Estimation of atmospheric dispersion sources from sensor data with Bayesian methods

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Aron Ward
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1 Etimatio of atmohri dirio our from or data with Baia mthod Lif Pro Swdih Df Rarh Ag FOI Diviio for CBRN Df ad Surit Umå Swd Abtrat Etimatio for our haratriti for a atmohri rla i oidrd whr or data ad mtorologial iformatio ar tak ito aout uig a Baia tatitial mthodolog. Kword: atmohri dirio or data our timatio Itrodutio Dirio modl ar ud to timat th air otratio from a rla of a toi ubta whih i ar to b abl to timat th halth haard for od ol. Howvr dirio modl rquir th our loatio ad our trgth to b ifid. I a th oitio ad/or trgth of a our i ukow but otratio maurmt ar availabl from a or twork w ma u a ivr dirio modl to timat th our trgth ad oitio. Thi i a tial art of a haard maagmt tratg for atmohri rla of airbor toi ubta if ar or twork ar to b ud. Thi ar rt a fat ivr dirio modl bad o Gauia lum modl ad omutatio of robabilit diti for th ought our aramtr. A robabiliti framwork for atmohri dirio modllig A atmohri dirio ro i imml oml du to th wid rag of al ivolvd ad th haoti haratr of th atmohri turbul. H th o of a atmohri dirio modl i limitd i it aot b grall modlld a a lod tm. Som hiloohial imliatio of thi i diud i []. Havig thi i mid ad bau of th urtait ad ar of th iut data a tatitial mthodolog i alld for. Eah dirio modl or a tatitial mbl of ituatio haratrid b rtai aramtr whih ma b laifid ito Sour haratriti oitio trgth t of agt t. olltivl dotd b. Thi might ilud ditributd our lik ool of irrgular ha t. I th Gauia modl oidrd hr th our i haratrid b a our trgth a our oitio ad a "virtual our" oitio whih dtrmi th iitial ha of th lum. Sor twork haratriti or oitio avragig tim t olltivl dotd b. Som of th or twork od might rrt oit whr variou otratio avrag ar to b timatd.g. for rik timatio whilt othr twork od might rrt hial or to b ud for modl amt. Eviromt haratriti mtorolog ad trrai haratriti olltivl dotd b. Th Gauia modl oidrd hr aum flat horiotall homogou trrai

2 haratrid b o aramtr th rough lgth ad horiotall homogou mtorologial oditio haratrid b ma wid dirtio ma wid d ad Paquill tabilit la. Mor oml modl might u thr--dimioal dirtid high--rolutio mtorologial fild from umrial wathr forat modl ad trrai data from gograhial databa. Not that diffrt dirio modl ar bad o diffrt mbl ad h u diffrt aramtr t. Th outut of a dirio modl i idall th oditioal robabilit dit for th modlld or data. Thi dit tll u th robabilit of a rtai t of otratio maurmt giv th our haratriti th twork haratriti ad th mtorologial ad toograhial oditio. I rati ol rtai fatur of th oditioal robabilit dit ar timatd i a fw a b fild rimt mot ommol th tatio ma but alo th varia ad highr ordr tatitial momt. H th robabilit dit i modlld b a vr mall aml from th mbl ad alwa roviioal but rfltig th urrt "tat of th art". 3 A Baia robabiliti framwork for ivr dirio modllig A ihrt fatur of a ivr dirio modl i it ill-od i.. itivit to iut data whih mak it ar to al om rgulariatio tratg ad whih t a ihrt limit o th aura of th ivr modl. Baia tatitial mthod rovid o uh rgulariatio tratg f. []. Th ida of Baia tatiti i th followig. Duall oidrd a a futio of th oditioal robabilit dit for i alld th liklihood futio l ad o wa of timatig i to dtrmi th maimum of l maimum liklihood timatio. Th otrior dit i obtaid from th liklihood futio b Ba' thorm l Whr i th rior oditioal robabilit dit. H u to a roortioalit otat th otrior robabilit dit i th rodut of two fator th liklihood futio rovidd b th dirio modl ad th rior robabilit dit a ubtiv robabilit whr th ur ma od a riori blif about th our haratriti. I a th ur i idiffrt about th our haratriti th o-iformativ rior ma b ud i whih a th otrior robabilit i roortioal to th liklihood. Th margial dit d i a ormaliatio fator iddt of th our haratriti. Thi fator i ot dd if Mot Carlo timatio.g. Markov hai Mot Carlo or qutial Mot Carlo i ud. To aout for dttio limit of th or ad to b abl to u alo ro otratio data w modl th robabilit dit futio for b a otiuou art for otratio abov a dttio limit ad a igular art ituatd at ro otratio i.. w hav a robabilit dit futio of th form q δ { > } r ddig o a giv dttio lvl >. Aordig to th gralid Ba formula.g. [3] th otrior oditioal robabilit dit futio i giv b q r {} [ q d r d Similarl for vral or aumig tatitial idd w hav

3 d r q r q 4 A tatitial Gauia lum dirio modl W aum that th or aramtr ar th or oitio ad a avragig tim whih i uffiitl log for timatig th ma otratio aroimatl miut i... Th our aramtr ar th otat our oitio YZ ad th otat our trgth Q o YZQ. Th viromtal aramtr ar odd i th varia futio ad ad th ma wid d U i.. U. For imliit w aum that th oordiat tm i aligd with th ma wid dirtio o that th ma wid i i th dirtio. W aum that th avrag otratio i giv b a Gauia lum modl > } { Z Z Y U Q π Th lid ormal ditributio with aramtr ma ad tadard dviatio of th udrlig ormal ditributio ad th liig lvl i dfid b th robabilit dit futio R Q P δ Whr Q ad > R { } Ad ad Ф dot th tadard ormal robabilit dit futio ad it umulativ ditributio futio rtivl i.. / rf π To omut Ф ffiitl for 6 / < w u th ri aroimatio for rf i [4]. W gt th firt ad od tatitial momt ] [ ] [ d P E d P E Giv amld timat of E[] ad of E[ ] for a or w ma ow timat b olvig th oliar tm of quatio for

4 F G If ol i giv a b timatd from tial valu of th dimiol momt M / o th lum trli f. [5]. Th Jaobia of th maig F G a b omutd liitl i trm of ad Ф o w a ffiitl olv for b hooig ad al Nwto--Raho itratio. If. w rgulari b hooig Figur how thti data a lum from a our rd quar with thr or loatio markd with llow dot. Figur how th otrior robabilit dit futio for horiotal our oitio bad o rturbd thti or data with or loatio markd with llow dot ad th tru our loatio markd with a rd quar. 5 Coludig rmark W hav dmotratd th faibilit of a Baia mthodolog bad o th imlt kid of dirio modl Gauia lum modl. Th tatitial modl i roviioal ad rv mail to rgulari th ivr modl. To validat th tatitial modl furthr tudi mut b do for diffrt or ad mtorologial oditio i th radom variatio of th maurd otratio i a ombiatio of radom of th uobrvabl radom otratio fild ovtd b th turbult wid ad radom i th maurmt ro i th or. Howvr th lid ormal ditributio i a ommo aroimatio i th litratur f. [6-]. Limitatio of thi modl ar oitd out i [-]. Erimtal data for otratio flutuatio ar dribd i.g. i [5] [3]. Rfr [] Ork N. Shradr-Frhtt K. ad Blit K. Vrifiatio Validatio ad Cofirmatio of Numrial Modl i th Earth Si. Si Fbruar 994 Vol [] Etig Ia G. Ivr Problm i Atmohri Cotitut Traort. Cambridg Uivrit Pr. [3] Litr R. S. ad Shirav A. N. Statiti of Radom Pro I Gral Thor. Srigr.

5 [4] Tllambura C. ad Aamalai A. Effiit Comutatio of rf for Larg Argumt. IEEE Traatio o Commuiatio Aril Vol. 48 No [5] Nil M. t al. Cotratio Flutuatio i Ga Rla b Idutrial Aidt Fial Rort. Thial Rort Riø R 39EN Ma Riø Natioal Laborator Rokild Dmark. [6] Johao G. t al. Dami Baia Modl via Mot Carlo A Itrodutio with Eaml. Thial Rort UCRL TR Lawr Livrmor Natioal Laborator U.S.A. [7] Lwll W. S. ad Sk R. I. Aali of Cotratio Flutuatio from Lidar Obrvatio of Atmohri Plum. Joural of Alid Mtorolog Augut 986 Vol. 5 No [8] Mol N. Som Itrtio btw Turbult Dirio ad Statiti. Eviromtri 99 Vol. No [9] Robi P. t al. A Probabiliti Chmial Sor Modl for Data Fuio. I 5 8 th Itratioal Cofr o Iformatio Fuio 5 Vol.. 6. [] Robi P. ad Thoma P. No Liar Baia CBRN Sour Trm Etimatio. I 5 8 th Itratioal Cofr o Iformatio Fuio 5 Vol.. [] Chatwi P. C. Som Rmark o Modllig th PDF of th Cotratio of a Dirig Salar i Turbul. Euroa Joural of Alid Mathmati Vol [] Chatwi P. C. Sigular PDF of a Dirig Salar i Turbul. Flow Turbul ad Combutio 4 Vol [3] Lug T. t al. Maurmt ad Modllig of Full Sal Cotratio Flutuatio O fild Erimt uig Krto 85 ad Ttrahdrothio a Trar. Agrarthih Forhug Vol. 8 No.. E5 E5.

POSTERIOR ESTIMATES OF TWO PARAMETER EXPONENTIAL DISTRIBUTION USING S-PLUS SOFTWARE

POSTERIOR ESTIMATES OF TWO PARAMETER EXPONENTIAL DISTRIBUTION USING S-PLUS SOFTWARE Joural of Rliabilit ad tatistial tudis [IN: 0974-804 Prit 9-5666 Oli] Vol. 3 Issu 00:7-34 POTERIOR ETIMATE OF TWO PARAMETER EXPONENTIAL DITRIBUTION UING -PLU OFTWARE.P. Ahmad ad Bilal Ahmad Bhat. Dartmt