Stochastic Hydrology

Transcription

1 GEO4-44 Sochsc Hydrology Prof. dr. Mrc F.P. Beres Prof. dr. Frs C. v Geer Deprme of Physcl Geogrphy Urech Uversy

2 Coes Iroduco Why sochsc hydrology? Scope d coe of hese lecure oes....3 Some useful defos for he followg chpers Descrpo of model ccordg o sysem s heory Noo... 4 Descrpve sscs Uvre sscs Bvre sscs....3 Exercses Probbly d rdom vrbles Rdom vrbles d probbly dsrbuos Elemes of probbly heory Objecvsc defos Subjecvsc defo Bref revew of elemery probbly heory Mesures of couous probbly dsrbuos Momes Chrcersc fucos Some well-ow probbly dsrbuos d her properes Two or more rdom vrbles Exercses Hydrologcl sscs d exremes Iroduco The lyss of mxmum vlues Flow duro curve Recurrece mes Mxmum vlues d he Gumbel dsrbuo Fg he Gumbel dsrbuo Esmo of he T-yer eve d s cofdece lms Po d Oher dsrbuos for mxmum vlues Mmum vlues... 66

3 4.3 Some useful sscl ess Tesg for depedece Tesg for reds Tesg presumed probbly dsrbuo Exercses Rdom fucos Defos Types of rdom fucos Sory rdom fucos Src sory rdom fucos Ergodc rdom fucos Secod order sory rdom fucos Relos bewee vrous forms of sory Irsc rdom fucos Iegrl scle d scle of flucuo Codol rdom fucos Specrl represeo of rdom fucos Specrl desy fuco Forml complex specrl represeo Esmg he specrl desy fuco Specrl represeos of rdom spce fucos Locl vergg of sory rdom fucos Exercses Tme seres lyss Iroduco Defos Dscree sory me seres Momes d Expeco Dscree whe ose process Rules of clculus Prcple of ler uvre me seres models Auoregressve processes AR process ARp process Movg verge processes MA process... 3

4 6.5. MAq process Auoregressve Iegred Movg Averge processes Auoregressve Movg Averge processes No-ero me ARMA processes d o-guss me seres Auoregressve Iegred Movg Averge processes Sesol ARIMA processes Modellg specs Geerl Idefco Esmo Dgoscs Trsfer fuco/ose models Formulo of Trsfer fuco models Formulo of rsfer fuco/ose models Modelg specs of TF/ose models Use of TF/ose models; exmple Exercses Geosscs Iroduco Descrpve spl sscs Spl erpolo by Krgg Smple rgg Ordry rgg Bloc rgg Esmg he locl codol dsrbuo Mulvre Guss rdom fucos Log-orml rgg Krgg orml-score rsforms Geosscl smulo More geosscs Exercses Forwrd sochsc modellg Iroduco Explc fucos of oe rdom vrble Explc fucos of mulple rdom vrbles Spl, emporl or spo-emporl egrls of rdom fucos

5 8.5 Vecor fucos Dfferel equos wh rdom vrble Sochsc dfferel equos Exercses... 9 Se esmo d d ssmlo Iroduco Formulo of Klm flerg Se equo d mesureme equo Se esmo lgorhm Klm Flerg d me seres Klm Fler lgorhm for Trsfer fuco/ose model Properes of he ler Klm Fler ppled o smple Trsfer fuco/ose model Exmple groudwer hed me seres Klm Flerg d spl dsrbued sysems Exmple ucery for oe dmesol groudwer flow Exmple wo dmesol groudwer flow Evluo of moorg sregy Exercses... 8 Refereces... Appedx: Exm Sochsc Hydrology

6 Iroduco. Why sochsc hydrology? The erm sochscs derves from he Gree word Sochscos Στοχαστικός whch ur s derved from Sochesh Στοχάζεσαι, whch s derved from Sochos Στόχος. The word Sochos mes rge, whle he word Sochesh hs he followg megs: o shoo rrow rge, b o guess or cojecure he rge, c o mge, h deeply, beh, coemple, coge, mede fer Kousoys,, p. 95. I he moder sese sochsc sochsc mehods refers o he rdom eleme corpored hese mehods. Sochsc mehods hus m predcg he vlue of some vrble o-observed mes or o-observed locos, whle lso sg how ucer we re whe mg hese predcos Bu why should we cre so much bou he ucery ssoced wh our predcos? The followg exmple Fgure. shows me seres of observed wer ble elevos peomeer d he oucome of groudwer model hs loco. Also ploed re he dffereces bewee he d d he model resuls. We c observe wo feures. Frs, he model me seres seems o vry more smoohly he he observos. Secodly, here re osy dffereces bewee model resuls d observos. These dffereces, whch re clled resduls, hve mog ohers he followg cuses: observo errors. I s rrely possble o observe hydrologcl vrble whou error. Ofe, exerl fcors fluece observo, such s emperure d r pressure vros durg observo of wer levels; errors boudry codos, l codos d pu. Hydrologcl models oly descrbe pr of rely, for exmple groudwer flow lmed rego. A he boudres of he model vlues of he hydrologcl vrbles such groudwer heds or fluxes hve o be prescrbed. These boudry vlues co be observed everywhere, so here s lely o be some error volved. Also, f model descrbes he vro of hydrologcl sysem me, he he hydrologcl vrbles me sep ero mus be ow s deermes how he sysem wll be evolve ler me seps. Ag, he l vlues of ll he hydrologcl vrbles ll locos re o excly ow d re esmed wh error. Flly, hydrologcl models re drve by pus such s rfll d evporo. Observg rfll d evporo for lrger res s very cumbersome d wll usully be doe wh cosderble error; uow heerogeey d prmeers. Properes of he ld surfce d subsurfce re hghly heerogeeous. Prmeers of hydrologcl sysems such s surfce roughess, hydrulc coducvy d vegeo properes re herefore hghly vrble spce d ofe lso me. Eve f we were ble o observe hese prmeers whou error, we co possbly mesure hem everywhere. I my hydrologcl models prmeers re ssumed homogeeous,.e. represeed by sgle vlue for he ere or pr of he model rego. Eve f models e ccou of he heerogeey of prmeers, hs heerogeey s usully represeed by some erpoled mp from few locos where he prmeers hve bee observed. 6

7 Obvously, hese mperfec represeos of prmeers led o errors model resuls; scle dscrepcy. My hydrologcl models coss of umercl pproxmos of soluos o prl dfferel equos usg eher fe eleme or fe dfferece mehods. Oupu of hese models c bes be erpreed s verge vlues for elemes or model blocs. The oupus hus gore he wh eleme or wh bloc vro of hydrologcl vrbles. So, whe compred o observos h represe verges for much smller volumes vrully pos, here s dscrepcy scle h wll yeld dffereces bewee observos d model oucomes Beres e l., ; model or sysem errors. All models re smplfed versos of rely. They co co ll he rce mechsms d ercos h opere url sysems. For sce, sured groudwer flow s descrbed by Drcy s Lw, whle rely s o vld cse of srogly vryg veloces, res of prly olmr flow e.g. fuls or res of very low permebly d hgh coceros of solves. Aoher exmple s whe surfce wer model uses emc wve pproxmo of surfce wer flow, whle rely suble slope grdes surfce wer levels dome he flow. I such cses, he physcs of rely dffer from h of he model. Ths wll cuse ddol error model resuls. I cocluso, pr from he observo errors, he dscrepcy bewee observos d model oucomes re cused by vrous error sources our modelg process. 4 Wer ble cm surfce resduls cm Dy umber dy s Jury 985 Groudwer model Observos Resduls Fgure. Observed wer ble dephs d wer ble dephs predced wh groudwer model he sme loco. Also show re he resduls: he dffereces bewee model oucome d observos. 7

8 There re wo dsc wys of delg wh errors hydrologcl model oucomes: Deermsc hydrology. I deermsc hydrology oe s usully wre of hese errors. They re e o ccou, ofe prmve wy, durg clbro of models. Durg hs phse of he modelg process oe res o fd he prmeer vlues of he model e.g. surfce roughess or hydrulc coducvy such h he mgude of he resduls s mmed. Afer clbro of he model, he errors re o explcly e o ccou whle performg furher clculos wh he model. Errors model oucomes re hus gored. Sochsc Hydrology. Sochsc hydrology o oly res o use models for predcg hydrologcl vrbles, bu lso res o qufy he errors model oucomes. Of course, prcce we do o ow he exc vlues of he errors of our model predcos; f we ew hem, we could correc our model oucomes for hem d be olly ccure. Wh we ofe do ow, usully from he few mesuremes h we dd e, s some probbly dsrbuo of he errors. We wll defe he probbly dsrbuo more precsely he ex chpers. Here suffces o ow h probbly dsrbuo ells oe how lely s h error hs cer vlue. To me hs dfferece more cler, Fgure. s show. Cosder some hydrologcl vrble, sy sol mosure coe, whose vlue s clculed some loco d some me by usured oe model. The model oupu s deoed s. We he cosder he error e. Becuse we do o ow excly we cosder s so clled rdom vrble chper 3 E oce he use of cpls for rdom vrbles whose exc vlue we do o ow bu of whch we do ow he probbly dsrbuo. So cse of deermsc hydrology modelg effors would oly yeld upper fgure of Fgure., whle sochsc hydrology would yeld boh d he probbly dsrbuo of he rdom error E lower fgure of Fgure.. 8

9 b Probbly desy e Fgure. Sochsc Hydrology s bou combg deermsc model oucomes wh probbly dsrbuo of he errors Fgure., or lervely, cosderg he hydrologcl vrble s rdom d deermg s probbly dsrbuo d some bes predco Fgure.b. Mos of he mehods used sochsc hydrology do o cosder errors model oucomes explcly. Ised s ssumed h he hydrologcl vrble self s rdom vrble. Ths mes h we cosder he hydrologcl vrble e.g. sol mosure s oe for whch we co ow he exc vlue, bu for whch we c clcule he probbly dsrbuo see Fgure.b. The probbly dsrbuo of Fgure.b hus ells us h lhough we do o ow he sol mosure coe excly, we do ow h s more lely o be roud.3 he roud. or.5. Models h provde probbly dsrbuos of rge vrbles sed of sgle vlues re clled sochsc models. Bsed o he probbly dsrbuo s usully possble o ob so clled bes predco ẑ, whch s he oe for whch he errors re smlles o verge. Icdelly, he vlue of he bes predco does o hve o be he sme s he deermsc model oucome. Box. Sochsc models d physcs A wdespred mscocepo bou deermsc d sochsc models s h he former use physcl lws such mss d momeum coservo, whle he ler re lrgely emprcl d bsed erely o d-lyss. Ths of course s o rue. Deermsc models c be eher physclly-bsed e.g. model bsed o S-Ve equos for flood roug d emprcl e.g. rg curve used s deermsc model for predcg sedme lods from wer levels. Coversely, y physclly-bsed model becomes sochsc model oce s pus, prmeers or oupus re reed s rdom. There re umber of cler dvges g he ucery model resuls o ccou,.e. usg sochsc sed of deermsc models. The exmple of Fgure. shows h model oucomes ofe gve much smooher pcure of rely. Ths s becuse models re ofe bsed o deled represeo of rely wh smple processes d homogeous prmeers. However, rely s usully messy d rugged. Ths my be problem whe eres s focused o exreme vlues: deermsc models Probbly desy ẑ 9

10 ypclly uderesme he probbly of occurrece of exremes, whch s rher uforue whe predcg for sce rver sges for dm buldg. Sochsc models c be used wh echque clled sochsc smulo see chpers herefer whch s ble o produce mges of rely h re rugged eough o ge he exreme sscs rgh. As sed bove, he vlue of he bes predco ẑ does o hve o be he sme s he deermsc model oucome. Ths s prculrly he cse whe he relo bewee model pu e.g. rfll, evporo or model prmeers e.g. hydrulc coducvy, mg coeffce d model oupu s o-ler hs s he cse lmos ll hydrologcl models d our deermsc ssessme of model pus d prmeers s o error free lso lmos lwys he cse. I hs cse, sochsc models re ble o provde he bes predco usg he probbly dsrbuo of model oucomes, whle deermsc models co d re herefore less ccure. If we loo closely he resduls Fgure. c be see h hey re correled me: posve resdul s more lely o be followed by oher posve resdul d vce vers. Ths correlo, f sgfc, mes h here s sll some formo prese he resdul me seres. Ths formo c be used o mprove model predcos bewee observo mes, for sce by usg me seres modelg chper 6 or geosscs chper 7. Ths wll yeld beer predcos h he deermsc model loe. Also, urs ou h f he resduls re correled, clbro of deermsc models whch ssume o correlo bewee resduls yeld less ccure or eve bsed wh sysemc errors clbro resuls whe compred wh sochsc models h do e ccou of he correlo of resduls e Sroe, 995. By explcly ccoug for he ucery our predco we my fc be ble o me beer decsos. A clssc exmple s remedo of pollued sol, where sochsc mehods c be used o esme he probbly dsrbuo of pollu cocero some o-vsed loco. Gve crcl hreshold bove whch regulo ses h remedo s ecessry, s possble o clcule he probbly of flse posve decso we decde o remede, whle rely he cocero s below he hreshold d h of flse egve we decde o o remede whle rely he cocero s bove he hreshold. Gve hese probbles d he ssoced coss of remedo d helh rs s he possble for ech loco o decde wheher o remede such h he ol coss d helh rs re mmed. There re bud sochsc mehods where relo s esblshed bewee he ucery model oucomes d he umber of observos me d spce used o eher prmeere or clbre he model. If such relo exss, c be used for moorg ewor desg. For exmple, groudwer exploro wells re drlled o perform pumpg ess for he esmo of rsmssves d o observe hydrulc heds. The rsmssvy observos c be used o me l mp of rsmssvy used he groudwer model. Ths l mp c

11 subsequely be upded by clbrg he groudwer model o hed observos he wells. Cer sochsc mehods re ble o qufy he ucery groudwer hed predced by he model relo o he umber of wells drlled, her loco d how ofe hey hve bee observed e.g. Beres, 6. These sochsc mehods c herefore be used o perform moorg ewor opmo: fdg he opml well locos d observo mes o mme ucery model predcos. The ls reso why sochsc mehods re dvgeous over deermsc mehods s reled o he prevous oe. Sochsc mehods eble us o rele he ucery model oucomes o dffere sources of ucery errors pu vrbles, prmeers d boudry codos. Therefore, usg sochsc lyss we lso ow whch error source corbues he mos o he ucery model oucomes, whch source comes secod ec. If our resources re lmed, sochsc hydrology hus c gude us where o sped our moey how my observos for whch vrble or prmeer o cheve mxmum ucery reduco mmum cos. A excelle boo o hs vew o ucery s wre by Heuvel Scope d coe of hese lecure oes These oes m preseg overvew of he feld of sochsc hydrology roducory level. Ths mes h wde rge of opcs d mehods wll be reed, whle ech opc d mehod s oly reed bsc level. So, he boo s me s roduco o he feld whle showg s bredh, rher h provdg deph rese. Refereces re gve o more dvced exs d ppers for ech subjec. The boo hus ms echg he bscs o hydrologss who re seeg o pply sochsc mehods. I c be used for oe-semeser course hrd yer udergrdue or frs yer grdue level. The lecure oes re bsc opcs h should be he core of y course o sochsc hydrology. These opcs re: descrpve sscs; probbly d rdom vrbles; hydrologcl sscs d exremes; rdom fucos; me seres lyss; geosscs; forwrd sochsc modelg; se predco d d-ssmlo. A umber of more dvced opcs h could cosue eough merl for secod course re o reed. These re, mog ohers: smplg d moorg; verse esmo; ordry sochsc dfferel equos; po processes; upsclg d dowsclg mehods, ucery d decso mg. Durg he course hese dvced opcs wll be shorly roduced durg he lecures. Sudes re requred o sudy oe of hese opcs from exemplry ppers d wre reserch proposl bou..3 Some useful defos for he followg chpers

12 .3. Descrpo of model ccordg o sysem s heory My mehods sochsc hydrology re bes udersood by loog hydrologcl model from he vewpo of sysem s heory. Wh follows here s how model s defed sysem s heory, s well s defos for se vrbles, pu vrbles, prmeers d coss. pu vrbles se vrbles prmeers coss oupu vrbles model boudry Fgure.3 Model d model properes ccordg o sysem s heory Fgure.3 shows schemc represeo of model s used sysem s heory. A model s smplfed represeo of pr of rely. The model boudry sepres he pr of rely descrbed by he model from he res of rely. Everyhg h s o ow bou he pr of rely descrbed by he model cer me s coed he se vrbles. These re vrbles becuse her vlues c chge boh spce d me. The vro of he se vrbles s cused by he vro of oe or more pu vrbles. Ipu vrbles re lwys observed d orge from ousde he model boudry. Cosequely, pu vrbles lso clude boudry codos d l codos such s used whe solvg dfferel equos. If he se vrbles re ow, oe or more oupu vrbles c be clculed. A oupu vrble rverses he model boudry d hus flueces he pr of rely o descrbed by he model. Boh pu vrbles d oupu vrbles c chge spce d me. The se vrbles re reled o he pu vrbles d oupu vrbles hrough prmeers. Prmeers my chge spce bu re vr me. Becuse hey re cos me, prmeers represe he rsc properes of he model. Flly, model my hve oe or more coss. Coss re properes of model h do o chge boh spce d me wh he cofes of he model. Exmples of such coss re he grvy cos d he vscosy of wer desy depede groudwer flow cos emperure.

13 p A r v q Fgure.4 Illusro of model properes followg sysem s heory wh model of cchme; v: se vrble, sorge surfce wer cchme [L 3 ]; q: oupu vrble, surfce ruoff from cchme [L 3 T - ]; p: pu vrble, precpo [LT - ]; : prmeer, reservor cos [T - ]; r : prmeer, flro cpcy [LT - ]; A: cos, re of he cchme [L ]. Becuse he descrpo bove s rher bsrc, we wll ry o llusre wh he exmple show Fgure.4. We cosder model descrbg he dschrge from surfce ruoff q [L 3 T - ] from cchme cused by he verge precpo p [LT - ] observed s verges over dscree me seps,.e. q d p represe he verge dschrge d precpo bewee - d. The model boudry s formed by geogrphcl boudres such s he cchme boudry.e. he dvde o he sdes, he cchme s surfce below d few meers bove he cchme s surfce bove, d lso by he vrul boudry wh everyhg h s o descrbed by he model such s groudwer flow, sol mosure, chemcl rspor ec. Obvously, precpo s he pu vrble d surfce ruoff he oupu vrble. The se vrble of hs model s he mou of wer sored o he cchme s surfce: v [L 3 ]. The se vrble s modeled wh he followg wer blce equo: v v { A [ p r] q }. where r [LT - ] s he flro cpcy. The superscrp s dded o [p-r] o deoe h f p < r we hve [p-r]. The oupu vrble q s reled o he se vrble v he prevous me sep wh he followg equo: q v. Through subsuo of. o. we c clcule he developme me of he se vrble drecly from he pu vrble s: v [ ] v A [ p r].3 3

14 Two model prmeers c be dsgushed: he flro cpcy of he sol r [LT - ] whch reles he pu vrble wh he se vrble d he cchme prmeer [T - ] relg he oupu vrble o he se vrble. The cos A [L ] s he re of he cchme..3. Noo The cocep of rdom vrbles d rdom fucos wll be expled del he followg chpers. However, s useful o defe he oo coveos brefly he begg. Reders c hus refer bc o hs subseco whle sudyg he res of hs boo. Coss re deoed rom, e.g. he cos g for grvy ccelero, or A for he re. Vrbles d prmeers re deoed lcs: e.g. h for hydrulc hed d for hydrulc coducvy. The dsco bewee deermsc d rdom sochsc vrbles s mde by deog he ler s cpl lcs. So, h sds for he deermsc groudwer hed ssumed compleely ow d H for groudwer hed s rdom vrble. Vecors d mrces re gve bold fce oo. Vecors re deoed s lower cse, e.g. h vecor of groudwer heds he odes of fe dfferece model, whle mrces re deoed s cpls, such s K for esor wh coducves vrous drecos. Uforuely, s dffcul o me dsco bewee sochsc d deermsc vecors d mrces. Therefore, f o cler from he coex, wll be dced explcly he ex wheher vecor or mrx s sochsc or o. Spl co-ordes x,y, re deoed wh he spce vecor x, whle s reserved for me. Dscree pos spce d me re deoed s x d respecvely. Rdom fucos of spce, me d spce-me re hus deoed s exmple wh H: Hx, H, Hx,. Oucomes from deermsc model re deoed s exmple wh h: h. Opml esmes of deermsc prmeers, coss or vrbles re deoed wh h exmple wh : ˆ, whle opml predcos of relos of rdom vrble deoed by Kˆ. Noe h he erm esme s reserved for deermsc vrbles d predco for rdom sochsc vrbles. To deoe spl or emporl or spo-emporl verge of fuco overbr s used, e.g. h f hydrulc hed s deermsc d H f s sochsc. So, H ˆ x sds for he predco of he spl verge of he rdom fuco Hx. 4

15 Descrpve sscs I hs chper d furher o hs boo we me use of syhec bu exremely llusrve d se Wler le d se h hs bee cosruced by Jourel d Deusch 998. The d se s used o show how some smple sscs c be clculed.. Uvre sscs Le us ssume h we hve mde 4 observos of some hydrologcl vrble e.g. hydrulc coducvy m/d. Fgure. shows plo of he smple locos wh he grey scle of he dos ccordg o he vlue of he observo. Fgure. Smples of hydrulc coducvy To ob sgh o our dse s good prcce o me hsogrm. To hs ed we dvde he rge of vlue foud o umber sy m of clsses -, - 3, 3-4,, m - m d cous he umber of d vlues fllg o ech clss. The umber of observos fllg o clss dvded by he ol umber of observos s clled he All of he lrger umercl exmples show hs chper re bsed o he Wler-le d se. The geosscl lyses d he plos re performed usg he GSLIB geosscl sofwre of Deusch d Jourel

16 relve frequecy. Fgure. shows he hsogrm or frequecy dsrbuo of he - d. From he hsogrm we c see how he observos re dsrbued over he rge of vlues. For sce, we c see h pproxmely 33% of our d hs vlue of hydrulc coducvy bewee - m/d. Fgure. Hsogrm or frequecy dsrbuo of hydrulc coducvy Aoher wy of represeg he dsrbuo of d vlues s by usg he cumulve frequecy dsrbuo. Here we frs sor he d scedg order. Nex d re gve r umber,,..,, wh he ol umber of observos our cse 4. Afer h, he d vlues re ploed gs he r umber dvded by he ol umber of observos plus o: /. Fgure.3 shows he cumulve frequecy dsrbuo of he hydrulc coducvy d. 6

17 Fgure.3 Cumulve frequecy dsrbuo of hydrulc coducvy The cumulve frequecy dsrbuo shows us he percege of d wh vlues smller h gve hreshold. For sce, from.3 we see h 64% of he observos hs vlue smller h 5 m/d. Noe, h f he 4 smples were e such wy h hey re represeve of he re e.g. by rdom smplg h he cumulve frequecy dsrbuo provdes esme of he frco of he reserch re wh vlues smller or equl o cer vlue. Ths my for sce be relev whe mppg polluo. The cumulve frequecy dsrbuo he provdes mmedely esme of he frco of err wh coceros bove crcl hresholds,.e. he frco h should be remeded. To me couous curve he vlues bewee he d pos hve bee lerly erpoled. Fgure.4 shows he relo bewee he hsogrm d he cumulve frequecy dsrbuo. I shows h oce he cumulve frequecy dsrbuo fuco s cosruced from he d 5 d vlues for hs smple exmple c be used o cosruc hsogrm by dffereo. 7

18 Vlues: : 5 R : derved hsogrm d 3 d d d 3 d d Fgure.4 The relo bewee he Cumulve frequecy dsrbuo lef d he hsogrm To descrbe he form of frequecy dsrbuo umber of mesures re usully clculed. The me m s he verge vlue of he d d s mesure of locly,.e. he ceer of mss of he hsogrm. Wh he umber d d he vlue of he h observo we hve m The vrce s s mesure of he spred of he d d s clculed s: s mx m.. The lrger he vrce he wder s he frequecy dsrbuo. For sce Fgure.5 wo hsogrms re show wh he sme me vlue bu wh dffere vrce. 8

19 smll vrce lrge vrce me me Fgure.5 Two hsogrms of dses wh he sme me vlue bu wh dffere vrces Sdrd devo The sdrd devo s lso mesure of spred d hs he dvge h s hs he sme us s he orgl vrble. I s clculed s he squre-roo of he vrce: s s m x.3 Coeffce of vro To ob mesure of spred h s relve o he mgude of he vrble cosdered he coeffce of vro s ofe used: CV s m.4 Noe h hs mesure oly mes sese for vrbles wh srcly posve vlues e.g. hydrulc coducvy, sol mosure coe, dschrge. Sewess The sewess of he frequecy dsrbuo ells us wheher s symmercl roud s cerl vlue or wheher s symmercl wh loger l o he lef < or o he rgh > CS s m Fgure.6 shows wo hsogrms wh he sme vrce, where oe s egvely d oe s posvely sewed. 9

20 Sewess < Sewess > Fgure.6 Two frequecy dsrbuos wh he sme vrces bu wh dffere coeffces of sewess. Cuross The cuross mesures he peedess of he frequecy dsrbuo see Fgure.7 d s clculed from he d s: CC s m Cuross < Cuross > Fgure.7 Frequecy dsrbuos wh posve d egve cuross The vlue 3 s deduced Equo.6 becuse for orml Guss dsrbuo see lso chper 3, he frs pr of Equo.6 s excly equl o 3. So by CC we compred he peedess of he dsrbuo wh h of orml dsrbuo, beg more peed whe lrger h ero d fler whe smller h ero. Fgure.8 shows some ddol mesures of locly d spred for he cumulve frequecy dsrbuo fuco.

21 ..9 Ierqurle rge: Q 3 -Q percele hrd qurle: Q 3. 5-percele frs qurle: Q 5-percele med secod qurle: Q 9-percele.9-qule Fgure.8 Some ddol mesures of locly d spred bsed o he cumulve dsrbuo fuco. The f-percele or f/-qule of frequecy dsrbuo s he vlue h s lrger h or equl o f perce of he d vlues. The 5-percele or.5-qule s lso clled he med. I s ofe used s lerve mesure of locly o he me cse he frequecy dsrbuo s posvely sewed. The me s o very robus mesure h cse s s very sesve o he lrges or smlles vlues he dse. The 5-percele, 5-percele d 75-percele re deoed s he frs, secod d hrd qurles of he frequecy dsrbuo: Q, Q, Q 3 respecvely. The erqurle rge Q 3 -Q s lerve mesure of spred o he vrce h s preferbly used cse of sewed dsrbuos. The reso s h he vrce, le he me, s very sesve o he lrges or smlles vlues he dse. A effce wy of dsplyg locly d spred sscs of frequecy dsrbuo s mg Box-d-whser plo. Fgure.9 shows exmple. The wdh of he box provdes he erqurle rge, s sdes he frs d hrd qurle. The le he mddle represes he med d he cross he me. The whsers legh s re equl o he mmum d he mxmum vlue crcles s log s hese exremes re wh.5 mes he erqurle rge e.g. lower whser Fgure.9, oherwse he whser s se equl o.5 mes he erqurle rge e.g. upper whser Fgure.9. Observos lyg ousde.5 mes he erqurle rge re ofe defed s oulers. Box-d-whser plos re covee wy of vewg sscl properes, especlly whe comprg mulple groups or clsses see Fgure. for exmple of observos of hydrulc coducvy for vrous exure clsses.

22 Mmum vlue me Mxmum vlue lower whser Q med Q 3 upper whser Fgure.9 Compoes of box-d-whser plo Fgure. Box-d-whser plos re covee wy o compre he sscl properes of mulple groups or clsses from Beres, 996. Bvre sscs Up o ow we hve cosdered sscl properes of sgle vrble: uvre sscl properes. I hs seco sscs of wo vrbles re cosdered,.e. bvre sscs. I cse we re delg wh wo vrbles mesured smuleously sgle loco or sgle me, ddol sscs c be obed h mesure he degree of co-vro of he wo d ses,.e. he degree o whch hgh vlues of oe vrble re reled wh hgh or low vlues of he oher vrble. Covrce The covrce mesures ler co-vro of wo dses of vrbles d y. I s clculed from he d s:

23 3 y y y m m y m y m C.7 Correlo coeffce The covrce depeds o he cul vlues of he vrbles. The correlo coeffce provdes mesure of ler co-vro h s ormled wh respec o he mgudes of he vrbles d y: y m y m y m y m s s C r y y y.8 A covee wy of clculg he correlo coeffce s s follows: y y y y y r.9 So, oe clcules Σ, y Σ, Σ, y Σ d y Σ d evlues.9. Fgure. shows so clled scerplo bewee he -vlues observed he 4 locos of Fgure. d he y-vlues lso observed here e.g. could for sce be hydrulc coducvy d y sd frco %. The correlo coeffce bewee he - d y-vlues equls.57. Fgure. shows exmples of vrous degrees of correlo bewee wo vrbles, cludg egve correlo lrge vlues of oe exs ogeher wh smll vlues of he oher. Bewre h he correlo coeffce oly mesures he degree of ler covro.e. ler depedece bewee wo vrbles. Ths c lso be see Fgure. lower rgh fgure, where obvously here s srog depedece bewee d y, lhough he correlo coeffce s ero.

24 5 5 y-vlue vlue Fgure. Scer plo of - d y-d showg covro. The correlo coeffce equls.57 ρ Y < ρ Y < ρ Y y - < ρ Y < ρ Y - y ρ Y y y y y Fgure. Scer plos showg covro d he ssoced correlo coeffces bewee wo ses of vrbles h hve bee observed smuleously. 4

25 .3 Exercses Cosder he followg d se: y y Me hsogrm of wh clss-wdhs of 5 us. Wh frco of he d hs vlues bewee 5 d?. Cosruc he cumulve frequecy dsrbuo of d y 3. Clcule he me, he vrce, he sewess, he qules, he medum d he erqule rge of d y. 4. Drw box-d-whser plo of he - d y-vlues. Are here y possble oulers? 5. Suppose h s he cocero of some pollu he sol mg/g. Suppose h he smples hve bee e rdomly from he se of eres. If he crcl cocero s 5 mg/g d he se s 8 m. Approxmely wh re of he se hs bee cleed up? 6. Clcule he correlo coeffce bewee d y? 7. Wh frco of he d hs -vlue smller h 5 d y-vlue smller h? 8. Wh frco of he d hs -vlue smller h 5 or y-vlue smller h? 5

26 3 Probbly d rdom vrbles 3. Rdom vrbles d probbly dsrbuos A rdom vrble s vrble h c hve se of dffere vlues geered by some probblsc mechsm. We do o ow he vlue of sochsc vrble, bu we do ow he probbly wh whch cer vlue c occur. For sce, he oucome of hrowg de s o ow beforehd. We do however ow he probbly h he oucome s 3. Ths probbly s /6 f he de s o mpered wh. So he oucome of hrowg de s rdom vrble. The sme goes for he oucome of hrowg wo dce. The probbly of he oucome beg 3 s ow /8. A rdom vrble s usully wre s cpl e.g. D for he uow oucome of hrowg wo dce d cul oucome fer he dce hve bee hrow wh lower cse e.g. d. The expeced vlue or me of rdom vrble c be clculed f we ow whch vlues he rdom vrble c e d wh whch probbly. If D s he oucome of hrowg wo dce, he probbly dsrbuo Prd s gve he followg ble: Tble 3. Probbles of oucomes of hrowg wo dce D Prd /36 /36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 /36 /36 The me or expeced vlue s clculed s N d he umber of possble oucomes d d oucome : N d E [ D] d Pr[ d ] / 36 3 / / Th he expeced vlue equls 7 mes h f we were o hrow he wo dce very lrge umber of mes d clcule he verge oucomes of ll hese hrows we would ed up wh umber very close o 7. Ths mes h we could e smple of oucomes d j of rdom vrble D d esme s me wh equo such s.: E ˆ [ D] 3. d j j The me s he ceer of mss of he probbly dsrbuo d ells us wh would be he verge of geerg my oucomes. The vrce s mesure of spred. I ells us somehg bou he wdh of he probbly dsrbuo. Also, ells us how dffere he vrous geered oucomes hrows of he dce re. A lrger vrce mes h he probbly dsrbuo s wde, he vro mog oucomes s lrge d herefore we re more ucer bou he oucome of he rdom vrble. Fgure.5 shows wo probbly dsrbuos wh he sme me, bu wh dffere vrces. The vrce of rdom vrble s clculed from he probbly dsrbuo s: 6

27 Vr[ D] E[ D E[ D] ] d N d / E[ D] Pr[ d / ] / The vrce c be esmed from rdom smple of oucomes hrows of wo dce d j s: Vr ˆ [ D] d Eˆ[D] 3.4 Whe we compre equo 3.4 wh he vrce formul gve chper Equo. we see h here we dvde by - sed of. Ths s becuse hs cse we provde esmor of he vrce cse he me s o ow d mus be esmed from he d. To ob ubsed esme for he vrce whou sysemc error we hve o ccou for he ucery bou he me. Hece we dvde by -, ledg o slghly lrger vrce. The umber - s lso clled he degrees of freedom. Aoher wy of loog hs s h we hve o hd oe degree of freedom s we lredy used o esme he me! Ised of he vrce, oe ofe uses s squre roo s mesure of spred. Ths squre roo s clled he sdrd devo. Gree symbols used for he me, vrce d sdrd devo re µ,σ d σ respecvely. The cocep of rdom vrble s used o express ucery. If we re ucer bou he cul vlue of some propery e.g. he cocero of pollu or he umber of dvduls populo, hs propery s modeled s rdom vrble. The more ucer we re bou he cul bu uow vlue, he lrger he vrce of he probbly dsrbuo of hs rdom vrble. f Fgure 3.. A probbly desy fuco 7

28 The oucome of hrowg dce s dscree propery. I c oly e lmed umber of couble vlues. If he propery s couous c e y rel vlue bewee cer bouds e.g. lude, hydrulc coducvy, cocero. To descrbe he probbly of cer oucome of rel vlued rdom vrble, sed of dscree probbly dsrbuo, couous fuco clled he probbly desy fuco f s used see Fgure 3.. The probbly desy self does o provde probbly. For sce, we co sy Pr[ ] f! Ised, he probbly desy gves he probbly mss per u. So, he probbly h les bewee wo boudres c be clculed from he probbly desy by g he egrl: Pr[ < ] f d 3.5 Equo 3.5 c ow be used o rrve more forml defo of probbly desy by g he followg lm: f Pr[ < d] lm d d 3.6 A ddol codo ecessry for f o be probbly desy fuco pdf s h he re uder s equl o : f d 3.7 The probbly h s smller h cer vlue s gve by he cumulve probbly dsrbuo fuco cpdf, lso smply clled dsrbuo fuco: F Pr[ ] f d 3.8 From 3.8 lso follows h he pdf s he dervve of he cpdf: f df 3.9 d I rs lyss oe s ofe eresed clculg he probbly h cer crcl hreshold c s exceeded. Ths c be clculed from boh he pdf d he cpdf s: Pr[ > c] c f d F c 3. 8

29 Smlrly, he probbly h s bewee wo vlues c be clculed wh he pdf Equo 3.5, bu lso wh he cpdf: Pr[ < ] F F Elemes of probbly heory The bsc rules used sochsc lyss sem from elemery probbly heory. Logclly, we would le o sr ou wh defo of probbly. As urs ou hs s o srghforwrd s here exs dffere oos bou probbly. A frs subdvso h c be mde s bewee objecvsc d subjecvsc oos of probbly e.g. Chrsos, Objecvsc defos There re hree dffere defos here. Cerl o hese defos s he oo of some eve A e.g. eve c be he de fllg o 5, flood occurrg or he vlue of coducvy beg he 5- m/d rge. The clsscl defo Ths s he oldes oo of probbly d c for sce be used o deduce he probbly dsrbuos of hrowg wo dce. The probbly PrA of eve A s deermed pror whou expermeo wh he ro: N A Pr A 3. N wh N he umber of possble oucomes d N A ll he oucomes resulg eve A, provded h ll oucomes re eqully lely. A problem wh hs defo of course s h s o lwys possble o deduce N especlly s N s fe such s couous vlued eves. Moreover, he defo cos he erm eqully lely, whch s self probbly seme. The relve frequecy defo Ths oo of probbly uses he followg defo. The probbly PrA of eve A s he lm of performg probblsc expermes: A Pr A lm 3.3 where A he umber of occurreces of A d he umber of rls. Ths frequesc vew of probbly s uvely ppelg becuse provdes ce l bewee 9

30 probbly d he relve frequecy descrbed chper. However, here re some problems, such s he fc h s prcce o possble o perform fe rls. The xomc defo Ths defo, whch c be rbued o Kolmogorov 933, uses se heory o defe probbly. We mge experme, whch he eve A s he oucome of rl. The se of ll possble oucomes of rl s clled he smplg spce or he cer eve S. The uo {A B} of wo eves A d B s he eve h A or B occurs. The xomc defo of probbly s bsed erely o he followg hree posules:. The probbly of eve s posve umber ssged o hs eve: Pr A 3.4. The probbly of he cer eve he eve s equl o ll possble oucomes equls : Pr S If he eves A d B re muully exclusve he: Pr A B Pr A Pr B 3.6 Fgure 3. shows schemclly usg so clled Ve dgrms he cer eve S wh eves A d B h re muully exclusve lef fgure d o muully exclusve rgh fgure. Some more derved rules bsed o he xomc probbly defo wll be gve herefer. A B A B S Fgure 3. Exmple of Ve dgrms showg wo muully exclusve eves A d B d wo eves h re o muully exclusve. Geerlly, he xomc oo of probbly s deemed superor o he ohers. For exesve descrpo o he suble dffereces d peculres of he vrous defos of probbly we refer o Ppouls 99. S 3

31 3.. Subjecvsc defo I he subjecvsc defo, probbly mesures our cofdece bou he vlue or rge of vlues of propery whose vlue s uow. The probbly dsrbuo hus reflecs our ucery bou he uow bu rue vlue of propery. The probbly desy fuco he mesures he lelhood h he rue bu uow vlue s bewee cer lms. So, hs subjecvsc defo of probbly we do o hve o h bou frequeces, populo ses or eves. We re fced wh some propery h s o ow excly, eher becuse we c oly mesure wh some rdom mesureme error or becuse we co mesure ll, or oly prly. Th for sce bou hydrulc coducvy heerogeeous geologcl formo. I s mpossble o mesure everywhere resoble coss, so prcce we c oly mesure lmed umber of locos ofe wh mesureme error, becuse g udsurbed sedme cores d perform Drcy expermes s very dffcul prcce. If we hve qufer wh o observos, bu we do ow h cosss of sds, we ow h he rue vlue some loco s more lely o be close o md - h. md - or md -. Bsed o hs experece from elsewhere observos oher qufers we c he defe pror probbly dsrbuo h mesures he lelhood of he vrous possble vlues our uow loco. Wh we do he bc of our md s collecg ll he formo we hve o sdy qufers he res of he world d propose h her coducves re smlr o he oe hd. We c he use observos from hese oher qufers o cosruc pror dsrbuo fuco. If subsequely observos re beg colleced h re specfc o he qufer hd, we my use hese observos o rrow he pror probbly dsrbuo dow, by corporg he observed vlues. Wh resuls s so clled poseror probbly dsrbuo h hs smller vrce, such h we re more cer bou he uow coducvy uobserved loco he we were before he observos. The subjecvsc probbly does o eed y observo o defe. I c be defed from he op of our hed, hus expressg our ucery or cofdece bou uow vlue. Ths wy of vewg probbly d he possbly o upde such probbly wh observos s clled Byes sscs see herefer d hs led o much debe d coroversy he sscl commuy, especlly bewee people who ccep Byes sscs d people who vew probbly s xomc or frequesc. I sochsc hydrology, whch s ppled scefc dscple, he vrous oos of probbly hve ever bee rel ssue, bu sghs hve bee borrowed from he vrous probbly coceps: probbly s mosly vewed s subjecvsc excep mybe Hydrologcl sscs chper 4 whch s more frequesc ure; pror probbly dsrbuos re ofe o relly subjecvsc bu bsed o observos e oher mes or locos he sme re of eres; updg of he pror probbly dsrbuos o poseror dsrbuos mes use of Byes heorem, whch s fc bes defed usg xomc probbly rules. 3

32 Box : Abou ucery d rely Ofe oe c red ppers semes le: hydrulc coducvy s ucer, or he ucer behvor of rver dschrge s modeled s... Such semes seem o sugges h rely self s rdom. Wheher hs s rue or o s rher phlosophcl queso. The mos commo vew s h ure s deermsc, excep mybe he subomc level. We wll dhere o hs vew hs boo d use he followg oo of rely d ucery, whch reles o subjecvsc vew o probbly: Rely s compleely deermsc. However, we do o hve perfec owledge of rely, becuse we oly hve lmed formo o. We c oly observe prly, observe wh error or do o excly ow he uderlyg process descrpo. I s becuse of hs h we my perceve prs of rely s rdom d fd h rdom vrble or rdom process d he ssoced coceps of probbly cosue useful model of rely. Therefore, rdomess s o propery of rely bu propery of he sochsc model h we use o descrbe rely d our ucery bou Bref revew of elemery probbly heory Eve hough he defo of probbly my be subjecvsc oe, o relly perform clculos wh probbles requres rules derved from he xomc defo. Here we wll revew some of hese rules. Ths revew s bsed o Vmrce 983. The bsc xoms of probbly re gve by 3.4 d 3.6. As sed bove, he uo of eves A d B s he eve h eher A or B occurs d s deoed s {A B}. The jo eve {A B} s he eve h boh A d B occur. From he Ve dgrm Fgure 3.3 follows drecly h he probbly of he uo of eves d he jo eve re reled s follows: Pr A B Pr A Pr B Pr A B 3.7 If eves A d B re muully exclusve Fgure 3. lef fgure c be see h Pr A B Pr A Pr B d Pr A B. If he mulple eves A, A, A M re muully exclusve, he probbly of he uo of hese eves s he sum of her probbles: M A... A M Pr A Pr A 3.8 I he specl cse h ll eves S re muully exclusve d h hey cosue ll possble eves hey re sd o be collecvely exhusve he follows h her probbles sum o : 3

33 M Pr A 3.8 Muully exclusve d collecvely exhusve eves re lso clled smple eves. For c c M smple eves mply h Pr A Pr A wh A he compleme of A {A B} A B {A B} S Fgure 3.3 Ve dgrm showg he relo bewee he uo of eves d jo eves. The degree of probblsc depedece bewee wo eves s mesured by he so clled codol probbly of A gve B: Pr A B Pr A B 3.9 Pr B Of course A d B c be erchged so h Pr A B Pr A B P B Pr B A P A 3. Two eves A d B re sd o be ssclly depede f he probbly of her jo eve s equl o he produc of he probbles of he dvdul eves: Pr A B Pr APr B 3. Ths lso mples h Pr A BPr B Pr A d Pr B APr A Pr B. Ths mes h owledge bou B does o hve effec o he ucery bou A d vce vers. Flly, f we cosder se of smple eves A erseced by eve B, we c deduce from he Ve dgrm Fgure 3.4 d Equo 3. he followg reloshp: 33

34 M M Pr B Pr A B Pr B A Pr A 3. Ths shows h he probbly of B s he weghed sum of he probbly of B gve A wh he probbly of A s wegh. Ths reloshp s ow s he ol probbly heorem. A exmple o how o use hs s s follows: suppose h we hve from prevous d-lyses for ech exure clss,.e. sd, cly, sl d pe, he probbly dsrbuo of hydrulc coducvy. The, f we hve esmed some uvsed loco he probbles o sd, cly, sl d pe from borehole d see for sce chper 7, we re ble o derve probbles of hydrulc coducvy from hese usg 3.. The codol probbly of A, gve B c be clculed by combg Equos 3.9 d 3.: Pr B A Pr A j j j Pr A B M 3.3 Pr B A Pr A Ths reloshp s ow s Byes heorem. As expled before, c be used o upde pror dsrbuos usg d. For sce, suppose h we hve from formo elsewhere he pror probbly of sol mosure coe some oobserved loco, sy PrA. Le B represe he oucomes of observos roud he o-observed loco. The probbly PrA B s clled he poseror probbly,.e. he probbly of sol mosure coe he uobserved loco gve he observed vlues roud. To clcule he poseror probbly we eed he so clled lelhood PrB A,.e. he probbly of observg B gve h sol mosure coe A s rue. A A A A M {A B} S Fgure 3.4 Ve dgrm showg he erseco bewee eve B d se of muully exclusve d collecvely exhusve eves A,,..,M. 34

35 3.3 Mesures of couous probbly dsrbuos I Chper we roduced umber of mesures of frequecy dsrbuos, whch re reled o dses d her hsogrm form. Smlr o hsogrm, he locly d form of probbly desy fucos c be descrbed by umber of mesures. These mesures re le Equos 3. d 3.3, bu s we re ow worg wh couous vrbles, he sums re replced by egrls. Before roducg hese mesures we sr wh he defo of he expeced vlue. Le g be fuco of some rdom vrble. The expeced vlue of g s defed s: E[ g ] g f d 3.4 For dscree rdom vrbles D he expeced vlue gd s defed s E [ g D] g d p d D 3.5 where p D d s he probbly mss fuco of dscree rdom vrble e.g. he probbles Tble 3.. So we see h c be vewed s he weghed sum of g over he dom of wh he probbly desy of s wegh. If we e g we ob he me or expeced vlue of he couous verso of 3.. E[ ] f µ d 3.6 If we e g - we ob he vrce couous verso of 3.3: E[ µ ] µ f σ d 3.7 The esmors of he me d he vrce re he sme s Equos 3. d 3.4 wh d j replced wh j. The sdrd devo s gve byσ σ d he coeffce of vro by CV σ µ

36 The followg rules pply o me d vrce f d b re deermsc coss: E [ b] be] 3.9 Vr [ b] b Vr[ ] 3.3 The sewess s defed s: CS 3 E[ µ ] 3 µ f 3 3 σ σ d 3.3 d he cuross s: CC µ 4 E[ µ ] σ σ f d Sewess d cuross c be esmed wh equos.5 d.6 wh replced by - f he me d he vrce hve bee esmed s well. 3.4 Momes The h mome µ of rdom vrble s defed s: E[ ] f µ d 3.33 Ofe, oe wors wh he cerl momes defed s: M E[ ] µ f µ d 3.34 Momes d cerl momes re reled o he more sdrd mesures of probbly dsrbuos s: 36

37 µ µ σ CS CC M M σ 3 3 M σ µ µ Chrcersc fucos There re umber of rsformos of probbly dsrbuos h re useful whe worg wh rdom vrbles. We sr wh he mome geerg fuco, whch s defed s: M s s s E[ e ] e f d 3.36 The mome-geerg fuco s reled o he Lplce rsform. The mome geerg fuco c be used o clcule he momes s: E[ d ] M s µ 3.37 ds s Te for sce he egve expoel dsrbuo: f λ e λ, 3.38 The mome geerg fuco of hs dsrbuo s: M s s s e e λ λ λ λ d λ e d 3.39 λ s From hs we c clcule he momes: d λ µ ds λ s λ λ s s s λ

38 d λ λ µ 3 ds λ s λ s s λ s 3.4 So he vrce s equl o /λ. Aoher rsformo ofe used s he chrcersc fuco: ϕ [ s ] s s E e e f d wh 3.4 The chrcersc fuco s o he Fourer rsform. The verse of 3.4 c lso be defed: f s e ϕ s π ds 3.43 Ths mes h f wo rdom vrbles hve he sme chrcersc fuco hey re declly dsrbued. Le he mome geerg fuco he chrcersc fuco c be used o clcule he momes: d ϕ s µ E[ ] 3.44 ds s If we expd he expoel exps Tylor seres roud we ob: 3 e s s s s By g expecos o boh sdes we ob expresso relg he chrcersc fuco o momes of : 3 ϕ [ s s E e ] se[ ] s E[ ] s E[ 6 3 sµ s µ s µ ] Or wre more geerlly: s ϕ s µ 3.47! 38

39 I c be prove h he pdf of s compleely defed by s chrcersc fuco. From 3.47 c lso be see h f ll momes exs d f 3.47 coverges, h he chrcersc fuco d hrough 3.43 lso he pdf s compleely defed. Ths s he cse for mos of he pdfs ecouered prcce. Ths mes h for ll prccl purposes oe c pproxme he pdf hrough suffce umber of momes. Some ddol properes of he chrcersc fuco: If d re wo depede rdom vrbles we hve Grmme d Srer, 98: ϕ s ϕ s ϕ 3.48 s The sme relo holds for he mome geerg fuco. Also we hve h for vrble Y b he chrcersc fuco becomes Grmme d Srer, 98: ϕ s e sb ϕ s 3.49 Y From 3.48 we c lso deduce h f we hve he sum of M declly dsrbued vrbles wh chrcersc fuco ϕ s M Y... M, 3.5 he chrcersc fuco of Y s gve by: M [ ϕ s ]. ϕ s 3.5 Y The form of 3.5 smules he roduco of he logrhm of he chrcersc fuco. Ths s clled he cumul fuco d s defed s: K s lϕ s 3.5 From 3.5 d 3.5 he follows h he cumul of he sum Y of M declly dsrbued vrbles wh cumul fuco s s by: K KY s MK s 3.53 The seres expso of he cumul fuco s gve by: s K s κ 3.54 where κ re clled he cumuls whch re reled o he cumul fuco s: 39

40 4 s ds s K d κ 3.55 The cumuls re coveely reled o he momes of he pdf, such h we c clcule momes from cumuls d vce vers: µ µ µ µ µ µ µ µ µ µ µ σ µ µ µ 3.56 Up o ow we hve oly led bou couous rdom vrbles. The mome geerg fuco d he chrcersc fuco lso exs for dscree rdom vrbles. I hs cse we hve wh D d p he probbly mss fuco: ] [ D D d p sd e sd e E s M 3.57 ] [ D D d p sd e sd e E s ϕ 3.58 Apr from hese fucos, dscree rdom vrbles c lso be chrcered wh usg he probbly geerg fuco: ] [ D D d p d s D s E s G 3.59 Ths rsformo s reled o he -rsform d oly exss for dscree vrbles. Noe h Pr d G D d D G. Some useful properes Grmme d Srer, 98: s G s G s G D D D D 3.6 ] [ s ds s dg D E D 3.6 ] [ ] [ s ds s G d D E D E D 3.6

41 3.6 Some well-ow probbly dsrbuos d her properes There re my dffere models for probbly dsrbuos. Whch model o choose for whch vrble depeds o s ype. My hydrologcl vrbles re srcly posve e.g. hydrulc coducvy, rfll esy d requre herefore probbly desy fucos pdfs for posve vrbles. Also, cer vrbles, such s he umber of r sorms rrvg fxed ervl, re dscree, whle ohers re couous. I hs seco we wll provde overvew of umber of probbly desy fucos d her properes. Tble 3. gves he pdfs d expressos for he me d he vrce erms of he dsrbuo prmeers. Also gve he ls colum re umber of hydrologcl vrbles h my be descrbed wh he vrous pdfs. Fgure 3.5 provdes plos for umber of he couous pdfs of Tble 3. d Tble 3.3 gves expressos for he ssoced geerg fucos. Some words should be spe o he mos fmous of dsrbuos: he orml or Guss dsrbuo. Ths s he dsrbuo h urlly rses for rdom vrbles h re hemselves he resul of he sum of lrge umber of depede eves. The uderlyg rule s clled he Cerl Lm Theorem d reds: Le,,, N be se of N depede rdom vrbles h hve rbrry probbly dsrbuo wh me µ d vrce σ. The he orml form rdom vrble Y orm N N N σ µ 3.63 hs lmg cumulve dsrbuo fuco h pproches he orml sdrd Guss dsrbuo Typclly error dsrbuos, very relev o sochsc hydrology, re Guss dsrbued, becuse errors re ofe he sum of my depede error sources. If N s very lrge d he dvdul vrbles re mldly depede he urs ou prcce h he summed vrble s pproxmely Guss. A exmple of hydrologcl vrble h c ofe be descrbed wh Guss dsrbuo s freely flucug groudwer level H h flucues uder pulses of precpo surplus P precpo mus evporspro. Usg smple wer blce of he sol colum s possble o wre he groudwer level some me s he sum of precpo surplus eves Koers d Beres, 999: M h α P

42 If we vew he rfll surplus seres s rdom vrbles, he he groudwer level wll be pproxmely Guss dsrbued f M d α re lrge eough. Tble 3.4 provdes cumulve dsrbuo ble Fχx Pr[χ x] for he sdrd orml rdom vrble χ, wh me ero µ d sdrd devo equl o σ. A umber of ofe used qules of he dsrbuo re gve Tble 3.5. Aoher dsrbuo ofe used hydrology h s worh meog s he logorml dsrbuo. A vrble hs logorml or logguss dsrbuo f s url logrhm Yl s Guss dsrbued. A well-ow exmple s hydrulc coducvy. Whe smpled rdomly spce, hydrulc coducvy obeys logorml dsrbuo Freee, 975. Ths ssumpo hs bee recofrmed by my observos herefer. Some useful rsformo formule bewee he mes d vrces of he orml d he logorml dsrbuo re: µ µ Y σ Y / e 3.65 µ Y σ Y σ σ e e Y 3.66 Tble 3. Some well-ow dscree d couous probbly desy fucos Dsrbuo Probbly desy/mss Expeced Vrce Exmple of Hydrologcl vlue pplco Boml BN,p N 4. The umber of flood N p p 5. Np Np-p eves wh probbly p occurrg N me seps,,,.., N Geomerc Gp p p p The umber of me seps ul flood eve wh p p probbly p occurs. Posso Pλ λ e λ The umber of r sorms λ λ occurrg gve me! perod. Uform U,b b b b No-formve pror dsrbuo of b hydrologcl prmeer provded o prmeer esmo mehod Expoel Eλ λe λ The me bewee wo r sorms λ λ Guss/Norml [ µ / σ ] My pplcos: pror Nµ,σ e µ πσ σ dsrbuo for prmeer opmo; modelg of lognorml Lµ,σ Gmm Γ, λ oe: R [ l / e µ σ ] µ πσ σ λ λ e Γ λ λ errors; lelhood fucos Hydrulc coducvy Sum of depede rdom vrbles h re expoelly dsrbued 4

43 Be p, q β Γ p q p q Gumbel G,b Exreme vlue dsrbuo Type I Webull W λ, β Exreme vlue dsrbuo ype III Γ p Γ q p >, q >, be exp e b b β λ β exp[ λx β β ] p p q pq p q p q.577 b π 6b Γ λ β A λ B A Γ β B Γ β wh prmeer λ; Iseous u hydrogrph of ler reservors seres; pdf of rvel mes cchme; very flexble dsrbuo for srcly posve vrbles. Very flexble dsrbuo for vrbles wh upper d lower boudres; used s pror dsrbuo Byes lyss d prmeer esmo Yerly mxmum dschrge used for desg of dms d des Yerly mmum dschrge used low flow lyss. 6. Tble 3.3 Chrcersc fucos for umber of probbly dsrbuos Dsrbuo Probbly geerg fuco Mome geerg fuco Chrcersc fuco Boml B,p Geomerc Gp p ps ps p s s p pe s pe p e s s p pe s pe p e Posso Pλ s e λ e s e λ e e λ s Uform U,b - bs s bs s Expoel Eλ - Guss/orml N µ, σ, λ e e s b λ λ s - s σ s Gmm Γ - e e e s b λ λ s µ µ s σ s λ s λ e λ s λ s 43

44 Fgure 3.5 Plos for some well-ow probbly desy fucos 3.7 Two or more rdom vrbles If wo rdom vrbles d Y re smuleously cosdered e.g. hydrulc coducvy d porosy we re eresed he bvre probbly desy fuco f Y,y h c be used o clcule he probbly h boh d Y re bewee cer lms: Pr[ < y < Y y ] f, y ddy 3.67 A more forml defo of he bvre pdf s: y y Y f Y Pr[, y lm d dy < y ddy < Y y ]

45 The bvre cumulve dsrbuo fuco s F Y,y s gve by: F Y, y Pr[ Y y] 3.69 The desy fuco d he dsrbuo fuco re reled s: y F, y f, y ddy 3.7 Y Y f Y, y FY, y y 3.7 The mrgl dsrbuo of c be obed from he bvre dsrbuo by egrg ou he Y vrble: f f Y, y dy 3.7 The codol probbly c be obed from he dsrbuo fuco s: F Y y y Pr{ Y 3.73 whch hus provdes he probbly h s smller h gve h Y es he vlue of y. The codol pdf c be derved from hs by dffereo: df Y y f Y y 3.74 d The codol desy ssfes: f Y y d 3.75 The relo bewee he bvre pdf d he codol pdf s gve by see lso 3..3: f Y Y Y Y Y, y f y f y f y f 3.76 The ol probbly heorem erms of desy fucos reds: Y y f Y y f f ` d

46 d Byes heorem becomes: f f y f Y, y Y f Y y 3.78 fy y f y f d Y A mesure of ler sscl depedece bewee rdom vrbles d Y s he covrce s defed s: Cov, Y] E[ µ Y µ ] µ y µ f, y ddy 3.79 [ Y Y Y The covrce bewee wo d ses c be esmed usg Equo.7, where we hve o replce he umber of observos by - f he respecve me vlues of d Y hve bee esmed oo. The followg relos bewee vrce d covrce exs d b coss: Vr[ by] Vr[ ] b Vr[ Y] bcov[, Y] Vr[ by] Vr[ ] b Vr[ Y] bcov[, Y] 3.8 Ofe he correlo coeffce s used s mesure of ler sscl depedece: ρ T Cov[, Y ] 3.8 σ σ Y The followg should be oed. If wo rdom vrbles re ssclly depede hey re lso ucorreled: Cov[Y,] d ρ Y. However ero correlo coeffce does o ecessrly me h Y d re ssclly depede. The covrce d correlo coeffce oly mesure ler sscl depedece. If o-ler relo exss, he correlo my be bu he wo vrbles my sll be ssclly depede, s s show he lower rgh fgure of Fgure.3. Fgure 3.6 shows surfce plos d soplos of he bvre Guss Dsrbuo: f Y, y πσ σ Y ρ exp ρ Y Y µ σ µ Y σ Y µ σ 3.8 µ Y σ Y 46

47 where he lef plos show he suo for whch ρ T d he rgh plos for whch ρ T.8. We c see h he soles form ellpse whose form s deermed by he roσ / σ Y d s prcple dreco by ρ T. Fgure 3.6 Surfce plos d soplos of he bvre Guss dsrbuo of depede lef d depede rdom vrbles d Y. From he reloshp f Y y f Y, y / fy y we c derve h he codol Guss desy f Y Y hs Guss dsrbuo N µ, σ wh y σ µ µ ρ Y y µ Y d σ σ ρy 3.83 σ Y From hese expressos we ler h f we hve wo depede rdom vrbles d we mesure oe of hem hs cse Y h our pror dsrbuo s upded o ew poseror dsrbuo d h our ucery bou hrough he vrce hs decresed. Of course, f boh vrbles re depede we see h µ µ d σ σ. 47

48 Flly, useful propery of he Guss dsrbuo s h y ler combo of Guss rdom vrbles wh deermsc weghs Y N 3.84 wh me µ,,..,n, vrce σ,,..,n d ρ j,,j,..,n correlo coeffces bewee rdom vrbles d j, s self Guss dsrbued N µ Y, σ Y wh me d vrce gve by: N µ µ 3.85 Y σ Y N N j ρ σ σ j j 3.86 We wll ed hs chper wh some oes o mulvre dsrbuos. All he reloshps gve here for bvre dsrbuos c be esly geerled o probbly dsrbuos of mulple vrbles mulvre dsrbuos: f,..,. N.. N A dsrbuo ofe used sochsc hydrology o prmeere mulvre dsrbuos s he mulvre Guss dsrbuo. I c be defed s follows: Le,,, N be colleco of N rdom vrbles h re collecvely Guss dsrbued wh me µ,,..,n, vrce σ,,..,n d ρ j,,j,..,n correlo coeffces bewee vrbles d j. We defe sochsc vecor,,, N T d vecor of me vlues µ µ,µ,,µ N T superscrp T sds for rspose. The covrce mrx C s defed s E[-µ-µ Τ ]. Τhe covrce mrx s N N mrx of covrces. Eleme C j of hs mrx s equl o ρ j σ σ j. The mulvre Guss probbly desy fuco s gve by: µ T C µ f,..., e... N N N / / π C 3.87 wh C he deerm of he covrce mrx d C he verse. 48

49 Tble 3.4 Cumulve dsrbuo ble for he sdrd orml Guss dsrbuo N,; Fχx Pr[χ x], e.g. F χ.6.79; oe Fχ-x - Fχx Tble 3.5 Seleced qules of he sdrd orml dsrbuo N,;oe h q-p - qp 49

50 3.8 Exercses 3. Cosder he esy of oe-hour rfll whch s ssumed o follow λ expoel dsrbuo: λe. Wh λ., clcule: Pr[>]. f 3. Cosder he followg probbly desy fuco descrbg verge sol mosure coe he roo oe of some sol see lso he Fgure: f Probbly desy Sol mosure coe Gve he expresso for he cumulve probbly dsrbuo. b Clcule he probbly h verge sol mosure exceeds.3. c Clcule he me µ d he vrce σ of sol mosure coe. 3.3 Hydrulc coducvy some uobserved loco s modeled wh log-orml dsrbuo. The me of YlK s. d he vrce s.5. Clcule he me d he vrce of K? 3.4 Hydrulc coducvy for qufer hs logorml dsrbuo wh me m/d d vrce m /d. Wh s he probbly h o-observed loco he coducvy s lrger h 3 m/d? 3.5 Bsed o geologcl lyss we exrced he followg probbles of exure clsses occurrg some qufer: Pr[sd].7, Pr[cly]., Pr[pe].. The followg ble shows he probbly dsrbuos of coducvy clsses for he hree exures: 5

51 Tble: probbles of coducvy clsses m/d for hree exure clsses Texure Sd Cly Pe Clcule he probbly dsrbuo of coducvy for he ere qufer use he ol probbly heorem for hs. 3.6 Cosder wo rdom vrbles d wh me d 5 d vrces 3 d 45 respecvely. The correlo coeffce bewee boh vrbles equls.7.. Clcule he covrce bewee d. b. Clcule he expeced vlue of Y. c. Clcule he vrce of Y. 3.7 For he sme wo vrbles of 3.5: Assume h hey re bvre Guss dsrbued d:. Clcule he probbly Pr[ < 3] b. Clcule he probbly Pr[ < 5] c. Wre he expresso for he probbly Pr[ < 3 < 4] d. Wre he expresso for he probbly Pr[ < 3 < 4] 3.7 * Derve equos 3.48, 3.49, 3.5 d

52 4 Hydrologcl sscs d exremes 4. Iroduco The feld of sscs s exesve d ll of s mehods re probbly pplcble o hydrologcl problems. Some elemery descrpve sscs ws lredy reed chper, whle geosscs wll be reed chper 7. The feld of hydrologcl sscs s mly cocered wh he sscs of exremes, whch wll be he m opc hs chper. I he frs pr we wll mly loo he lyss of mxmum vlues, prculr o flood esmo. I he secod pr we wll cover some of he sscl ess h should be ppled o seres of mxm before y furher lyss o exremes s possble. 4. The lyss of mxmum vlues 4.. Flow duro curve We sr ou wh me seres of over yers of dly verged flow d of he Rver Rhe Lobh pproxmely he loco where he Rhe eers he Neherlds. Fgure 4. shows plo of hs me seres. To sy somehg bou he flow frequecy so -clled flow duro curve c be mde. Such curve s show Fgure 4.. The flow frequecy my be formve cer cses, e.g. for 5 % of he me he dschrge wll be bove 5 m 3 /s. However, whe comes o buldg dms or des, oly he frequecy of dvdul floods or dschrge pes re mpor. To elbore: f oe eeds o fd he requred hegh of de, he he mxmum hegh of pe s mos mpor d o lesser exe s duro. So our gol s o lye flood pes. Geerlly, wo dffere mehods re used o cover seres of dschrges o seres of pes: oe s bsed o defco of he mxmum dschrge per yer d he oher o lyg ll dschrge vlues bove gve hreshold. We wll cocere o he frs mehod here, d brefly re he secod. 4.. Recurrece mes To ob seres of mxmum vlues we smply record for ech yer he lrges dschrge mesured. Ths resuls he sme umber of mxmum dschrges s recorded yers. Somemes, f mxmum dschrges occur oe pr of he seso wer, my be wse o wor wh hydrologcl yers Aprl o Mrch 3 s Norher Europe. Ths hs bee doe wh he Rhe dschrges. The plo wh mxm s show Fgure 4.3. To furher lyses hese mxmum vlues he followg ssumpos re mde:. he mxmum vlues re relos of depede rdom vrbles;. here s o red me; 3. he mxmum vlues re declly dsrbued. 5

53 Fgure 4. Dly verge dschrges m 3 /s of he rver Rhe Lobh Fgure 4. Flow duro curve of he rver Rhe Lobh 53

54 4 Q mx m 3 /s Yer Fgure 4.3 Yerly mxmum vlues of dly verge dschrge of he rver Rhe Lobh I seco 4.3 we wll descrbe some sscl ess o chec wheher hese ssumpos re lely o be vld. If he mxmum vlues re deed depede rdom vrbles, he frs sep of lyss would be o clcule he cumulve frequecy dsrbuo d use hs s esme for he cumulve dsrbuo fuco F y Pr Y y. The probbly h cer vlue y s exceeded by he mxmum vlues s gve by he fuco P y Pr Y > y F y. Flly, he recurrece me or reur perod Ty whe ppled o yerly mxmum vlues s he me or expeced umber of yers bewee wo flood eves h re lrger h y d s clculed usg eher Fy or Py s: T y 4. P y F y To esme he recurrece me, frs he cumulve dsrbuo fuco mus be esmed. The mos commo wy of dog hs s by rrgg he vlues scedg order d ssgg r umbers from smll o lrge: y,,..,n, wh N he umber of yers. Ordes of he cumulve dsrbuo fuco re he esmed by: ˆ F y 4. N 54

55 There hve bee oher esmors proposed e.g. Grgore: F ˆ y.44 /., bu for lrger N dffereces bewee hese re smll. Tble 4. shows pr of he lyss performed o he mxmum dschrge vlues for he rver Rhe. As c be see, he mxmum recurrece perod h c be lyed wh he rw d s N yers 3 he exmple. However, my dms d des re desged lrger reur perods. For sce, rver des he Neherlds re bsed o :5 yer floods. To be ble o predc flood ses belogg o hese lrger recurrece mes we eed o somehow exrpole he record. Oe wy of dog hs s o f some probbly dsrbuo o he d d use hs probbly dsrbuo o predc he mgudes of floods wh lrger recurrece mes. As wll be show he ex seco, he Gumbel dsrbuo s good cdde for exrpolg mxmum vlue d. Tble 4. Alyss of mxmum vlues of Rhe dschrge Lobh for recurrece me Y R Fy Py Ty Mxmum vlues d he Gumbel dsrbuo The dded ssumpo wh regrd o he record of mxmum vlues s h hey re rdom vrbles h follow Gumbel dsrbuo,.e. he followg pdf d dsrbuo fuco: f b b be exp e 4.3 b F exp e

56 Here we wll show h he dsrbuo of he mxmum vlues s lely o be Gumbel dsrbuo. Le,,, N be N depede d declly dsrbued vrbles wh dsrbuo fuco F. Le Y be he mxmum of hs seres Y mx,,, N. The dsrbuo fuco of Y s he gve by: F y Pr Y Y y Pr Pr F y N y, y Pr y,.., N y y Pr N y 4.5 We co derve he probbly dsrbuo from 4.5 loe, becuse F y N s so clled degeerve dsrbuo: f N he F y. To ob odegeerve dsrbuo of Y we eed o reduce d ormle he mxmum vlues. Now suppose h he vrbles hve expoel dsrbuo: F exp b wh b>. We pply he followg rsformo of he mxmum Y: N X l N b Y b 4.6 The dsrbuo fuco of hs vrble becomes: Pr X l N x Pr b Y x Pr Y b b F x l N b exp x N N N x l N exp x l N N 4.7 Tg he lm yelds: lm Pr X N N exp x x x lm exp e 4.8 N N whch s he ormled Gumbel dsrbuo. So he lm dsrbuo of X f hs expoel dsrbuo s exp e x. If we defe log N / b he XbY-. So for lrge N we hve: 56

57 Pr[ X x] exp e x Pr[ b Y x] exp e x Pr[ Y y] Pr[ b Y b y ] exp e b y 4.9 So, flly we hve show h he mxmum Y of N depedely d declly expoelly dsrbued rdom vrbles I hs Gumbel dsrbuo wh prmeers d b. For fe N hs dsrbuo s lso used, where d b re foud hrough fg he dsrbuo o he d. I c be show smlr wy s for he expoel dsrbuo h he Gumbel lm dsrbuo s lso foud for he mxmum of depede vrbles wh he followg dsrbuos: Guss, logguss, Gmm, Logsc d Gumbel self. Ths s he reso why he Gumbel dsrbuo hs bee foud o be suble dsrbuo o model probbles of mxmum vlues, such s show Fgure 4.3. Of course, we hve o ssume h hese mxmum vlues hemselves re obed from depede vrbles wh yer. Ths s clerly o he cse s s show Fgure 4.. However, s log s he mxmum vlues re pproxmely depede of oher mxmum vlues, urs ou h he Gumbel dsrbuo provdes good model f d b c be fed Fg he Gumbel dsrbuo To be ble o use he Gumbel dsrbuo o predc lrger recurrece mes hs o be fed o he d. There re severl wys of dog hs: usg Gumbel probbly pper; ler regresso; 3 he mehods of momes; 4 mxmum lelhood esmo. The frs hree mehods wll be show here. Gumbel probbly pper Usg equo 4.4 c be show h he followg relo holds bewee gve mxmum y d he dsrbuo fuco Fy y l l F y 4. b So by plog -l-lfy gs he mxmum vlues y we should ob srgh le f he mxmum vlues follow Gumbel dsrbuo. If Gumbel probbly pper s used he double log-rsformo of he dsrbuo fuco s lredy mplc he scle of he bscss. O hs specl pper plog mxmum vlues y wh r gs T y /[ / N ] wll yeld srgh le f hey re Gumbel dsrbued. The Rhe dschrge mxmum vlues of ble 4. hve bee ploed Fgure 4.4. Oce hese d hve bee ploed srgh le c be fed by hd, from whch he prmeers d /b c clculed. The fed le s lso show Fgure

58 Fgure 4.4 Gumbel probbly plo of Rhe dschrge mxmum vlues d fed ler relo; ˆ 564, bˆ.5877 Ler Regresso Alervely, oe c plo y gs l[ l / N ] o ler scle. Ths wll provde srgh le cse he y re Gumbel dsrbued. Fg srgh le wh ler regresso wll yeld he ercep l[ l / N ] d he slope /b. Mehods of momes The eses wy o obg he Gumbel prmeers s by he mehod of momes. From he probbly desy fuco c be derved h he me µ Y d vrce σ Y re gve by see lso Tble 3.3:.577 µ 4. Y b π σ Y 4. 6b Usg sdrd esmors for he me m Y / Y d he vrce s Y / Y my he prmeers c hus be esmed s: 58

59 6s Y ˆ b π ˆ my bˆ 4.4 Fgure 4.5 shows plo of he recurrece me versus mxmum dschrge wh prmeers fed wh he mehod of momes. Q mx m 3 /s Recurrece me Yers Fgure 4.5: Recurrece me versus yerly mxmum of dly verged dschrge of he rver Rhe Lobh: ˆ 56, bˆ.6674 The mehod of momes yelds bsed esme of he prmeers. More sophsced esmors re ubsed d more ccure re he probbly weghed momes mehod Ldwehr e l, 979 d debsed mxmum lelhood mehods e.g. Hosg, 985. These esmo mehods re however much more complex, whle he resuls re o lwys very dffere Esmo of he T-yer eve d s cofdece lms Oce he prmeers of he Gumbel-dsrbuo hve bee esmed we c esme mgude he T-yer eve, whch s he bss for our desg. Ths eve s esmed by oe hs s bsed esme: 59

60 ˆ y T T ˆ l l 4.5 bˆ T Alervely, oe could of course red drecly from he Gumbel plo o probbly pper. Whe ppled o he Rhe dse d he prmeers obed from he mehod of momes we ob for sce for T5 hs s he curre desg orm 3 y l l m / s. Wh he prmeers obed from ler regresso we ob 7736 m 3 /s. Whe prmeers re obed hrough ler regresso he 95%-cofdece lms of y T c be obed hrough: yˆ s T y yˆ s 4.6 T 95 ˆ T T 95 ˆ T Y Y wh 95 he 95-po of he sude s -dsrbuo, d s Yˆ he sdrd error of he regresso predco error g ccou of prmeer ucery whch s esmed s: s x T x 4.7 N Yˆ T y yˆ N x T x N wh x T l[ l T / T ] he rsform of recurrece ervl T of eve wh r from he N prs T, y T he Gumbel plo. I Fgure 4.4 he 95% cofdece lms re lso gve. The 95% cofdece lms of he 5 yers eve re {783, 889} m 3 /s. A pproxme esme of he esmo vrce of y T cse he prmeers re esmed wh he mehod of momes s gve by Sedger e l., 993:..5x.6x r ˆ yˆ 4.8 bˆ N V T wh x l[ l T / T]. I fgure 4.5 he 95%-cofdece lms re gve ssumg he esmo error of ŷt o be Guss dsrbued: y ˆT ±.96 Vr yˆ T. For T5 yers hese lms re: {586,96} m 3 /s. I V Mofor 969 cofdece lms re gve for he cse whe he prmeers re obed wh mxmum lelhood esmo. The umber of T-yer eves gve perod 6

61 I should be sressed h he T-yer flood eve does o excly rse every T yers! If we hd very log perod of observos sy τ yers, wh τ he order of yers or so he he T-yer eve would occur o verge τ/t mes. The probbly h N yers T-yer flood eve occurs mes follows boml dsrbuo: y N N p p N Pr eves T yers 4.9 wh p / T. So he probbly of yer flood eve occurrg he ex yer s.. The probbly of excly oe flood eve occurrg he comg yers s 9 Preves y yers The probbly h oe or more flood eves occur yers s -Pro eves Gog bc o our Rhe exmple: he probbly of les oe T5 yers eve occurrg he ex yers s , whch s sll lmos 8%! The me ul he occurrece of T-yer eve The umber of yers m ul he ex T-yer eve s rdom vrble wh geomerc dsrbuo wh p /T : Pr m yersuleve y T m p p 4. The recurrece me T s he expeced vlue of hs dsrbuo: E [ m] / p T. The probbly of T-yer flood occurrg he ex yer s p s expeced Po d A lerve wy of obg flood sscs from me seres of dschrges s hrough prl duro seres or pe over hreshold po d. The de s o choose hreshold bove whch we mge h dschrge pes would be clled flood eve. We he cosder oly he dschrges bove hs hreshold. Fgure 4.6 shows he po-d for he Rhe cchme for hreshold of 6 m 3 /s. If here re exceedces d f he mgude of he exceedces re depede d declly dsrbued wh dsrbuo fuco F h he probbly of he mxmum Y of hese exceedces s gve by y d y f 6

62 Pr[ Y y] Pr[ Pr[ y y] Pr[ y... y] Pr[ y y] F y 4. Becuse he umber of exceedeces N s lso rdom vrble from Equo 3. we fd: F y Pr[ Y y] Pr[ N ] 4.3 I urs ou h f he hreshold s lrge eough h he umber of exceedces hs Posso dsrbuo. Thus we hve for 4.3: Pr[ Y y] e λ λ F! λ[ F y] e 4.4 If he exceedces obey expoel dsrbuo he follows h he mxmum of he Po-d hs Gumbel dsrbuo: Pr[ Y λ[ ] F y] by y e exp[ λ e ] b y lλ exp[ l by e λ e exp[ e b ] 4.5 So cocluso: he mxmum vlue of Posso umber of depede d expoelly dsrbued exceedces follows Gumbel dsrbuo. 6

63 4 Q over m 3 /d Yer Fgure 4.6 Pe over hreshold d of dly verge Rhe dschrge Lobh The lyss of he po d s rher srghforwrd. The T-yer exceedce s gve by: ˆ y T l λ T l l b b T 4.6 whλ he me umber of exceedces per me u d /b he verge mgude of he exceedces whch c boh be esmed from he po d. The po d of he Rhe d Lobh yeld: 5.786d 8m 3 /s. The 5 yer exceedce s hus gve by: 49 yˆ T 8 l l l 9 m 3 /s The :5 yer eve he s obed by ddg he hreshold: 69 m 3 /s Oher dsrbuos for mxmum vlues The Gumbel dsrbuo s o he oly probbly dsrbuo h s used o model mxmum vlues. I fc, he Gumbel dsrbuo s specfc cse of he so clled Geerl Exreme Vlue dsrbuo GEV: 63

64 / { [ b y ] } exp F y 4.8 exp{ exp b y } Where s form prmeer. As c be see he GEV revers o he Gumbel dsrbuo for. The Gumbel dsrbuo herefore s ofe clled he Exreme vlue dsrbuo of ype I EV ype I. If > we ob EV ype III dsrbuo or Webull dsrbuo whch hs fe upper boud. If < s clled he EV ype II dsrbuo whose rgh hd l s hcer h he Gumbel dsrbuo, such h we hve lrger probbly of lrger floods. Fgure 4.7 shows he hree ypes of exreme vlue dsrbuos o Gumbel probbly pper. As c be see, he hree dsrbuos cocde for T /-e.58 yers. Apr from he GEV dsrbuo oher dsrbuos used o model mxmum vlues re he logorml dsrbuo see 3. d he log-perso ype 3 dsrbuo. The prmeers of he logorml dsrbuo c be obed hrough for sce he mehod of momes by esmg he me d he vrce of he log-mxmum vlues µ l Y d σ ly. The Perso ype 3 dsrbuo does o hve closed form soluo bu s buled. The p-qule y p of he probbly dsrbuo vlue of y for whch Pr Y y p s lso buled usg so clled frequecy fcors K p CS Y : y p µ σ K CS 4.9 Y Y p Y wh µ Y,σ Y, CSY he me, sdrd devo d coeffce of sewess respecvely. The Log-Perso ype 3 dsrbuo s obed by usg he me, vrce d sewess of he logrhms: µ l Y, σ l Y, CSl Y. Dffereces bewee exrpolos wh hese dsrbuos c be subsl f he me seres legh s lmed, whch cse he choce of he dsrbuo my be very crcl. To be sfe, somemes severl dsrbuos re fed d he mos crcl e s desg orm. Fgure 4.8 shows wo dffere dsrbuos fed o he Rhe d. 64

65 Fgure 4.7 GEV-dsrbuos o Gumbel probbly pper. 8 Logorml 6 4 Gumbel Q mx m 3 /s Recurrece me Yers Fgure 4.8 Gumbel d logorml dsrbuos fed o he sme mxmum vlues. 65

66 4..8 Mmum vlues Up o ow we hve oly bee cocered wh mxmum vlues. However he lyss of, for sce, low flows we my be cocered wh modellg he probbly dsrbuo of yerly mmum dschrge. Oe smple wy of delg wh mmum vlues s o rsform hem o mxmum vlues, e.g. by cosderg y or /y. Also, he GEV ype III or Webull dsrbuo s suble for modellg mmum vlues h re bouded below by ero. 4.3 Some useful sscl ess Esmo of T-yer eves from records of mxmum vlues s bsed o he ssumpo h he mxmum vlues re depede, re homogeous here s o chge me d vrce over me d hve cer probbly dsrbuo e.g. Gumbel. Here we wll provde some useful ess for checg mxmum vlue d o depedece, reds d ssumed dsrbuo. For ech of he ssumpos here re umerous ess vlble. We do o presume o be complee here, bu oly prese oe es per ssumpo. 66

67 4.3. Tesg for depedece Vo Neum s Q-ssc c be used o es wheher seres of d c be cosdered s relos of depede rdom vrbles: Q Y Y Y Y 4.3 where Y s he mxmum vlue of yer d Y s he mxmum vlue of yer ec., d he legh of he seres. I c be show h f he correlo coeffce bewee mxmum vlues of wo subseque yers s posve ρ >, whch s usully he cse for url processes, h Q wll be smller f ρ s lrger. If we ccep h f he mxmum vlues re depede h her correlo wll be posve, we c use Q s es ssc for he followg wo hypohess: H : he mxmum vlues re depede; H : hey re depede. I h cse we hve lower crcl re. Tble 4. shows lower crcl vlues for Q. Uder he ssumpo h y re depede d from he sme dsrbuo we hve h E[Q]. If Q s smller h crcl vlue for cer cofdece level α here eher.,. or.5 we c sy wh ccurcy of α h he d re depede. For sce, g he mxmum vlues of he ls yers Lobh we ob for h Q.6. From Tble 4. we see h Q s o he crcl re, so here s o reso o rejec he hypohess h he d re depede. Noe h for lrge we hve h Q Q ' 4.3 / s pproxmely sdrd Guss dsrbued. I cse of he Rhe d se we hve for 3 h Q.7 d herefore Q.366 whch s fr from he crcl.5 vlue of.96: he ero-hypohess of depede mxmum vlues co be rejeced. 67

68 Tble 4. Lower crcl vlues for vo Neum s es of depedece. α α 4.3. Tesg for reds There re my ess o reds, some of hem specfclly focused o ler reds or sep reds or bsed o specl ssumpos bou he dsrbuo of he d sequece. Here oprmerc es ssc s preseed he M-Kedll es for reds h mes o ssumpos bou he ype of red ler or o-ler or he dsrbuo of he d. The red mus however be moooous d o perodc. Cosder he seres of ul mxmum vlues Y,..,. Ech vlue Y,.., s compred wh ll he subseque vlues Y j j,..,- o compue he M-Kedll es ssc: 68

69 T j sg Y Y j 4.3 wh f Y > Yj sg Y Yj f Y Yj 4.33 f Y < Yj For lrge >4 d o serl depedece of he d he ssc T s sympoclly Guss dsrbued wh me E[ T] d vrce Vr [ T] [ 5]/8. Ths mes h he es H : he mxmum vlues hve o red; H : hey hve red, usg he es ssc T' 8T /[ 5] hs wo-sded crcl re wh crcl levels gve by he qules of he sdrd orml dsrbuo: χα / d χ α / wh α he sgfcce level. Performg hs es for he Rhe d se of mxmum vlues yelds: T689 d T.99. Ths mes h he Hypohess of o red s rejeced. The vlue of T suggess posve red 5% ccurcy Tesg presumed probbly dsrbuo A very geerl es o dsrbuo s Perso s ch-squred χ es. Cosder d se wh mxmum vlues. The vlues re clssfed o m o-overlppg clsses o ob hsogrm. For ech clss we clcule he umber of d fllg hs clss m oe h. Usg he presumed dsrbuo fuco F Y y he expeced umber of d fllg o clss c be clculed s: wh y up s gve s: e d y F y F y 4.34 low Y up Y low he upper d lower boudres of clss respecvely. The es ssc Χ m e e 4.35 Noe h f he prmeers of he presumed dsrbuo re esmed wh he mehod of momes some bs s roduced. 69

70 The ssc Χ hs χ m dsrbuo,.e. ch-squred dsrbuo wh m- degrees of freedom. Tble 4.3 provdes upper crcl vlues for he χ dsrbuo for vrous degrees of freedom d sgfcce levels. Applco o he Rhe d se gves for clsses of wdh 5 from 5 o 3 m 3 /d d he ssumed Gumbel dsrbuo wh 56 d b leds o Χ Ths s ousde he crcl re for 9 degrees of freedom d ccurcy of 5%. We herefore coclude h he ssumpo of he mxm beg Gumbel dsrbued co be rejeced. Performg he sme es wh he logorml dsrbuo yelds Χ 4. 36whch fs eve beer. 7

71 Tble 4.3 Upper crcl vlues for he χ dsrbuo Probbly of exceedg he crcl vlue ν Probbly of exceedg he crcl vlue ν

72 4.4 Exercses 4. Cosder he followg yerly mxmum dly srem flow d of he Meuse rver he Eysde so he Duch-Belg border dly verge dschrge m 3 /s Plo he d o Gumbel-probbly pper d deerme he prmeers of he Gumbel dsrbuo. Esme he -yer flood? b. Deerme he prmeers of he Gumbel dsrbuo by ler regresso of Q mx gs l[- l{/}] wh he r of he h smlles mxmum. Esme he -yer flood d s 95% cofdece lms. c. Deerme he prmeers of he Gumbel dsrbuo wh he mehod of momes. Esme he -yer flood d s 95% cofdece lms. d. Esme he -yer flood ssumg logorml dsrbuo of he mxmum vlues. e. Tes wheher hese mxmum vlues c be modeled s depede sochsc vrbles. f. Tes wheher he d c be cosdered s oucomes of Gumbel dsrbuo. g. Tes wheher he d c be cosdered s oucomes of logorml dsrbuo. h. Wh s he probbly h he -yer flood occurs les oce wh he ex 4 yers?. Wh s he probbly h he -yer flood occurs wce he ex yers? 7

73 5 Rdom fucos I chper 3 rdom vrbles were reed. I hs chper hese coceps re exeded o rdom fucos. Oly he bsc properes of rdom fucos re reed. Elbore reme of rdom fucos c be foud sdrd exboos o he subjec such s Ppouls 99 d Vmrce 983. A very bsc d excelle roduco o rdom fucos c be foud Iss d Srvsv Defos Fgure 5. shows he schemclly he cocep of rdom fuco RF. Cosder some propery h vres spce e.g. hydrulc coducvy, me e.g. surfce wer levels or spce d me e.g. groudwer deph. Excep locos or mes where he vlue of propery s observed, we do o ow he exc vlues of. I order o express our ucery bou we dop he followg cocep. Ised of usg sgle fuco o descrbe he vro of spce d/or me, fmly of fucos s chose. Ech of hese fucos s ssumed o hve equl probbly of represeg he rue bu uow vro spce d/or me. The fmly of eqully probbly fucos s clled he esemble, or lervely rdom fuco. As wh rdom vrble, rdom fuco s usully deoed wh cpl. If s fuco of spce s lso referred o s rdom spce fuco RSF or rdom feld: x x 3, x, y R or x, y, R x If s fuco of me s referred o s rdom me fuco RTF, rdom process or sochsc process:, R If s fuco of spce d me s referred o s rdom spce-me fuco RSTF or spce-me rdom feld: 3 x,, x R / R, R I hs chper we wll re mos of he heory of rdom fucos usg he emporl frmewor. The spl frmewor x s used for coceps h re oly defed spce; b c be beer expled spce c cse cer defos he spl frmewor re dffere from he emporl frmewor. I Fgure 5. four fucos esemble members re show. Oe prculr fuco or esemble member ou of he my s clled relo d s deoed wh he lower cse. Depedg 73

74 o he ype of rdom fuco see herefer he umber of possble relos mg up rdom fuco c be eher fe or fe. Fgure 5. Schemc represeo of rdom fuco Aoher wy of loog rdom fuco s s colleco of rdom vrbles oe every loco spce or po me h re ll muully ssclly depede. If we reur o he exmple of Fgure 5.: every po me rdom vrble s defed. A ech he rdom vrble s descrbed wh probbly desy fuco pdf f ; whch o oly depeds o he vlue of, bu lso o he vlue of. Ths pdf could be esmed by smplg ll relos cer po sy me or loco d clculg he hsogrm of he smples. Fgure 5. shows h he vro mog he relos s lrger po h po, ledg o probbly desy fuco wh lrger vrce h. So, we re less cer bou he uow vlue h we re. A ech po me or loco we c clcule he me- or loco- depede me d vrce: d he vrce s defed s E[ ] f ; µ d 5. µ σ E[{ µ } ] { } f ; d. 5. Also, he rdom vrbles re usully correled me or spce for spl rdom fucos: Cov [, ]. The covrce s geerlly smller whe rdom vrbles re cosdered locos furher pr. The covrce s defed s: 74

75 Cov[, ] E[{ µ }{ { µ }{ µ µ } f, }] ;, d d 5.3 For he covrce equls he vrce 5.. For se of N dscree rdom vrbles he jo or mulvre probbly dsrbuo Pr[d,d,,d N ] descrbes he probbly h N rdom vrbles hve cer vlue,.e. h D d d D d d.d D N d N. Smlrly, for N couous rdom vrbles he mulvre probbly desy f,,. N s mesure of N rdom vrbles,,..,n vrbles hvg cer vlue. The logue for rdom fuco s he probbly mesure h relo of se of N locos,,..,n hs he vlues bewee < d, < d,.., N < N N dn respecvely. The ssoced mulvre probbly desy fuco pdf s deoed s f,,. N ;,,. N d defed s: f,,. N N Pr ;,,. N <, <,.., L lm N,..., N < N N N 5.4 Becuse reles o rdom vrbles dffere pos or locos, he mulvre pdf of rdom fuco s someme referred o s mulpo pdf. Theoreclly, rdom fuco s fully chrcered we ow ll here s o ow bou f he mulvre probbly dsrbuo for y se of pos s ow. 5. Types of rdom fucos Rdom fucos c be dvded o ypes bsed o wheher her fucol vlues re couous e.g. hydrulc coducvy spce or dschrge me or dscree e.g. he umber of floods gve perod. Aoher dsco s bsed o he wy he dom of he rdom fuco s defed see Fgure 5.. For sce, couous vlued rdom fuco c be: defed ll locos me, spce or spce me:, x or x,; b defed dscree pos me, spce or spce me, where,,..,k s referred o s rdom me seres d x, j y,,..,i; j,..,j s lce process; c defed rdom mes or rdom locos spce d me: T, X or X,T. Such process s clled compoud po process. The occurrece of he pos rdom coordes spce d me s clled po process. If he occurrece of such po s ssoced wh he occurrece of rdom vrble e.g. he occurrece of huder sorm cell cer loco spce wh rdom esy of rfll s clled compoud po process. 75

76 Nurlly, oe c me he sme dsco for dscree-vlued rdom fucos, e.g. D, D or DT; Dx, D x, j y or DX ec. Fgure 5. Exmples of relos of dffere ypes of rdom fucos bsed o he wy hey re defed o he fucol dom; rdom me seres; b lce process; c couous-me rdom fuco; d compoud po process. 5.3 Sory rdom fucos 5.3. Src sory rdom fucos A specl d of rdom fuco s clled src sory rdom fuco. A rdom fuco s clled src sory f s mulvre pdf s vr uder rslo. So for y se of N locos d for y rslo we hve h f,,, N ; f,,,.,., N N ; ', ',, N ', ' 5.5 Thus, we c hve y cofguro of pos o he me xs d move hs cofguro he whole cofguro, o oe po he me of pos forwrd d bcwrds me d hve he sme mulvre pdf. For he spl dom we hve o sress h src sory mes vr pdf uder rslo oly, o roo. So for src sory rdom fuco wo dmesos, he wo ses of locos he lef fgure of 5.3 hve he sme mulvre pdf, bu 76

77 hose he rgh fgure o ecessrly so. A rdom fuco whose mulvre pdf s vr uder roo s clled ssclly soropc rdom fuco. y y x x Fgure 5.3. Trslo of cofguro of pos whou roo lef fgure d wh roo rgh fgure 5.3. Ergodc rdom fucos Oe could s why he propery of sory s so mpor. The reso les esmg he sscl properes of rdom fuco such s he me, he vrce d he covrce. I cse of rdom vrble, such s he oucome of hrowg dce, we c do seres of rdom expermes cully hrowg he dce d esme he me d vrce from he resuls of hese expermes. Ths s o he cse wh rdom fucos. To esme he sscl properes of rdom fuco, we should be ble o drw lrge umber of relos. However, prcce, we oly hve oe relo of he rdom fuco, mely rely self. So we mus be ble o esme ll he relev sscs of he rdom fuco from sgle relo. I urs ou h hs s oly possble for sory rdom fucos. The reso s h src sory cully sys h ll sscl properes re he sme, o mer where you re. For sce, suppose we w o esme he me µ Ζ cer po. The orml procedure would be o e he verge of my relos po, whch s mpossble becuse we oly hve oe relo rely. However, f he rdom fuco s sory he pdf f ; y loco s he sme d herefore lso he me. Ths lso mes h wh y sgle relo we hve every loco smple from he sme pdf f ;. So, he me c lso be esmed f we e suffce umber of smples from sgle relo, such s our rely. Ths s llusred furher Fgure 5.4. Ths propery of rdom fuco,.e. beg ble o esme sscl properes of rdom fuco from lrge umber of smples of sgle relo s clled ergodcy. Apr from he rdom fuco beg src sory, here s oher codo ecessry for ergodcy o pply. The smples from he sgle relo should be e from lrge eough perod of me or, he spl cse, lrge eough re. A more forml defo of rdom fuco h s ergodc s me s: lm d f d µ T ; 5.6 T T 77

78 So he egrl over he probbly dsrbuo he esemble c be replced by emporl or spl or spo-emporl egrl of very lrge ervl T or re or volume. Smlrly, rdom fuco s sd o be covrce-ergodc f: lm τ d { µ }{ µ } f, ;, τ dd T T 5.7 T Cov[, τ ] τ µ µ Fgure 5.4. Ergodc rdom fucos: he verge of observos from sgle relo s he sme s he verge of my relos gve loco Secod order sory rdom fucos A weer form of sory s secod order sory. Here we requre h he bvre or wo-po probbly dsrbuo s vr uder rslo: f, ;, f, ; ', ',,τ 5.8 Ofe, he umber of observos vlble s oly suffce o esme he me, vrce d covrces of he rdom fuco. So prcce, we requre oly ergodcy d herefore oly sory for he me, vrce d covrces. Hece, eve mlder form of sory s usully ssumed whch s clled wde sese sory. For wde sese sory rdom fucos lso clled homogeous rdom fucos he me d vrce do o deped o or x d he covrce depeds oly o he sepro dsce bewee wo pos me or spce: µ µ s cos σ σ s cos d fe Cov[, ] C C τ

79 The grph descrbg he covrce s fuco of he dsce τ - lso clled lg s clled he covrce fuco. I cse of wde sese sory, he covrce fuco for s equl o he vrce d decreses o ero whe he dsces - becomes lrger. Ths mes h rdom vrbles suffcely lrge dsces re o correled. For rdom spce fucos we sese sory mes h Cov x, x ] C x x C 5. [ h where h x - x s he dfferece vecor bewee wo locos see Fgure 5.5. If he rdom fuco s lso soropc we hve h Cov x, x ] C x x C h C 5. [ h where lg h h s he orm legh of he dfferece vecor. If he covrce fuco s dvded by he vrce we ob he correlo fuco: ρ τ C τ / σ d ρ h C h / σ. y x 3 x h y h x -x x h h h x h y h x Fgure 5.5 I cse of spl rdom fucos he covrce depeds o he lg vecor hx -x wh legh h. I cse of soropc RSF he covrce bewee x d x s he sme s bewee x d x 3, whch mples h he covrce oly depeds o he legh of he vecor h. For d h re regulrly posoed me or spce he covrce fuco c be esmed s: C ˆ ˆ ˆ µ µ 5. wh he lg us of me or spce d he me/dsce bewee observos. Ths esmor s ofe used me seres lyss see chper 6. I cse of rregulrly posoed d spce he covrce fuco c be esmed s: x h C ˆ [ ˆ ][ ˆ h x µ x h ± h µ ] h

80 where h s he lg whch s vecor spce, h s lg-olerce whch s eeded o group umber of d-prs ogeher o ge sble esmes for gve lg d h re he umber of d-prs h re dsce d dreco h ± h pr. I Fgure 5.6 wo dffere hydrologcl me seres re show. Fgure 5.6 shows me seres of mxmum dschrge for he rver Rhe he Lobh Gugg so. The vlues re ucorreled me. Also gve Fgure 5.6b s he heorecl correlo fuco h f hese d. The uderlyg sochsc process h belogs o hs heorecl correlo fuco s clled whe ose, whch s specl process cossg of ucorreled Guss deves every wo mes o mer how smll he lg τ bewee hese mes. Fgure 5.6c shows relo of hs couous process. If he process s smpled he sme dscree mes s he mxmum dschrge seres oce yer we ob seres h loos smlr s h show Fgure 5.6. Fgure 5.6d shows me seres of groudwer deph d observed oce dy for he yers ow clled De Bl The Neherlds. The correlo fuco esmed from hese d s show Fgure 5.6e. Also show s fed correlo fuco belogg o he couous process show Fgure 6.6f. Ag, whe smples he sme dscree mes re e s h of he groudwer hed seres we ob dscree process wh smlr sscl properes s he orgl seres. Ths shows h dscree me seres c be modeled wh dscree rdom fuco, s wll be show chper 6, bu lso s dscree smple of couous rdom fuco. 8

81 4. d Mxmum dschrge m 3 /d Phrec surfce cm surfce Yer Dy umber..8.6 b e Correlo Lg yers correlo Lg dys c Smuled Phrec surfce cm surfce f Fgure 5.6 Tme seres of yerly mxmum vlues of dly dschrge for he Rhe rver Lobh; b esmed correlo fuco d fg correlo fuco of couous rdom fuco: whe ose; c relo of whe ose; d me seres of wer ble dephs De Bl; e esmed correlo fuco from groudwer level me seres d fed correlo fuco of couous RF; e relo of hs Rdom Fuco.. There re umerous models for modellg he covrce fuco of wde sese sory processes. Tble 5. shows four of hem for soropc rdom fucos. The prmeer s clled he correlo scle or egrl scle of he process, d s mesure for he legh over whch he rdom vrbles wo locos of he rdom fuco RF re sll correled. I cse of soropc rdom fucos, for sce hree spl dmesos wh egrl scles x, y,, he sme model c be used s hose show Tble 5. by replcg h/ wh he followg rsformo: Yer Tme sep dys

82 h h x x h y y h 5.4 Ths form of soropy, where oly he degree of correlo vrous wh dreco bu o he vrce of he process s clled geomerc soropy. Tble 5. A umber of possble covrce models for wde sese sory rdom fucos h/ Expoel covrce Guss covrce Sphercl covrce Hole effec wve model Whe ose model * C h σ C h σ e e h / h, > h, > 3 3 h h σ C h h, > bs h / C h σ h, > h ρ h h h > f h < f h * The whe ose process hs fe vrce, so srcly speg s o wde sese sory. Here, we hus oly provde he correlo fuco h does exs Relos bewee vrous forms of sory A src sese sory rdom fuco s lso secod order sory d s lso wde sese sory, bu o ecessrly he oher wy roud. However, f rdom fuco s wde sese sory d s mulvre pdf s Guss dsrbuo Equo 3.87, s lso secod order sory 3 d lso src sese sory rdom fuco. More mpor, wde sese sory rdom fuco h s mulvre Guss d hus lso src sory s compleely chrcered by oly few sscs: cos me µ µ d covrce fuco C - h s oly depede o he sepro dsce. So o recpule rrow mes mples I geerl: Type of sory: Src sese Secod order Wde sese Propery: Mulvre pdf Bvre pdf rslo vr rslo vr Me d vrce rslo vr If he mulvre pdf s Guss: Type of sory: Wde sese Secod order Src sese Propery: Me d vrce Bvre pdf Mulvre pdf 3 Ofe he lerure he erm secod order sory s used whe fc oe mes wde sese sory. 8

83 83 rslo vr rslo vr rslo vr Irsc rdom fucos A eve mlder form of sory rdom fucos re rsc rdom fucos. For rsc rdom fuco we requre we show he spl form here:, ] [ x x x x E 5.5, ] [ x x x x x x E γ 5.6 So he me s cos d he expeced qudrc dfferece s oly fuco of he lg-vecor h x -x. The fuco x - x γ s clled he semvrogrm d s defed s: ] [, x x x x x x E γ γ 5.7 The semvrogrm c be esmed from observos s smlrly he emporl dom: ± } { ˆ h h h x x h h γ 5.8 Tble 5. shows exmples of couous semvrogrm models h c be fed o esmed semvrogrms. Tble 5. A umber of possble semvrce models for rsc rdom fucos Expoel model, ; ] [ / > c h e c h h γ Guss model ; ] [ / > h e c h h γ Sphercl model < h c h h h c h f f 3 3 γ, ; > c h Hole effec wve model, ; / s > c h h h b c h γ Pure ugge model > > c h c h h γ Power model ; ; > b h h h b γ The semvrogrm d he covrce fuco of wde sese sory rdom fuco re reled s follows: x x x x C σ γ 5.9

84 Ths mes h he semvrogrm d he covrce fuco re mrror mges wh c σ s c be see Fgure 5.7. Ths lso mes h where he covrce fuco becomes ero for lrge eough sepro dsces, he semvrogrm wll rech pleu clled he sll of he semvrogrm h s equl o he vrce. The dsce whch hs occurs clled he rge of he semvrogrm s he dsce beyod whch vlues o he rdom fuco re o loger correled. The frs fve models of ble 5. re semvrogrm models h mply wde sese sory fucos wh c σ. For he sxh model, he power model, hs s o he cse. Here, he vrce does o hve o be fe, whle he semvrce eeps o growg wh cresg lg. Ths shows h f rdom fuco s wde sese sory, s lso rsc. However, rsc rdom fuco does o hve o be wde sese sory,.e. f he semvrogrm does o rech sll. σ γ C h h sll h x - x rge Fgure 5.7. Covrce fuco d semvrogrm for wde sese sory rdom fuco Iegrl scle d scle of flucuo The egrl scle or correlo scle s mesure of he degree of correlo for sory rdom processes d s defed s he re uder he correlo fuco. I ρ τ dτ 5. For he expoel, Guss d sphercl correlo fucos he egrl scles re equl o, π / d 3/8 respecvely. Gve h he correlo fucos of wde sese sory processes re eve fucos,.e. ρ τ ρ τ, oher mesure of correlo s he scle of flucuo defed s: ρ τ d τ I 5. For D d 3D rdom spce fuco he egrl scles re defed s: 84

85 I / / x ρ h, h dhdh ; I x ρ h, h, h3 dhdh dh3 5. π π 5.4 Codol rdom fucos I hs seco we vesge wh hppes f observos re doe o rdom fuco. Suppose h we hve sory rdom fuco me h s observed umber of locos. Suppose for he mome h hese observos re whou error. Fgure 5.8 shows umber of relos of couous me rdom fuco h s observed four locos whou error. I c be see h he relos re free o vry d dffer bewee he observo pos bu re cosr,.e. codoed, by hese pos. Ths c be see whe comprg he pdfs wo locos d. I c be see h ucery s lrger furher from observo h close o observo. Ths s uvely correc becuse observo s ble o reduce ucery for lmed ervl proporol o he egrl scle of he rdom fuco. A dsce lrger h he rge, he rdom vlues re o loger correled wh he rdom vrble he observo loco d he ucery s s lrge he vrce of he pdf s lrge s h of he rdom fuco whou observos. Fgure 5.8. Relos of rdom fuco h s codol o umber of observos; dshed le s he codol me. The rdom fuco h s observed umber of locos d/or mes s clled codol rdom fuco d he probbly dsrbuos locos d codol probbly desy fucos cpdfs: f ; y,.., ym, f ; y,.., ym. Where y,, y m re he observos. The complee codol rdom fuco s defed by he codol mulvre pdf: f,,. N ;,,. N y,.., ym. The codol mulvre pdf c heory be derved from he ucodol mulvre pdf usg Byes rule. However, hs s usully very cumbersome. A lerve wy of obg ll he requred sscs of he codol rdom fuco s clled sochsc smulo. I chpers 7 d 8 some mehods re preseed for smulg relos of boh ucodol d codol rdom fucos. 85

86 The codol dsrbuo of s,, or s me vlue see dshed le d vrce, c lso be obed drecly hrough geosscl predco or rgg chper 7 d se-spce predco mehods such s he Klm fler chper 9. These mehods use he observos d he sscs e.g. semvrogrm or covrce fuco of he rdom fuco sscs esmed from he observos o drecly esme he codol dsrbuo or s me d vrce. 5.5 Specrl represeo of rdom fucos The correlo fuco of he me seres of wer ble deph De Bl Fgure 5.6 hs oly bee lyed for wo yers. Hd we lyed loger perod we would hve see correlo fuco wh perodc behvor, such s he Hole-effec model Tbles 5. d 5.. Fgure 5.9 shows he correlo fuco of he dly observos of dschrge of he Rhe Rver Lobh. A cler perodc behvor s observed s well. The perodc behvor whch s lso ppre he me seres see Fgure 4. s cused by he fc h evporo whch s drve by rdo d emperure hs cler sesol chrcer hgher d lower ludes d empere clmes. I rcc clmes he emperure cycle d ssoced sow ccumulo d mel cuse sesoly, whle he sub-ropcs d sem-rd clmes he occurrece of rfll s srogly sesol. I cocluso, mos hydrologcl me seres show sesol vro. Ths mes h o lye hese models wh sory rdom fucos requres h hs sesoly s removed see for sce chper 6. The occurrece of sesoly hs lso spred he use of specrl mehods sochsc modellg, lhough mus be sressed h specrl mehods re lso very suble for lyg sory rdom fucos Correlo Lg dys Fgure 5.9 Correlo fuco of dly verged dschrge of he rver Rhe Lobh. 86

87 5.5. Specrl desy fuco We wll herefore sr by specrl represeo of sory rdom fuco. Such preseo mes h he rdom fuco s expressed s sum of s me µ d K susods wh cresg frequeces, where ech frequecy hs rdom mplude C d rdom phse gle Φ : µ K K C cos ω Φ 5.3 wh C C, Φ Φ d ω ± ω Fgure 5. shows schemclly how sory rdom fuco s decomposed o rdom hrmocs. ω K ω 3 ω ω µ µ C Φ C Φ 3 3 C Φ C Φ µ ± C cos ω Φ ± C cosω Φ ±... ± C cos Kω Φ K K Fgure 5. Schemc of he specrl represeo of sory rdom fuco s decomposo of he rdom sgl o hrmocs of cresg frequecy wh rdom mplude C d rdom phse gle Φ : Bsed o hs represeo c be show see Vmrce pp h he followg relos hold: 87

88 C S τ S ωcos ωτ dω ω C τ cos ωτ π 5.4 dτ 5.5 These relos re ow s he Wer-Khche relos. The fuco S ω s ow s he specrl desy fuco of he rdom process d Equos 5.4 hus show h he covrce fuco s Fourer rsform of he specrum d vce vers: hey form Fourer pr. The physcl meg of he specrl desy fuco c bes be udersood by seg he lg τ equl o ero 5.4. We he ob: C σ S ω dω 5.6 I c be see h he vrce of he rdom fuco s equl o he egrl over he specrl desy. Ths mes h he vrce s weghed sum of vrce compoes, where ech compoe cosss of rdom hrmoc fuco of gve frequecy. The specrl desy he represes he wegh of ech of he rbug rdom hrmocs,.e. he relve mporce of ech rdom hrmoc explg he ol vrce of he rdom fuco. I s esy o see he logy wh he elecromgec specrum where he ol eergy of elecromgec rdo whch s logous o he vrce of our rdom sgl s foud by he re uder he specrum d c be rbued o relve corbuos from dffere wveleghs. I ble 5.3 expressos re gve of specrl desy fucos belogg o some of he covrce fucos gve Tble 5.. Fgure 5. From Gelhr, 993 shows ypcl relos of he rdom fucos volved, her correlo fuco d he ssoced specrum. Wh c be see from hs s h he specrum of whe ose s horol le, mplyg fe vrce ccordg o Equo 5.6. Ths shows h whe ose s mhemcl cosruc, d o fesble physcl process: he re uder he specrum s mesure for he ol eergy of process. Ths re s fely lrge, such h ll he eergy he uverse would o be suffce o geere such process. I prcce oe ofe ls bou wde bd processes, where he specrum hs wde bd of frequeces, bu ecloses fe re. Tble 5.3 A umber of possble covrce fuco d ssoced specrl desy fucos τ - Expoel / Rdom hrmoc: cos ω φ,ω where re deermsc coss d φ s rdom C τ σ > e S σ ω π ω C τ cos ω τ S ω δ ω ω 4 τ,, ω > 88

89 Hole effec wve model Whe ose model τ / C τ σ / e > ρ τ τ τ > S ω S ω c 3 σ ω π ω ρτ c Fgure 5. Schemc exmples of covrce fuco-specrl desy prs dped from Gelhr, 993. The specrl desy fuco s eve fuco s s he covrce fuco: S ω S ω. Ths moves he roduco of he oe-sded specrl desy fuco G ω S ω, ω. The Weer-Khche relos he become: C τ G ωcos ωτ dω 5.7 G ω C τ cos ωτ dω 5.8 π 89

90 Somemes s covee o wor wh he ormled specrl desy fucos, by dvdg he specr by he vrce: s ω S ω / σ d g ω G ω / σ. For sce, from 5.5 we c see h here s relo bewee he ormled specrl desy fuco s ω d he scle of flucuo. Seg ω Equo 5.5 we ob: s ρ τ ω π d 5.9 π 5.5. Forml complex specrl represeo Ofe more forml defo of he specrl desy s used he lerure bsed o he formulo erms of complex clculus. Here he rdom fuco s defed s he rel pr of complex rdom fuco * : K * ω ω Re{ } Re µ X e Re µ e dx ω 5.3 K K wh ω ω he frequecy, d X complex rdom umber represeg he mplude. Ths equo els h he complex rdom process s decomposed o lrge umber of complex hrmoc fucos e ω cos ω s ω wh rdom complex mplude, Gve hs represeo s possble derve he Weer-Khche equos s Vmrce, 983, p. 88: C S τ S ω e ω C τ π ωτ d ω 5.3 ωτ d τ 5.3 I c be show Vmrce, 983, p.94 h Equos 5.3 d 5.3 re mhemclly equvle o Equos 5.4 d 5.5 respecvely Esmg he specrl desy fuco For wde sese sory rdom fucos he specrl desy c be esmed from esmed covrce fuco s: S M ω λ C λcˆ τ cos ω 5.33 π 9

91 wh ω π / M,,.., M. The weghs λ re ecessry o smooh he covrces before performg he rsformo. Ths wy smoohed specrl desy fuco s obed dsplyg oly he relev feures. There re umerous ypes of smoohg weghs. Two frequely used expressos re he Tuey wdow d he Pre wdow π λ cos,,,..., M 5.34 M M / M M λ M / M M The hghes frequecy h s lyed s equl o f mx ωmx / π.5. Ths s he hghes frequecy h c be esmed from me seres,.e. hlf of he frequecy of he observos. Ths frequecy s clled he Nyqus frequecy. So f hydrulc hed s observed oce per dy h he hghes frequecy h c be deeced s oe cycle per wo dys. The smlles frequecy lrges wvelegh h c be lyed depeds o he dscreo M: f m π /πm / M, where M s lso he cuoff level mxmum lg cosdered of he covrce fuco. The wdh of he smoohg wdows s djused ccordgly. As exmple he specrum of he dly dschrge d of he Rhe rver Lobh Fgure 4. d Fgure 5.9 for he correlo fuco ws esmed usg Pre wdow wh M9. Fgure 5. shows he ormled specrl desy fuco so obed. Clerly, smll frequeces dome wh smll pe bewee 4 d 5 yers. Mos prome of course, s expeced, here s pe frequecy of oce yer, whch exemplfes he srog sesoly he me seres Specrl represeos of rdom spce fucos If we exed he prevous o wo dmesos he sory rdom fuco x,x c be expressed erms of rdom hrmocs s: 9 K K x, x C cos ω x ω x Φ K K µ 5.36 wh C dφ rdom mplude d phse gle belogg o frequecy ω,, : ω ω ± 5.37

92 3 Normlsed Specrl desy sf Frequecy f cycles/dy Fgure 5. Normled specrl desy fuco of dly verged dschrge of he rver Rhe Lobh. The Weer-Khche equos he become, h S ω, ω cos ωh ωh dωd, ω C h, h cos ω h ωh dh π C h ω 5.38 S ω dh 5.39 The vrce s gve by he volume uder he D specrl desy fuco. S ω, ω dωdω σ 5.4 T T If we use vecor-oo we hve: ω ω, ω, h h, h dωh ωh ω h. A shor hd wy of wrg 5.38 d 5.39 resuls: C S h S ωcos ω h dω ω C hcos ω h π 5.4 dh 5.4 9

93 These equos re vld for hgher dmesol rdom fucos, where D /π D dmeso processreplces/ π 5.4. The more forml defo usg complex clculus he gves: x Re e ω x µ dx ω 5.43 The Weer Khche equos become C S h S ω e ω h d ω 5.44 ω h C h ω d exp h D π Locl vergg of sory rdom fucos Cosder sory rdom fuco d cosder he rdom fuco T h s obed by locl movg vergg see Fgure 5.: T T T / T / τ dτ 5.46 T µ T Vr σ [ T ] V T T µ Fgure 5. Locl movg vergg of sory rdom fuco 93

94 94 Locl vergg of sory rdom process wll o ffec he me, bu does reduce he vrce. The vrce of he verged process c be clculed s whou loss of geerly we c se he me o ero here: T T T T T T T T T T T d d C T d d E T d T d T E d T d T E Vr / / / / ] [ ] [ s sory becuse ] [ ] [ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ 5.47 A ew fuco s roduced h s clled he vrce fuco: ] [ T Vr T V σ 5.48 The vrce fuco hus descrbes he reduco vrce whe vergg rdom fuco s fuco of he vergg ervl T. From 5.47 we c see h he vrce fuco s reled o he correlo fuco s: T T d d T T V τ τ τ τ ρ 5.49 Vmrce 983, p 7 shows h Equo 5.49 c be smplfed o: T d T T T V τ τ ρ τ 5.5 I Tble 5.4 umber of correlo fucos d her vrce fucos re gve. If we exme he behvor of he vrce fuco for lrge T we ge see Vmrce, 983: T T V T lm 5.5 where s scle of flucuo. I ble 5. he scle of flucuo s lso gve for he hree correlo models. The scle of flucuo ws lredy roduced erler s mesure of spl correlo of he rdom fuco d c lso be clculed usg he correlo fuco Equo 5. or he specrl desy Equo 5.9. Equo 5.5 hus ses h for lrger vergg ervls he vrce reduco hrough vergg s versely proporol o he scle

95 95 of flucuo: he lrger he scle of flucuo he lrger T should be o cheve gve vrce reduco. I prcce, relo 5.5 s lredy vld for > T Vmrce, 983. Tble 5.4 Vrce fucos d scle of flucuo for hree dffere correlo fucos τ - Expoel frs order uoregressve process, / > e τ τ ρ τ e T T T V T / Secod order Auoregressve process see chper 6, / > e τ τ τ ρ τ e T e T T V T T 4 / / Guss correlo Fuco, / > e τ τ ρ τ π π e T Erf T T V T / The covrce of he verged process s gve by: T T d d C T T C,, τ τ τ 5.5 Geerlly s o esy o ob closed form expressos for 5.5. However s show chper 7, s relvely esy o ob vlues for hs fuco hrough umercl egro. We ed hs seco by gvg he equos for he D-spl cse, where s srghforwrd o geerle hese resuls o hgher dmesos. The locl verge process s for re AL L defed s: / / / /,, L x L x L x L x T du du u u L L x x 5.53 The vrce fuco s gve by:,, L L dh dh h h L h L h L L L L V ρ 5.54 The lm of he vrce fuco defes he spl scle of flucuo or chrcersc re α:,, lm L L L L V L L α 5.55 where α c be clculed from he correlo fuco s follows:

96 ρ u, u dudu α 5.56 The chrcersc re α c lso be obed hrough he specrl represeo by seg ω ω he Weer-Khche relo 5.39 s, S,/ σ ρ h, h dh dh 5.57 π Combg equos 5.56 d 5.57 he leds o: α π s, I Tble 5.5 he vrous wy of obg he scle of flucuo d he chrcersc re summred: Tble 5.5 Three wys of obg he scle of flucuo me d chrcersc re D spce fer Vmrce, 983 Scle of flucuo Chrcersc re α lmtv T L L V L, L T lm L L ρ τ dτ ρ u, u dudu π s 4π s, Flly, he covrce of he verged rdom fuco wo dmesos s gve by:, L L L h L h C L, L; h, h C x, y, x, y dxdydx dy 5.59 L L The covrce of he splly verged rdom fuco s frequely used geosscl mppg, s expled chper 7, where s vlues re pproxmed wh umercl egro. To lm he ool burde, Equo 5.59 s usully wre vecor oo wh T T T x x, y, x x, y, h h, h d AL L : 5.7 Exercses C h h A; h A C x, x dxdx 5.6 A A h 5.. Gve exmples of hydrologcl vrbles h c be modeled wh couous-vlued d dscree-vlued rdom seres, lce process, 3 couous-me process, 4 96

97 couous-spce process, 5 couous spce-me process, 6 me compoud po process, 7 spce-compoud po process. Noe h 4 combos re sed for. 5.. The covrce of rdom fuco s descrbed by: C h exp h / 3, d cos me. The pdf gve loco s he Gmm dsrbuo. Is hs process: wde sese sory; b secod order sory; d src sory? 5.3. The covrce of rdom fuco s descrbed by: C h exp h / 3, d cos me. The pdf gve loco s he Guss dsrbuo. Is hs process: wde sese sory; b secod order sory; c src sory? 5.4. The covrce of rdom fuco s descrbed by: C h exp h / 3, d cos me. The mulvre pdf of y se of locos s he Guss dsrbuo. Is hs process: wde sese sory; c secod order sory; d src sory? 5.5. Cosder he followg soropc covrce fuco of rdom fuco x: 3 3 h h 5 f h < C h f h lg h The rdom fuco hs cos me.. Wh ype of sory c be ssumed here? b. Wh s he vrce of he rdom fuco? c. Clcule he vlues of he correlo fuco for lgs h, 5,, 5. d. Clcule he egrl scle of he rdom fuco Cosder sory rdom fuco whose specrl desy s gve by he followg equo: S ω / ω e Wh s he vrce of he rdom fuco? 5.7. Show h he egrl scle of he expoel correlo fuco of Tble 5. s equl o prmeer d of he sphercl correlo fuco s equl o 3/ Cosder rdom fuco wh scle of flucuo 5 dys, expoel covrce fuco d σ. Plo he relo bewee he vrce of he verged process T wh T cresg from o dys. 97

98 6 Tme seres lyss 6. Iroduco My dymc vrbles hydrology re observed more or less regulr me ervls. Exmples re rfll, surfce wer sge d groudwer levels. Successve observos from prculr moorg so observed regulr ervls re clled me seres. I he coex of sochsc hydrology we should loo me seres s relo of rdom fuco. I he ermology of Chper 5 me seres c eher be vewed s rel-vlued dscree-me rdom fuco Fgure 5. or rel-vlued couous-me rdom fuco h hs bee observed dscree mes Fgure 5.6. Irrespecve of hs vew of rely, seprely from he heory of rdom fucos, hydrologss hve bee usg echques mosly comg from ecoomercs speclly desged o lye d model hydrologcl me seres. The mos commo of hese echques, collecvely ow s me seres lyss, wll be reed hs chper. The m resos for lyg hydrologcl me seres re:. Chrcero. Ths cludes o oly he lyss of properes le verge vlues d probbly of exceedg hreshold vlues, bu lso chrcerscs such s sesol behvor d red.. Predco d forecsg. The m of predco d forecsg s o esme he vlue of he me seres o-observed pos me. Ths c be predco me fuure forecsg, or predco o-observed po me he ps, for exmple o fll gps he observed seres due o mssg vlues. 3. Idefy d qufy pu-respose relos. My hydrologcl vrbles re he resul of umber of url d m-duced flueces. To qufy he effec of dvdul fluece d o evlue wer mgeme mesures, he observed seres s spl o compoes whch c be rbued o he mos mpor flueces. The focus of hs chper s o me seres models, expressg he vlue of he me seres s fuco of s ps behvor d me seres of flueces fcors e.g. pu vrbles h fluece he hydrologcl vrble h s lyed. I hs chper we resrc ourselves o ler me seres models s descrbed by Box d Jes 976. Ths mes h he vlue of he vrble uder cosdero s ler fuco of s ps d of he relev fluece fcors. Exesve dscussos o me seres lyss c, mogs ohers, be foud boos of Box d Jes 976, Presly 989 d Hpel d McLeod Defos Smlr o he lyss d modelg of spl rdom fucos by geosscs Chper 7, he me seres lyss lerure uses s ow ermology whch my be slghly dffere from he more forml defos used chpers 3 d 5. Cosequely, some defos of properes of rdom me seres wll be repeed frs. Also he symbols used my be slghly dffere h used he prevous chpers, lhough we ry o eep he oo s close s possble o h used elsewhere he boo. 98

99 6.. Dscree sory me seres As sed before, mos hydrologcl vrbles, le rver sges, re couous s me. However, f we cosder he vrble regulr ervls me, we c defe dscree me seres see fgure Fgure 6. Schemc of couous rdom fuco observed dscree mes The vlues of he couous me seres he regulr mes ervls re:,...,,,,...,. 6. The seres s clled dscree me seres. I he me seres lerure ofe he subscrp s used sed of he subscrp.,...,,,,...,. 6. I he remder of hs chper we wll use he subscrp. Noe h he subscrp s r umber rher h vlue of he rug me. 6.. Momes d Expeco A sgle me seres s cosdered o be sochsc process h c be chrcered by s cerl sscl momes see chper 3. I prculr he frs d secod order momes re relev: he me vlue, he vrce d he uocorrelo fuco. For sscl sory process he me vlue d he vrce re: µ E ] 6.3 [ σ Vr [ ] E[{ µ }{ µ }]

100 The uocovrce s mesure of he reloshp of he process wo pos me. For wo pos me me seps pr, he uocovrce s defed by: Cov[, ] E[{ µ }{ µ }],...,,,,..., 6.5 Ofe s clled he me lg.. oe h for Cov[, ] [. oe h Cov, ] Cov[, ] Cov[, ] I me seres lyss we ofe use he uocorrelo fuco ACF, defed by: σ Cov[, ] E[{ µ }{ µ }] ρ,,...,,,,..., 6.7 Cov[, ] σ I c be prove h he vlue of he ACF s lwys bewee d -. A vlue of or - mes perfec correlo, whle vlue dces he bsece of correlo. From he defo follows h he ACF s mxmum for ρ. Jus le for he uocovrce follows from he defo h:, ρ ρ,...,,,,..., 6.8,, The grphcl represeo of he ACF s clled he uocorrellogrm see fgure 6.. Becuse he ACF s symmercl roud, oly he rgh posve sde s show. The dymc behvor of me seres s chrcered by s vrce d ACF. Ths s vsuled fgure 6. for ero me me seres.

101 low ACF hgh,,8,6,4,,,8,6,4, ACF ACF ACF low Vrce hgh -3-3 Fgure 6. Dffere chrcerscs of ero me me seres due o hgh d low vlues for he vrce d he ACF. Alogous o he uocovrce d he ACF, h expresses he relo of me seres wh self, he relo bewee wo dffere me seres s expressed by he cross covrce d he crosscorrelo fuco CCF. The cross covrce d CCF for he me seres d X s defed s: ρ Cov [ X, ] E[{ X µ x }{ µ }] 6.9 E[{ X µ }{ µ }] x X, 6. σ xσ Alogous o he ACF c be prove h he vlue of he CCF s lwys bewee d -. However he hghes vlue of he CCF does ecessrly occur, d he CCF s o symmercl roud ρ 6. X, ρx, From he defo follows h

102 ρ 6. X, ρx, I he rel world we do' ow he process excly d we hve o esme ll formo from observos. I bsece of observo errors, he observo equls he vlue of he process. Suppose we hve observos of he process from me sep ul me sep. Th he codol expeco of he process me sep τ s deoed s: ˆ τ 6.3 E [,..., ] τ We wll use wo operors: he bcshf operor B d he dfferece operor bl The bcshf operor s defed s: Ad herefore B 6.4 B B 6.5 The dfferece operor s defed s: Ad Dscree whe ose process A mpor clss of me seres s he dscree whe ose process. Ths s ero me me seres wh Guss probbly dsrbuo d o correlo me. E[ ] E[ σ f 6.8 ] f Becuse of he bsece of correlo me, he dscree whe ose process me sep does o co y formo bou he process oher me seps Rules of clculus Clculo rules wh expecos re summred s:

103 E[ c] c E[ c ] c E[ ] E[ E[ X ] E[ ] E[ X X ] E[ ] E[ X ] ] Cov[, X ] 6.9 Where X d re dscree me seres d c s cos. 6.3 Prcple of ler uvre me seres models The geerl cocep of ler me seres models s o cpure s much formo s possble he model. Ths formo s chrcered by he me vlue, he vrce d he ACF. We cosder he me seres s ler fuco of whe ose process see fgure 6.3. Tme seres model Fgure 6.3 Schemc represeo of me seres model. Becuse he ACF of he whe ose process equls ero for y, ll formo of he uocorrelo s cpured he me seres model. I he followg we wll descrbe he dffere ypes of rdom processes h uderly he mos commoly used me seres models. For ech process, we sr wh roducg he recurre equo defg he process, followed by he relo bewee he process prmeers d sscl properes of he process, showg how such processes c be predced f observos re e d edg wh umercl exmple. 6.4 Auoregressve processes 6.4. AR process Defo The frs clss of ler me seres processes dscussed hs chper s formed by he uoregressve AR processes. The mos smple uoregressve process s he AR process AuoRegressve process of order. A ero me AR process s defed s: φ 6. Prmeer Deermo As sed before, he whe ose process s ero me, ucorreled process. Therefore he AR process cos wo uows: - he frs order uo regressve prmeer φ d 3 - he vrce of he whe ose processσ

104 These uows hve o be deermed from he chrcerscs of he me seres, prculr he vrce d he ACF. By mulplyg boh sdes of equo 6. wh - d g he expeco we ob E [ ] E[ φ ] E[ φ ] E[ ] 6. The process s depede of fuure vlues of he whe ose process. Therefore, he vlue of he process - s depede of he whe ose me sep d he secod erm he rgh hd sde of 6. equls ero. Dvdg 6. by he vrce σ yelds: E[ 6. σ ] E[ ] φ σ Ad becuse E[ ] d E σ he frs order uo regressve prmeer s: [ ] φ ρ, 6.3 The vrce of he whe ose process s deermed by g he expecos of he squre of boh sdes of equo 6.. E[ ] E[{ φ } ] E[ φ φ ] φ E[ ] E[ ] 6.4 From 6.4 follows h: σ σ 6.5 φ σ σ σ φ Properes of AR process Sory: Becuse he me seres s emporlly correled process, he ps vlues of he process co formo bou he fuure behvor. I hydrology d my oher felds of pplco s well f we go furher o he fuure mosly he fluece of ps vlues eveully dsppers. I oher words, he process hs lmed memory. To esure hs, he me seres process should be sory. I order for AR process o be sory he bsolue vlue of he model prmeer should be smller h. φ < 6.6 Noe h f he codo 6.6 s o fulflled, from 6.5 follows h he vrce of he process does o exs. I hs cse, he process s sd o be o-sory. ACF of AR process: By repeve use of equo 6., c be expressed s: 4

105 5 φ φ φ φ φ 6.7 Mulplyg boh sdes by - d g he expecos yelds: ] [ ] [ ] [ E E E φ φ 6.8 Sce - s depede of fuure vlues of he whe ose process oly he frs erm he rgh hd sde s o-ero. Dvdg boh sdes by ] [ E σ yelds for he ACF: E E E E, ] [ ] [ ] [ ] [ φ φ ρ 6.9 From 6.9 c be see h he ACF of sory AR process s expoel fuco. Forecs of AR process Suppose we hve observos up o me,,l. From 6. follows h he forecs for s: ˆ ˆ ˆ φ 6.3 Sce ˆ d ˆ 6.3 follows h ˆ φ 6.3 I geerl he forecs for me l equls l l l l l l l l ˆ ˆ ˆ ˆ ˆ φ φ φ φ 6.33 The forecs error for me l s: l l e ˆ l l l l l l l φ φ φ φ 6.34 Of course he forecs error self s uow, bu we c clcule he forecs error vrce s mesure of he forecs ucery. Tg he expecos of he squre of boh sdes he vrce of he forecs error for me l s:

106 E[{ e σ e l l σ } ] E l φ l φ l l φ E[ l ] 6.35 Noe h he error vrce for forecs fr he fuure pproches he vrce of he process. l ; l φ φ d hus σ σ e σ 6.36 l φ Exmple of AR process Le be ero me AR process, wh φ. 9 d σ : From 6.9 follows h he ACF of he process equls: ρ,. 9 The uocorrellogrm of he process s gve fgure Fgure 6.4 Auocorrellogrm of he AR process The forecs of he AR process d he forecs error vrce re gve by 6.33 d 6.35: l l l. 9 d σ l e ±.96 σ ˆ re gve fgure 6.5. l ˆ The forecs d he correspodg 95% cofdece ervl 6

107 7 p p φ φ φ K Fgure 6.5. Forecs of AR process wh correspodg 95% cofdece ervl. As c be see from fgure 6.5, he forecs of he AR process grdully pproches ero for forecs furher o he fuure. The decy curve of he forecs reflecs he memory of he AR process. Cosequely, he cofdece ervl grdully pproches s mxmum vlue ±.96 σ for forecss furher he fuure ARp process Defo The geerl form of ero me AuoRegressve process of order p ARp s defed s: 6.4 The prmeers φ,, p re clled he uo regressve prmeers of he order. Prmeer deermo Smlr o he prmeer deermo of AR process, he prmeers of he ARp process c be expressed erms of he vrce d uocorrelo of he process. Boh sdes of 6.4 re mulpled by -,,p. Ths yelds he se of equos: p p p p p p p p p p p φ φ φ φ φ φ φ φ φ L M L L 6.4 Tg expecos d dvdg boh sdes by he vrce σ yelds: - cof. ervl forecs observed

108 ρ ρ M ρ,,, p φ φ φ ρ ρ φ, φ, p ρ, φ L φ L φ ρ, p p p ρ ρ, p, p L φ p 6.4 Wrg hs se of equos mrx form yelds: ρ M ρ,, p ρ ρ,, p L L O L ρ ρ, p, p M φ ρ φ ρ M M φ p ρ,,, p 6.43 The AR-prmeers c be solved from he se of equos 6.43 whch s ow s he Yule- Wler Equos. For exmple he prmeers of uo regressve model of order re: ρ φ, ρ ρ ρ, ρ φ ρ,,,, 6.44 Noe h ule he prmeer he AR process, for he AR processφ ρ. Properes of ARp process Usg he bcshf operor B defed wh B, ARp process c be wre s: φ φ φ 6.45 p B B L pb Defg: B φ B φ B Lφ 6.46 Φ B p p The geerl form of ARp process s: Φ B Sory. Alogous o he AR process, he vlues of he prmeers re lmed order for he ARp process o be sory. Whou proof s sed here h ARp model s sory f ll complex roos of he equo: Φ B

109 le ousde he u crcle. For exmple, for AR process he roos of he fuco Φ B φ B φ B 6.49 mus le ousde he u crcle. Ths mples h he prmeers φ d φ mus le he rego defed by: φ φ < φ φ < < φ < 6.5 ACF of ARp process. Mulplco of equo 6.4 by -, g he expecos d dvdg boh sdes by he vrce σ yelds:, φρ, φρ, φpρ, p ρ L 6.5 As exmple, he uocorrellogrm for he AR process fgure 6.6. s gve Fgure 6.6. Auocorrellogrm of AR process The ACF s by defo. From he ACF d ACF fgure 6.6, he effec of he secod order uo regressve prmeer c be see. Smlr o he AR model, he uo correlo for lrger me lgs decys grdully. Ths grdul decy s geerl chrcersc of AR-models. Forecs of ARp process Forecsg ARp process s smlr o h of AR process. The ARp process some po me s depede o he p me seps before. Therefore, he forecs of ARp process 9