ON THE RELIABILITY OF DATA OBTAINED BY KRIGING

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1 Buhad Schaffi ON HE RELIABILIY OF AA OBAINE BY RIGING Buhad SHAFFRIN patmt of ivil ad Eviomtal Egiig ad Godtic Scic h Ohio Stat Uivit olumbu OH 43 USA Schaffi@oudu Woig Goup I/4 EY WORS Outli ttig liabilit igig patial data ABSRA Igulal ditibutd data ma b itptd a a ampl daw fom a udlig patial poc who popti ca b aumd to b ow o uow dpdig o th paticula ituatio I ca of a lativl mooth poc o of th vaiou igig mthod could b mplod to div giddd poit data with compaabl accuac povidd that o outli a pt Othwi th hav to b limiatd bfohad o at lat thi ifluc mut b ducd to th lvl of adom uctait Mau of liabilit to dcib th pottial fo idtifig outli i upiciou ampl poit ad to quatif th ffct of a udtctd outli wll-ow fo th Gau-Maov Modl will b itoducd fo th ca of a patial poc wh th ampld data a uppodl colatd at lat i th patial I thi tud w hall coid Simpl a wll a Odia igig which i tiall idtical to lat-qua collocatio with ow p uow cotat td INROUION A REVIEW OF HE GAUSS-MAROV MOEL Oigiall fo th u i godtic two W Baada 968 had itoducd a ttig pocdu fo outli that i ow ow a data oopig Shotl aftwad a tho of liabilit wa dvlopd b W Baada 976 which foud a umb of applicatio both i god ad photogammt W ol f to th wo b H lz 98 S F El-Haim 98 ad W Föt 983; 985; fo a ovviw coult g R och 988 chapt 44 o B Schaffi 988 h oigial tho wa bad o a Gau-Maov Modl A ξ xm wh i th vcto of obvatioal icmt; ξ i th m vcto of uow paamt icmt; A i th m matix of cofficit with Aq mim; i th vcto of uow adom obvatio o; Q - i th poitiv-dfiit cofacto matix; i th uow vaiac compot h wightd LEat-Squa Solutio LESS ca b ta fom th omal quatio N ξ c fo N c A A but ma ot b uiqu ul qm h gal olutio pac ma thu b ptd a ξ N c NN N N i N i g-iv of N 3 alog with th dipio matic Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam 893

2 Buhad Schaffi ξ N NN N N N NN N N 4 I a ca th idual vcto A ξ I AN A Q 5 will b uiqu with th dipio matix AN A Q ad o will b th vaiac compot timat q q c ξ 6 Now lt u aum that a outli ha occud i th -th obvatio So th tu modl hould itad hav b Aξ 8 with a -th uit vcto ad a uow iz of th outli oqutl th omal quatio hould hav ad N A A ξ c with th modifid olutio pac ξ ξ N A NN N N ad th timatd iz of th outli Q Q which tu out to b uifoml ubiad a log a Q Not that pt th idual vcto 5 fom th u-modifid Gau-Maov Modl which ca b itptd a a combiatio of modl 8 with th cotait I mot al applicatio th obvatioal wight matix wa aumd to b diagoal aml iagp p 3 which lad to th paticulal appalig fomula / p 4 with th dudac umb Q Q 5 h am obvioul f to th fact that t Q q 6 ild th total dudac of th modl Moov i thi pcial ca w hav Q Q 7 which ow i th -th diagoal lmt of a mmtic idmpott matix Fo uch matic w ow that all diagoal lmt li btw th maximum igvalu ad th miimum igvalu thu Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam

3 Buhad Schaffi fo all 8 Not that thi popt i lot i th ca of colatd obvatio a wa poitd out b J Wag/Y h 994 ad B Schaffi 997 But th w loo th impl latio 4 btw th timatd outli iz ad th copodig u-modifid idual a wa ad w hav to tu to fomula If w loo at th chag of th copodig idual howv that i caud b th pc of a outli accodig to 8 w obtai Q 9 fom 5 aft placig b Appatl thi fomula hold tu v fo o-diagoal thb tllig u that th total ffct of th outli o th copodig idual i cald b th facto hfo wh bcom almot o v bigg a outli of th am iz bcom mo ad mo viibl i th pctiv idual ad covl wh i too mall o v gativ th outli ma b hidd A lag thu hould ica th poibilit fo u to tud th ffct of a outli o that paticula obvatio Ufotuatl thi do ot gall ma that it would bcom ai to dtct thi outli whv i lag ic fo th dtctio w hav to l o th timatd outli iz ath tha it tu iz A a ult w ca coid a mau of liabilit ol i th ucolatd ca wh latio 4 hold I thi ca aml th o-ctalit paamt fo th altativ hpothi i outli i th -th obvatio H a bcom ϑ p / ad thi paamt ha to upa a ctai thhold dpdig o th o pobabilit α ad th cho pow of th tt β i od to bcom paabl fom th ull hpothi i o outli i th -th obvatio H W ta it fom that a lag hlp i thi pct; thu th dudac umb ma wll v a idicato fo th i liabilit I th gal ca of colatd obvatio howv w follow B Schaffi 997 ad appl th omalizd liabilit umb itad which a dfid a th atio Q 3 h a tu galizatio a fo diagoal p w d up with th oigial agai Accodig to fomula cal th ifluc of o th timatd iz of th outli I additio lt u dfi th out liabilit b th iz of th ffct that th maximum o-dtctibl outli would hav o th timatd paamt maud i a popl wightd om I aalog to fomula w adil obtai ξ ξ N AN A i gal ad i th ca of ucolatd obvatio p Q ξ ξ 5 N 4 idpdt of th cho g-iv N H dot th maximum outli that caot b dtctd at th α - lvl with pow β wh α ad β a to b pcifid i advac I th followig w hall galiz th abov tho to alo cov patial poc which a tochatic i atu - uli th vcto ξ - ad patiall colatd Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam 895

4 896 Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam RELIABILIY MEASURES FOR SIMLE RIGING H w bgi with th dfiitio of ou modl of a tatioa patial poc at locatio S 6 with th poc ma aumd to b ow h poc ha b ampld at th locatio i i with th ult 7 wh i th obvatio vcto i th uow vcto of adom ffct i th uow vcto of obvatioal o Futhmo dot th adom dviatio of th poc fom it ma aumd to b ucolatd with th vcto vwh hu w hav fo a ummatio vcto 8 wh x x x mm M O L L 9 Followig N A i 993 o R Rao/H outbug 995 g th Simpl igig olutio pt th Bt ihomogoul LIa dictio ihom BLI of ad ca b gatd b wightd lat-qua h olutio i obtaid i th ampl poit a 3 ad ca b how to b wal ubiad hu th ma qua pdictio o matix ca b computd via MSE 3 Futhmo th two idual vcto bcom 3 ad 33 with thi dipio matic ad th covaiac matix 36 A poibl vaiac compot timat ca b obtaid though 37 but ma ot tu out to b ubiad Buhad Schaffi

5 897 Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam Now if w aum a outli i th -th ampl poit ou modifid modl will ith ad 38 wh th outli i attibutd to fault obvatio o 39 wh th pio ifomatio ma valu i coidd fault h fit ca ca b hadld alog imila li a dvlopd fo th Gau-Maov Modl ladig to th modifid olutio 4 with th timatd iz of th outli 4 which fo a diagoal wight matix duc to / p 4 H th liabilit umb a dfid b / / < 43 ad ma v to idicat th i liabilit I cotat th out liabilit would b quatifid b 44 i gal ad i th ca of ucolatd obvatio b p 45 wh agai dot th maximum o-dtctibl outli fo giv α ad β a xplaid i ctio Fo o-diagoal i th abov fomula would hav to b placd b 46 fo 4 47 I th cod ca w obtai th modifid olutio 48 with th timatd iz of th outli 49 h omalizd liabilit umb i aalog to B Schaffi 997 thu bcom < 5 to idicat i liabilit whil th out liabilit i giv b 5 Buhad Schaffi

6 898 Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam with dotig th maximum o-dtctibl outli ow aumd to affct th ma valu of th -th ampl poit i E without big oticd Not that bcau of 33 w hav th dualit btw ad with 46 ad 5 idicatig wh to locat th outli ai wh it occud 3 RELIABILIY MEASURES FOR ORINARY RIGING I thi ctio w hall div imila fomula fo a patial poc with cotat but uow ma h modl udlig Odia igig i compact fom ad 5 wh th ow valu ha b placd b th uow With fc to N A i 993 th Odia igig olutio pt th Bt homogoul Lia wal Ubiad dictio homblu of ad ca b obtaid fom th w modl 5 b wightd latqua ladig to th omal quatio i dual tm fom g 53 o to th pimal tm ψ 54 with ψ g 55 ad 56 h ma qua pdictio o matix ult i MSE I ψ 57 with I Futhmo w obtai th two idual vcto a 6 ad 6 with thi dipio matic 6 63 ad th covaiac matix Buhad Schaffi

7 899 Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam 64 A poibl though ot cail ubiad timat of th vaiac compot would ow b 65 I th followig lt u fit aum a outli i th -th obvatio ladig to th modifid modl 66 h w ducd omal quatio aft limiatig would th ad 67 fom which w obtai I viw of 68 a appopiat liabilit umb ma b dfid a 7 which i th pctag of th timatd outli iz foud i th copodig idual thb mauig th i liabilit Fo th out liabilit w comput th wightd dviatio 73 followig 7 xcpt that th timatd outli iz ha b placd b th maximum o-dtctibl outli fo giv α ad β ; ctio fo mo dtail Uig 6 thi fomula ca b futh implifid to 74 with th obviou ductio i ca of a diagoal wight matix I th dual ca of a outli i th ma fo th -th poit i E w tat fom th modifid modl 75 but aiv at a imila t of omal quatio aft limiatig fit 76 fom which w obtai th immdiat copodc Buhad Schaffi

8 Buhad Schaffi ad 77 Howv ic w ow ipct ath tha to dtct th outli w hav to u latio 6 to pt th timatd outli iz a 78 thu povidig u with th omalizd liabilit umb 79 which idicat th pctag of th timatd outli iz a i th copodig idual It i ow taight-fowad to div th ffct of th maximum o-dtctibl outli i th wightd om alog imila li a bfo ladig to 8 4 ONLUSIONS W hav divd omalizd liabilit umb fo patial poc that idicat th i liabilit i th pottial to dtct outli ith b ipctig th obvatioal idual o th idual fo th poc ma i th ampl poit B a impl latio mau fo th out liabilit ca alo b gaid fom thm It appa that gall paig outli ca b dtctd mo ail i th cotxt of Simpl igig tha fom Odia igig hi lo i itivit d to b tudid i th futu REFERENES Baada W 968 A tig ocdu fo U i Godtic Ntwo Nth Godtic ommiio ubl o God Nw Si Vol No 5 lft/nl Baada W 976 Rliabilit ad pciio of two VII Itl ou fo Suvig Egiig amtadt/gma i N A 993 Statitic fo Spatial ata Wil Nw Yo tc El-Haim S F 98 ata oopig with wightd obvatio Itl Achiv of hotogammt III-4 pp 6-33 Föt W 983 Rliabilit ad dicibilit of xtdd Gau-Maov modl Gma Godtic omm A-68 Muich Föt W 985 h liabilit of bloc tiagulatio hotog Egg & Rm S 5 pp och R 988 aamt Etimatio ad Hpothi tig i Lia Modl Spig Nw Yo/Bli tc lz H Som citia fo th accuac ad liabilit of two Gma Godtic omm B-5 Muich pp Rao R ad H outbug 995 Lia Modl Lat-Squa ad Altativ Spig Nw Yo/Bli tc Schaffi B 988 ciio Rliabilit ad Sitivit i Italia; i F oilla/l Muio d ogttazio Ottimizzazio dl Rilivo opogafico Fotogammtico d otollo Itl t of Mchaical Sci ISM Udi/Ital pp 9-59 Schaffi B 997 Rliabilit mau fo colatd obvatio J of Suvig Egg 3 No 3 pp 6-33 Wag J ad Y h 994 O th liabilit mau of obvatio Acta Godat t atogaph Siica Eglih Editio pp Itatioal Achiv of hotogammt ad Rmot Sig Vol III at B4 Amtdam

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