. d McCorc,. J. (99), Boluecece o fro Oce Su Irrdce, Appl. Opc, Vol.3, o.6, pp Copyrgh 999 by ASM
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