Quantitative Assessment on Fitting of Gumbel and Frechet Distributions for Extreme Value Analysis of Rainfall

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015 IJSRST Volume 1 Issue Prt ISSN: 395-6011 Ole ISSN: 395-60X Themed Secto: Scece Quattatve Assessmet o Fttg of umbel ad Frechet Dstrbutos for Extreme Value Aalyss of Rafall ABSTRACT N.

1 015 IJSRST Volume 1 Issue Prt ISSN: Ole ISSN: X Themed Secto: Scece Quattatve Assessmet o Fttg of umbel ad Frechet Dstrbutos for Extreme Value Aalyss of Rafall ABSTRACT N. Vvekaada Cetral Water ad Power Research Stato, Pue, aharashtra, Ida Rafall frequecy aalyss plays a mportat role hydrologc ad ecoomc evaluato of water resources projects. It helps to estmate the retur perods ad ther correspodg evet magtudes thereby creatg reasoable desg crtera. Depedg o the sze, lfe tme ad desg crtera of the structure, dfferet retur perods are geerally stpulated for adoptg Extreme Value Aalyss (EVA) results. Ths paper llustrates the use of quattatve assessmet o fttg of umbel (EV1) ad Frechet (EV) probablty dstrbutos to the seres of aual 1-day maxmum rafall (AR) data usg oodess-of-ft (of) ad dagostc tests. Order Statstcs Approach (OSA) s used for determato of parameters of the dstrbutos. Based o of (usg Aderso- Darlg ad Kolmogorov-Smrov) ad dagostc (usg D-dex) test results, the study detfes the EV1 dstrbuto s better suted for EVA of rafall for Fatehabad ad Hssar. Keywords: Aderso-Darlg, D-Idex, Frechet, umbel, Kolmogorov-Smrov, Rafall I. INTRODUCTION Estmato of rafall for a desred retur perod s a prerequste for plag, desg ad operato of varous hydraulc structures such as dams, brdges, barrages ad storm water draage systems. Depedg o the sze, lfe tme ad desg crtera of the structure, dfferet retur perods are geerally stpulated for adoptg Extreme Value Aalyss (EVA) results. For arrvg at such desg values, a stadard procedure s to aalyse hstorcal aual 1-day maxmum rafall (AR) data over a perod of tme (yr) ad arrve at statstcal estmates. I probablstc theory, the Extreme Value Dstrbutos (EVDs) clude eeralsed Extreme Value (EV), umbel (EV1), Frechet (EV) ad Webull (EV3) s geerally adopted for EVA of rafall [1-3]. EVDs arse as lmtg dstrbutos for the sample of depedet, detcally dstrbuted radom varables, as the sample sze creases. Out of umber of parameter estmato methods, Order Statstcs Approach (OSA) s appled for determato of dstrbutoal parameters because of the OSA estmators are havg mmum varace. I ths paper, EV ad EV3 dstrbutos are ot cosdered for EVA of rafall due to o-exstece of OSA for determato of dstrbutoal parameters. Number of studes carred out dfferet researchers llustrated that there s o uque dstrbuto s avalable for EVA of rafall for a rego or coutry [4-10]. Ths apart, whe dfferet dstrbutos are used for estmato of rafall, a commo problem s ecoutered as regards the ssue of best model fts for a gve set of data. Ths ca be aswered by quattatve assessmet usg oodess-of-ft (of) ad dagostc tests; ad the results are quatfable ad relable [11]. For quattatve assessmet o rafall wth the recorded rage, Aderso-Darlg (A ) ad Kolmogorov-Smrov (KS) tests are appled for checkg the adequacy of fttg of EV1 ad EV dstrbutos to the seres of AR data. A dagostc test of D-dex s used for the selecto of sutable probablty dstrbuto for estmato of rafall. I ths paper, quattatve assessmet o fttg of EV1 ad EV probablty dstrbutos s made to detfy the best sutable dstrbuto for estmato of rafall for Fatehabad ad Hssar. IJSRST1514 Receved: 5 Jue 015 Accepted: 30 Jue 015 ay-jue 015 [(1): 68-73] 68

2 II. ETHODS AND ATERIALS The Cumulatve Dstrbuto Fuctos (CDFs) of EV1 ad EV dstrbutos are expressed by: R (R) e e F,, >0 (for EV1) (1) F(R) ( F RF ) e F, F, F >0 (for EV) () Here, ad are the locato ad scale parameters of EV1 dstrbuto. The rafall estmates (R ) adoptg EV1 dstrbuto are computed from R α Y β wth l( l(1 (1/T))). T Smlarly, F ad F are the scale ad shape parameters of EV dstrbuto. Based o extreme value theory, EV dstrbuto ca be trasformed to EV1 dstrbuto through logarthmc trasformato. Uder ths trasformato, the rafall estmates (R F ) adoptg EV dstrbuto are computed from R F Exp(R ), βf Exp(α ) ad λf 1/ β [1]. Theoretcal Descrptos of OSA OSA s based o the assumpto that the set of extreme values costtutes a statstcally depedet seres of observatos. The parameters of EV1 dstrbuto are gve by: α α Y T r r α ; β r β r β (3) where r ad r are proportoalty factors, whch ca be obtaed from the selected values of k, ad usg the relatos r k/ N ad r / N. Here, N s the sample sze of the basc data that are dvded to k sub groups of elemets each leavg remaders; ad N ca be wrtte the form of N=k+. I OSA, α ad β are the dstrbuto parameters of the groups ad α ad β are the parameters of the remaders, f ay. These ca be computed from the followg equatos: α β where 1 k α S 1 1 kβ S 1 k S R j, 1 ad α ad β α R (4) 1 1 β R (5) j=1,,3,..,. Here, R s the th observato the remader group havg elemets, R j s the th observato the j th group havg elemets. Table 1 gves the weghts of α ad β used determato of parameters of the dstrbutos [13]. The parameters are further used to estmate the rafall for dfferet retur perods. The Stadard Error (SE) o the estmated rafall s computed by: SE [Var(R )] r 1 k k N T 1/ ad ad r N Var(R ) r R T r R (6) Here, R ad R are defed by the geeral form R A Y B Y C as T T β. Here R T deotes the estmated rafall by ether R or R F. The values of A, B, ad C are gve Table. TABLE 1 WEIHTS OF AND FOR COPUTATION OF DISTRIBUTIONAL PARAETERS TABLE VARIANCE DETERINATORS FOR R N A B C oodess-of-ft Tests The adequacy of fttg of probablty dstrbutos to the seres of recorded AR s evaluated by quattatve assessmet usg of tests statstc. Theoretcal descrpto of A test statstc s as follows: N ( 1) l(z ) A N 1 N (7) 1 N 1 l(1 Z ) Here Z F(R ) for =1,,3,,N wth R 1<R <..<R N, F(R ) s the CDF of th sample ( R ) ad N s the sample sze. The theoretcal value ( A ) of A statstc for dfferet sample sze (N) at 5% percet sgfcace level s computed from 0.757(1 (0./ N). A C C Iteratoal Joural of Scetfc Research Scece ad Techology ( 69

3 The KS statstc s defed by: N KS ax ( F (R ) F (R )) 1 e D (8) F s Here F e X s the emprcal CDF of X ad D X the computed CDF of X (Zhag, 00). The theoretcal value KS statstc for dfferet sample sze (N) at 5% sgfcace level s avalable the techcal ote o oodess-of-ft Tests for Statstcal Dstrbutos book [14]. Test crtera: If the computed values of of tests statstc gve by probablty dstrbuto are less tha that of theoretcal values at the desred sgfcace level the the dstrbuto s cosdered to be acceptable for EVA of rafall at that level. Dagostc Test The selecto of a sutable probablty dstrbuto for EVA of rafall s performed through D-dex test [1], whch s defed as below: 6 D-dex = 1 R R R (9) 1 Here, R s the average value of the recorded data whereas R ad R are the hghest recorded ad correspodg estmated values by EV1 ad EV. The dstrbuto havg the least D-dex s cosdered as better suted dstrbuto for rafall estmato [15]. III. APPLICATION I ths paper, a study was carred out to estmate the rafall for dfferet retur perods for Fatehabad ad Hssar adoptg EV1 ad EV dstrbutos (usg OSA). Daly rafall data recorded at Fatehabad for the perod 1954 to 011 ad Hssar for the perod 1969 to 011 was used. From the scruty of the daly rafall data, t was observed that the data for the termttet perod for Fatehabad ad Has (1966 ad 1967) ad Hssar (00) are mssg. So, the AR for the mssg years were mputed by the seres maxmum value of 140 mm (for Fatehabad) ad 56.5 mm (for Hssar) accordace wth Atomc Eergy Regulatory Board gudeles ad used for EVA. Table 3 gves the descrptve statstcs of AR recorded at Fatehabad ad Hssar. Rego TABLE 3 DESCRIPTIVE STATISTICS OF AR Descrptve statstcs R (mm) SD (mm) Skewess Kurtoss Fatehabad Hssar SD: Stadard Devato IV. RESULTS AND DISCUSSIONS By applyg the procedures as descrbed above, a computer program was developed ad used to ft the AR recorded at Fatehabad ad Hssar. The program computes the rafall estmates for dfferet retur perods adoptg EV1 ad EV dstrbutos (usg OSA), of tests statstc ad D-dex values. Table 4 gves the rafall estmates (ER) together wth Stadard Error (SE) adoptg EV1 ad EV dstrbutos for the statos uder study. From Table 4, t may be oted that the estmated rafall by EV dstrbuto s relatvely hgher tha the correspodg values of EV1 for Fatehabad ad Hssar. Iteratoal Joural of Scetfc Research Scece ad Techology ( 70

4 TABLE 4 ESTIATED RAINFALL WITH STANDARD ERROR ADOPTIN EV1 AND EV DISTRIBUTIONS (USIN OSA) FOR FATEHABAD AND HISSAR Retur Estmated rafall (mm) wth stadard error (mm) for perod Fatehabad Hssar (yr) EV1 EV EV1 EV ER SE ER SE ER SE ER SE Aalyss Based o of Tests For quattatve assessmet o fttg of EV1 ad EV dstrbutos to the recorded AR data, of tests statstc values were computed from Eqs. (7) ad (8), ad gve Table 5. TABLE 5 COPUTED AND THEORETICAL VALUES OF OF TESTS STATISTIC Rego A KS Computed values Theoretcal value at Computed values Theoretcal value at EV1 EV 5% level EV1 EV 5% level Fatehabad Hssar From the of tests results gve Table 5, t may be oted that the KS test cofrmed the use of EV1 ad EV dstrbutos (usg OSA) for EVA of rafall (Fatehabad ad Hssar). Smlarly, A test cofrmed the use of EV1 dstrbuto for EVA of rafall for Fatehabad. As regards EVA of rafall for Hssar, A test suggested the EV1 ad EV dstrbutos were ot acceptable. Aalyss Based o Dagostc Test For the selecto of a sutable probablty dstrbuto, D- dex values of EV1 ad EV dstrbutos are computed from Eq. (9) ad gve Table 6. From the results, t may be oted that the D-dex values of EV1 dstrbuto are mmum whe compared wth the correspodg values of EV for the statos uder study. TABLE 6 D-INDEX VALUES OF EV1 AND EV Rego D-dex EV1 EV Fatehabad Hssar Based o quattatve assessmet usg of ad dagostc tests, the study showed that the EV1 dstrbuto s better suted for estmato of rafall for Fatehabad ad Hssar. Fgures 1 ad gve the plots of recorded ad estmated rafall usg EV1 (OSA) wth cofdece lmts at percetage level for Fatehabad ad Hssar. Iteratoal Joural of Scetfc Research Scece ad Techology ( 71

5 (where ea deotes the estmated rafall ad SE the Stadard Error) values of about 75 mm (for Fatehabad) ad 48 mm (for Hssar) computed from EV1 (OSA) dstrbuto were suggested for desg purposes. ACKNOWLEDEENTS The author s grateful to the Drector, Cetral Water ad Power Research Stato, Pue, for provdg the research facltes to carry out the study. The author s thakful to /s Nuclear Power Corporato of Ida Lmted, umba ad Ida eteorologcal Departmet, Pue, for supply of rafall data. FIURE 1: RECORDED AND ESTIATED 1-DAY AXIU RAINFALL USIN EV1 (OSA) DISTRIBUTION WITH PERCENT LOWER AND UPPER CONFIDENCE LIITS FOR FATEHABAD FIURE : RECORDED AND ESTIATED 1-DAY AXIU RAINFALL USIN EV1 (OSA) DISTRIBUTION WITH PERCENT LOWER AND UPPER CONFIDENCE LIITS FOR HISSAR V. CONCLUSIONS The paper preseted the procedures volved quattatve assessmet o fttg of EV1 ad EV dstrbutos (usg OSA) for EVA of rafall for Fatehabad ad Hssar. The KS test results cofrmed the fttg of EV1 ad EV dstrbutos to the seres of AR recorded at the statos uder study. The A test results suggested the use of EV1 dstrbuto for EVA of rafall for Fatehabad. The dagostc aalyss showed that the EV1 dstrbuto s better suted for estmato of rafall for Fatehabad ad Hssar. By cosderg the desg-lfe of the structure over the etre teded ecoomc lfetme, the yr retur perod ea+se REFERENCES [1] E.J. umbel, Statstc of Extremes, d Edto, Columba Uversty Press, New York, [] J.A. Carta ad P. Ramrez, Aalyss of twocompoet mxture Webull statstcs for estmato of wd speed dstrbutos, Joural of Reewable Eergy, Vol. 3, No.3, pp , 007. [3] N. ujere, Flood frequecy aalyss usg the umbel dstrbuto, Joural of Computer Scece ad Egeerg, Vol. 3, No. 7, pp , 011. [4].C. Casas, R. Rodrguez,. Prohom, A. azquez, ad A. Redao, Estmato of the probable maxmum precptato Barceloa (Spa), Joural of Clmatology, Vol. 31, No. 9, pp , 011. [5] E. Baratt, A. otaar, A. Castellar, J.L. Salas, A. Vgloe, ad A. Bezz, Estmatg the flood frequecy dstrbuto at seasoal ad aual tme scales, Hydrologcal Earth System Scece, Vol. 16, No. 1, pp , 01. [6] A. Peck, P. Prodaovc ad S.P. Smoovc, Rafall testy durato frequecy curves uder clmate chage: cty of Lodo, Otaro, Caada, Caada Water Resources Joural, Vol. 37, No. 3, pp , 01. [7] L.S. Esteves, Cosequeces to flood maagemet of usg dfferet probablty dstrbutos to estmate extreme rafall, Joural of Evrometal aagemet, Vol. 115, No. 1, pp , 013. [8] B.A. Olumde,. Sadu, ad A. Oluwasesa, Evaluato of best ft probablty dstrbuto Iteratoal Joural of Scetfc Research Scece ad Techology ( 7

6 models for the predcto of rafall ad ruoff volume (Case Study Tagwa Dam, a- Ngera), Joural of Egeerg ad Techology, Vo1. 3, No., pp , 013. [9] V. Rahma, S.L. Hutchso, J..S. Hutchso ad A. Aadh, Extreme daly rafall evet dstrbuto patters Kasas, Joural of Hydrologc Egeerg, Vol. 19, No. 4, pp , 014. [10] Zhalg L, Zhaje L, We Zhao, ad Yuehua Wag, Probablty odelg of Precptato Extremes over Two Rver Bass Northwest of Cha, Advaces eteorology, Vol. 13, Artcle ID 37417, pp. 1-13, 015. [11] J. Zhag, Powerful goodess-of-ft tests based o the lkelhood rato, Joural of Royal Statstcal Socety, Vol. 64, No., pp , 00. [1] A.H. Ag, ad W.H. Tag, Probablty cocepts egeerg plag ad desg, Vol., Joh Wley & Sos, [13] Atomc Eergy Regulatory Board (AERB), Extreme values of meteorologcal parameters, AERB Safety ude No. NF/S/ S-3, 008. [14] P.E. Charles As, oodess-of-ft Tests for Statstcal Dstrbutos, [ egeerg.com/goodess.html], 009. [15] Uted States Water Resources Coucl (USWRC), udeles for determg flood flow frequecy, Bullet No. 17B, Iteratoal Joural of Scetfc Research Scece ad Techology ( 73

Rainfall Frequency Analysis using Order Statistics Approach of Extreme Value Distributions

Rainfall Frequency Analysis using Order Statistics Approach of Extreme Value Distributions SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September 014 Rafall requecy Aalyss usg Order Statstcs Approach of Extreme Value Dstrbutos N Vvekaada Cetral Water ad Power Research Stato,