2020-04-10T12:58:39Z
Acceptance in OA@INAF
þÿMonte Carlo evaluation of a CZT 3D spectrometer suitable for a Hard X- and soft-³
rays polarimetry balloon borne experiment
Title
CAROLI, EZIO; DE CESARE, GIOVANNI; Curado da Silva, R. M.; Abbene, L.;
AURICCHIO, NATALIA; et al.
Authors
10.1109/NSSMIC.2015.7582278
DOI
http://hdl.handle.net/20.500.12386/23984
Handle
PROCEEDINGS..IEEE NUCLEAR SCIENCE SYMPOSIUM AND MEDICAL
IMAGING CONFERENCE
0RQWH&DUORHYDOXDWLRQRID&=7'VSHFWURPHWHU
VXLWDEOHIRUD+DUG;DQGVRIWȖUD\VSRODULPHWU\
EDOORRQERUQHH[SHULPHQW
(&DUROL*'H&HVDUH50&XUDGRGD6LOYD/$EEHQH1$XULFFKLR&%XGW]-¡UJHQVHQ6'HO6RUGR
3)HUUDQGR-/*DOYq]0+HUQDQ]-,VHUQ,.XYYHWOL3/DXUHQW2/LPRXVLQ
-00DLD00RLWD13URGXLW-%6WHSKHQ$=DSSHWWLQL
Abstract– 7RGD\, WKHPHDVXUHPHQWRIWKHSRODUL]DWLRQVWDWXVRIFRVPLF VRXUFHV KLJKHQHUJ\ HPLVVLRQ LV UHFRJQL]HG DV D NH\ REVHUYDWLRQDO SDUDPHWHU WR XQGHUVWDQG WKH DFWLYH SURGXFWLRQ PHFKDQLVP DQG LWV JHRPHWU\ 7KHUHIRUH D PDQGDWRU\ UHTXLUHPHQW IRU QHZ LQVWUXPHQWDWLRQV RSHUDWLQJ LQ WKLV HQHUJ\ UDQJH ZLOO EH WR SURYLGH KLJK VHQVLWLYLW\ IRU SRODULPHWULF PHDVXUHPHQWV,QWKLVIUDPHZRUNZHKDYHSUHVHQWHGWKHFRQFHSW RIDVPDOOKLJKSHUIRUPDQFHLPDJLQJVSHFWURPHWHURSWLPL]HGIRU SRODULPHWU\EHWZHHQDQGNH9VXLWDEOHIRUDVWUDWRVSKHULF EDOORRQERUQH SD\ORDG DQG DV D SDWKILQGHU IRU D IXWXUH VDWHOOLWH PLVVLRQ7KHGHWHFWRUZLWK'VSDWLDOUHVROXWLRQLVEDVHGRQD&=7 VSHFWURPHWHU LQ D KLJKO\ VHJPHQWHG FRQILJXUDWLRQ GHVLJQHG WR RSHUDWH VLPXOWDQHRXVO\ DV D KLJK SHUIRUPDQFH VFDWWHULQJ SRODULPHWHU +HUHLQ ZH UHSRUW UHVXOWV RI D 0RQWH &DUOR VWXG\ GHYRWHG WR RSWLPL]H WKH FRQILJXUDWLRQ RI WKH GHWHFWRU IRU SRODULPHWU\ZLWKSDUWLFXODUIRFXVRQHYHQWVHOHFWLRQILOWHUDEOHWR LQFUHDVHWKHSRODULPHWULFSHUIRUPDQFH7KLVSUHOLPLQDU\DQDO\VLV VKRZVWKDWDSURFHGXUHWRRSWLPL]HWKHSRODUL]DWLRQUHVSRQVHRID ' VSHFWURPHWHUV VKRXOG ILUVW RI DOO GHWHUPLQH WKH EHVW WUDGHRII EHWZHHQ WKH VWDWLVWLFDO VLJQLILFDQFH DQG WKH TXDOLW\ RI WKH PRGXODWLRQIDFWRU
, ,1752'8&7,21
7+(PHDVXUHPHQWVRIWKHSRODUL]DWLRQVWDWXVRIKDUG;DQG
VRIW Ȗ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
0DQXVFULSWUHFHLYHG...
( &DUROL * 'H &HVDUH 1 $XULFFKLR DQG - % 6WHSKHQ DUH ZLWK ,1$),$6)%RORJQD 9LD *REHWWL , %RORJQD ,WDO\ SKRQH HPDLOFDUROL#LDVIERLQDILW
50 &XUDGRGD 6LOYD -00DLDDQG0 0RLWDDUHZLWK/,3 'HSGH )tVLFD 8QLYHUVLGDGH GH &RLPEUD 3 &RLPEUD 3RUWXJDO SKRQH HPDLOUXLVLOYD#FRLPEUDOLSSW
/$EEHQHLVZLWKWKH'L)&8QLYHUVLW\RI3DOHUPR9LD$UFKLUDILQ, 3DOHUPR ,WDO\ WHOHSKRQH HPDLO OHRQDUGRDEEHQH#XQLSDLW &%XGW]-¡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
,, 6&$77(5,1*32/$5,0(75<:,7+3,;(/'(7(&7256 ,Q WKH KDUG ; DQG VRIW ȖUD\V UHJLPH SRODULPHWULF PHDVXUHPHQWV DUH SRVVLEOH XVLQJ &RPSWRQ VFDWWHUHG HYHQWV 7KH .OHLQ1LVKLQD .1 HTXDWLRQ FURVVVHFWLRQ H[KLELWV DQ D]LPXWKDO GHSHQGHQF\ LQ WKH VFDWWHULQJ GLUHFWLRQ IRU OLQHDUO\ SRODUL]HGSKRWRQV>@IRUSRODUL]HGSKRWRQVWKHSUREDELOLW\
6'HO6RUGRLVZLWK,1$),$6)3DOHUPR9LD8JR/D0DOID, 3DOHUPR,WDO\SKRQHHPDLOGHOVRUGR#LIFLQDILW
3 )HUUDQGR 3 /DXUHQW DQG 2 /LPRXVLQ DUH ZLWK &($ 6DFOD\ '60,UIX6HUYLFH G¶$VWURSK\VLTXH /RUPH GHV PHULVLHUV ) *LIVXU <YHWWH &HGH[ )UDQFH SKRQH HPDLO ROLYLHUOLPRXVLQ#FHDIU
-/*DOYq]0+HUQDQ]DQG-,VHUQDUHZLWK,&(&6,&,((&&DPSXV 8$% ( %HOODWHUUD 6SDLQ SKRQH HPDLO +HUQDQ]#LHHFXDEHV
1 3URGXLW LV ZLWK ,6'& 'DWD &HQWUH IRU $VWURSK\VLFV 8QLYHUVLW\ RI *HQHYD &+ 9HUVRL[ 6ZLW]HUODQG SKRQH HPDLO 1LFRODV3URGXLW#XQLJHFK $=DSSHWWLQLLVZLWK&15,0(03DUFR$UHDGHOOH6FLHQ]H$, 3DUPD,WDO\SKRQHHPDLO]DSS#LPHPFQULW
7
)LJ 7KH .6 VFDWWHULQJ SUREDELOLW\ GLDJUDP IRU SRODUL]HG SKRWRQVYVWKHD]LPXWKDODQJOHMIRU qVFDWWHULQJDQJOH7KHYHFWRU(JLYHV WKH GLUHFWLRQ RI WKH SKRWRQV SRODUL]DWLRQSODQHGLVWULEXWLRQ RI WKH VFDWWHULQJ DQJOH ZLOO SUHVHQW D PD[LPXP DORQJWKHGLUHFWLRQRUWKRJRQDOWRWKHSRODULVDWLRQSODQHDQGD PLQLPXPLQWKHSDUDOOHORQH)LJ
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
)RU D SL[HO GHWHFWRU D VLPSOH ILJXUH RI PHULW DV VFDWWHULQJ SRODULPHWHUWKH4IDFWRULVGHILQHGE\ x y x y N N Q N N ZKHUH Nx DQG Ny DUH WKH VXPV RI VFDWWHUHG HYHQWV LQ WZR
RUWKRJRQDOGLUHFWLRQV,QIDFW4LVDSHULRGLFDO SIXQFWLRQ RIWKHD]LPXWKDODQJOHM7KH4UHDFKHVDPD[LPXPLQDEVROXWH YDOXHZKHQRQHRIWKHWZRRUWKRJRQDOGLUHFWLRQVLVSDUDOOHOWR WKHSRODUL]HGSKRWRQHOHFWULFILHOGGLUHFWLRQ
)LJ 6FDWWHU PDSV RI D SRODUL]HG EHDP FROOLPDWHGLQWRRQH RIWKH FHQWHUSL[HOVZLWKWKHSRODUL]DWLRQGLUHFWLRQLQFOLQHGqZLWKUHVSHFWWRGHWHFWRU VLGHVOHIWPPWKLFN&=7GHWHFWRU32/&$ZLWKîSL[HOVRIPP SLWFKULJKW&DOLVWHPRGXOHHTXLSSHGZLWKPPWKLFN&=7SL[HOVSHFWURPHWHU îSL[HOVZLWKPPSLWFK ,QVXFKW\SHRIGHWHFWRUVWKHSRODUL]DWLRQVWDWXVRIWKHLQFRPLQJ IOX[ FDQ EH HYDOXDWHG E\ EXLOGLQJ WKH GLVWULEXWLRQ RI WKH WZR LQWHUDFWLRQSRVLWLRQVGHSRVLWLQJ(DQG(UHVSHFWLYHO\)LJ VKRZVWZRH[DPSOHVRIUHFRQVWUXFWHGVFDWWHULQJPDSVREWDLQHG ZLWKWZRGLIIHUHQW&=7SL[HOGHWHFWRUV
,,, '&=7&'7(63(&7520(7(56)256&$77(5,1*
32/$5,0(75<
7KH SHUIRUPDQFH RI D SRODULPHWHU EDVHG RQ D KLJKO\ VHJPHQWHG GHWHFWRU LV VWURQJO\ GHSHQGHQW RQ LWV JHRPHWULFDO FRQILJXUDWLRQ 7R DFKLHYH JRRG SRODULPHWULF SHUIRUPDQFH IRU KDUG ;VRIW Ȗ UD\V WKH GHWHFWRU VKRXOG IXOILO WKH IROORZLQJ UHTXLUHPHQWV
KLJK VFDWWHULQJ HIILFLHQF\ DFKLHYDEOH ZLWK LQFUHDVHG WKLFNQHVV
KLJKPRGXODWLRQIDFWRUDFKLHYDEOHZLWKGHFUHDVHGWKLFNQHVV DQGSL[HOVL]H
ILQH VSHFWURVFRS\ DOORZLQJ HIIHFWLYH EDFNJURXQG UHMHFWLRQ DQGJRRGHYHQWVHOHFWLRQ
$Q HIILFLHQW DQVZHU WR DOO WKLV UHTXLUHPHQWV LV RIIHUHG E\ VSHFWURLPDJHUV ZLWK WKUHH GLPHQVLRQ VSDWLDO UHVROXWLRQ ' VSHFWURPHWHUV ,Q SDUWLFXODU FRQFHUQLQJ &=7&G7H GHWHFWRU WHFKQRORJ\DWOHDVWWZRGLIIHUHQW VROXWLRQV DUH XQGHU VWXG\ %RWK DUH EDVHG RQ WKH 37) 3ODQDU 7UDQVYHUVH )LHOG LUUDGLDWLRQ FRQILJXUDWLRQ )LJ LQ ZKLFK
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
DE
)LJD3L[HOGHWHFWRUXVHGLQ37)7KHEDVLFXQLWLVDVLQJOHî PP &G7H FU\VWDO PP WKLFN ZLWK D SODQDU FDWKRGH DQG ZLWK WKH DQRGH VHJPHQWHGLQDîSL[HODUUD\ZLWKPPSLWFKE'ULIWVWULSGHWHFWRUXVHG LQ37)7KHEDVLFXQLWLVDVLQJOHîPP&=7FU\VWDOPPWKLFN
&=7&G7HVHQVRUVXQLWVUHJDUGOHVVRIWKHFRQILJXUDWLRQFDQ EH DVVHPEOHG RQ WKLQ LQVXODWLQJ VXSSRUWV WR EXLOG OLQHDU PRGXOHV7KHUHIRUHWKHOLQHDUPRGXOHVLWVHOIFRXOGEHVWDFNHG WRJHWKHUDVLQWKHVFKHPHVKRZQLQ)LJ.
:H SRLQW RXW WKDW VXFK W\SH RI ' GHWHFWRU PRGXOHV LV HTXLYDOHQWWRVWDFNRIWKLQQHUOD\HUVEXWZLWKRXWDQ\SDVVLYH PDWHULDOZLWKLQ 7KH ILQDO VL]H RI ' SL[HOV LH YR[HOV LQ WKHVH PRGXOHV GHSHQGV RQERWKWKHFRQILJXUDWLRQ DQG WKH VHJPHQWDWLRQ RI HOHFWURGHVWKH&G7H&=7 VHQVRUV XQLWV DQG FDQ UDQJH EHWZHHQ PP DQG KXQGUHGVRIPLFURPHWHUV VFDOHV )LJ 37) LUUDGLDWLRQ FRQILJXUDWLRQ WKH SKRWRQV DUH LPSLQJLQJ RQWR WKH GHWHFWRU RUWKRJRQDOO\WRWKHFROOHFWLQJILHOG GLUHFWLRQ )LJ$SRVVLEOHDVVHPEOLQJVFKHPHIRU D'VWDFNPRGXOH
,9 7+(&ȝ63%$//221%251(3$</2$'287/,1( ,QRUGHUWRUHDOL]HWKHGHWHFWRURI&ȝ63ZHSURSRVHWREXLOG DPRVDLFRIîRIWKHDERYH'VWDFNPRGXOHV(DFKPRGXOH ZLOOEHURWDWHGZLWKUHVSHFWWRWKHDGMDFHQWRQHVWRPLQLPL]H WKHV\VWHPDWLFHIIHFWRQWKHSRODUL]DWLRQPRGXODWLRQLQGXFHGE\ WKH YR[HO JHRPHWU\ HJ YR[HO GLPHQVLRQV DV\PPHWU\ GLDJRQDOHIIHFW>@7DEOH,UHSRUWVWKHPDLQFKDUDFWHULVWLFVRI SRVVLEOHFRQILJXUDWLRQIRUWKH&ȝ63GHWHFWRU 7KHGHWHFWRULVFRXSOHGZLWKDVLPSOHLHVDPHVL]HRIWKH GHWHFWRU&RGHG0DVNPDGHE\WKLQKLJK=RSDTXHHOHPHQWVWR SURYLGHLPDJLQJXSaNH97KHLPDJLQJDWcOHYHOZLOO EHXVHGWRFRQWUROWKHVRXUFHSRLQWLQJDQGWKHFDOLEUDWLRQRIWKH LQVWUXPHQWLWVHOI,QWKHRSSRVLWHWKHPDVNVKDOOEHFRPHDOPRVW FRPSOHWHO\ WUDQVSDUHQW DERYHa NH9 WR H[SRVH WKH IXOO VHQVLWLYHGHWHFWRUDUHDLQFUHDVLQJWKHSRODUL]DWLRQVHQVLWLYLW\
7KH PDLQ REMHFWLYH RI WKH &ȝ63 EDOORRQ SD\ORDGZLOOEH WR DVVHVVGHILQLWLYHO\WKHSHUIRUPDQFHRIKLJKO\VHJPHQWHGKLJK= ' VSHFWURPHWHUV DV VFDWWHULQJ SRODULPHWHUV E\ SHUIRUPLQJ D PHDVXUHPHQWRIWKH&UDEVRXUFHLQWKHUDQJHEHWZHHQ NH97KHVXFFHVVRIVXFKEDOORRQH[SHULPHQWZLOOFRQILUPWKH DGYDQWDJHVDQGWKHVXLWDELOLW\RIXVLQJWKLVW\SHRIGHWHFWRUIRU IXWXUHVSDFHPLVVLRQEDVHGRQQHZKLJKHQHUJ\IRFXVLQJV\VWHPV
7$%/(,&63'(7(&7250$,1&+$5$&7(5,67,&6$1'5(6285&(6
DDQGEUHIHUVWR37)SL[HODQG37)GULIWVWULSVHQVRUXQLWVUHVSHFWLYHO\ 6HQVLWLYHDUHD FP 'FRQILJXUDWLRQ îPRVDLFRIVWDFNPRGXOHV 6WDFNPRGXOH OLQHDUPRGXOHîFP /LQHDUPRGXOH &=7PPWKLFNFU\VWDOV &=7&G7HXQLW îFPPPWKLFN 9R[HOVL]H îîPPJHRPHWULF 9R[HOVQXPEHU aDE 3RZHU$)(( aDaE: $FWLYH:HLJKW aNJ ,QVWUXPHQWW\SH 6LPSOH&RGHG0DVN 0DVN'HWHFWRU6HSDUDWLRQ FP 6KLHOGLQJ 3DVVLYH 'HWHFWRU2SHUDWLYH5DQJH NH9 3RODULPHWU\5DQJH NH9
7KH PDLQ REMHFWLYH RI WKH &ȝ63 EDOORRQ SD\ORDGZLOOEH WR DVVHVVGHILQLWLYHO\WKHSHUIRUPDQFHRIKLJKO\VHJPHQWHGKLJK= ' VSHFWURPHWHUV DV VFDWWHULQJ SRODULPHWHUV E\ SHUIRUPLQJ D PHDVXUHPHQWRIWKH&UDEVRXUFHLQWKHUDQJHEHWZHHQ NH97KHVXFFHVVRIVXFKEDOORRQH[SHULPHQWZLOOFRQILUPWKH DGYDQWDJHVDQGWKHVXLWDELOLW\RIXVLQJWKLVW\SHRIGHWHFWRUIRU IXWXUHVSDFHPLVVLRQEDVHGRQQHZKLJKHQHUJ\IRFXVLQJV\VWHPV $PHDVXUHRIWKHSRODULPHWULFFDSDELOLW\RIDQLQVWUXPHQWLV JLYHQ E\ WKH 0LQLPXP 'HWHFWDEOH 3RODUL]DWLRQ 0'3 GHILQHGDV>@
F FA
S
B
MDP
A
S Q
T
H
H
%DVHG RQ WKH FKRVHQ GHWHFWRU FRQILJXUDWLRQ DQG RQ WKH 4 YDOXHV REWDLQHG ZLWK &=7 SL[HO SURWRW\SH LQ WKH GLIIHUHQW H[SHULPHQWV DW WKH (65) *UHQREOH ZH H[SHFW WRDFKLHYHZLWKWKH&X63LQVWUXPHQWD0'3OHYHOEHWZHHQDQG IRUDVREVHUYDWLRQVRYHUWKHUDQJHNH9>@
9 0217(&$5/2678'<2)7+(32/$5,0(75,&'
'(7(&7255(63216(
0RQWH &DUOR PRGHOOLQJ UHSUHVHQWV DQ HVVHQWLDO VWHS LQ WKH RSWLPLVDWLRQ RI SL[HO GHWHFWRU SRODULPHWULF UHVSRQVH DQG LQ GHILQLQJ WKHEHVWWUDGHRII EHWZHHQ SHUIRUPDQFHDQG GHWHFWRU GHVLJQ FRPSOH[LW\ 8VLQJ *($17 >@ ZH DUH SHUIRUPLQJ QXPHULFDO VLPXODWLRQV XVLQJ DVLPSOH PRGHO EDVHG RQD &=7 EORFNFPîFPîFPWKDWFDQEHVHJPHQWHGLQ'DW DQ\GHILQHGYR[HOVL]HE\D6:WRROGHYHORSHGLQ0$7/$% >@ 7KLV WRRO UHDG WKH HYHQW ILOHV JHQHUDWHG E\ *($17 VLPXODWLRQUXQVDQGDOORZWRSHUIRUPDQGLPSOHPHQWGLIIHUHQW DQDO\VLVE\DQHDV\SDUDPHWULVDWLRQRIGLIIHUHQWSDUDPHWHUV
7KHVLPXODWLRQVUHODWHGWRWKHSUHVHQWZRUNDUHEDVHGRQWKH IROORZLQJ FKDUDFWHULVWLFV IRU ERWK WKH GHWHFWRU DQG WKH LPSLQJLQJSKRWRQIOX[ 3DUDOOHODQGRQD[LVIOX[LQFLGHQFHDQJOH qLPSLQJLQJ XQLIRUPO\RQWKHRYHUDOOGHWHFWRUWRSVXUIDFHFP ,PSLQJLQJIOX[OLQHDUO\SRODUL]HG 3KRWRQVVSHFWUXPî(SKRWRQVNH9FPVLQWKH NH9UDQJH&UDEOLNHVSHFWUXPIRUDWRWDORISKRWRQV 6HJPHQWDWLRQRIWKHGHWHFWRUYROXPHPPîPPî PPYR[HOV
'HWHFWRU YROXPH HTXLYDOHQW WR D ' L M N DUUD\ RI îîHOHPHQWV
7KHVLPXODWHGQXPEHURISKRWRQVLVHTXLYDOHQWWRDVRXUFH REVHUYDWLRQWLPHRIaKRXUV7KLVWLPHLVIXOO\FRPSDWLEOH ZLWK WKHW\SLFDOOHQJWKRI DQ REVHUYDWLRQ SRLQWHGWRWKH &UDE VRXUFHIURPDPLGODWLWXGHEDOORRQERUQHH[SHULPHQWODXQFKHG LQWKHQRUWKHUQKHPLVSKHUH
7DEOH ,, UHSRUW WKH YDOXHV RI GRXEOH HYHQWV HIILFLHQF\ LQ HYHQO\ VSDFHG ORJDULWKPLF HQHUJ\ EDQG EHWZHHQ DQG NH9WRJHWKHUZLWKWKHPRGXODWLRQIDFWRU4REWDLQHGXVLQJ(T E\VXPPLQJWKHFRXQWVRYHUWKHFHQWUDOURZDQGWKHFHQWUDO FROXPQ RI WKH VFDWWHULQJ PDS 7KLV GUDIW HYDOXDWLRQ RI 4 LV SRVVLEOH EHFDXVH ZH NQRZ WKH RULHQWDWLRQ RI WKH SRODUL]DWLRQ YHFWRU RI WKH LPSLQJ SKRWRQV %RWK GRXEOH HYHQW HIILFLHQFLHV DQG WKH 4 YDOXHV DUH JLYHQ WKH ILUVW WLPH FRQVLGHULQJ DOO WKH GRXEOH HYHQWV Z=freeDQG LQ WKH VHFRQG FDVH FRXQWLQJ RQO\ GRXEOHHYHQWVLQZKLFKHDFKKLWVKDYHWKHVDPH=FRRUGLQDWH Z=cost7KLVVHOHFWLRQWHQGWRLQFUHDVHWKH4IDFWRUEHFDXVHLV HTXLYDOHQWWRVHOHFWVFDWWHULQJDQJOHLQDYHUDJH PRUHFORVHWR q 7$%/(,,'28%/((9(176()),&,(1&<$1' 5(/$7,9(02'8/$7,21)$&7254 '( NH9 'RXEOH
Z=free 4IUHH 'RXEOHZ=cost 4FRVW
,QJHQHUDOZLWKPPVHJPHQWDWLRQVFDOHLQ'WKHVHOHFWLRQ RIGRXEOHHYHQWVZLWKZ=costDOORZDVPDOOLPSURYHPHQWRIWKH 4IDFWRULQDOPRVWDOOHQHUJ\EDQGV+RZHYHUWKLVVHOHFWLRQLV
SDLGE\VWURQJORVVLQVWDWLVWLFVGXHWRWKHGHFUHDVHRIGRXEOH HYHQWV HIILFLHQF\ IURP WR PRUH WKDQ RI DOO GRXEOH HYHQWV ,Q IDFW WKH VPDOO LPSURYHPHQW LQ 4 GRHV QRW FRPSHQVDWH WKH UHGXFWLRQ RI WKH GRXEOH HYHQW GHWHFWLRQ HIILFLHQF\İEHFDXVHWKHGHWHFWLRQHIILFLHQF\DQGWKH4YDOXH ZHLJKWDV1/(Q×sqrt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x DQG Ny UHTXLUHG E\ (T 7KH UHG GDUN VTXDUH FRQFHQWULF
SHULPHWHUVH[HPSOLI\WKHDSSOLHGDGMDFHQF\ILOWHU )LJ ULJKW VKRZV WKH PHWKRG LPSOHPHQWHG LQ WKH 0$7/$%GDWDDQDO\VLVWRROWREXLOGWKHPRGXODWLRQFXUYH7KH WZRRUWKRJRQDODQJXODUVHFWRUqDSHUWXUHDUHURWDWHGE\D GHILQHGDQJXODUVWHSqLQWKHSUHVHQWZRUNWRFRYHUq 7KHUHIRUHIRUHDFKSKDVHDQJOHWKHPRGXODWLRQIDFWRU4ĮLV HYDOXDWHGE\XVLQJ(T7KHUHVXOWVRIWKLVSURFHGXUHDOOWKH GRXEOH HYHQWV UHFRUGHG LQ WKH HQHUJ\ EDQG NH9 LV SORWWHGLQ)LJIRUWKHWZRFDVHVGRXEOHHYHQWVZLWKZ=free DQGZ=costUHVSHFWLYHO\ 7KHVKDSHVRIERWKPRGXODWLRQFXUYHVLQ)LJH[KLELWDTXLWH HYLGHQWODUJHVFDOHGLVFUHWL]DWLRQZLWKWKHSKDVHDQJOH7KLVLV WKHHIIHFWRIWKHILUVWRUGHUDGMDFHQF\FRXQWVWKDWDUHGRPLQDQW LQWKHVWDWLVWLFV7KHVFDWWHULQJLQWKHILUVWRUGHUDGMDFHQF\SL[HO WHQG WR OLPLWV WKH SKDVH DQJOH VHQVLWLYLW\ WR q LH VXUURXQGLQJSL[HOVVWHSV
,QRUGHUWRRYHUFRPHWKLVHIIHFWZHKDYHEXLOWWKHPRGXODWLRQ FXUYH DSSO\LQJ D ILOWHU LQ WKH DGMDFHQF\ RUGHU RI VFDWWHUHG HYHQWV ,Q SDUWLFXODU DV D ILUVW WULDO ZH KDYH VHOHFWHG GRXEOH HYHQWVZKRVHGLVWDQFHEHWZHHQWKHKLWVLVFRPSULVHGEHWZHHQ DQGSL[HOV7KHUHVXOWLVSUHVHQWHGLQ)LJIRUWKHXVXDOFDVH RIZ=freeDQGZ=cost7KHLPSOHPHQWDWLRQRIWKLVHYHQWILOWHU SURGXFHVDPDUNHGLPSURYHPHQWRIERWKWKHPD[LPXP4DQG WKHVKDSHRIWKHPRGXODWLRQFXUYHV,QWKHFDVHRIZ=freeGRXEOH HYHQWVWKH PD[LPXP 4UHDFKDYDOXHRIZLWKUHVSHFWWR IRU WKH QR DGMDFHQF\ ILOWHU FDVH 7KH LPSURYHPHQW RI
PD[LPXP4LVHYHQJUHDWHUZKHQWKHILOWHULVDSSOLHGWRZ=cost GRXEOHHYHQWVUHDFKLQJIRUWKHQRDGMDFHQF\ILOWHU FDVH2IFRXUVHDVLWLVSRVVLEOHWRLQIHUIURPWKHVDPHILJXUH WKLVLPSURYHPHQWLVSDLGDOVRLQWKLVFDVHE\DGUDPDWLFORVVLQ WKHVWDWLVWLFDOVLJQLILFDQFHRIWKHFDOFXODWHGPRGXODWLRQYDOXHV ,QIDFWWKHVHUURUVEHFRPHTXLWHODUJHLQSDUWLFXODUIRUWKH
Z=cost GRXEOH HYHQWV VHOHFWLRQ GHFUHDVLQJ GUDVWLFDOO\ WKH
UHOLDELOLW\RIWKHUHVXOW7KLVLPSURYHPHQWLVPDLQO\GXHWRWKH IDFW WKDW H[FOXGLQJ WKH ILUVW RUGHU DGMDFHQF\ RQO\ D YHU\ ORZ SHUFHQWDJHRIGRXEOHHYHQWVLVOHIWVHH)LJOHIWLQSDUWLFXODU LQ WKH GLUHFWLRQ RUWKRJRQDO WR WKH SRODUL]DWLRQ SODQH 7KLV WHQGHQF\LVFRQILUPHGLIZHDSSO\DQDGMDFHQF\ILOWHUVHOHFWLQJ RQO\GRXEOHHYHQWVZKRVHKLWVDUHDWDGLVWDQFHEHWZHHQDQG SL[HOV7KHPD[LPXP4DJDLQLPSURYHIRUZ=costEXW WKH HUURUV EHFRPH VR ODUJHU WKDW WKH HVWLPDWHG YDOXHV KDYH DOPRVWQRVWDWLVWLFDOVLJQLILFDQFH $VH[SHFWHGWKHPRGXODWLRQFXUYHVH[KLELWDVPRRWKHUDQG PRUHFRQWLQXRXVVKDSHDVIXQFWLRQRIWKHSKDVHDQJOHWKHHIIHFW RIWKHVTXDUHGLVFUHWL]DWLRQRIWKHGHWHFWRULVOHVVHYLGHQW )LJ7KHPRGXODWLRQFXUYHZLWKQRDGMDFHQF\ILOWHU4IRUDOOWKHGRXEOH HYHQWVz=freeLVZKLOHIRUWKHz=costGRXEOHHYHQW4 7KLVYDOXHVDUHFRQVLVWHQWZLWKWKHRQHVUHSRUWHGLQ7DEOH,, )LJ7KHPRGXODWLRQFXUYHREWDLQHGXVLQJGRXEOHHYHQWVZLWKDGMDFHQF\ EHWZHHQDQGZLWKZ=costDQGZ=freeUHVSHFWLYHO\7KHHUURUVDUHDWV OHYHO %HFDXVHWKHVLJQLILFDQFHRIWKHUHVXOWVVWURQJO\GHSHQGVRI WKHHYHQWVVWDWLVWLFVZHKDYHDOVRLPSOHPHQWHGLQWKHDQDO\VLV WRRO WKH SRVVLELOLW\ WR HDVLO\ GHILQH WKH ZLGWK RI WKH DQJXODU DSHUWXUHRIWKHRUWKRJRQDOVHFWRUVXVHGWRHYDOXDWHWKHNxDQG
WKHNy FRXQWVUHTXLUHGIRUWKHHYDOXDWLRQRI4YVWKHSKDVHDQJOH
VHH )LJ ULJKW ,QFUHDVLQJ WKH DPSOLWXGH RI WKH DQJXODU DSHUWXUH RI WKHVH VHFWRUV ZH FDQ LQFUHDVH WKH VWDWLVWLFV RI FRXUVH ORRVLQJ DJDLQ LQ VRPH RWKHU FKDUDFWHULVWLFV OLNH WKH PD[LPXP4YDOXHVDQGWKHVHQVLWLYLW\WRSKDVHDQJOH,Q)LJ ZHSUHVHQWWKHPRGXODWLRQFXUYHVREWDLQHGZLWKWZRGLIIHUHQW DQJOH DSHUWXUH RI WKH LQWHJUDWLRQ VHFWRUV q DQG q UHVSHFWLYHO\ XVLQJ GRXEOH HYHQWV ZLWKLQ WR SL[HO DGMDFHQFLHV7KHGRXEOLQJRIWKHFRXQWLQWHJUDWLRQDUHDRYHUWKH VFDWWHULQJ PDWUL[ GRHV QRW KDYH DQ\ DSSDUHQW HIIHFW LQ WKH TXDOLW\RIWKH4IDFWRU,QIDFWWKHRQO\HIIHFWQRWUHSRUWHGLQ WKHILJXUHLVWKHLQFUHDVHRIWKHVWDWLVWLFDOVLJQLILFDQFHRIHDFK SRLQWLQWKHPRGXODWLRQFXUYHRIaξʹXVLQJqDSHUWXUHZLWK UHVSHFW WR q 7KH DSSDUHQW GLVFRQWLQXLW\ HDFK q LV UHODWHG ZLWKWKHVTXDUHVKDSHRISL[HOV:KHQSKDVHDQJOHSDVVWKURXJK WKH GLDJRQDO RI SL[HOGHWHFWRU WKH QXPEHU RI SL[HOV WKDW FRQWULEXWH WR FRXQWV FKDQJH VXGGHQO\ DQG WKLV UHIOHFWV LQ WKH YDOXHRIWKHPRGXODWLRQ
)LJ)LOWHUDGMDFHQF\RUGHUIRUZ=costDQGWZRDQJOHDSHUWXUHRIWKH LQWHJUDWLRQVHFWLRQRYHUWKHVFDWWHULQJPDS
$V D ILUVW FRQFOXVLRQ WKLV SUHOLPLQDU\ DQDO\VLV VKRZV WKDW D SURFHGXUH WR RSWLPL]H WKH SRODUL]DWLRQ UHVSRQVH RI D ' VSHFWURPHWHUVIRUHDFKJHRPHWULFDOFRQILJXUDWLRQVKRXOGILUVWRI DOO GHWHUPLQH WKH EHVW WUDGHRII EHWZHHQ WKH VWDWLVWLFDO VLJQLILFDQFHDQGWKHTXDOLW\RIWKHPRGXODWLRQIDFWRU
9, 21*2,1*$&7,9,7,(6$1'3(563(&7,9(
7KH&ȝ63EDOORRQERUQHSD\ORDGLVDSDWKILQGHUIRUIXWXUH VSDFH LQVWUXPHQWDWLRQ EDVHG RQ QHZ KDUG ; DQG VRIW ȖUD\V IRFXVLQJRSWLFV0XOWLOD\HU&RQFHQWUDWRUVDQGEURDGEDQG/DXH OHQVRUIRUDVPDOOVDWHOOLWHPLVVLRQLQWUXPHQW
,Q SDUWLFXODU ZH SODQ WR IRFXV IXWXUH DFWLYLWLHV RQ WKH IROORZLQJPDLQOLQHV
9DOLGDWLRQ RI WKH 0RQWH &DUOR FRGH E\ FRPSDULVRQ ZLWK H[SHULPHQWDOGDWDHJWKH(65)UHVXOWVRQWKH&DOLVWH PRGXOH ,PSOHPHQWDWLRQRIDPRGHORIWKHUHDGRXWGHWHFWRUUHVSRQVH HJDUHDOLVWLFHQHUJ\UHVROXWLRQPRGHO ,QYHVWLJDWLRQRIWKHHIIHFWRISL[HOVL]HRQWKHSRODULPHWULF FDSDELOLWLHVIXWXUHVSDFHLQVWUXPHQWDWLRQHJ/DXH/HQV 7HOHVFRSH
)XUWKHU LQYHVWLJDWLRQ RQ WKH SRODULPHWULF SHUIRUPDQFH RSWLPL]DWLRQWKURXJKDFFXUDWHGDWDDQDO\VLVDQGILQHWXQLQJ RIWKHGRXEOHHYHQWVVHOHFWLRQILOWHUV
5()(5(1&(6
>@ :HLVVNRSI 0 & HW DO ³7KH SURVSHFWV IRU ;UD\ SRODULPHWU\ DQG LWV SRWHQWLDO XVH IRU XQGHUVWDQGLQJ QHXWURQ VWDUV´ LQ 1HXWURQ 6WDUV DQG 3XOVDUV$VWURSK\V6SDFH6FL/LEUDU\YROS
>@ 'HDQ $- HW DO ³3RODUL]HG *DPPD5D\ (PLVVLRQ IURP WKH &UDE´ 6FLHQFHYROSS >@ )RURW0HWDO³3RODUL]DWLRQRIWKH&UDESXOVDUDQGQHEXODDVREVHUYHG E\WKH,QWHJUDO,%,6WHOHVFRSH´$VWURSK\V-YROS >@ /DXUHQW3HWDO³3RODUL]HGJDPPDUD\HPLVVLRQIURPWKHJDODFWLFEODFN KROH&\JQXV;´6FLHQFHYROS >@ *|W]'HWDO³*5%$WKHPRVWGLVWDQWSRODUL]HGJDPPDUD\ EXUVW´0RQWKO\1RWLFHVRIWKH5R\DO$VWU6RFYROS >@ <RQHWRNX ' HW DO ³0DJQHWLF 6WUXFWXUHV LQ *DPPD5D\ %XUVW -HWV
3UREHG E\ *DPPD5D\ 3RODUL]DWLRQ´ 7KH $VWURSK\V -RXUQDO /HWWHUV YROS/
>@ 3HDUFH 0 HW DO ³%DOORRQERUQH KDUG ;UD\ SRODULPHWU\ ZLWK 3R*2/LWH´,(((1XFO6FL6\PS&RQI5HFRUGVS >@ 0F&RQQHOO0/HWDO³&XUUHQWVWDWXVRIWKH*5$3(EDOORRQSURJUDP´ 3URFRI63,(6SDFH7HOHVFRSHVDQG,QVWUXPHQWDWLRQ8OWUDYLROHWWR *DPPD5D\YROS3
>@ 2UVL6HWDO³7KH32/$5JDPPDUD\EXUVWSRODULPHWHURQERDUGWKH &KLQHVH 6SDFHODE´ 3URF RI 63,( 6SDFH 7HOHVFRSHV DQG ,QVWUXPHQWDWLRQ 8OWUDYLROHW WR *DPPD 5D\ YRO S 0
>@ 0&KDXYLQHWDO³2EVHUYDWLRQRISRODULVHGKDUG;UD\HPLVVLRQIURP WKH &UDE E\ WKH 3R*2OLWH 3DWKILQGHU´ VXEPLWWHG WR 015$6 DU;LYY>DVWURSK,0@
>@ ³+DUG [UD\VRIW JDPPDUD\ WHOHVFRSH GHVLJQV IRU IXWXUH DVWURSK\VLFV PLVVLRQV´ 3URF 63,( 2SWLFV IRU (89 ;5D\ DQG *DPPD5D\ $VWURQRP\9,YROS
>@ ) )URQWHUD HW DO ³6FLHQWL¿F SURVSHFWV LQ VRIW JDPPDUD\ DVWURQRP\ HQDEOHGE\WKH/$8(SURMHFW´3URFRI63,(2SWLFVIRU(89;5D\ DQG*DPPD5D\$VWURQRP\9,YROS >@ )/HL$-'HDQDQG*/+LOOV³&RPSWRQSRODULPHWU\LQJDPPDUD\ DVWURQRP\´6SDFH6FLHQFH5HYLHZYROS >@ (&DUROLHWDO³$&G7HSRVLWLRQVHQVLWLYHVSHFWURPHWHUIRUKDUG;DQG VRIWȖUD\SRODULPHWU\´1XFO,QVWUDQG0HWKLQ3K\V5HV$YRO SS± >@ &DUROL(HWDO³$3RODULPHWULF([SHULPHQW:LWKD/DXH/HQVDQG&=7 3L[HO'HWHFWRU´,(((7UDQVRQ1XFO6FLYROS >@ &XUDGRGD6LOYD50HWDO³3RODUL]DWLRQ'HJUHHDQG'LUHFWLRQ$QJOH (IIHFWVRQD&G=Q7H)RFDO3ODQH3HUIRUPDQFH´,(((7UDQVRQ1XFO 6FLYROS
>@ $QWLHU 6 HW DO ³+DUG ;UD\ SRODULPHWU\ ZLWK &DOLVWH D KLJK SHUIRUPDQFH &G7H EDVHG LPDJLQJ VSHFWURPHWHU´ ([SHULPHQWDO $VWURQRP\YROSS±
>@ *iOYH] - / HW DO ³'HYHORSPHQW DQG 3HUIRUPDQFH RI D *DPPD5D\ ,PDJLQJ'HWHFWRU´3URFRI63,(8OWUDYLROHWWR*DPPD5D\YRO S
>@ .XYYHWOLHWDO³$'&=7KLJKUHVROXWLRQGHWHFWRUIRU;DQGJDPPD UD\ DVWURQRP\´ 3URF RI 63,( +LJK (QHUJ\ 2SWLFDO DQG ,QIUDUHG 'HWHFWRUVIRU$VWURQRP\9,YROS;
>@ 1 $XULFFKLR HW DO ³'HYHORSPHQW VWDWXV RI D &=7 VSHFWURPHWHU SURWRW\SH ZLWK ' VSDWLDO UHVROXWLRQ IRU KDUG ; UD\ DVWURQRP\´ 3URF 63,(+LJK(QHUJ\2SWLFDODQG,QIUDUHG'HWHFWRUVIRU$VWURQRP\9YRO S6
>@ &DUROL(HWDO³$'&=7KDUG[UD\SRODULPHWHUIRUDEDOORRQERUQH SD\ORDG´ 3URF RI 63,( 6SDFH 7HOHVFRSHV DQG ,QVWUXPHQWDWLRQ 8OWUDYLROHWWR*DPPD5D\YROS2
>@ $OOLVRQ-HWDO³*HDQWGHYHORSPHQWVDQGDSSOLFDWLRQV´,(((7UDQVRQ 1XFO6FLYROS:HEVLWHKWWSJHDQWFHUQFK >@ 0$7/$%ZHEVLWHKWWSZZZPDWKZRUNVFRPSURGXFWVPDWODE