A Hierarchical Bayes Model for Combining Precipitation Measurements from Different Sources

Transcription

1 A Herarchcal Bayes Model for Combnng Precpaon Measuremens from Dfferen Sources Tamre P. Cardoso and Peer Guorp NRCSE T e c h n c a l R e p o r S e r e s NRCSE-TRS No. 8 March, 7

2 A Herarchcal Bayes Model for Combnng Precpaon Measuremens from Dfferen Sources Tamre P. Cardoso and Peer Guorp ABSTRACT Surface ran rae s an mporan clmac varable and many enes are neresed n obanng accurae ran rae esmaes. Ran rae, however, canno be measured drecly by currenly avalable nsrumenaon. A herarchcal Bayes model s used as he framework for esmang ran rae parameers hrough me, condonal on observaons from mulple nsrumens such as ran gauges, ground radars, and dsromeers. The herarchcal model ncorporaes relaonshps beween physcal ranfall processes and colleced daa. A key feaure of hs model s he evoluon of drop-sze dsrbuons (DSD) as a hdden process. An unobserved DSD s modeled as wo ndependen componen processes: ) an AR() me-varyng mean wh GARCH errors for he oal number of drops evolvng hrough me, and ) a me-varyng lognormal dsrbuon for he sze of drops. From he modeled DSDs, precpaon parameers of neres, ncludng ran rae, are calculaed along wh assocaed uncerany. Ths model formulaon devaes from he common noon of ran gauges as ground ruh ; raher, nformaon from he varous precpaon measuremens s ncorporaed no he parameer esmaes and he esmae of he hdden process. The model s mplemened usng Markov chan Mone Carlo mehods. INTRODUCTION Surface ranfall s an mporan envronmenal varable ha s ncorporaed across many areas of sudy, ncludng meeorology, clmaology, agrculure, land use, and hydrology, for example. The varous felds of sudy requre esmaon of precpaon a a range of emporal and spaal scales. Hydrologss and land use planners may be neresed n shor-erm ranfall n relavely small regonal areas for sudes nvolvng flood and flash flood forecasng. Researchers n clmaology or agrculure may be neresed n clmac sudes ha focus on weekly, monhly, or annual oals over large spaal exens. Alhough of consderable neres, surface precpaon raes and amouns are dffcul o esmae. Unlke oher amospherc varables, ranfall can dsplay exreme heerogeney n space and me. Boh he occurrence of precpaon and he rae a whch falls may be hghly varable, even whn a sngle ran even. As such, due o he mporance of and dffcully wh esmaon, here has been much research n he area of precpaon measuremen and esmaon. Precpaon s measured usng many dfferen nsrumens. Some nsrumenaon measures ranfall drecly, whle ohers make ndrec measuremens of quanes ha can be relaed o ranfall. All of he nsrumenaon, however, make ndrec measuremens of he usual

3 quany of neres, namely, ran rae. The mos common nsrumens nclude ran gauges and ground-based scannng radar. Dsromeer n su measuremens and saelle deployed nsrumenaon are also used, alhough less commonly. Each of he nsrumens has nheren srenghs and weaknesses, whch affec he observed measuremens and subsequen esmaon of surface ranfall. There exs many algorhms, each wh s own se of assumpons, for calculang ranfall relaed parameers from he nsrumenal observaons. Many mes he observaons obaned from one nsrumen are used n he calculaon of esmaes based on he observaons from anoher nsrumen. Ran gauge daa are ofen consdered o be ground ruh and are used o adjus he esmaes based on oher nsrumens. Ths s done despe he known fac ha observaons from ran gauges are nherenly prone o errors and ha hey are no more rue han he observaons from oher nsrumens. Surface precpaon can be defned by a populaon of fallng drops. The dsrbuon of he sze dameers of he drops characerzes he populaon behavor; hus, he drop-sze dsrbuon (DSD) forms a basc descrpor n he modelng of ran mcrophyscs. The DSD s used o compue a varey of derved properes of ranfall hrough mahemacal relaonshps; common compued properes nclude waer conen, ran rae, and reflecvy. DSD daa, however, are no colleced on a roune bass due o he expense of dsromeer nsrumens and he lmed spaal coverage of an ndvdual nsrumen. Ran gauge measuremens and ground radar mages are much more readly avalable. The leraure s full of modelng approaches for he esmaon of surface ranfall, ncludng emprcal sascal models (Sern and Coe 984; Skaugen e al. 996, for example) and a varey of sochasc models (Bell 987; Rodrguez-Iurbe and Eagleson 987; Cox e al. 988; Rodrquez-Iurbe e al. 988; Smh 993, for example). More recenly, herarchcal Bayesan models have been used n a number of envronmenal applcaons (Royle e al. 998, Wkle 998, Berlner, Berlner e al. a, Wkle e al., Hrafnkelsson 3). In hs arcle we presen a generalzed model for surface ranfall ha accommodaes daa from mulple nsrumens and produces esmaes of precpaon parameers and assocaed unceranes. Usng a herarchcal Bayesan approach, he evoluon of DSDs are modeled as an unobserved process, whle also ncorporang nformaon from daa gahered hrough commonly deployed nsrumens. Ths formulaon does no depend on havng expensve DSD daa and depars from

4 he noon of gauge daa as beng ground ruh. Raher, an esmae of he drop-sze dsrbuon a each pon n me, wh aendan uncerany measures, can be used o calbrae all he nsrumens smulaneously. HIERARCHICAL MODEL Surface ranfall parameers hrough me are esmaed usng a flexble fve-sage herarchcal Bayesan formulaon ha s based on a model for generang ranfall drop-sze dsrbuons. Esmaes of he unobserved process, denoed by N(D), are he prmary quany of neres. The frs sage of he herarchy specfes a measuremen error model for he observaonal daa, denoed by and G, boh of whch are observaons of funcons of N(D), wh error. The second sage of he herarchy allows for a me seres formulaon of he unobserved DSD process. Tme seres parameers and emporal dynamc erms of he DSD evoluon process are modeled n he hrd sage. The herarchcal Bayes formulaon s compleed n sages four and fve by specfyng prors on model parameers. Emprcal Analyss of DSD Daa Exploraory daa analyss was conduced on a se of one-mnue drop sze specra colleced usng a Joss-Waldvogel dsromeer (JWD) from a se n Eureka, Calforna from January hrough March 999. Ran raes were calculaed for each of he one-mnue specra o denfy ranfall evens. Seven evens rangng from o 4 hours n duraon were chosen for evaluaon. Barplos of he bnned drops and me seres of he oal number of drops/mnue were evaluaed for each even. Two general observaons were made:. The dsrbuons of drop-szes were generally unmodal and rgh-skewed. Based on vsual nspecon and fve number summares, he dsrbuons of he shapes of he onemnue DSD daa were smlar boh whn and across ranfall evens. Fgure shows barplos for some one-mnue DSDs.. The oal number of drops from one-mnue o he nex was hghly varable boh whn and across evens. Log-ransformed me seres daa suggesed an underlyng auoregressve process. Fgure shows some me-seres plos for oal number of drops. Unobserved Process A DSD, denong he number of drops per mm dameer bn nerval and per defned as 3 m of ar s well ( ) N D = ( ) ( ) n D AV D D () 3

5 where ( ) n D s he observed number of drops (raw couns) n each dameer bn nerval samplng nerval ; A s he sample surface area n D for m ; s he lengh of sample collecon n seconds; V ( D ) s he ermnal fall speed n sll ar n m/s; and, D s he drop dameer nerval n mm. Now we defne he number of drops over dameers D as ( ) ( ) n D = n P D () where n s he oal number of raw couns over all drop dameers for he me nerval and s assumed ndependen of P( D ), a probably densy funcon defnng he shape of he DSD over he nerval ; P( D ) gves he probably of fndng a drop whn a dameer D o D + dd. Subsung equaon no equaon yelds ( ) N D = n P( D ) ( ) AV D D (3) Nex, defne N as he oal number of drops per he DSD formulaed n erms of oal couns as 3 m of ar. Subsuon no equaon 3 gves ( ) N D = N ( ) P D D (4) For a ranfall even of duraon T, he unobserved saes N and N ( D ) are esmaed hrough models for P( D ) ; =,,T. To be conssen wh commonly colleced dsromeer daa, each represens a one-mnue nerval. Model for Toal Number of Drops, Le N N denoe he oal number of drops a me mnues, =,,,T. To accoun for he seral correlaon and burs varably observed n many of he me seres of oal number of drops n conguous DSD specra, we nroduce a model composed of a me-varyng mean process negraed wh a GARCH(,) condonal varance process. Le TN = ln N = +. N 4

6 The s are an ndependen sequence of correlaed errors from an unknown dsrbuon. The values, condonal on all prevous values are assumed o be normally dsrbued:,, N,# ( ). Under a GARCH(,) model = a + a # + b # where he parameers a, a, and b are subjec o he consrans a >, a, b, a <, and a + b (Alexander 998). By subsuon, ( ) ( ) N,a + a + b, N,a + a + b #, #,,, N (,a + a # + b # ) Gven he condonal dsrbuon of,,,,, he jon probably for s: #$,,, % & = # $,, % # &,, % $ 3 & #$ % & # % = (' # $ +,,,, % &) #$ % & $ = & #,. % = (. ' exp - ( * ) ) ( = $ /. + ). * exp, - ( ) & / Assume ha condonal on ndependen and are dsrbued as N (, ) Se N and he parameers N { a, a, b, # } N TN # $ = TN,,TN T N,, NT ;% ' & = ' ' TN N T = =, he ln N s are. Then he jon dsrbuon of ln N s:,,% &T ( ) + - exp,* TN )% & % * N.- & =. Ths reduces ( ) - / /. T, = # $( P o he consan erm % ' exp & $ TN # $ N (' # /# - ( ( exp,* TN - )% $ & % * N.- & ( ) # and yelds: ) ' * 3 +' # ( ) / - - 5

7 Prors on GARCH parameers Snce he GARCH parameers are consraned o be greaer han zero, each coeffcen s assgned a lognormal pror: a a b : Lognormal, : Lognormal, : Lognormal, ( # ) ( # ) ( # ) The hyperparameers,,,, and, # # # are aken as fxed, wh values ha may vary by applcaon. Mean process for number of drops The mean for N N N N TN, = a + where varance N, s modeled as a frs-order auoregressve process. Le N s an ndependen random normal process wh mean and. The parameers are { a N,#} =. Condonal on a N and, N N a N N #$ % &' N a (,( N N ) ); ( ) IG ( ),* ) The parameer a N s esmaed assumng a N N ( N, N ). The hyperparameers,#,, and # are aken as fxed, wh values ha may vary by applcaon. N N DSD shape, P( D ) The DSD shape s modeled as a lognormal dsrbuon wh me varyng parameers and. Condonal on and, he D s are assumed ndependen yeldng he jon dsrbuon for D as ( ln d & ) T $ % ( D, ) ' ', - = exp *& + = # d '. '/ where N (, ); # IG ( $ #,% # ). The hyperparameers fxed, wh values ha may vary by applcaon. # $ % are aken as,,, and 6

8 Daa Componens The herarchcal model ncorporaes wo common daa sources, namely, ran gauge and ground radar observaons. Snce gven a DSD he nsananeous (rue) ran rae and a derved (rue) radar reflecvy can be calculaed, we develop a measuremen error model for boh he ran gauge and radar observaons. Gauge Observaons, G Ran gauges measure he amoun of ranfall accumulang over me a a fxed locaon n space. If he amoun of ranfall measured a a pon s measured whou error, he gauge measuremen, G, can be equaed o he approprae me negral of ran rae. The predomnan error n gauge measuremens s a sysemac bas nduced by wnds, whch ofen resuls n an underesmae of he surface ranfall. Thus, defne a measuremen error model for gauges as a funcon of he rue ran rae and wnd speed (akng oher sysemac bases as neglgble). Defne a gauge observaon as $ % & & G = ' c w + R ds ( + ) * ( ) s G & # & where R s he derved nsananeous ran rae a me ; ( ) c w are gauge ype-specfc coeffcens ( c ) prmarly based on wnd speed, w; and errors modeled as ( N, G) ; G IG( G, G) :. G are ndependen measuremen Gven a DSD, he nsananeous ran rae can be derved based on meeorologcal prncples as a funcon of he hrd power of drop dameer (Baan 973). The ran rae, R, for a gven, s esmaed as 3 = R ( ) ( ) 6 # R c D V D N D dd where D s he drop dameer n mm; V(D) s a deermnsc funcon for he ermnal velocy for drops wh dameer D; per N ( D ) s he DSD gvng he number of drops per mm dameer D 3 m a me ; and c R s an addonal consan o accoun for uns. 7

9 Gven he model for ( ) N ( D ) yelds: where P( D ). N D (equaon 4), summng over all D 3 R 6 N s he oal drop concenraon a me and and subsung N P( D) for R = c ' D V ( D) # % N f ( D) $ & dd (5) f ( D ) s he pdf for D defned hrough Assumng an ndependen Gaussan measuremen error model, and condonal on ran rae as derved from a DSD and parameer { G G} case G (, N m G G) : where measuremens, of he nsananeous ran rae defned above. Ground Radar Observaons, =, he elemens of G are ndependen and n each m G s he negral over he me nerval beween gauge The equvalen radar reflecvy observaons from ground radar,, are a funcon of he average reurned power and range of he radar scan (Baan 973). If here are no measuremen errors, wll conform o he reflecvy derved from a DSD. To accommodae varous facors ha can lead o errors when esmang, we consruc a smple measuremen error model for equvalen radar reflecvy as a funcon of he derved reflecvy and random errors; D s he meeorologcally derved reflecvy a me and errors modeled as ( N, ); IG(, ) :. The reamen of derved reflecvy, = + where D are ndependen measuremen D, s smlar o ha for ran rae. Gven a DSD, D can be derved as he sxh momen of he DSD (Baan 973). Derved reflecvy a me s esmaed as 6 D = ( ) c D N D dd where D s he drop dameer n mm; N ( D ) s he curren drop specra gvng he number of drops per mm dameer D per 3 m a me ; and c s a consan relaed o sample volume and 8

10 3 equals one for a one- m sample. As for gauge observaons, summng over all subsung N P( D) for ( ) N D yelds: D and 6 D = ( ) & c D $ N f D # % dd where varables are as prevously defned. In pracce, D s esmaed as a sum over bnned drop szes. Assumng hs ndependen Gaussan measuremen error model, and condonal on D derved from a DSD and parameer { } each case N ( D, ). Model Summary =, he elemens of are ndependen and n We use he componens presened above o consruc he herarchcal model formulaon. A he hghes level we assume, based on a seres of condonal ndependence assumpons, ha gven a DSD ( N( D ) ), he gauge (G) and radar () observaons are condonally ndependen wh he followng facorzaon: ( ),, = ( ), ( ) # N D G $ # N D $ G N D # $ The componens G N ( D) and N ( D) ( ) ( ) ( ) = # N D $ G N D N D # $ # $ # $ # $ represen lkelhoods based on he gauge and radar daa, respecvely. The frs componen, N ( D) # $ represens he pror probably for he DSD process. The condonal probables for he full model, ncludng he parameers and hyperparameers, are summarzed n Table. 9

11 Table. Herarchcal Model Summary. Varables Observaonal Daa Hdden Process Temporal Dynamcs for mean of TN Model Parameers b G N ( D ) # $ G % ( ) N D # $ % TN N, # $ N % $ D D #% # $ N % Condonal Probables for Model Componens a c G,, N, D, H G # H G # H N # H N # H D # $ # % = $ % $ % $ % $ H $ % D % where H s a collecon of hyperparameers and { } ; G = G = $ G G # % & G ' { } ; = = $ # % & ' $ N = { a, a, b,% } = ' a ; & ( ' a ; & ( ' # # ; & ( # '% ( ) * ) * ) * ) * # {, } ; ; = an $ = & a N N % ' & $ N % ' ( ) ( ) # {, } ; D = = ' $ ( ' % ; & ( ) * ) * Hyperparameers & H = { G, ' G,,',,,, ',',', N,', #,'#, $,( $,%, %} N a The bracke noaon # $ s used as a shor hand for denong condonal dsrbuons. b N P D. c N ( D ) s general noaon for ( ) TN = ln N.

12 Parameer Esmaon We used a Markov Chan Mone Carlo (MCMC) mplemenaon o sample from he poseror dsrbuon of he herarchcal model. The approprae Markov chan s consruced usng a sngle-componen Meropols-Hasngs algorhm based on he herarchcal srucure of he model, and on he condonal probables assocaed wh he varous model componens. Sarng values, hyperparameer values, he lengh of burn-n, and he number of eraons mus be deermned for each applcaon. Sarng values are random, based on he pror dsrbuons of he parameers; he dsrbuons can be alered by he choce of he fxed hyperparameers. The requred number of eraons for a gven applcaon depends on he lengh of he burn-n, he auocorrelaon of he parameers, and he number of samples needed for parameer characerzaon. Convergence s monored usng vsual analyss of ploed parameers n wo ways: ) by performng mulple runs usng dfferen random sars; and ) runnng mulple chans usng he same sarng values. APPLICATION TO EUREKA RAINFALL Daa The Eureka se consss of hree sources of daa: dsromeer, ran gauge, and ground radar daa. The dsromeer daa were colleced usng a JWD nsrumen. The dsromeer was locaed near he Eureka radar se a 4 48'N/4 9'W. Each daa fle conaned one hours worh of raw bn couns colleced a one-mnue nervals. The raw bn couns were convered o one-mnue DSD specra couns 3 N ( m mm ) by adjusng raw couns for he nsrumen collecon area, D me, ermnal fall speed, and drop dameer. A se of processed -mnue dsromeer daa was also avalable. The -mnue daa are drop specra ha have been averaged over conguous one-mnue perods. The observed -mnue drop specra were used o calculae ran raes and reflecvy values o use n place of gauge and radar observaons for model verfcaon. Hourly gauge accumulaon daa (TD-34) were obaned from he Naonal Clmac Daa Cener (NCDC) on-lne servce. The daa ncluded measuremens from 4 gaugng saons n he vcny of Eureka, Calforna over he perod January 999 hrough 3 March 999, of whch nne saons were n he vcny of he radar and dsromeer. Daa were colleced usng Fscher-Porer precpaon gauges wh auomaed readous. Precpaon daa from he NCDC daabase are qualy checked and eded, as necessary, by an auomaed and a manual ed

13 (Hammer and Seurer 998). To concde wh he me doman only mplemenaon of he model, gauge daa were lmed o measuremens from he gaugng saon locaed closes o he dsromeer a 4 49'N/4 'W. Level II WSR-88D radar scans were ordered from NCDC for he Eureka radar se, whch s locaed a 4 9'54''N/4 7'3''W, for perods concdng wh ranfall evens denfed by gauge and dsromeer daa. There was a radar scan approxmaely every sx mnues for several elevaon ncremens. The sphercal coordnae Level II daa were nerpolaed o a km x km Caresan coordnae sysem of equvalen reflecvy (db) values usng sandard programs for NEXRAD radar. The nerpolaed daa were only avalable for one of he denfed ranfall evens; he avalably of radar daa defned, and lmed, he me perod over whch he model was mplemened. Approprae reflecvy values needed o be exraced from he km x km grdded values o correspond wh he fxed spaal locaon of he dsromeer and closes ran gauge. Reflecvy n he me doman was exraced from he spaal daa by frs denfyng he closes pxel locaed vercally above he dsromeer and nearby ran gauge. Gven he uncerany n he vercal flow of ranfall above he surface of he earh o a pon on he ground, an equvalen reflecvy was calculaed as he aeral average of frs and second order neghbors usng he denfed pxel and he egh neghborng pxels. Issues of beam blockage along he pah of he radar o he locaon of he dsromeer were consdered o be neglgble due o he gradually ncreasng naure of he coasal erran of Norhern Calforna. Therefore, he reflecvy values were exraced from he lowes radar beam (l elevaon of.5 o ). We mplemened he herarchcal Bayes model usng daa from a nne hour (55 mnues) ranfall even on 6 February 999. For he course of he model run here were nne hourly gauge readngs and 9 ground radar observaons. Specfcaon of Hyperparameers, Inal Values, and Consrans on Parameers We used exploraory analyss of dsromeer daa from fve February ranfall evens o come up wh nformave means and varances for he fxed hyperparameers. The consan hyperparameer values are summarzed n Table wh model resuls.

14 In addon o he hyperparameer values, we specfed several = sarng values as follows: = = =. We based nal condons for he remanng,., and N 3. parameers on random draws from he approprae pror dsrbuons. We nroduced cuoffs for wo ypes of consrans; namely, consrans due o ranfall dynamcs/nsrumenaon and consrans due o sascal consderaons. Consrans based on ranfall dynamcs ncluded upper bounds for hree parameers:. We lmed he oal number of drops n a one-mnue nerval o 6., abou.5% above he maxmum value observed from he dsromeer daa. Ths corresponds o a oal of exp 6 43 raw coun drops when summed over he drop dameer bns. We ( ) appled hs cuoff o he parameers TN and N.. We lmed he upper bound for drop dameer sze such ha P( D 5).. Ths lmaon was mposed snce he ran even of neres represened lgh ranfall. The dsromeer daa had no raw couns beyond a mean drop dameer of 3.5 mm. Proposals for parameer updang are drawn from unform dsrbuons cenered a he curren parameer values. Ths mplemenaon made possble o draw negave-valued proposals, dependng on he curren value of he parameer. Consrans based on sascal consderaons hus ncluded lower bounds on varance parameers ( ) model consrans were also placed on he hree GARCH coeffcens. Esmang m G and D > and >. Requred A each me perod for whch here were gauge or radar observaons, he daa componens of he model requred compuaon of he wo quanes m G and D, he rue equvalen gauge value and he rue derved reflecvy, respecvely. Boh of hese values are deermnsc quanes based on he curren sae of he parameers ha defne he hdden DSD process. We calculae he rue equvalen gauge value as he sum over he prevous 6 mnues of each of he nsananeous ran raes n mm/mn derved from he sae of he DSD before each pon n me. The nsananeous ran raes defned by equaon 5 are approxmaed by summng over he drop dameer bns ha would ypcally be observed usng a JWD nsrumen as follows: { TN } $ P( D ) 3.6 exp # R D V D D &( ') 3 = * & ' ( )% 6 = % D 3

15 where ( ) P D s based on he cdf of a lognormal dsrbuon wh parameers ermnal fall speed, V ( D ) drop dameer for bn. Then, was calculaed as and. The D (Smh 993), where D s he mean = 6+ m G = True derved reflecvy s calculaed as an average, over he prevous mnues, of he sxh momen of he DSD. As wh he equvalen gauge calculaon, we approxmae derved reflecvy by summng over he drop dameer bns usng 6 R Then D, n uns of { TN }# P( D ) D D %' &( exp 6 = ) % & $ = $ D = + D = 6 3 mm m, are convered o db as log ( D ). Tunng, Thnnng, and Convergence Tunng he chan was opmzed by makng many small runs (5 o 5 eraons n lengh) and monorng he accepance raes for each of he parameers. Adjusmens o he accepance raes were made by eher ncreasng or decreasng he upper and lower bounds on he unformly dsrbued parameer proposals. Afer much expermenaon, bounds on he unform proposals were adjused o yeld accepance raes rangng from abou.4 on he low end o abou.85 on he hgh end. Mos parameers had accepance raes n he range of Whle hese values exceed he commonly recommended values of. o.5 for updang muldmensonal componens (Robers 996), we found ha mxng and convergence performed beer a hese hgher levels. We used hnnng o oban roughly ndependen samples from he jon poseror dsrbuon. We used gbbs (Rafery and Lews 995) o ge esmaes of he deal spacng beween eraons and he number of burn-n eraons, as well as he number of eraons for a desred precson. In mos cases a hnnng value of 5 was more han suffcen o oban 4

16 approxmaely ndependen samples, alhough on occason, a parameer hnnng value would h 3 or so. We monored convergence by vsual nspecon of ploed oupu. The number of burnn eraons recommended by gbbs was generally low and always < 5. Convergence was also assessed by lookng a he pah of several runs usng boh he same and dfferen sarng values. Resuls Model Verfcaon There s no way o specfcally valdae he model n erms of comparng oupus o rue values. Pror o runnng he fully mplemened model, however, we conduced model verfcaon by assessng he ably of he model o realscally reproduce argeed sgnals and negraed parameers assocaed wh surface ranfall. We replace NEXRAD radar observaons wh derved reflecvy calculaed from -mnue dsromeer daa. Ran gauge daa are omed for hs assessmen. To maxmze he npus we use a -mnue me sep, hus nsurng a daa value a each me sep. The verfcaon run ncludes 49 derved reflecvy observaons compued from he one se of dsromeer daa. The MCMC run was ended afer 8, eraons followng, burn-n eraons. Based on graphcal oupu, all parameers converged whn he, burnn perod. Resuls for a ypcal model verfcaon run showng me ndependen parameer esmaes and me varyng parameer esmaes (usng he 8 MCMC samples generaed afer applyng a hnnng value of 35) are shown n Tables 3, respecvely. Furher, usng he poseror parameer esmaes for he hdden process we can compare calculaed negraed ranfall parameers wh values obaned from he -mnue dsromeer daa. Fgures 3 and 4 show he comparsons for ran rae and reflecvy. Noe ha alhough we compare he model oupu wh quanes derved from dsromeer daa, we do no consder he dsromeer daa as ground ruh. Dsromeer daa are subjec o measuremen errors. The poseror means of he hree GARCH coeffcens are all lower han hose specfed for he pror dsrbuons. The varably for he a and b coeffcens s slghly less han ha for he prors. The mean of he AR() coeffcen s slghly hgher han ha specfed n he 5

17 pror. The varance parameer for he AR() errors,, s consderably hgher han ha specfed n he pror, as s he measuremen error varance for radar ( ). The mean values for TN and N are abou he same and poseror esmaes of TN are close o expeced couns alhough he model ends o underesmae he couns. In general, he DSD dsrbuon parameers and are more varable han esmaes based on he dsromeer daa. On average, he poseror esmaes of dsromeer daa. The model ends o slghly underesmae are smlar o he values esmaed from he The poseror esmaes for boh negraed parameers (ran rae and reflecvy) are more varable han values calculaed from he dsromeer daa realzaon. Ths s expeced snce hese esmaes nclude he dsromeer daa realzaon wh some uncerany. Asde from he exra varably, he model esmaes are a good mach wh he negraed quanes calculaed from dsromeer observaons. The model does a good job of capurng he overall means and he sgnals presen n he me seres. Full Model Run The full model run ncludes 9 hourly ran gauge observaons and 9 effecve reflecvy observaons from ground radar. The MCMC run was ended afer 45, eraons followng a burn-n perod of 35, eraons. Afer applyng a hnnng value of 35, here were 7 MCMC samples for poseror esmaes. Convergence for he full model run s much slower han ha for he verfcaon run, akng up o abou 6, eraons for a few parameers. Wh he excepon of he AR() varance erm, eraons. The., all parameers appear o converge by 6, varance hardly moves unl 6, eraons, a whch pon he value sars bouncng around and hen seles down a abou.5, followed by more nermen bouncng. Ths paern could be due o slow mxng or denfably ssues. Resuls for a ypcal full model run showng me ndependen parameer esmaes and me varyng parameer esmaes are shown n Tables 3, respecvely, along sde he verfcaon run resuls. Tme seres resuls for negraed parameers calculaed from he esmaed parameers for he hdden DSD process are shown n Fgures 5 8. Excep for a, he poseror esmaes for he GARCH parameers are smlar o hose from he verfcaon run. The generally larger sandard errors reflec he greaer uncerany n 6

18 he one-mnue model, whch does no have daa a every me sep. The mean error erms,, are no sgnfcanly dfferen from zero based on 95% confdence nervals for he means. The poseror dsrbuons are relavely symmerc and approxmaely normal, whch s n lne wh he GARCH model for. The whn me poseror means for TN and N are essenally he same. The poseror dsrbuons for TN are nearly symmercal and appear o be approxmaely normally dsrbued. The represenave dsrbuons all have means close o he mean for he pror dsrbuon. The poseror dsrbuons are farly symmerc; values for are varable, rangng from -.8 o. and do conan he observed esmaes based on he one realzaon of dsromeer daa. The represenave dsrbuons all have means close o.8, he mean of he pror dsrbuon. The poseror dsrbuons are rgh-skewed; values for are varable, rangng from.5 o.3 and do conan he observed esmaes calculaed from he dsromeer daa. The n-seres varably for TN,, and s greaer han ha for he observed dsromeer daa. Excep for, he me seres values esmaed from he dsromeer daa are nearly cenered n he poseror model resuls. As wh he verfcaon run, he full model underesmaes he DSD shape parameer,. From he poseror parameer esmaes for he hdden DSD process, we can esmae he negraed values for ran rae and derved reflecvy. The me seres of calculaed ran raes show hgher varably han ha produced from he one realzaon of dsromeer daa. The large varably makes dffcul o compare he sgnals, hough here appears o be some amoun of agreemen n he gross sgnal feaures for ran rae (Fgure 5). The me seres plos of derved reflecvy also show hgher varably han ha produced from one realzaon of dsromeer daa, hough no o he same exen as for ran rae (Fgure 6). The model oupu conssenly overesmaes dsromeer derved values. A mean normalzed plo for reflecvy (Fgure 7) removes he bas and clearly shows some smlary n he sgnals compared wh he dsromeer based calculaons. Whle he cause of he bas s unknown, s possble ha he observed bas ndcaes poor calbraon of he radar. 7

19 An alernae vsualzaon of he calculaed ran rae parameer based on model oupu dsplays hsograms for ran raes from pons along he me seres (Fgure 8). Correspondng values calculaed from he dsromeer daa are also shown. The ndvdual hsograms conan he dsromeer-based ran raes, alhough he dsromeer-based values are n he lower ends of he dsrbuons for mos me pons. Table. MCMC Summary Sascs for Tme Independen Parameers. MCMC Poseror Mean (SE) Model Parameer Pror Mean Pror SD Verfcaon Run Full Run GARCH Coeffcens a (.) (.) a (.8) (.) b (.99) (.3) AR() Process a (.8) (.) N (.3) Measuremen Errors.478 (.33) G (.44) (.3) (.865) 8

20 Table 3. MCMC Summary Sascs for Tme Varyng Parameers for hree me pons. For he verfcaon run, mes are = 9, 4, 39. For he full run, = 9, 35, 39. Verfcaon Run Full Model Run Model Parameer Observed Value Poseror Mean (SE) Observed Value Poseror Mean (SE)n Toal # drops TN (.6) (.56) TN TN (.9) 6.9 (.8) (.55) 5.4 (.5) Mean # drops N N N 3 6. (.6) 6.34 (.8) 6.3 (.6) 5. (.5) 5.5 (.49) 5.5 (.48) Errors for TN 3 -. (.8) -.5 (.8) -. (.) -.7 (.4) -.6 (.).3 (.7) Error Varance 3.5 (.5).7.36).8 (.4).3 (.77).39 (.84).39 (.75) Lognormal Scale Parameer Pror mean: -.5 SD:.4 3 Lognormal Shape Parameer Pror mean:.8 SD: a a a Esmaed from -mnue dsromeer daa. b Esmaed from -mnue dsromeer daa (.9) -.46 (.9) -.49 (.9).55 (.8).76 (.3).66 (.3) -.56 b b (.5) -.54 (.5) -.53 (.5).8 (.3).9 (.3).9 (.4) 9

21 DISCUSSION The model resuls for he Eureka ranfall are promsng. Usng herarchcal Bayes mehods we are able o negrae nformaon from mulple daa sources for purposes of esmang ran rae and relaed parameers. Whle we are unable o valdae he poseror esmaes of TN,, and, he man parameers for he unobserved DSD process, we can compare her values wh he one avalable dsromeer realzaon. For TN, many of he MCMC realzaons, hough no all, produce me seres sgnals smlar o ha observed for he dsromeer realzaon wh model oupu ha s always more varable. Ths hgher varably s due, n par, o he hgher uncerany ha s nroduced when observaons from wo dfferen daa sources are consdered. The DSD scale and shape parameers capure he gross feaures of he drop specra, bu he me seres for dsromeer esmaes. and are many mes more varable han hose produced from A comparson of auocorrelaons for hese parameers based on dsromeer daa versus model oupu suggess ha here s more srucure n he me evoluon of he DSD shape han wha s accouned for n he curren model. Ulmaely he esmaed DSD parameers are used o produce esmaes of he negraed quanes ran rae and derved reflecvy. The verfcaon runs do a good job of capurng he feaures of hese quanes; he full model runs capure he gross feaures, bu agan produce hgher varable me seres oupu and overesmaes of derved reflecvy. The bas seen n derved reflecvy could be due o a number of facors; mos noably, dfferences may be relaed o samplng errors assocaed wh dsromeer daa. Derved reflecvy s more adversely affeced by small dfferences n drop-sze dsrbuons snce funcon of D s a funcon of 6 D whle R s a 3 D. Eher overesmaon of larger drops by he model, or under recordng of larger drops by he dsromeer could lead o hese observed dfferences. Verfcaon runs produced beer conssency beween model derved reflecvy and dsromeer derved values; use of - mnue dsromeer daa helps compensae for dsromeer samplng errors ha arse due o he small sample volume colleced n a one-mnue nerval. There s no measurable ruh when comes o deermnng ran raes a he earh s surface. For each nsrumen deployed n he feld, here are clams and belefs as o how well hey perform her respecve asks, n lgh of uncerany. Insead of usng ran gauge daa as ground ruh, he poseror model dsrbuons capure mulple sources of uncerany ha

22 reflec uncerany assocaed wh one or more sources such as: he nably o drecly measure quanes of neres; measuremen and samplng errors assocaed wh he mulple nsrumens; sparse gauge daa; and, ncorrec or nadequae assumpons and specfcaons relaed o equaons used o represen he underlyng physcal processes. As a proof of concep, he herarchcal Bayes model s able o generae reasonable esmaes for mos of he modeled parameers when appled o he Eureka daa se. Usng exsng model runs as baselnes we can furher characerze he observed unceranes by makng sepwse modfcaons o model sages or by ncorporang addonal daa. In he shor erm, a emporal dynamc sage should be developed for he evoluon of DSD hrough me. Also, we should run he model for oher ranfall evens. The parcular Eureka even conssed of relavely lgh ranfall. Deermnng how well he model performs for heaver ranfall evens wll provde some dea as o he porably/usably of hs model under a varey of ranfall scenaros. In he longer erm, some deas for expandng and usng hs modelng approach nclude:. If applcable, ncorporae wnd correcons for gauge observaons n fuure model runs. Gauge cachmens are nherenly affeced by wnds; a wnd effec would be one componen of he overall uncerany observed n he model oupu.. Incorporae a vercal correcon model componen for radar reflecvy observaons, based on an underlyng process model, o compensae for he fac ha ground-based radar provdes nformaon abou precpaon above he earh s surface. Include daa from vercal ponng radar o augmen he vercal correcon model componen. 3. Expand he me only model o a space-me model. Such a model could provde a ool ha allows for he creaon of spaal ranfall maps wh uncerany esmaes, whle also provdng he opporuny for more expanded analyss of he componen unceranes. 4. Use model oupu o esmae beer -R relaonshps dynamcally. 5. Use model oupu o assess he value added by usng oher ypes of precpaon measuremens (or by removng some nsrumens). ACKNOWLEDGEMENTS We would lke o hank Dr. Sandra Yuer, who provded he dsromeer and NEXRAD radar daa ses, for her help wh daa pre-processng and her nvaluable nsghs relaed o nsrumenaon, meeorologcal properes of precpaon, and relaonshps beween he wo. REFERENCES Alexander C. Volaly and correlaon: measuremen, models, and applcaons. In Alexander C, edor, Rsk Managemen and Analyss. Vol. : Measurng and Modellng Fnancal Rsk. John Wley Sons Ld, 998.

23 Baan LJ. Radar observaon of he amosphere. Unversy Chcago Press, Chcago, revsed edon, 973. Bell TL. A space-me sochasc model of ranfall for saelle remoe-sensng sudes. Journal of Geophyscal Research, 9(D8): , 987. Berlner LM. Herarchcal Bayesan modelng n he envronmenal scences. Allgemenes Sasshe Archv, 84:4-53,. Berlner LM, Levne RA, and Shea DJ. Bayesan clmae change assessmen. Journal of Clmae, 3():385-38, a. Cox DR, S. RF, and IshamV. A smple spaal-emporal model of ranfall. Proc. R. Soc. Lond., 45:37-38, 988. Hammer G and Seurer P. Daa documenaon for hourly precpaon daa TD-34. Techncal repor, Naonal Clmac Daa Cener, February, 998. Hrafnkelsson B. Herarchcal modelng of coun daa wh applcaon o nuclear fall-ou. Journal of Envronmen and Ecologcal Sascs, :79-, 3. Rafery A and Lews S. Implemenng MCMC. In Glks W, Rchardson S, and Spegelhaler D, edors, Markov Chan Mone Carlo n Pracce, pages 5-3. Chapman Hall, London, 996. Robers GO. Markov chan conceps relaed o samplng algorhms. In Glks W, Rchardson S, and Spegelhaler D, edors, Markov Chan Mone Carlo n Pracce, pages Chapman Hall, London, 996. Rodrguez-Iurbe I, Cox DR, F.R.S., and Isham V. A pon process model for ranfall: furher developmens. Proc. R. Soc. Lond., 47(83-98), 988. Rodrguez-Iurbe I and Eagleson PS. Mahemacal models of ransorm evens n space and me. Waer Resources Research, 3():8-9, 987. Royle JA, Berlner LM, Wkle CK, and Mllff R. A herarchcal spaal model for consrucng wnd felds from scaeromeer daa n he Labrador Sea. In Case Sudes n Bayesan Sascs IV, pages Sprnger-Verlag, 998. Skaugen T, Creun JD, and Goschald L. Reconsrucon and frequency esmaes of exreme daly areal precpaon. Journal of Geophyscal Research, (D): , 996. Smh JA. Marked pon process models of randrop-sze dsrbuons. Journal of Appled Meeorology, 3:84-96, 993. Sern R and Coe R. A model fng analyss of daly ranfall daa. Journal of he Royal Sascal Socey A, 47(Par ):-34, 984. Wkle CK, Mllff RF, Nychka D, and Berlner M. Spaoemporal herarchcal Bayesan modelng: ropcal ocean surface wnds. Journal of he Amercan Sascal Assocaon, 96(454):38-397,. Wkle CK. Herarchcal Bayesan space-me models. Envronmenal and Ecologcal Sascs, 5:7-54, 998.

24 Fgure. Barplos for nne consecuve mnues of dsromeer DSD daa from a Eureka ranfall even; n s he oal number of drops n each one-mnue nerval. Fgure. Tme seres of he one-mnue oal number of drops (lef) and log ransformed oal number of drops (rgh) for each of fve Eureka ranfall evens. 3

25 Sample: Sample: 5 log(rr) (mm/hr) log(rr) (mm/hr) me (mnues) me (mnues) Sample: 3 Sample: 45 log(rr) (mm/hr) log(rr) (mm/hr) me (mnues) me (mnues) Sample: 5 Sample: 7 log(rr) (mm/hr) log(rr) (mm/hr) me (mnues) me (mnues) Fgure 3. Tmes seres plos of poseror log ran rae esmaes calculaed for some represenave samples from he valdaon run (lgh lnes) wh ran rae esmaes from he dsromeer daa (heavy lnes). Sample: Sample: 5 Reflecvey (db) 3 Reflecvey (db) me (mnues) me (mnues) Sample: 3 Sample: 45 Reflecvey (db) 5 3 Reflecvey (db) me (mnues) me (mnues) Sample: 5 Sample: 7 Reflecvey (db) 3 Reflecvey (db) me (mnues) me (mnues) Fgure 4. Tmes seres plos of poseror reflecvy esmaes calculaed for some represenave samples from he valdaon run (lgh lnes) wh reflecvy esmaes from he dsromeer daa (heavy lnes). 4

26 Sample: 4 log(rr) (mm/hr) me (mnues) Sample: 365 log(rr) (mm/hr) me (mnues) Sample: 4 log(rr) (mm/hr) me (mnues) Fgure 5. Tme seres plos of full model run poseror log ran rae esmaes calculaed for seleced samples (lgh lnes) wh logged ran rae values based on dsromeer daa (heavy lnes). Sample: 4 Reflecvy (db) me (mnues) Sample: 365 Reflecvy (db) me (mnues) Sample: 4 Reflecvy (db) me (mnues) Fgure 6. Tme seres plos of full model run poseror reflecvy esmaes calculaed for seleced samples (lgh lnes) wh reflecvy values based on dsromeer daa (heavy lnes). 5

27 Sample: 4 Reflecvy (db) me (mnues) Sample: 365 Reflecvy (db) me (mnues) Sample: 4 Reflecvy (db) me (mnues) Fgure 7. Tme seres plos of full model run poseror reflecvy (normalzed o mean ) esmaes calculaed for seleced samples (lgh lnes) wh reflecvy values based on dsromeer daa (heavy lnes). Fgure 8. Frequency hsograms of he full model run poseror dsrbuon of log ran rae esmaes for seleced me pons whn he 55 mnue modeled even. All hsograms are scaled wh equvalen y-axes. The whe crcles correspond o he values of ran rae calculaed from one realzaon of dsromeer daa. 6