Application of the stochastic self-training procedure for the modelling of extreme floods

Transcription

1 The Exremes of he Exremes: Exraordary Floods (Proceedgs of a symposum held a Reyjav, Icelad, July 000). IAHS Publ. o. 7, Applcao of he sochasc self-rag procedure for he modellg of exreme floods VADIM KUZMIN Deparme of Hydrology, Russa Sae Hydromeeorologcal Uversy, Maloohs 98, S Peersburg 9596, Russa e-mal: uzm@solars.ru PIETER VAN GELDER Delf Uversy of Techology, Faculy of Cvl Egeerg, Room 3.87, PO Box 5048, 600 GA Delf, The Neherlads e-mal: pvagelder@c.udelf.l HAFZULLAH AKSOY Isabul Techcal Uversy, Cvl Egeerg Faculy, Hydraulcs Dvso, 8066 Ayazaga, Isabul, Turey e-mal: hasoy@u.edu.r ISMAIL KUCUK Elecrcal Power Resources, Survey ad Developme Admsrao, Essehr Yolu 7. Km, 0650 Aara, Turey e-mal: ucu@ee.gov.r Absrac Modellg echques vald for ordary hydrologcal codos are ofe adequae for caasrophc ruoff codos, as he majory of he parameers cluded he modellg schemes are o ecessarly applcable o flood codos. To overcome hs problem, usg a sochasc self-rag procedure (SSTP) s proposed. The procedure s based o he real waer balace equao ad he sascal model of he waer balace. A applcao of he procedure wh daa from he Flyos Rver (Turey) s preseed hs sudy. Key words exreme floods; rver flow forecasg; sochasc models; sochasc self-rag procedure; Flyos Rver bas, Turey; emac wave forecasg INTRODUCTION Several ypes of ruoff forecasg echques are avalable he leraure. These echques wor well ordary hydrologcal codos bu hey become adequae for exraordary hydrologcal codos such as caasrophc floods. Exreme rver flows cao be forecased accuraely by usg echques based upo daa from ordary rver flow codos. A mehodology ca be chose as a forecasg ool f s able o do reasoable forecass. A crero for hs s he perceage of forecass fallg o he cofdece erval cossg of a upper lm ad a lower lm. A wdely used crero for a evaluao of a mehod s he rao of S/σ, whch S s he roo mea square error (RMSE) of forecass, defed as: S = m ( Q Q ) = ()

2 38 Vadm Kuzm e al. ad σ s he sadard devao of successve values of ruoff, defed as: σ = ( ) = where Q ad Q are he observed ad he compued ruoffs a he h me erval, respecvely, s he umber of forecass, m s he umber of parameers he forecas equao, = Q Q τ so-called aural forecas error, =, s he = umber of days havg he decal edecy of ruoff chagg, τ s he legh of forecas. The mehod s good f S/σ < 0.5 ad s accepable f S/σ < 0.8. Parameers are deermed so ha S/σ s mmum. I hs sudy, a sochasc self-rag procedure (SSTP) s proposed as a forecasg echque for modellg exreme floods. The procedure s appled o daa ses from rver flows Turey. () STOCHASTIC SELF-TRAINING PROCEDURE (SSTP) I order o evaluae he role of a umber of secodary or umeasured facors, le us cosder a example of a forecasg model based o he waer balace equao. Cosder wo equaos ha forecas he value of he ruoff Q. The frs s he real waer balace expresso: Q τ τ τ τ τ ( x ) F ( x ) + F ( x ) + + F ( x ) = F ( x ) + = F (3) The secod equao descrbes a sascal model of he waer balace, whch cludes a fe umber of predcors: τ τ τ τ Q = f ( x ) + f ( x ) f ( x = f ( x = τ I equaos (3) ad (4), ( ) s a real physcally-based fuco of Q τ, ( ) (4) F x f x s a sascally defed relaoshp of forecased ruoff Q o predcors x τ, C s a sascally defed cosa, whch parally reflecs he fluece of secodary facors, Q s he error of a sgle forecas, he lower dex defes he predcor, he upper dex s he relaxao me of he h predcor, s he oal umber of real facors ad s he umber of predcors. The dscharge a me sep s depede of he observaos pror o maxτ. Each erm of equao (3) correspods o a physcal quay whch ca be measured. The erms of equao (4) however do o correspod o physcal quaes. Equao (3) cludes all possble facors (,...,,..., ) ad equao (4) cludes oly he mos evde predcors (,..., ), whch are used a gve hydrologcal eve. Cosder he dfferece of he correspodg erms of equao (3) ad (4). By omg he upper dces: ( x ) F ( x ) + + f ( x ) F ( x = F ( x ) + F ( x ) f (5) =

3 Applcao of he sochasc self-rag procedure for he modellg of exreme floods 39 s obaed. The dfferece f ( x ) F ( x ) ca be wre brefly as ( ) equao (5) ca be rewre as: f ( x ) F ( x = f ( x = F ( x ) = = = = + Equao (6) ca also be wre as: Q F = = ( x ) + f ( x = f x. Thus, Equao (7) shows ha wh formao abou he physcally-based parameers of he model, s ecessary o observe boh he evoluo of resduals (Kachroo, 99) ad he curre devaos of all parameers. O he oher had, equaos relaed o he mode, Q mod ad mahemacal expecao, m of he dscharge are ow. For example, he followg formula ca be used: Qmod m µ 3 / µ = (8) Equao (8) s vald oly for he hree-parameer gamma dsrbuo (X mod = m X β whch β he scale parameer), bu s possble o calculae he dscharge correspodg o he mode of may oher ypes of forecasg probably dsrbuos. Le us fd a probablsc erpreao for he erms of equao (8): µ 3 = ( x / µ = f (9) or aoher way: µ 3 F = + ( x ) / µ = (0) The frs mome m s forecas by he usual mehod. Depedg o he ype of he model, he curre chages of dfferece m Qmod are foud: by smulaeous aalyss of f ( x ) = appled (oe ha C s sascally defed as cosa), by smulaeous aalyss of F ( x ) = + (6) (7) ad Q, f some sascal models are ad Q, f all predcors ca be measured. Aeo should be pad o he possbly of usg smple models for he forecasg of exreme flood cases, whch are o que correspodg o codos whch he model s used. For example, he model of emacs wave for he calculao of a flood wave rasformao ca be used. The dffusve erms of a more dealed model (gored he emac wave model) are calculaed by F ( x ) = + Geerally mehods of forecasg are based o he aalyss of he era properes of processes ad predcors. The eress of he dfferece Q mod ad m, whch s defed as r = m Qmod s cosdered ex. If r s cosa (ad cosequely he hrd ceral mome µ 3 s cosa) durg he legh τ of he forecas, he τ τ µ / µ = = ad Qmod = m are recommeded. Here, -τ s he error of 3.

4 30 Vadm Kuzm e al. he prevous forecas. Ths suao occurs exremely seldom ad herefore he τ correspodg mehodology becomes almos useless. If µ 3 s cosa (.e. µ 3 = µ 3 ), bu µ s o, he a more complex equao ca be obaed as: τ τ µ Qmod = m () µ τ τ τ where µ = µ 3 /. The fac ha µ 3 / µ = follows from he comparso of equaos (7) ad (8). The erm ca be cosdered he al. Havg a absoluely correc model whch accous for all facors, = 0. All uaccoued facors form he al ad herefore hs al expresses he error of forecas. A prcple for praccal calculao of he mode of he probably dsrbuo s derved hs way. There are may varas o he calculao of µ ad µ 3. Oe of hem s based o he assumpo ha µ ca be defed as a lear fuco of he square of dscharge dffereces over legh τ. I exreme cases he followg smple formula may be useful (Kuzm, 998): where τ ( m ) τ a Qup, + c τ Q m mod = () τ a Qup, ( m ) + c τ τ s he error of he prevous forecas, Q up, s he sum of dscharges observed upper saos, a = / s a ormalzg cosa, c = Q / s a ormalzg cosa (Q s mea dscharge of he observed flood, whch s obaed by emprcal τ equaos). fro of he fraco equao () expresses he error of he prevous forecas. The procedure s called self-rag sce hs error erm s updaed as me proceeds. The descrbed smplfed mehod gves que good resuls, eve dffcul suaos (Kuzm, 998). The flood geerag causes are refleced by calculao of m. Furher vesgao of he sochasc self-rag modellg deals wh he sudy of fucos for µ ad µ 3. I CASE STUDY The descrbed mehodology of SSTP wll be appled o he Flyos Rver bas locaed he weser Blac Sea rego of Turey. The Flyos Rver flooded recely o 0 May of 998, causg a lo of damage (Yasar, 998). A schemac ewor of he Flyos Rver bas s depced Fg., showg he ma chael (Flyos Rver wh gaugg saos 37, 34, 336 ad 335), he wes rbuary Devre (wh saos 334 ad 3) ad he eas rbuary Arac (wh sao 333). The elevao of he uppermos sao (37) s 4 m whereas he elevao of 335 s m. The dsace bewee he wo saos s 59.4 m, correspodg o a slope of The secos C B ad E D have average slopes of ad 0.004, respecvely. Daly flow daa from every sao are avalable from he Elecrcal Power Resources, Survey

5 Applcao of he sochasc self-rag procedure for he modellg of exreme floods 3 ad Developme Admsrao (EIEI) Aara, Turey, for he las 0 40 years. I hs case sudy oly he daly flow daa are used from he year 979 (sce daa from he 998 flood are sll offcally o ye publshed). Fgure shows he performace of he SSTP wh respec o compued (legh of forecas s day) ad observed daly ruoff from December 978 o Jue 979 a he ed sao (335) of he rver bas. The rao S σ appears o be ha shows ha he mehod s accepable accordg o Seco (as he rao s smaller ha 0.8). A Fg. A schemac vew of he Flyos Rver bas ad ma gaugg saos.

6 3 Vadm Kuzm e al. comparso s also made wh emac wave forecasg mehod (MKW, used maly for moua rvers) by deermg S BASIC ad comparg wh S SSTP accordg o [(S SSTP S BASIC )/S BASIC ]00%. A value of 4% s obaed for he Flyos Rver favour of he SSTP (a rao S σ of 0.80 was fouds for he MKW model ad wh SSTP). The performace of SSTP was furher examed by applcao o he Volga Rver, Oa Rver, Norh Dva Rver, Pechora Rver, Irsh Rver (Russa), ad he Mssour Rver (USA). Alhough floods hese rvers are geeraed by dffere facors he SSTP gave good resuls (o show here) every case. Fg. Comparso bewee observed ad compued dscharges a sao 335, December 978 Jue 979. CONCLUSION I exreme flood forecasg, he sochasc self-rag procedure (SSTP) ca mprove resuls sgfcaly as he case sudy of floods o he Flyos Rver has show. Acowledgeme The auhor H. Asoy was suppored by ITU ad TUBITAK as a pos-docoral researcher a he Uversy of Calfora, Davs. REFERENCES Kachroo, R. K. (99) Rver flow forecasg. Par. A dscusso of he prcples. J. Hydrol. 33, 5. Kuzm, V. (998) Forecasg of large rver floods wh a robus self-rag model he absece of hydromerc daa. I: Proceedgs of he NATO Advaced Research Worshop o Sochasc Hydrologcal Processes (Moscow, Russa, 3 7 November 998), Waer Problems Isue, Moscow. Yasar, K. (998) Weser Blac Sea floods May 998 ( Tursh). I: Waer Egeerg Problems Semar ( 3 Sepember 998). Sae Waer Wors, Fehye, Turey.