Theoretical, practical, and economic difficulties in sampling for trace constituents

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1 Thtic, ctic, d cmic diffictis i smig f tc cstitts by F.F. itd* Sysis My idstis bs thi dcisis th ssyig f tiy ytic sb-sms. Th bm is tht mst f th tim sv smig d sb-smig stgs qid bf th bty vids its timt ssys sig dvcd chmic d hysic mthds f ysis. As g s ch smig d sb-smig stg is th bjct f d diigc sig th thy f smig it is iky tht th itgity f th sght ft ifmti hs t b td d th gtd dtbs is sti cb t ffi its ifmtiv missi. Uftty, m ft th t, wss f th bsic tis f htgs mtis cmbid with th wss f stigt qimts istd i th thy f smig, d t th ccsi tht mssiv disccis my b bsvd btw th xsiv tcm f g chi f smig d ytic cstdy, d ity. Th s tht m vb t sch misft th smig d ssyig f tc mts f cstitts f itst i th vimt, i high ity mtis, i cis mts xti, fd chi, chmics, d hmctic dcts. Witht th vtiv sggstis f th thy f smig sis diffictis my is wh mkig Gssi ximtis v gm mitis i th sbsqt itttis. A cmmty dstdig f iss csss ijctd i th thy f smig my gty h th ctiti dstd stct smig bms d vt ftt mistks fm big td v d v ti cisis is chd. This sts vviw f th thtic, ctic d cmic diffictis ft vsty dstimtd i th sch f qtifyig tc mts f vb wcm cmts. Sc Th htgity f tc cstitts i ts t b smd f th dtmiti f thi ctts hs b th bjct f xtsiv wk by my ths i th st. Th sc f this is t fcs ttti th wks d by Gy1 3, Igms4 7 d itd4, Liks btw th wks f ths ths ivstigtd, d -t-dt sttgy t sv smig diffictis is sggstd. Th chg is t vid dqt, istic sm d sb-sm mss t smig d sb-smig stgs, th wy t th bc m t th ssyig bty. M ft th t, mtig thy f smig (TOS) bsic qimts t k th vic f th fdmt smig (FSE) withi sb imits byd cmic ch, t st i c. Thf, wh ths diffictis igd f ctic ss, wss bcms th y t t diss t shw th ssib csqcs. Sch wss mst b y mgd, which is th imy bjctiv f this. F th w d, TOS fs t Gy s wk cmbid with cmtib d sitiv ctibtis md by ths. TOS is dymic kwdg tht shd b cmmtd by xistig d ft ctibtis, which is th missi f WCSB i my wys. Dfiitis d ttis Th gth f this big imitd, th d is fd t txtbks f sm dfiitis d ttis (Gy1 3; itd8; Igms d itd4). Oy th ssti s istd bw. Lti tts ctt f cstitt f itst FSE fdmt smig GSE gig d sggti IDE icmt dimitti IEE icmt xtcti IH ivit f htgity IE icmt ti IWE icmt wightig M mss wight f sm t t b smd mb f w fqcy istd gis f giv cstitt f itst s ximt stimt f stdd dviti Y gig fct Z sggti fct * Fcis itd Smig Cstts, USA. Th Sth Afic Istitt f Miig d Mtgy, SA ISSN X/ This ws fist bishd t th SAIMM Cfc, Fth Wd Cfc Smig & Bdig, Octb Th f Th Sth Afic Istitt f Miig d Mtgy VOLUME 110 UNE

2 Thtic, ctic, d cmic diffictis i smig f tc cstitts Gk tts θ vg mb f cstitt f itst gis sm μ vg mb f cstitt f itst gis sm i imy smig stg wh tw csctiv smig stgs itdc iss css σ t kw v f stdd dviti γ mst bb st Idstis tht shd b ccd Rgdss f wht th cstitt f itst is i mti t b smd, it wys cis cti mt f htgity. My idstis ccd bt sch stct ty. Sm idstis sig mtis f mi igi sch s mtgy, cmt, c, gss, cmics, im, d s, chgd vy dy t qtify ctts f citicy imtt mts. Ths diffictis ch xysm wh ths mts st i tc mts. Th my th simi xms i th gict, fd,, chmic, d hmctic idstis. Th is th stig xm i smig f tc cstitts i th vimt; cmis stggig t mt gty qimts hv gt ccs bt th cbiity t cct sttiv sms tht wi b ssyd f tc cstitts. A ths xms jst th ti f th icbg. A gic ch sggstd by th thy f smig Th thy f smig is by dfiiti vtiv t f wkig i th idsty t fid wys t miimiz th gtiv ffcts f th htgity cid by citicy imtt cmts. Sch htgity gts vibiity i sms, thf vibiity i dt tht t ctd. Th fwig sts ssti f th dfiiti f gic d sccssf smig tc. Th discssi is imitd t th smig f z-dimsi, mvb ts. F -dimsi ts th d is fd t Chsttistics (itd12); f tw- d th-dimsi ts th d is fd t m i-dth dig f th TOS (Gy1 3; itd8; Esbs d Mikki13; ts14; Dvid15). Migic d micscic bsvtis At th y stg f y smig jct it is mdty t cd with thgh migic stdy micscic stdy tht my shw hw giv tc cstitt bhvs i th mti t b smd. Th ccsis f sch stdy my t b sttiy i distc tim, vthss thy giv id bt th dicti tht my g wh chig th it wh ximt mst b dsigd t ms th tyic htgity f th cstitt f itst. Ths imtt stdis mst mi w fcsd. F xm, i th gd idsty it is t t s migic stdy f th gd fmd f giv f giv miig jct. Th, th fi t my csist f 49 gs btig th my mis st i th, d y g f gd which is by f th mst vt cstitt; w-fcsd sbstc shd b th ssc. Htgity tsts My vsis f htgity tsts hv b sggstd by vis ths. F xm, Gy sggstd bt th vsis, Fçis-Bgç sggstd t st tw, itd sggstd sv, Vism sggstd, d Igms sggstd sv. Thy hv smthig i cmm: thy sy tid t w-fcsd bjctiv d thy hv thi mits withi tht ctxt. It is imtt t f t Fçis-Bgç s wks16 19 bcs f his wdcmtd chs. It is th viw f this th tht f tc cstitts, ximts sggstd by Vism20 d Igms vid th cssy ifmti t mk imtt dcisis, bt smig tcs, th ittti f th ximt sts, d th ittti f ft dt cctd i simi mtis; this is sciy t t fid mthds t vcm y svb smig bms bcs f th cmic imct f id smig tcs. Rsctig th cdi s f smig cctss Lt s b vy c citicy imtt iss: if y smig tc y smig systm ds t by th cdi s f smig cctss istd i th Thy f Smig, th miimizd smig s dig t cctb v f ctity g xist withi chb dmi. I th wds, if icmt dimitti s (IDE), icmt xtcti s (IEE), icmt wightig s (IWE) d icmt ti s (IE) t ddssd i sch wy tht thi m is g cs t z, w swy v th dmi f smig d t th dmi f gmbig. I this th ssmti is md tht th m f ths bis gt s is z. I th vtity y bysss smig cctss f sm ctic s, stis g sid i th wd f wisdm d gtd dt simy ivid d thic. It is th bffig tht my stdds cmmitts smig sti t dds with th s f smig cctss. Qtifyig th fdmt smig Ems mts f wk hv b d by Gy, Fçis- Bgç, d itd th my wys t cct th vic f th fdmt smig. F th cd th thy f smig ffs vy difft chs d fms f th fwig css: Th d, cssic mtic ch1 wh sh fct, tic siz distibti fct, migic fct, d ibti fct mst b stimtd A m scitific ch3 ivvs th gb dtmiti f th cstt fct f cstitti htgity (i.., IHL) A tty difft ch22 fcss th siz, sh, d siz distibti f th ibtd, ibtd, v i sit gis f cti cstitt f itst A sci cs wh th mhsis f smig is th dtmiti f th siz distibti f mti2,22. Th cf cmbiti f css 3 d 4 c cty vid vy sim, ctic d cmic sttgy tht my hv b vkd by my smig ctitis. 314 UNE 2010 VOLUME 110 Th f Th Sth Afic Istitt f Miig d Mtgy

3 Thtic, ctic, d cmic diffictis i smig f tc cstitts Miimizig th gig d sggti Th gig d sggti GSE is chctizd by th fwig tis f its m d vic: If th vic f GSE is th dct f th fcts, this wd sggst tht th ccti f y fct cd imit GSE. It is t ssib t cc th vic f FSE ss th sm is th ti t, which is t th bjctiv f smig. Hwv, it shd b miimizd d w kw hw t d this. It is t ssib t cc Y ss w cct sm by cctig -fgmt icmts t dm t tim. This is t ctic; hwv, it is d i cmmdd mthds by Gy d itd f th ximt dtmiti f IH L. I ti smig tc, th ight sttgy is t cct s my sm icmts s cticy ssib s th fct Y c b dsticy miimizd; this vs t b by f th mst ffctiv wy t miimiz th vic f GSE. It is t ssib t cc th fct Z which is th st f tsit sggti. A hmgizig csss hv thi wksss d ft wishf thikig csss; this vs t b th mst iffctiv wy t miimiz th vic f GSE. Th chgs f ity Rity ft shws tht btw wht is sggstd by Gy s thy d wht th ct immtd tcs, th is bysm diffc d w shd dstd th ss f sch ftt shtcmig; th cd b sv ss: Rqimts fm Gy s thy dismissd s imctic d t xsiv. Th TOS is t dstd, dig t th imssi th TOS ds t cv sm ci bms wh it mst ctiy ds. Th ctiti ds t kw hw t g d sm ssmtis md i sm ts f th TOS wh imittis f ths ssmtis hv b w ddssd d cd wh cssy. tcs bsd st xic fm smbdy s. T mgmt ds t dstd th ik btw hidd cst d smig. Nm gm sttistics id withi dmis wh thy d t bg. iss csss vsty misdstd d igd. hv ïv dfiiti f wht ti is, tc. Igms s wk t th sc Cy, w d difft ch i d t mk TOS m tb t my ctitis d this is wh th wk f Igms c gty h. Igms s ch c h smig ctitis t btt dstd th bhvi f bd dt, s mgmt c btt b cvicd tht ft, Gy s vtiv ch is th wy t g, v if it sms xsiv t fist gc; i this sttmt th is itic d sychgic sbtty tht hs ctd bis f th TOS f my ys, d bkig this bi ws th ti ssc f itd s thsis22. Fm Vism t Igms Mst f th vb wk f Igms is bsd Vism s smig thy. It is t th itti f this t ijct Vism s wk i th TOS. Wht is mst vt is Igms s wk iss distibtis tht c b sd s cvit t t shw th isks ivvd wh th vic f FSE is gig t f ct: it ct b mhsizd stgy gh tht th ivsi f y dtbs by iss csss c ty hv ctsthic cmic csqcs i y jct sch s xti, fsibiity, cssig, vimt, d g ssssmts. Agi, t s mk it vy c, y dtbs ivdd by iss css bcs f th smig d sb-smig cds tht w sd is dict, fgt dt fm th d diigc ctics i y jct. Yt, smtims w d t hv th xy f chic, sch s th smig f dimds; th wss is th ssc. Limittis f m d gm sttistic mds At tim, scitists bcm cvicd tht th Gssi distibti ws ivsy icb, d vwhmig mjity f ictis f sttistic thy bsd this distibti. A cmm hs b t jct tis tht ct b md t fit th Gssi md sm mdificti f it s th gm md. Th tdcy, sd by sm gsttisticis, hs b t mk th dt fit ccivd md istd f schig f md tht fits th dt. O this iss, Whitt qt21 t sd d mdifid by Mich Dvid15, ws sb: th mthmtic mds tht c cim divi ight t st vigm. It is w t tht tis ft th mst imtt dt its i giv dt-st, d gd dstdig f iss csss is cvit wy f dstdig hw d why thy ctd. iss s csss iss d db iss csss6 8,22 xi why highy skwd distibti f ssy vs c cc. Th gd d cti f idivid it ssy, which fws sig db-iss distibti, wi hv vity tishi, d it wi b imssib t ssig gd th th th m v t mib sm-siz bcks. Simi diffictis c cc with th ssssmt f imity ctts i vb cmmditis. Nw, th is itt sbtty bt iss csss, s sm my sy th siti f gi cst f gis is v cmty dm s xid i gsttistics; this is t th it. Th it is tht, m ft th t, th vm f bsvti w s my itsf gt th iss css; th is diffc. Th f Th Sth Afic Istitt f Miig d Mtgy VOLUME 110 UNE

4 Thtic, ctic, d cmic diffictis i smig f tc cstitts Th sig iss css Th iss md is imit cs f th bimi md wh th ti f th cstitt f itst is vy sm (.g., fcti f 1%, m b), whi th ti q = 1 f th mti sdig th cstitt f itst is cticy 1. Exic shws tht sch cstitt my cc s, tiy gis, tivy t tims, d thy my my t b ibtd; thy my v b i sit. Smig ctitis mst xit th digm f kig t ibtd gis xcsivy; th bm is mch wid th tht. As th sm bcms t sm, th bbiity f hvig gi sfficit mt f thm i sctd sm dimiishs dsticy. F i sit mti th sm c b cd by imgiy vm f bsvti t y giv c. Wh gi cst is st, th stimt S f L bcms s high tht it is ft csidd s ti by th ixicd ctiti whs it is th mst imtt fidig tht shd idd is ttti. A this is w-kw bm f ths ivvd with th smig f dimds. Lt s c (x = ) th dm bbiity x f wfqcy istd cs gis ig i sm, d θ is th vg mb f ths gis sm; s divti f th fwig fm i it itt8,22. [1] If m is th mb f tis (i.., sctd, ict sms), th vic f th iss distibti is θ = mq m sic q is cs t 1. F ctic ss, th m v f th iss distibti is θ m. As cy shw i th divti f Eqti [1] w cd ssm i fist d ximti tht θ m. Th db iss css Wh imy sms tk fm th dsit cti th cstitt f itst i imitd (.g., ss th 6)4,15 vg mb μ f disct gis csts f sch gis (i.., [y=]), d thy sb-smd i sch wy tht th sb-sms s cti disct gis f dcd siz i imitd (.g., ss th 6)4,15 vg mb θ (i.., [x=]), db iss distibti f th ssy vs is iky. Th bbiity f gis f mi ig i y sb-sm is dtmid by th sm f th bbiitis f gis big gtd fm sms with gis. Lt s dfi th ti f: With θ = μ f θ = f f ch ssibiity, th qti f th stig, cmdd bbiity f th db iss distibti is: [2] A xm f this cs is giv i itd s thsis22. This is th bbiity f btiig sm with gis f th cstitt f itst. Th qti cd b mdifid sig imvd Stiig ximtis f fctis, f xm: I ctic, ds t sy ct gis; ccttis msd. Th cvsi fct fm mb f gis t, ct X f xm, is C, th ctibti f sig vg gi. Sic th vic f sig iss distibti is q t th m: [5] Thf: Bt vics f dm vibs dditiv, th f db iss distibti w wd hv: [7] Th dt vib sy ssys i % mt, gm/t, m b. Thy td by th qti: wh x i is th ssy v f tic sm, i % f xm; H is th w m hmgs bckgd cctti i % f xm, which is si t sm; i is th mb f mi gis i th sm; c is th ctibti f gi t th ssy i % f xm: Ths th bbiity f sm hvig ssy v f x i qs th bbiity f th sm hvig i gis wh H is tivy cstt. Th m v f st f ssys c b shw t b: [4] [6] [8] [9] [10] F sig iss distibti this qti wd b: [11] wh x is stimt f th kw vg ctt L f th cstitt f itst. Assmig smig is cct, d f th sk f simicity, i th fwig t f this w shd sbstitt x with L. Th: [12] th: [13] [14] [3] Sbstittig Eqti [14] i Eqti [7]: [15] f = 0, 1, 2, 3, whc: 316 UNE 2010 VOLUME 110 Th f Th Sth Afic Istitt f Miig d Mtgy

5 Thtic, ctic, d cmic diffictis i smig f tc cstitts [16] Th bbiity tht th wi b diffict-t-sm gis f th cstitt f itst i dmy tk sbsm is fd by sbstittig = 0 i Eqti [3]: [17] If dt-st fits db iss distibti, th mts μ d θ f this distibti my b fd fm ittiv css, s fws: Mk imiy w stimt f H. Giv c bity w v. Cct imiy v f f fm Eqti [16], d f μ by gig Eqti [12]: [18] Sbstitt ths imiy stimts i Eqti [17]; vgig th wst (x=0) f th dt t bti w stimt f H. Icmt c d t ti bst fit is fd. If iss css is ivvd, which is t cssiy th cs, this icmt css f c wi idd cvg vy w. Nti f miimm sm wight Th is cssy miimm sm mss M smi i d t icd t st tic f th cstitt f itst bt 50% f th tim i th cctd sm which hs wh =1 i Eqti [1] wh =1 i Eqti [3]; Igms shws tht it c b cctd s fws: [19] F ict sms t vid my distibtd ti th cmmdd sm mss M Sc shd b t st 6 tims g th M Smi. As shw by Igms d itd4 d Dvid15 it tks t st =6 t miimiz th iss css t th it tht m my distibtd dt wi. Of cs th is mgic mb d shd cty b mch g th 6 t big bck th vic f th fdmt smig (FSE) it cctb v22. At this it th is imtt iss t ddss: qtis sggstd by Gy t stimt th it sm mss wh th htgity IH L cid by th cstitt f itst is ghy stimtd, shd b sd i sch wy tht w kw w sby withi dmi tht ds t cy y iss skwss. Igms sggstd tht th dd sm mss M Sc is bt 6 tims g th M Smi. Th cmmdd imit sggstd i Gy s y wk is %s FSE = ±16% tiv which hs t b v m stigt th th Igms s sggsti. Nti f timm sm wight I gic smig tc cmmis mst b fd btw th cssy sm mss qid f miimizig th vic f th FSE d th mb f sms tht dd t hv id bt th t vibiity d t ith sm-sc g-sc sggti. Sch timm sm mss M St ws fd by Igms d tstd i th it TOS ttis i itd s thsis22, d c b witt s fws: Wh s 2 s is c vic d t th sggti f th cstitt f itst i th t t b smd. [20] Cs stdy: stimti f th i ctt i highity mmim tgstt Th fwig cs stdy ivvs sig stg iss css d th cmic csqcs c dy b stggig bcs f th -sttiv ssssmt f th imity ctt f xtmy vb high ity mti. It shd b mhsizd tht th ytic tc tht ws sd ws ctgizd s fst, ch, d cvit. I th wds, it ws cd cst-ffctiv ytic mthd. A shimt f vb high-ity mmim tgstt sd i th fbicti f tgst cis i ight bbs ws ssyd by scifid si t cti bt 10 m i. Th ctct imit ws tht shimt shd cti m th 15 m i. Th cit s stimts sig g ssy sms w mch high th th si s stimts sig tiy 1-gm ssy sms. Th mximm tic siz f th dct ws 150-μm. T sv th dist cfy d 5000-gm sm, sttiv f th shimt, ws ssyd 80 tims sig th stdd 1-gm ssy sm wight sd t th si s bty. Tb I shws th ssy vs gtd f this ximt. A smmy f sts is s fws: Th stimtd vg x L f th 80 ssys ws 21 m. Th bst vic s2 = 378 m2 Th tiv, dimsiss vic sr 2 = 0.86 Th bst stdd dviti s = 19 m Th tiv, dimsiss stdd dviti s R = ±0.93 ±93%. Fm th TOS th fwig tishi c b witt: [21] A tms w dfid i th TOS. Th sbscit 1 fs t th ifmti tht is vib fm sm sm wighig 1 gm; it is i tht cs y fc tiv t th dscibd ximt. Th ffct f M L is gigib sic it is vy g tiv t M S. Th v f th vic s 2 GSE1 f th gig d sggti is t kw; hwv, th mti is w cibtd d th ss f t f sggti t tk c bcs th istd gis ctiig high i ctt hv bt th sm dsity s th th gis sic thi cmsiti is miy mmim tgstt. Thf it c b ssmd i this tic cs tht s 2 FSE1 s 2 GSE if ch 1-gm sm is md f sv dm icmts, s th v f IH L tht is cctd is y sighty ssimistic. Th y fct fit t iss md s shw i Fig 1 ws t th tim sfficit f tht th gig d sggti ws t th bm, d ws fth cfimd tt by th gd dcibiity btid by cctig mch g 34-gm sms. Th fwig qti c thf b witt: Th f Th Sth Afic Istitt f Miig d Mtgy VOLUME 110 UNE

6 Thtic, ctic, d cmic diffictis i smig f tc cstitts Tb I Smmy f 80 ict i ssys i high-ity mmim tgstt Sm mb m F Sm mb m F Sm mb m F Sm mb m F Thf, it c b ssmd tht IH L 0.86 g, sic th mss M S = 1 gm, jstifyig th wy Eqti [22] is witt. If th ttd stdd dviti f th FSE is ±16% tiv, th timm cssy sm mss M S c b cctd s fws: [22] [23] Obvisy, it is g wy fm th 1-gm tht ws sd f ctic ss. This mss f 34 gms is th miimm sm mss tht wi mk th gti f my distibtd ssys sts ssib. Ath mt tht c b btid is th w bckgd ctt H, which is iky d 4 m by kig t th histgm i Fig 1. This high-fqcy w v my smtims st y th wst ssib dtcti f th ytic mthd; thf cti is cmmdd wh th t w bckgd ctt f dct f giv imity is cctd. Ivstigti f th histgm Fig 1 istts th histgm f 80 ssys shw i Tb I. I this histgm it is c tht th fqcy f giv st chs mximm t g itvs, sggstig tht th cssificti f th dt i vis zs is ssib; z A with 27 sms shwig z gi f th i imity; z B with 29 sms shwig 1 gi; z C with 13 sms shwig 2 gis; z D with 5 sms shwig 3 gis; z E with 3 sms shwig 4 gis; z F with 1 sm shwig 5 gis; Z G with 6 gis shws vt; fiy z H with 7 gis shws vt, which my b my i th md f th distibti. Th st f sts s iss distibtd, d chctistic f th iss distibti is tht th vic is q t th m. Th fwig qivcs c b witt: [24] Th ssmti tht H = 4 m ds t b chckd. Th bbiity tht th wst ssy v sts H c b cctd. If th vg mb f gis f imity sm θ is sm, th is bbiity tht th wst ssys st H. Th bbiity tht sig cctd sm wi hv z gi is: If w c (x = 0) th bbiity f sccss, th th bbiity x f sccsss i N tis is giv by th bimi md: Wh is th bbiity f hvig sm with gi wh y sm is sctd, d (1- ) is th Fig 1 Histgm f ighty 1-gm ssys f i i mmim tgstt [25] [26] 318 UNE 2010 VOLUME 110 Th f Th Sth Afic Istitt f Miig d Mtgy

7 Thtic, ctic, d cmic diffictis i smig f tc cstitts bbiity f hvig t st gi wh y sm is cctd; th th bbiity f sccss (x 0) with N sms is: [27] Eqti [27] shws th bbiity tht f N sms is f fm w-fqcy imity gis. Th bbiity tht th wst ssy v sts H is: [28] Assmig tht H is t th ytic dtcti imit, w c b s tht th wst ssy sts H. Hvig fd tht th v θ = 1.18, w my cct th iss bbiitis f sms ctd i ch z isttd i Fig 1. Ths, by mtiyig ch bbiity by 80, w my cm th cctd distibti with th bsvd distibti. Rsts smmizd i Tb II. Th bsvd distibti is vy cs t th cctd distibti if w xcd th vy high st shwig 115 m, which shd t hv d with y 80 sms. A chctistic f th iss distibti is tht th vic s 2 f th ssys is q t th vg L. [29] [30] Bt, i ctic th mb f gis is t sd; istd ccttis sd sch s %, g/t, m, b. Lt s c C th cvsi fct d wit Eqti [30] y: [31] Ths, w my cct th ctibti C f sig vg imity gi t sig i ssy: [32] With th st f dt giv i Tb I st f tifici, g 10-gm sms md f Q=10 sm -gm sms c b ctd, d thy shw i Tb III. Vism smig qti With this ifmti Vism smig cstts A d B c b cctd; it is dstd tht Vism wd hv sggstd th ccti f g sms s w xid by itd i his thsis22: wh S is th ctity i th vg f N=80 ssys sms f idivid mss [35] A is th Vism hmgity cstt. It is th Gy s Cstt Fct f Cstitti Htgity IH L mtiid by th sq f th vg ctt f th t. Fm th vics d Vism s qti w bti: Fm Gy w sggstd i: Ths mbs vy cs d withi th vics cisi, thf this wd sggst th is m t cct th mt f sggti f i i th t. It is wis t ssm tht B, th Vism sggti cstt is: Tb II Cmis f th cctd distibti with th bsvd distibti Discssi f cctb mximm f th stdd dviti f th FSE Igms sggstd tht miimm f six f th gst gis, csts f tii gis it sig fgmt, f imity shd b st i sm f th ysis f this sm t b migf. Th bjctiv f sch sttmt is t imit th iss css fm dmgig th dtbs. If 1-gm sm ctis vg θ = 1.18 gis, th th miimm cmmdd sm mss is d 5 gms. Usig this mss d th v f IHL btid i w my wit: [33] [34] Bt, fwig Gy s cmmdtis23 34-gm sm is cmmdd i d t chiv ±16% tiv stdd dviti, which wd cti bt 41 gis; it is dstd tht f cti ictis, sch s smig f mti bc f cmmci sttmts, Gy s cmmdtis i his bictis w f m stigt (5% v 1%). I d t fth discss this diffc, t s cstct th sf Igms s smig digm. iss bbiity Cctd Obsvd f θ = 1.18 distibti distibti Tt Tb III I ctt f tifici g sms f mss q t 10 gms N sm mb Cmsitd sm I ctt i sms g sms Th f Th Sth Afic Istitt f Miig d Mtgy VOLUME 110 UNE

8 Thtic, ctic, d cmic diffictis i smig f tc cstitts This cfims th ii tht i i cibtd mmim tgstt gis hs s whtsv t sggt i sbstti wy, d th bsvd vibiity is d t th vic f FSE, s sggstd by y fct iss s fit isttd i Fig 1 shwig th bm ws sm mss t sm by m th d f mgitd. Th mst bb st Th mst bb st γ f th ssyig f i s fcti f ytic sm mss M S is cctd with th fwig Igms Eqti1 3, 22: [36] Vs f γ isttd i Fig 3, d it bsicy sts th cti f th md f th bbiity distibti tiv t th xctd ithmtic vg L. Nttis i Fig 3 d Igms ttis d th th gizs f this icvic d t th s f d sftw gm (i.., γ =Y, L = x, L= H ). A cf stdy f th γ cv i Fig 3 (i.., ight b cv) is th ky t cmt discssi f th diffc btw Igms s cmmdti d Gy s cmmdti f sggstd mximm v f th stdd dviti f FSE. It c b bsvd tht th cmmdd mss by Igms (i.., 6 gis i th sm %s SFE = ±41%) ds t cti f th md sti sbsttiy wy fm th xctd ithmtic vg L. It is t th cs with th cssy sm mss f 34 gms i d t bti %s SFE = ±16% s cmmdd by Gy. Igms s gg cctti Th w bckgd i ctt H stimtd i t b 4 m c b cctd sig th md fm sts fm sm sms d th md γ 2 fm sts fm g sms. Mds c b cctd sig th hmic ms h 1 d h 2 f th dt distibti f th sm d g sms. Th hmic m is cctd s fws: wh N is th mb f sms. Fm Eqti [36] w my wit: Th fm Eqtis [1], [2], d [3] th w bckgd ctt H c b cctd s fws: Rsts shw i Fig 2 d cfim th i stimt f 4 m. Fig 2 Sch f v f th w bckgd i ctt H [37] [38] [39] [40] Fig 3 Istti f th Igms s smig digm f i tcs i mmim tgstt ( =Y, L = x, L= H ) 320 UNE 2010 VOLUME 110 Th f Th Sth Afic Istitt f Miig d Mtgy

9 Thtic, ctic, d cmic diffictis i smig f tc cstitts If th d is itstd by th f divti f th bv fms, f t itd s dct thsis22. Ccsis Th ky t smig f tc mts f giv cstitt f itst is thgh micscic ivstigti f th wys sch cstitt is distibtd sm sc i th mti t b smd. Libtd t, th csst gis f sch cstitt mst b msd d cd it th ctxt f thi vg, c xctd gd. Th s f iss sttistics d th ccti f Igms smig digm c d t dfiig miimm smig fft dig th y hs f jct. Eqid with sch vb imiy ifmti, sm c cd with fsibiity stdy t immt cssy smig tc s sggstd by th TOS, with f dstdig f wht th csqcs wd b if d diigc is xcisd. Rcmmdtis As th smig qimts cssy t miimiz th vic f FSE s sggstd i th TOS my bcm cmbsm t my cmists, it bcms imtt t cd with imiy tsts i d t vid th cssy ifmti t ct vb isk ssssmts. Th fwig sts, i chgic d, cmmdd: Cfy dfi dt qity bjctivs. Awys sct th fdmt s f smig cctss s xid i th TOS. This st is t gtib. fm thgh micscic ivstigti f th mti t b smd i d t qtify th gis siz f th cstitt f itst, ibtd t. Emhsiz th siz f csts f sch gis if sch thig c b bsvd. cd with Vism s ximt, cct th Igms s mts, d dw ifmtiv Igms s smig digm. Shw th ifmti t xctiv mgs wh mst mk fsibiity stdy t jstify m fds t fm wis d cssy ch sig Gy s qimt t miimiz th vic f FSE. Rfcs 1. GY,.M. L Echtig ds Miis Vc (Smig f tict mtis). Vm 1. Rv d Idsti Mi, St. Eti, Fc. Nm Sci (Sci iss, y 15, 1967). 2. GY,.M. Smig f tict Mtis, Thy d ctic. Esvi Scitific bishig Cmy. Dvmts i Gmthmtics d GY,.M. Smig f Htgs d Dymic Mti Systms: This f Htgity, Smig d Hmgizig. Amstdm, Esvi INGAMELLS, C.O, d ITARD F. Aid Gchmic Aysis. Wiy Itscic Divisi, h Wiy d Ss, Ic., Nw Yk, INGAMELLS, C.O. d SWITZER,. A sd Smig Cstt f Us i Gchmic Aysis, Tt, v. 20, INGAMELLS, C.O. Ct f gchmic thgh smig d sbsmig digms. Gchim. Csmchim. Act, v. 38, INGAMELLS, C.O. Evti f skwd xti dt th ggt ffct. Gchim. Csmchim. Act, v. 45, ITARD, F. i Gy s Smig Thy d Smig ctic. CRC ss, Ic., 2000 Ct Bvd., N.W. Bc Rt, Fid Scd diti, y ITARD, F. Effcts f Rsid Vics th Estimti f th Vic f th Fdmt E, bishd i sci iss by Chmmtics d Itigt Lbty Systms, Esvi: 50 ys f i Gy s Thy f Smig, cdigs: Fist Wd Cfc Smig d Bdig -(WCSB1), ITARD, F. Bsth smig f gd ct: th my bms d stis. Smig My 2008, th, Asti. Th Astsi Istitt f Miig d Mtgy ITARD, F. Smig Cctss A Cmhsiv Gidi. Scd Wd Cfc Smig d Bdig Th Asti Istitt f Miig d Mtgy. bicti Sis N 4/ ITARD, F. Chsttistics A wf, gmtic, w scic f mtgists. Mtgic t Dsig d Otig sttgis (Mtt 2006) Stmb 2006 th, WA. Th Astsi Istitt f Miig d Mtgy ESBENSEN, K.H. d MINKKINEN,. (gst ds.). Chmmtics d Itigt Lbty Systms. vm 74, Sci iss: 50 ys f i Gy s Thy f Smig. cdigs: Fist Wd Cfc Smig d Bdig (WCSB1). Ttis Smig: Thy d ctic. 14. ETERSEN, L. i Gy s Thy f Smig (TOS) i ctic: Lbty d Idsti Didctics icdig fist fy it img ytic smig. ACABS Rsch G. Abg Uivsity Esbjg, Nis Bhs Vj 8, Dk Esbjg, Dmk. Thsis sbmittd t th Itti Dct Sch f Scic d Tchgy. y DAVID, M. Gsttistic O Rsv Estimti. Dvmts i Gmthmtics 2. Esvi Scitific bishig Cmy FRANÇOIS-BONGARÇON, D. Thy f smig d gsttistics: itimt ik. bishd i sci iss by Chmmtics d Itigt Lbty Systms, Esvi: 50 ys f i Gy s Thy f Smig, cdigs: Fist Wd Cfc Smig d Bdig (WCSB1), FRANÇOIS-BONGARÇON, D. Th mdig f th Libti Fct d its Cibti. cdigs: Scd Wd Cfc Smig d Bdig (WCSB2). bishd by Th AsIMM My 2005, Sshi Cst, Qsd, Asti FRANÇOIS-BONGARÇON, D. Th mst cmm i yig Gy s fm i th Thy f Mi Smig, d th histy f th ibti fct. Mgh Twd 2000, AsImm FRANÇOIS-BONGARÇON, D d GY,. Th mst cmm i yig Gy s Fm i th thy f mi smig d th histy f th Libti fct, i Mi Rsc d O Rsv Estimti. Th AsIMM Gid t Gd ctic. Th Astsi Istitt f Miig d Mtgy: Mb VISMAN,. A G Thy f Smig, Discssi 3, f Mtis, MLSA, v. 7,. 3, Stmb 1972, WHITTLE,. O sttiy csss i th. Bimtik, v ITARD, F. i Gy s Thy f Smig d C.O. Igms iss css Ach thwys t Rsttiv Smig d Ait Idsti Stdds. Dct thsis: Abg Uivsity, cms Esbjg, ISBN: GY,.M. Nmgmm d chtig, Smig Nmgm, bhm Nmgmm. Miis t Mtx, is (Ed.) Th f Th Sth Afic Istitt f Miig d Mtgy VOLUME 110 UNE