Rainfall Frequency Analysis using Order Statistics Approach of Extreme Value Distributions

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1 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, Pue, aharashtra, Ida ABSTRACT: Estmato of rafall for a desred retur perod s of utmost mportace for plag, desg ad maagemet of the hydraulc structures the project ste. Ths ca be acheved by fttg of probablty dstrbutos to the seres of aual 1-day maxmum rafall. Ths paper llustrates the use of extreme value dstrbutos for estmato of rafall for atehabad, Has, Hssar ad Tohaa statos. Order Statstcs Approach s used for determato of parameters of the extreme value dstrbutos. Goodess-of-t tests such as Aderso-Darlg ad Kolmogorov-Smrov are appled for checkg the adequacy of fttg of the dstrbutos to the recorded data. A dagostc test of D-dex s used for the selecto of a sutable probablty dstrbuto for rafall estmato. Based o Go ad dagostc tests results, the study shows the Gumbel dstrbuto s better suted for rafall estmato for the statos uder study. Keywords - rechet, Gumbel, Kolmogorov- Smrov, D-dex, Order Statstcs, Retur perod 1. INTRODUCTION Techcal ad egeerg apprasal of large frastructure projects such as dams, brdges, barrages, culverts, etc eeds to be carred out durg the plag ad formulato stages of such projects. I a hydrologcal cotext, t s well recogsed that whatsoever extreme the desgloadg, more severe codtos are lkely to be ecoutered ature [1]. or the reaso, Extreme Value Aalyss (EVA) of recorded rafall relatg to the geographcal rego where the project s located s a basc requremet for assessg such pheomea, ad arrvg at structural ad other desg parameters for the project. Depedg o the sze, lfe tme ad desg crtera of the structure, dfferet retur perods are geerally stpulated for adoptg EVA results. Atomc Eergy Regulatory Board (AERB) [] gudeles dcated that the 1000-year (yr) retur perod ea+se (where ea deotes the estmated rafall ad SE the Stadard Error) s cosdered to arrve at the desg rafall depth that a structure must wthstad durg ts lfetme. or arrvg at such desg values, a stadard procedure s to aalyse hstorcal rafall data over a perod of tme (yr) ad arrve at statstcal estmates. I probablstc theory, the extreme value dstrbutos (EVDs) clude Geeralsed Extreme Value (GEV), Gumbel, rechet ad Webull are geerally adopted for modellg the rafall ad stream flow data [3]. EVDs arse as lmtg dstrbutos for the sample of depedet, detcally dstrbuted radom varables, as the sample sze creases. I addto to the above, Atomc Eergy Regulatory Board (AERB) gudeles descrbed that the Order Statstcs Approach (OSA) ca also be adopted for determato of parameters of Gumbel ad rechet dstrbutos because of the OSA estmators are ubased ad havg mmum varace. I ths paper, GEV ad Webull dstrbutos are ot cosdered for EVA of rafall due to o-exstece of OSA for determato of dstrbutoal parameters. Number of studes has bee carred out by dfferet researchers o EVA of rafall ad the reports dcated that there s o uque dstrbuto s avalable for modellg the rafall data for a rego or coutry [4-6]. Ths paper descrbes the procedures volved rafall estmato adoptg Gumbel ad rechet probablty dstrbutos (usg OSA) for atehabad, Has, Hssar ad Tohaa statos. Goodess-of-t (Go) tests such as Aderso- Darlg (A ) ad Kolmogorov-Smrov (KS) are appled for checkg the adequacy of fttg of the dstrbutos to the recorded rafall data. A dagostc test of D-dex s used for the selecto of a sutable probablty dstrbuto for estmato of rafall. The methodology adopted determg the parameters of Gumbel ad rechet dstrbutos (usg OSA), computato of Go tests statstc ad D-dex are brefly descrbed the esug sectos.. ETHODOLOGY.1 Probablty Dstrbutos The Cumulatve Dstrbuto uctos (CDs) of Gumbel ad rechet dstrbutos are gve by: ISSN: Page 6

2 SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September 014 (R) e (R) e e R R G G G ( ), G, G >0 (Gumbel) (1),, >0 (rechet) () Here, G ad G are the locato ad scale parameters of Gumbel dstrbuto. The rafall estmates (R G ) adoptg Gumbel dstrbuto are computed from RG G Y T G ad YT l( l(1 (1/T))) Smlarly, ad are the scale ad shape parameters of rechet dstrbuto. Based o extreme value theory, rechet dstrbuto ca be trasformed to Gumbel dstrbuto through logarthmc trasformato [7]. Uder ths trasformato, the rafall estmates (R ) adoptg rechet dstrbuto are computed from R Exp(R G), Exp( G ) ad 1. /. Theoretcal Descrpto of OSA OSA s based o the assumpto that the set of extreme values costtutes a statstcally depedet seres of observatos. The parameters of Gumbel dstrbuto are gve by: G r r ; r r (3) G G Here, 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. Also, N s the sample sze cotas 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, ad are the dstrbuto parameters of the groups ad ad are the parameters of the remaders, f ay. These ca be computed from the followg equatos: Here, 1 ks 1 1 ks 1 k ad R (4) 1 ad R (5) 1 S j=1,,3,..,. R s the th R j, 1 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 Gumbel ad rechet dstrbutos [8]. Table 1. Weghts of α ad β for determato of dstrbutoal parameters α (or) β α α α α α β β β β β The parameters are used to estmate the expected rafall for dfferet retur perods. The stadard error (SE) o the estmated rafall s computed by: 1 k r ad r k N N (7) SE [Var(R T )] 1/ ad Var (R T) r R r R (6) Here, R T deotes the estmated rafall by ether R G or R. R ad R are defed by the geeral form ISSN: Page 7

3 SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September 014 as R AYT BYT C G. The values of A, B, ad C are gve Table. Table. Varace determators for R A B C Goodess-of-t Tests The adequacy of fttg of probablty dstrbutos to the recorded rafall s tested by Go tests usg A ad KS statstc, whch are defed by: N A ( N) (1/ N) ( 1)l(Z ) (N 1 ) l(1 Z ) (8) N 1 KS ax( (R ) (R )) 1 e D (9) Here, Z e(r) ( 0.44) /(N 0.1) s the emprcal CD of R wth R 1 <R <R 3,..,<R N ad D (R) s the computed CD of R [9]. If the computed values of Go tests statstc gve by the dstrbuto are less tha that of theoretcal value at the desred sgfcace level, the the dstrbuto s foud to be acceptable for modellg the rafall data..4 Dagostc Test The selecto of a sutable dstrbuto for rafall estmato s performed through D-dex, whch s defed by: 6 D-dex= ( 1/ R) R R (10) 1 Here, R s the average value of the seres of the recorded rafall, R s (for =1 to 6) are the sx hghest values the seres of recorded rafall ad R s the estmated rafall by probablty dstrbuto. The dstrbuto havg the least D-dex s cosdered as the better suted dstrbuto for rafall estmato [10]. 3. APPLICATION A attempt has bee made to ft the seres of aual 1-day maxmum rafall (AR) recorded at atehabad, Ha, Hssar ad Tohaa ra-gauge statos adoptg Gumbel ad rechet dstrbutos (usg OSA). Daly rafall data recorded at atehabad ad Has for the perod , Hssar for the perod ad Tohaa for the perod are used. The seres of AR s derved from the daly rafall data ad further used for EVA. Table 3 gves the summary statstcs of the AR recorded at the statos uder study. Table 3. Summary statstcs of AR Stato Average (mm) SD (mm) CV (%) Skewess Kurtoss atehabad Has Hssar Tohaa SD: Stadard Devato; CV: Coeffcet of Varato 4. RESULTS AND DISCUSSIONS By applyg the procedures descrbed above, a computer program was developed ad used to ft the AR recorded at atehabad, Has, Hssar ad Tohaa statos. The program computes the OSA estmators of the dstrbutos, Go tests statstc ad D-dex values. Tables 4 ad 5 gve the rafall estmates together wth stadard error for dfferet retur perods obtaed from Gumbel ad rechet dstrbutos for the statos uder study. Retur perod (yr) Table 4. Estmated rafall (mm) together wth stadard error (mm) usg Gumbel ad rechet dstrbutos for atehabad ad Has atehabad Has Gumbel rechet Gumbel rechet R G SE R SE R G SE R SE ISSN: Page 8

4 SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September Retur perod (yr) Table 5. Estmated rafall (mm) together wth stadard error (mm) usg Gumbel ad rechet dstrbutos for Hssar ad Tohaa Hssar Tohaa Gumbel rechet Gumbel rechet R G SE R SE R G SE R SE rom Tables 4 ad 5, t may be oted that the estmated rafall usg rechet dstrbuto s relatvely hgher tha the correspodg values of Gumbel for the retur perods of 5-yr ad above for the statos uder study. 4.1 Rafall requecy Curves (RCs) The parameters of Gumbel ad rechet dstrbutos were used to develop the plots of CDs ad preseted gure 1. Smlarly, the rafall estmates obtaed from Gumbel ad rechet dstrbutos were used to develop the RCs ad preseted gures ad 3. rom these fgures, t ca be see that the RCs usg Gumbel dstrbuto are the form of lear whereas the RCs usg rechet dstrbuto are the form of expoetal for the statos uder study. 4. Aalyss Based o Go Tests The values of A ad KS statstc for the seres of AR adoptg Gumbel ad rechet dstrbutos were computed from Eqs. (8-9) ad gve Table 6. Table 6. Computed ad theoretcal values of A ad KS statstc Stato A KS Computed values Theoretcal values Computed values Theoretcal values Gumbel rechet at 5% level Gumbel rechet at 5% level atehabad Has Hssar Tohaa rom A test results, t may be observed that the Gumbel dstrbuto s acceptable for modellg the AR of atehabad ad Tohaa whereas the KS test supports the use of both Gumbel ad rechet dstrbutos for all four statos uder study. The A test results also dd t support the use of rechet dstrbuto for modellg the AR recorded at the four statos. D-dex values gve by Gumbel dstrbuto are comparatvely mmum whe compared to the correspodg values of rechet for atehabad ad Tohaa. Also, from Table 7, t may be oted that the D-dex values of rechet are mmum whe compared to the correspodg values of Gumbel for Has ad Hssar. But, the A test results dd t support the use of rechet dstrbuto for modellg the AR of Has ad Hssar. 4.3 Aalyss Based o Dagostc Test or the selecto of a sutable probablty dstrbuto, D-dex values of the dstrbutos were computed by Eq. (10) ad gve Table 7. rom the dagostc test results, t may be oted that the ISSN: Page 9

5 SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September 014 Table 7. D-dex values of Gumbel ad rechet dstrbutos Stato Idces of D-dex Gumbel rechet atehabad Has Hssar Tohaa Based o Go ad dagostc tests results, the study showed that the Gumbel dstrbuto s better suted for estmato of rafall though KS test supports the use of Gumbel ad rechet dstrbutos for modellg the AR recorded at the statos uder study. gure 1: Plots of CDs usg Gumbel ad rechet dstrbutos ISSN: Page 10

6 SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September 014 gure. Rafall frequecy curves usg Gumbel dstrbuto gure 3. Rafall frequecy curves usg rechet dstrbuto 5. CONCLUSIONS The paper preseted a computer aded procedure for estmato of rafall for atehabad, Has, Hssar ad Tohaa. The A test results supported the use of Gumbel dstrbuto for modellg AR of atehabad ad Tohaa whereas KS test results cofrmed the use of both Gumbel ad rechet dstrbutos for all four statos. Based o Go ad dagostc tests results, the study showed that the Gumbel dstrbuto s better suted for estmato of rafall for the statos uder study. The study showed that the 1000-yr ISSN: Page 11

7 SSRG Iteratoal Joural of Cvl Egeerg (SSRG-IJCE) volume 1 ssue 4 September 014 retur perod ea+se (where ea deotes the estmated rafall ad SE the Stadard Error) values of about 19 mm for atehabad, 64 mm for Ha, 379 mm for Hssar ad 300 mm for Tohaa gve by Gumbel dstrbuto (usg OSA) could be adopted whle plag ad desg of hydraulc structures the respectve statos. ACKNOWLEDGENTS 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 also thakful to /s Nuclear Power Corporato of Ida Lmted, umba ad Ida eteorologcal Departmet, Pue, for the supply of rafall data. REERENCES [1] Iteratoal Atomc Eergy Agecy (IAEA), eteorologcal evets ste evaluato for Nuclear Power Plats IAEA Safety Gude, No. Ns-G-3.4, Iteratoal Atomc Eergy Agecy, Vea, 003. [] S.R. Bhakar, A.K. Basal, N. Chhajed ad R.C. Puroht, requecy aalyss of cosecutve days maxmum rafall at Baswara, Rajastha, Ida, ARPN joural of Egeerg ad Appled Sceces, 1(1), 006, [3] Atomc Eergy Regulatory Board (AERB) (008), Extreme values of meteorologcal parameters (Gude No. N/SG/ S-3). [4] L.S. Esteves, Cosequeces to flood maagemet of usg dfferet probablty dstrbutos to estmate extreme rafall, Evrometal aagemet, 115(1), 013, [5]. Ubold, A.N. Suls, C. Lussaa,. Cslagh ad. Russo, A spatal bootstrap techque for parameter estmato of rafall aual maxma dstrbuto, Hydrology ad Earth System Sceces Dscussos, 10(10), 013, [6] B.A. Olumde,. Sadu ad A. Oluwasesa, Evaluato of best ft probablty dstrbuto models for the predcto of rafall ad ruoff volume (Case Study Tagwa Dam, a-ngera), Egeerg ad Techology, 3 (), 013, [7] N. ujere, lood frequecy aalyss usg the Gumbel dstrbuto, Computer Scece ad Egeerg, 3(7), 011, [8] A.H. Ag ad W.H. Tag, Probablty cocepts egeerg plag ad desg, Vol., Joh Wley & Sos, [9] J. Zhag, Powerful goodess-of-ft tests based o the lkelhood rato, Royal Statstcal Socety, 64(), 00, [10] Uted States Water Resources Coucl (USWRC) (1981), Gudeles for determg flood flow frequecy, Bullet No.17B, ISSN: Page 1

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

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