Fourier Analysis Models and Their Application to River Flows Prediction

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1 The s Inernaonal Appled Geologcal ongress, Deparen of Geology, Islac Azad Unversy - Mashad Branch, Iran, 6-8 Aprl Fourer Analyss Models and Ther Applcaon o Rver Flows Predcon ohel Ghareagha Zare - Mohaad Hossen Kar Pasha Hosen edgh 3 (M.) suden n Waer cence and Engneerng, IAU-scence and research branch of Tehran-IRAN sohel.zare@gal.co (M.) suden n Waer cence and Engneerng, IAU-scence and research branch of Tehran-IRAN ar@yahoo.co 3 Prof. Faculy of Waer cence and Engneerng, IAU -scence and research branch of Tehran -IRAN hsedgh@yahoo.co Absrac The forecasng of hydrologc syses by usng he dae of he syse s one of he os poran advanages of e seres analyss (dynac syses) n waer scence. By usng e seres analyss ha s based on aheacal logcs and sascal soluons and recen elecronc soluon ehods, s possble o evaluae he syse's reacons n advance by usng he syses pas behavor characerscs. In os waer resource syses desgn and operaon sudes, he perodc phenoena have been represened by Fourer funcons. Afer Qupo (967), Fourer analyss has becoe a sandard ool n any hydrologc sudy concernng perodcy because Fourer analyss and odelng presen powerful ools o analyze dfferen perodc evens behavor. For analyss and desgn of waer resource syses, s soees useful o generae hgh- resoluon synhec rver flows. Modelng and sulaon of rver flow e seres s an poran sep n he plannng and operaonal analyss of waer resources syses. Thus, hs paper copares hree dfferen Fourer based odels n her capables and resuls. These hree odels are: Fourer PARMA odels, Adaped Fourer analyss wh alan fler(afam), Fourer seres ARIMA odel(fam). These odels es on shaharchay rver flow. I s prospeced ha he resuls show he bes way and s relably. Keywords: Fourer analyss, Fourer ransfors, rver flow, alan flerng. Inroducon Nuerous facors conrbued n envronenal changes whch ulaely resuled n creaon of varable envrons orphologcally as well as applcably. Aong he os val facors s eroson whch plays a crucal role n appearance and land use changes. gnfcan sors of eroson nclude wnd eroson, hydro-eroson as well as eroson due o huan applcaons. Waer wh a surprsng power s a ey coponen n eroson and sedenaon rverbeds and coasal lnes. Moreover, valleys and vas plans are fored due o waer eroson whch s anly assocaed wh waer flow. To easure he waer flow a quany called dscharge s appled. Thus, sudy of rver eroson s carred ou based on nforaon abou waer flow (dscharge easureen) n cobnaon wh geologcal properes. In hs regard, nvesgaon of colleced daa whch provde he e seres of rver dscharge s consdered as a raonal and applcable ehod o predc he fuure flow values. A naural rver flow process has sgnfcan perodc behavor n ean, sandard devaon, sewness, and seral dependence srucure. Ths sudy as o nvesgae and copare hree convenonal ehods appled o easure flow dscharge based on analyss of e seres whch nclude:.fourer 5

2 The s Inernaonal Appled Geologcal ongress, Deparen of Geology, Islac Azad Unversy - Mashad Branch, Iran, 6-8 Aprl eres ARIMA odels (FAM).Adaped fourer analyss wh alan fler(afam) 3.Fourer PARMA odels. These odels es and copare on naural rver flow. Fourer seres ARIMA odels (FAM) Mosly, s easer o show a funcon by a se of prary funcons called as base. o ha could be possble o llusrae he whole suded funcons as lnear coposes of prary funcons n base. Aong he applcable funcon are (sn) as well as (cos) funcons or xed ndces drecly applcable o frequency analyss. In 976, he suded odel was appled for he frs e by Bloofeld o analyss he e seres n hydrology. Afshar and Fah (996) provded a odel o predc he ranfall n Iran by cobnng Fourer odel wh ARIMA odels. Fourer odel s a aheacal srucure desgned of Fourer equaon n cobnaon wh Marov odel. F paern F paern s represened as follow: X n, n, Where, and are average coeffcen and sandard devaon of haronc seres, respecvely whch are copued by followng equaon: x ( A os B ) Where, s seres frequency perod. To copue he February seres expanson coeffcen ( and ), followng equaon s B A used: A B os In he above relaon, s average of npus n a dsnc nerval e (onh) durng he sascal perod and s he oal average of npus. x andard devaon of haronc seres, Where, ( A B A os os B ), could be easured as: = sandard devaon of npus n a dsnc nerval e (onh) durng sascal perod () () (3) (4) (5) (6) (7) 5

3 The s Inernaonal Appled Geologcal ongress, Deparen of Geology, Islac Azad Unversy - Mashad Branch, Iran, 6-8 Aprl Adaped Fourer analyss wh alan fler (AFAM) AFAM odel provded on he bass of a coplcaed aheacal srucure has been sulaed usng sasc, aheac as well as elecronc scences o predc he dynac syses. Ze chen (98) aenvely sulaed dscharge of flow n Gua and oloba rvers and hen predced he fuure levels for he suded rvers. Ths odel hen called as alan odel: ) (, ) ) (, ) Q( (8) Y ( ) (, ) Y ( ) (9) T T () K( ) H ( H (. ) H ( R( () Y ( Y ( ) K( Z( H( Y ( ) () I K( H ( ) There are hree essenal requreens n he odel desgned on he bass of alan fler odel: - sae varable vecor - Rule of ransfer he sae varable fro one e o a furher e 3- Prary sae vecor In hs odel, Fourer equaon s used as ode varable vecor. In oher words, he consdered equaon s he sulaor funcon for real values. To desgn he odel accordng o enoned above analyss, by copuaon of x ( ), followng equaon s acheved: ) A ( ) sn( sn ( ) B ( ) cos( cos ( ) ( (3) sn( sn( ( )), cos( cos( ( )) (4) M ( A ( B ( A ( B ( A ( B ( ) A ( ) B ( ) ( (5) M ( A ( B (. A ( B ( A ( B ( ) ) ) ) ) ) ) ) ( Maheacal llusraon of above arx s: (6) Y( (, ). Y ( ) W ( 5

4 The s Inernaonal Appled Geologcal ongress, Deparen of Geology, Islac Azad Unversy - Mashad Branch, Iran, 6-8 Aprl Fourer-Para Models In he area of sochasc hydrology, sandardzng or flerng s used o ransfor perodc e seres o saonary seres before fng saonary sochasc odels bu sandardzng or flerng of os rver flow seres wll no yeld saonary resduals due o perodc auocorrelaons. n hese cases, he resulng odel s s specfed. To odel such perodcy n auocorrelaons, perodc auoregressve ovng average (PARMA) odels can be eployed. n os cases, PARMA odels have been appled o e seres a a e scale of a onhs or ore. However, when he nuber of perods s large (e.g, weely daa), PARMA odels esaon of an exorban nuber of paraeers, hereo for ang PARMA odelng vrually praccal. he parsony n hese odels s acheved by expressng he perodc odel paraeers n erns of her dscree fourer ransfors. o fnd a parsony odel n e rver dscharge e seres, Tesfaye e. al(7) found fro her experence ha s pruden o nally f a PARMA (,) odel o he daa. for ore coplcaed PARMA (p,q) odels, he perodc ARMA process X ~ wh perod (denoed by PARMA (p,q))and he fourer seres represenaon of he paraeers ( l), ( l) and are X p a b Where, l l d such ha q ( ) X ( ) (7) r r r ~ X X ar br dr l O l O O r r r dr ar br l IN l IN IN r r r (8) (9) () and = sequence of rando varables wh ean zero and scale s ndependen and dencally dsrbued. = oal nuber of haroncs, whch s equal o or dependng on wheher s even or odd, respecvely. he valdaon of a e seres odel s anaoun o he applcaon of dagnosc checs o he odel resduals o see f hey reseble whe-nose. o es he whenose null hypohess he lung-box es has used. f he null hypohess of whe-nose resduals s no reeced and f he auocorrelaon and paral auocorrelaon funcon of he resduals show no evdence of seral correlaon, hen we udge he odel o adequae. oncluson In hs sudy, usng observaon nforaon abou hahr-chay Rver along wh Fourer, Fourer alan as well as Fourer PARMA odels, e seres of rver flow has been sulaed and appled o predc he fuure flow. Resuls showed ha every hree odels were srongly applcable n odelng of dynac e seres. Meanwhle, FAM odel due o applcaon of ARIMA odel and elascy of Fourer equaon has been locaed n a hgher poson raher han ARIMA lnear odels. In addon, AFAM odel was rearably able o 53

The s Inernaonal Appled Geologcal ongress, Deparen of Geology, Islac Azad Unversy - Mashad Branch, Iran, 6-8 Aprl rapdly reduce he errors and prove he resuls due o applcaon of alan fler.

Overall, ncreasng sasc regardng e seres of onhly flows hghly develop he applcably of hese odels n shor er predcons n order o be appled n blac box odelng.

5 The s Inernaonal Appled Geologcal ongress, Deparen of Geology, Islac Azad Unversy - Mashad Branch, Iran, 6-8 Aprl rapdly reduce he errors and prove he resuls due o applcaon of alan fler. Also, PARMA (, ) odel showed a sgnfcan accordance wh observaon nforaon due o applcaon of lower funcons. Overall, ncreasng sasc regardng e seres of onhly flows hghly develop he applcably of hese odels n shor er predcons n order o be appled n blac box odelng. fgure : Plo of observed and sulaed 6 onhly rver flows Daa for he hahar-chay rver Ura-IRAN References [] Tesfaye, Anderson, Meerschar (7), "Fourer PARMA odels and Ther Applcaon o Rver Flows"-Journal of Hydrologc Engneerng- AE. [] alas,j.d.,j.w.delleur.,v.yevevch.,w.l.lane (998)."Appled Modellng of Hydrologcal Te seres ", Waer Resources Publcaon. [3].K.Jan, A.Das, D.K.rvasava,(999) "Applcaon of ANN for reservor nflow predcon and operaon", AE ournal of Waer Research Plannng and Manageen 5(5). [4] Rosa Afshar, Fah (996), "Fourer seres ARIMA odels o ranfall predcon of IRAN", naonal waer organzaon of IRAN-Annual publcaon of power Adnsraon. [5] Pre, Rosa Afshar, "Applcaon of FAM and AFAM odels n ranfall and dscharge predcon", (M.c) hess, Ura unversy-ura-iran. [6] Box,G.E.P and Jenns,G.M (976)."Te eres Analyss Forecasng and onrol",end ed, Holden day, an Francsco. [7] We,W.W. (98). "Effec of yseac aplng on ARIMA odels", Theor & Mah. 54

THEORETICAL AUTOCORRELATIONS. ) if often denoted by γ. Note that

THEORETICAL AUTOCORRELATIONS. ) if often denoted by γ. Note that THEORETICAL AUTOCORRELATIONS Cov( y, y ) E( y E( y))( y E( y)) ρ = = Var( y) E( y E( y)) =,, L ρ = and Cov( y, y ) s ofen denoed by whle Var( y ) f ofen denoed by γ. Noe ha γ = γ and ρ = ρ and because