Rainfall Intensity-Duration-Frequency Analysis Using Order Statistics Approach of Gumbel and Frechet Distributions

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1 Internatonal Journal of Emergng Engneerng Research and Technology Volume 5, Issue 9, 017, PP ISS (Prnt) & ISS (Onlne) Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons. Vvekanandan Scentst-B, Central Water and Power Research Staton, Pune, aharashtra, Inda Correspondng Author:. Vvekanandan, Scentst-B, Central Water and Power Research Staton, Pune 41104, aharashtra, Inda ABSTRACT Ranfall ntenstes of varous frequences and duratons are the mportant parameter for the hydrologc desgn of storm sewers, culverts and many other hydraulc structures. Ths can be acheved by ranfall Intensty-Duraton-Frequency (IDF) relatonshp, whch s determned through Extreme Value Analyss (EVA) of hourly ranfall data. Ths paper detals a study on EVA of n-hourly maxmum ranfall for dfferent duraton of n such as 1-hour (hr), -hr, 3-hr, 6-hr, 1-hr, 18-hr, 4-hr, 48-hr and 7-hr for Bhavnagar and Jabalpur regons. Order statstcs approach s appled for determnaton of parameters of Gumbel and Frechet dstrbutons, whch are used for estmaton of ranfall for dfferent return perods. The estmated n-hourly maxmum ranfall for dfferent duraton of n s used to develop the IDF curves. The adequacy of Gumbel and Frechet dstrbutons used for development of IDF relatonshps s evaluated by model performance ndcators vz., correlaton coeffcent and root mean squared error. The paper presents the IDF relatonshp gven by Gumbel dstrbuton could be used for estmaton of ranfall ntensty for Bhavnagar and Jabalpur. Keywords: Correlaton coeffcent, Frechet, Gumbel, Intensty-Duraton-Frequency, ean squared error, Order statstcs approach, Ranfall ITRODUCTIO The ranfall Intensty-Duraton-Frequency (IDF) analyss s one of the most mportant tools n water resources engneerng to assess the vulnerablty of water resources structures as well as plannng, desgn and operaton. The IDF Relatonshp (IDFR) can be developed through Extreme Value Analyss (EVA) of hourly ranfall data that s used to estmate ranfall depth at a pont for a specfed exceedance probablty and duraton [1]. EVA s based ether on annual maxmum seres at a ste (at-ste analyss) or from several stes (regonal analyss). Ranfall n a regon can be charactersed f the ntensty, duraton and frequency of the dverse storms occurrng at that place are known [-3]. The frequency-data for storms of varous duratons, so obtaned, can be represented by IDF Curve (IDFC), whch gves a plot of ranfall ntensty (mm/hour) and duraton (hour). hat et al. [4-5] appled Gumbel dstrbuton for estmaton of ranfall for dfferent return perods for development of IDFR for Vetnam regon and Yodo catchments of Japan. Prodanovc and Smonovc [6] developed ranfall IDFC for London cty under dfferent clmatc condtons. Raford et al. [7] have developed IDF relatonshps (IDFR) and sopluval maps for the regon encompassng South Carolna, orth Carolna and Georga usng newly developed EVA methods. Km et al. [8] mproved the accuracy of IDFC by usng long and short duraton separaton technque. They derved IDFC by usng Cumulatve Dstrbuton Functon (CDF) of the nterestng ste and multobjectve genetc algorthm. Ben-Zv [9] proposed a procedure for basng IDFR on partal duraton seres whch are substantally larger than those commonly used for ths purpose. He concluded that the proposed procedure superor to the current ones where the use of large samples would reduce the senstvty of predcted ntenstes to samplng varatons. Bara et al. [10] appled smple scalng theory to the IDF characterstcs of short duraton ranfall. They have concluded that the IDFC, whch were deduced from daly ranfall, showed acceptable results n comparson wth the IDFC obtaned from at-ste short duraton ranfall data. Okonkwo and bajorgu [11] have developed IDFR usng graphcal and statstcal methods for southeastern gera and the results were compared. They have found that IDFR Internatonal Journal of Emergng Engneerng Research and Technology V5 I

2 Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons developed from graphcal and statstcal methods are very close to each other for the lower return perods of -year (yr) to 10-yr and dffer for hgher return perods of 50-yr to 100- yr, but the dfference was not sgnfcant at 5% level. Khaled et al. [1] appled L-moments and generalzed least squares regresson methods for estmaton of desgn ranfall depths and development of IDFR. Eman [13] appled sx probablty dstrbutons for modellng of hourly ranfall data for generaton of IDFC for Sna Pennsula n the northeast part of Egypt. Rashd et al. [14] appled Pearson Type-III dstrbuton for modellng of short duraton ranfall and development of IDFR for Sylhet Cty n Bangladesh. Dourte et al. [15] developed ranfall IDFR for varous regons of Andhra Pradesh for predcton of runoff and estmaton of groundwater recharge. In probablty theory, extreme value dstrbutons nclude Gumbel, Frechet and Webull are generally appled for frequency analyss of meteorologcal parameters. In the present study, Webull dstrbuton s not consdered for EVA because of non-convergence of the recorded data whle determnng the dstrbutonal parameters. In addton to the above, AERB (Atomc Energy Regulatory Board) gudelnes descrbed that Order Statstcs Approach (OSA) could be appled for determnaton of parameters of Gumbel and Frechet dstrbutons though number of methods s avalable for parameter estmaton. In ths paper, a study on ranfall IDF analyss adoptng Gumbel and Frechet dstrbutons (usng OSA) for Bhavnagar and Jabalpur regons s carred out. The estmated ranfall for dfferent return perods for dfferent duratons of n such as 1-hour (hr), - hr, 3-hr, 6-hr, 1-hr, 18-hr, 4-hr, 48-hr and 7- hr adoptng Gumbel and Frechet dstrbutons (usng OSA) are used for development of IDFR. odel Performance Indcators (PIs) vz., Correlaton Coeffcent (CC) and Root ean Squared Error (RSE) are used for selecton of sutable IDFR for estmaton of ranfall ntensty for the regons under study. ETHODOLOGY The methodology adopted n development of IDFR and computatons of PIs are brefly descrbed n the followng sectons. Probablty Dstrbutons The CDFs [F(R)] of Gumbel and Frechet dstrbutons are gven by: R G G e G F (R) e, G, G >0 (1) ( R F) F F F(R) e, F, F >0 () Where, G and G are locaton and scale parameters of Gumbel dstrbuton [16]. The parameters of the dstrbuton are computed by OSA and further used to estmate the ranfall (R G ) for dfferent return perods (T) usng the relaton R G =α G +Y T β G wth Y T =-ln(-ln(1-(1/t))). Based on extreme value theory, Frechet dstrbuton can be transformed to Gumbel dstrbuton through logarthmc transformaton usng natural logarthm of the actual varable. Under ths transformaton, the scale ( F ) and shape ( F ) parameters of Frechet dstrbuton are determned by OSA for estmaton of ranfall (R F ) usng R F =Exp(R G ), β F =Exp(α G ) and F =1/β G. The procedures adopted n determnaton of parameters of Gumbel dstrbuton by OSA [17] are as follows: r G r and r r G (3) Where, r and r are proportonalty factors, whch can be obtaned from the selected values of k, n and n usng the relatons as follows: r kn / and r n / (4) Here, s the sample sze contanng the basc data that are dvded nto k sub groups of n elements each leavng n remanders. and are the dstrbuton parameters of the groups. Lkewse, and are the parameters of the remanders, f any. These can be computed from the followng equatons: n 1 k ns, n 1 k R Here, S 1 1 n n 1 k ns, n 1 k R 1 k 1 n R, j=1,,3,..,n (5) 1 j Where The weghts of n and n used n determnng the parameters of Gumbel dstrbuton by OSA are presented n Table 1. Procedure for Development of IDFR IDFR s a mathematcal relatonshp between the ranfall ntensty, duraton, and return perod [18]. Intensty s defned as the tme rate of ranfall, whch s the depth per unt tme (mm/hr, or mm/day as the case may be). It can ether be the nstantaneous ntensty or the average ntensty over the duraton of ranfall. Theoretcally, the ntensty of storm n a regon vares wth duraton n such a way that hgh ntensty generally corresponds to short 14 Internatonal Journal of Emergng Engneerng Research and Technology V5 I9 017

3 Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons duraton, and low ntensty to longer duraton. The general form of emprcal formula used n development of IDFR s expressed by: I A T B d (6) where, I s the ranfall ntensty (mm/hr), T d s the ranfall duraton (hr) correspondng to return perod (T), and the terms A and B are constants. ethod of least squares s appled to compute the constant terms of emprcal formula. By applyng logarthm on both sdes of Eq. (6), we get ln(i)=ln(a)-bln(t d )Y=a-BX. Parameters A and B are computed from Eqs. (7 and 8) and are expressed by: Y X XY B X X 1 1 (7) A Exp Y BX (8) odel Performance Analyss The performance of IDFR gven by Gumbel and Frechet dstrbutons are evaluated by CC and RSE. Theoretcal descrptons of CC and RSE [19] are as follows: CC 1 1 I II I I I I I 1 (9) RSE 1 I I (10) where, I s the recorded ranfall ntensty of th event, I s the estmated ranfall ntensty of th event, I s the average recorded ranfall ntensty and I s the average estmated ranfall ntensty. APPLICATIO An attempt has been made to develop IDFR for dfferent return perods from -yr to 1000-yr for Bhavnagar and Jabalpur regons. Hourly ranfall data [0] recorded at Bhavnagar for the perod 1980 to 005 and Jabalpur for the perod 1969 to 1991 are used to compute the seres of n- hourly maxmum ranfall for dfferent duratons of n such as 1-hr, -hr, 3-hr, 6-hr, 1-hr, 18-hr, 4-hr, 48-hr and 7-hr. The seres was used to estmate the ranfall for dfferent return perods usng Gumbel and Frechet dstrbutons. The estmated ranfall values are consdered as a base values for development of IDFR usng Eq. (6). RESULTS AD DISCUSSIOS By applyng the procedures descrbed above, a computer program was developed and used. The program computes the parameters of dstrbutons, ranfall estmates and model parameters of IDFR and values of PIs. Table1. Weghts of n and n used n determnng the parameters of the dstrbutons by OSA n or n Estmaton of Ranfall usng PDFs The estmated ranfall for dfferent return perods for dfferent duraton of n obtaned from Gumbel and Frechet dstrbutons for Bhavnagar and Jabalpur are presented n Tables to 5. These estmates were further used to develop the IDFC. From Tables to 5, t may be noted that the estmated ranfall for return perods from 5-yr to 1000-yr by Frechet dstrbuton are hgher than the correspondng values of Gumbel. Internatonal Journal of Emergng Engneerng Research and Technology V5 I

4 Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons Table. Ranfall estmates for dfferent duratons of n gven by Gumbel dstrbuton for Bhavnagar Perod (yr) Estmated Ranfall (mm) for Dfferent Duratons of n Table3. Ranfall estmates for dfferent duratons of n gven by Frechet dstrbuton for Bhavnagar Perod (yr) Estmated Ranfall (mm) for Dfferent Duratons of n Table4. Ranfall estmates for dfferent duratons of n gven by Gumbel dstrbuton for Jabalpur Perod (yr) Estmated Ranfall (mm) for Dfferent Duratons of n Table5. Ranfall estmates for dfferent duratons of n gven by Frechet dstrbuton for Jabalpur Perod (Yr) Estmated Ranfall (mm) for Dfferent Duratons of n IDFR usng Emprcal Formula The estmated ranfall values were used to compute the ranfall ntensty for dfferent duratons usng the relaton of I=P/ T d, where P s the ranfall depth (.e., estmated ranfall) and T d the duraton. These ranfall ntenstes are used to develop IDFC. The parameters (A and B) of the IDF emprcal formula were determned from Eqs. (7 and 8) and presented n Tables 6 and 7. The values of CC and RSE gven by the ftted model were computed from 16 Internatonal Journal of Emergng Engneerng Research and Technology V5 I9 017

5 Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons Eqs. (9 and 10) and also presented n Tables 6 and 7. The plots of IDF curves for dfferent return perods gven by Gumbel dstrbuton for Bhavnagar and Jabalpur regons are presented n Fgures 1 and. Gumbel dstrbuton are lesser when compared to Frechet for dfferent return perods from 10- yr to 1000-yr for both the regons. Fgure1. IDF curves for dfferent return perods gven by Gumbel dstrbuton for Bhavnagar From Tables 6 and 7, t s noted that the RSE values on the estmated ranfall ntensty by Fgure. IDF curves for dfferent return perods gven by Gumbel dstrbuton for Jabalpur Therefore, the study suggested that the ftted IDFRs for dfferent return perods gven by Gumbel may be used for estmaton of ranfall ntensty for Bhavnagar and Jabalpur regons. Table6. Parameters of IDFR together wth CC and RSE (mm/hr) for dfferent return perods adoptng Gumbel and Frechet dstrbutons for Bhavnagar PIs and Parameters of IDFR Gumbel Frechet Perod (yr) odel Parameters PIs odel Parameters PIs A B RSE CC A B RSE CC Table7. Parameters of IDFR together wth CC and RSE (mm/hr) for dfferent return perods adoptng Gumbel and Frechet dstrbutons for Jabalpur Perod (yr) PIs and Parameters of IDFR Gumbel Frechet odel Parameters PIs odel Parameters PIs A B RSE CC A B RSE CC COCLUSIOS The paper presented a computer aded procedure for EVA of hourly ranfall data for development of IDFRs for Bhavnagar and Jabalpur. From the Internatonal Journal of Emergng Engneerng Research and Technology V5 I

6 Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons results of the data analyss, the followng conclusons were drawn from the study: ) The estmated ranfall by Gumbel and Frechet dstrbutons (usng OSA) for dfferent duratons of n such as 1-hr, -hr, 3-hr, 6-hr, 1-hr, 18-hr, 4-hr, 48-hr and 7- hr OSA was used for development of IDFR for dfferent return perods. ) The CC values obtaned from ftted IDFR adoptng Gumbel dstrbuton are very close the perfect correlaton value of 1 for Jabalpur. For Bhavnagar, the CC values based on ftted IDFR gven by Gumbel are vared from to ) The RSE values ndcated the IDFR developed through emprcal formula usng the estmated ranfall values of Gumbel dstrbuton provde better results when compared to Frechet. The study suggested that the IDFR gven by Gumbel dstrbuton may be useful for decson makers to compute ranfall ntensty for plannng and desgnng of any water resources projects n Bhavnagar and Jabalpur. ACKOWLEDGEETS The author s grateful to Dr. (rs.) V.V. Bhosekar, Drector, Central Water and Power Research Staton, Pune, for provdng the research facltes to carry out the study. The author s thankful to /s uclear Power Corporaton of Inda Lmted, umba for makng avalable the ranfall data used n the study. REFERECES [1] Burlando, P., and Rosso, R. (1996). Scalng and mult-scalng models of depth-duratonfrequency curves for storm precptaton. Journal of Hydrology, 187, [] Koutsoyanns, D., Kozons, D., and anetas, A. (1998). A mathematcal framework for studyng ranfall ntensty- duraton-frequency relatonshps. Journal of Hydrology, 06, [3] Bougads, J., and Adamowsk, K. (006). Scalng model of a ranfall ntensty-duratonfrequency relatonshp. Hydro-logcal Process, 0, pp [4] hat, L.., Tachkawa, Y., and Takara, K. (006). Establshment of ntensty-duratonfrequency curves for precp-taton n the monsoon area of Vetnam. Annuals of Dsaster Preventve Research Insttute, Kyoto Unversty, o. 49 B. [5] hat, L.., Tachkawa, Y., Sayama, T., and Takara, K. (006). Dervaton of ranfall ntensty-duraton-frequency relatonshps for short duraton ranfall from daly ranfall data. IHP Techncal Documents n Hydrology, o. 6, [6] Prodanovc, P., and Smonovc, S.P. (007). Report on Development of ranfall ntensty duraton frequency curves for the Cty of London under the changng clmate. Report o. 058, Unversty of Western Ontaro. [7] Raford, J.P., Azz,.., Khan, A.A., and Powell, D.. (007). Ranfall depth-duratonfrequency relatonshps for South Carolna, orth Carolna, and Georga. Amercan Journal of Envronmental Scence, 3, [8] Km, T., Shn, J., Km K., and Heo, J. (008). Improvng accuracy of IDF curves usng longand short-duraton separaton and multobjectve genetc algorthm. World Envronmental and Water Resources Congress, pp 1-1. [9] Ben-Zv, A. (009). Ranfall ntensty duraton frequency relatonshps derved from large partal duraton seres. Journal of Hydrology, 367, [10] Bara,., Gaal, L., Kohnova, S., Szolgay J., and Hlavcova, K. (009). Estmaton of IDF curves of extreme ranfall by smple scalng n Slovaka. Contrbutons to Geophyscs and Geodesy, 39, [11] Okonkwo, G.I., and bajorgu, C.C. (010). Ranfall ntensty-duraton- frequency analyses for South Eastern, gera. Agrcultural Engneerng Internatonal: CIGR Journal, 1, -30. [1] Khaled, H., Ataur, R., Jance G., and George, K. (011). Desgn ranfall estmaton for short storm duratons usng L-oments and generalzed least squares regresson-applcaton to Australan Data. Internatonal Journal of Water Resources and Ard Envronments, 1, [13] Eman, A.H.E. (011). Generaton of ranfall ntensty duraton frequency curves for ungauged stes le basn. Water Scence and Engneerng Journal, 4, [14] Rashd,.., Faruque, S.B., and Alam, J.B. (01). odellng of short duraton ranfall ntensty duraton frequency (SDRIDF) equaton for Sylhet Cty n Bangladesh. ARP Journal of Scence and Technology,, [15] Dourte, D., Shukla, S., Sngh, P., and Haman, D. (013). Ranfall ntensty-duraton-frequency relatonshps for Andhra Pradesh, Inda: Changng ranfall patterns and mplcatons for runoff and groundwater recharge. Journal of Hydrologc Engneerng, 18, [16] Vvekanandan,. (01). Predcton of seasonal and annual ranfall usng order statstcs approach of Gumbel and Frechet dstrbutons. Brtsh Journal of Engneerng and Technology, 1, Internatonal Journal of Emergng Engneerng Research and Technology V5 I9 017

7 Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons [17] AERB. (008). Extreme values of meteorologcal parameters. Atomc Energy Regulatory Board (AERB) Safety Gude o. F/SG/S-3. [18] Chowdhury, R., Alam, J.B., Das, P., and Alam,.A. (007). Short duraton ranfall estmaton of Sylhet: ID and USWB method. Journal of Indan Water Works Assocaton, 9, [19] Acar, R., Celk, S., and Senocak, S. (008). Ranfall ntensty-duraton-frequency model usng an artfcal neural network approach. Journal of Scentfc & Industral Research, 67, [0] ID. (010). Report on avalablty of meteorologcal data. Ctaton:. Vvekanandan, "Ranfall Intensty-Duraton-Frequency Analyss Usng Order Statstcs Approach of Gumbel and Frechet Dstrbutons", Internatonal Journal of Emergng Engneerng Research and Technology, vol. 5, no. 9, pp , 017. Copyrght: 017. Vvekanandan. Ths s an open-access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal author and source are credted. Internatonal Journal of Emergng Engneerng Research and Technology V5 I