Statistical Analysis of Rainfall Data for Estimation of Peak Flood Discharge Using Rational Formula

Transcription

1 Internatonal Journal of Emergng Engneerng Research and Technology Volume 5, Issue 5, 2017, PP 1-6 ISS (Prnt) & ISS (Onlne) Statstcal Analyss of Ranfall Data for Estmaton of Peak Flood Dscharge Usng Ratonal Formula. Vvekanandan 1, C. Ramesh 2 1 Scentst-B, Central Water and Power Research Staton, Pune, Maharashtra, Inda 2 Scentst-C, Central Water and Power Research Staton, Pune, Maharashtra, Inda *Correspondng Author:. Vvekanandan, Scentst-B, Central Water and Power Research Staton, Pune , Maharashtra, Inda Receved Date: Accepted Date: Publshed Date: ABSTRACT Estmaton of Peak Flood Dscharge (PFD) at a desred locaton on a rver s mportant for plannng, desgn and management of hydraulc structures. For ungauged catchments, ranfall depth becomes an mportant nput n dervaton of PFD. So, ranfall depth can be estmated through statstcal analyss by fttng probablty dstrbuton to the ranfall data. In ths paper, the seres of annual maxmum 1-day ranfall derved from daly ranfall data observed at Kasaul ran-gauge staton s used for estmaton of 1-day extreme ranfall adoptng Gumbel dstrbuton. Maxmum lkelhood method s used for determnaton of parameters of the dstrbuton. Kolmogorov-Smrnov test s appled for checkng the adequacy of fttng of the dstrbuton to the observed ranfall data. The 1-day extreme ranfall obtaned from Gumbel dstrbuton s used to compute the ranfall ntensty and consdered as an nput to estmate the PFD by ratonal formula adoptng CWC gudelnes. The study suggests the estmated PFD could be used for desgn of flood protecton works for dfferent trbutares of Srsa rver. Keywords: Kolmogorov-Smrnov test, Gumbel, Ranfall, Peak flood dscharge, Maxmum lkelhood method ITRODUCTIO Estmaton of desgn floods n ungauged catchments s frequently requred n hydrologcal practce and s of great economc sgnfcance. These nclude dfferent types of flood such as standard project flood, probable maxmum flood and desgn bass flood. In case of large rver basns, the hydrologcal and stream flow seres of a sgnfcant duraton are generally avalable. However, for ungauged catchments, more data s not avalable other than ranfall. The ranfall data s also of shorter duraton and may becomes an mportant nput n estmaton of Peak Flood Dscharge (PFD) [1]. For ths purpose, statstcal analyss nvolves fttng of probablty dstrbuton to the seres of Annual Maxmum 1-day Ranfall (AMDR) s carred out. Out of a number of probablty dstrbutons, the famly of Extreme Value Dstrbutons (EVDs) ncludes Generalzed Extreme Value, Extreme Value Type-1 (Gumbel), Extreme Value Type-2 (Frechet) and Extreme Value Type-3 (Webull) s wdely adopted for Extreme Value Analyss (EVA) of ranfall [2]. EVDs arse as lmtng dstrbutons for the sample of ndependent, dentcally dstrbuted random varables, as the sample sze ncreases. In the group of EVDs, Gumbel dstrbuton has no shape parameter as when compared to other dstrbutons and ths means that there s no change n the shape of Probablty Dstrbuton Functon (PDF). Moreover, the Gumbel dstrbuton has the advantage of havng only postve values, snce the data seres of ranfall are always postve (greater than zero); and therefore Gumbel dstrbuton s mportant n statstcs. Deka and Borah [3] have derved the best ftted dstrbuton among the fve extreme value dstrbutons used to descrbe the annual seres of maxmum ranfall data of nne dstantly located statons n north east Inda. Sharma and Sngh [4] analyzed the seres of annual, seasonal, monthly and weekly maxmum ranfall data of Pantnagar regon and dentfed the best ftted probablty dstrbuton among the sxteen dstrbutons used n the study. Mujere [5] appled Gumbel dstrbuton for modellng flood data for yanyadz Rver, Zmbabwe. Baratt et al. [6] carred out flood frequency analyss at seasonal and annual tme scales for Blue le Internatonal Journal of Emergng Engneerng Research and Technology V5 I

2 Rver adoptng Gumbel dstrbuton. Esteves [7] appled Gumbel dstrbuton to estmate the extreme ranfall depths at dfferent ran-gauge statons n southeast Unted Kngdom. Olumde et al. [8] appled normal and Log-Gumbel dstrbutons for predcton of ranfall and runoff at Tagwa dam ste n Mnna, gera. They have also expressed that the normal dstrbuton as better suted for ranfall predcton whle Log- Gumbel for runoff. Vvekanandan [9] appled Gumbel dstrbuton for modellng the seasonal and annual ranfall for Krshna and Godavar rver basns. Rasel and Hossan [10] appled Gumbel dstrbuton for development of ntensty duraton frequency curves for seven dvsons n Bangladesh. In vew of the above, Gumbel dstrbuton s used n the present study. Parameters of the Gumbel dstrbuton are determned by Maxmum Lkelhood Method (MLM) and used for estmaton of 1-day extreme ranfall. For quanttatve assessment on ranfall data wthn the observed range, Kolmogorov-Smrnov (KS) test s appled. The 1-day extreme ranfall obtaned from Gumbel dstrbuton s used to compute the ranfall ntensty and consdered as an nput to estmate the PFD for dfferent trbutares of rver Srsa. The methodology adopted n EVA of ranfall usng Gumbel dstrbuton, assessng the adequacy of fttng of Gumbel dstrbuton to the AMDR seres usng GoF test and estmaton of PFD usng ratonal formula are brefly descrbed n the ensung sectons. METHODOLOGY The methodology adopted n estmaton of PFD for dfferent trbutares of rver Srsa ncludes () assess the adequacy of fttng of Gumbel dstrbuton (usng MLM) to the seres of AMDR usng GoF test; () estmate the 1-day extreme ranfall for dfferent return perods; () compute the ranfall ntensty from the 1-day extreme ranfall usng CWC gudelnes; (v) estmate the PFD usng ratonal formula; and (v) analyze the results obtaned thereof. PDF and CDF of Gumbel Dstrbuton The PDF and Cumulatve Dstrbuton Functon (CDF) of the Gumbel dstrbuton are gven as: PDF : f (r) CDF : F(r) e e r / r / e e r / e, 0, where (r r 1,r2,r3,..., r ) (1) where, α and β are the locaton and scale parameters of the dstrbuton [11]. The parameters are computed by MLM through Equatons (2) and (3), and used to estmate the ranfall (R T ) for dfferent return perods from RT α YTβ. Here, Y T ln ln1 1/ T s called as a reduced varate for a gven return perod T (year). ln R SE(R 1 exp r r exp r exp r T) YT 1.1Y T (2) (3) (4) where r s the observed AMDR of th sample, R s the average value of AMDR and s the sample sze. The lower and upper confdence lmts (LCL and UCL) of the estmated extreme ranfall are obtaned from the lnear expressons vz.lcl=er-1.96(se) and UCL=ER+1.96(SE). Here, ER s the estmated Extreme Ranfall and SE the Standard Error. Goodness-of-Ft Test GoF test s essental for checkng the adequacy of probablty dstrbuton to the observed seres of AMDR. Out of a number GoF tests avalable, the wdely accepted GoF test s KS, whch s used n the study. The theoretcal descrpton of KS test statstc [12] s as follows: 1 KS Max F e r F D r where, r /( 1) D (5) F e s the emprcal CDF of r and F s the computed CDF of r. If the computed value of KS test statstc gven by Gumbel dstrbuton s less than that of the theoretcal value at the desred sgnfcance level then the dstrbuton s consdered to be acceptable for modellng the seres of AMDR. Applcaton In ths paper, a study on estmaton of PFD for dfferent return perods for dfferent trbutares of Srsa was carred out. The seres of Annual AMDR was extracted from the daly ranfall data observed at Kasaul ran-gauge staton durng the perod 1991 to 2015 and used. The descrptve statstcs such as average, standard devaton, coeffcent of varaton, coeffcent of skewness and coeffcent of kurtoss of the observed AMDR was determned as 93.8 mm, 52.1 mm, 55.5%, and respectvely. From the scrutny of ranfall data, t was observed that the ranfall data for the years 2004 and 2005 are not avalable. Therefore, the data for the mssng year was replaced wth the maxmum value of AMDR seres,.e, 262 mm. 2 Internatonal Journal of Emergng Engneerng Research and Technology V5 I5 2017

3 The AMDR seres wth mputed value was used for estmaton of 1-day extreme ranfall. Fgure 1 presents the tme seres plot of the observed AMDR. Fgure1. Tme seres plot of observed AMDR The estmated 1-day extreme ranfall obtaned from Gumbel dstrbuton (usng MLM) was used as an nput to estmate the PFD. The study nvolves flood protecton works for Srsa rver for whch the survey data of the rver was collected and used. Srsa s a trbutary to rver Satluj. There are 12 trbutares to Srsa that contrbute to flood flows. For estmatng the floods, catchment characterstcs (catchment area, length of stream, slope, etc.) area requred, whch was extracted from the Survey of Inda (SoI) toposheets of the regon. The catchment areas of the trbutares of Srsa rver were gven n Table 1. Table1. Catchment area of trbutares of Srsa rver Trbutary ID ame of the trbutary Catchment Area (km 2 ) 1 Srsa rver at RD (a) Srsa rver at RD (b) Srsa rer at RD (c) Srsa rer at RD (d) Srsa rver at RD Chkn Khad at RD Khera Khad at RD Bagbana Khad at RD Manpura Khad at RD Ratta man rver at RD Sandhol Khad at RD Pal Mahadev at RD RESULTS AD DISCUSSIOS By applyng the procedures of Gumbel dstrbuton, parameters were determned by MLM and used for estmaton of 1-day extreme ranfall. Estmaton of 1-day Extreme Ranfall Usng Gumbel Dstrbuton Table 2 gves the 1-day extreme ranfall estmates wth confdence lmts for dfferent return perods vary from 2-year to 100-year adoptng Gumbel dstrbuton. The observed and estmated AMDR are presented n Fgure 2 along wth confdence lmts. From Fgure 2, t can be seen that about 75 percent of the observed AMDR are wthn the confdence lmts of the estmated 1-day extreme ranfall usng Gumbel dstrbuton. Table2. Estmated 1-day extreme ranfall (mm) wth confdence lmts usng Gumbel dstrbuton Return Estmated Standard Confdence lmts perod Ranfall Error on at 95% level (yr) (ER) ER Lower Upper Analyss Based on GoF Test The adequacy of fttng of Gumbel dstrbuton to the seres of AMDR was performed by adoptng KS test, as descrbed above. From the result, t was observed that the computed value of KS test statstc s 0.199, whch s not greater than the theoretcal value of at 5% sgnfcance level [13], and at ths level, the KS test confrmed the sutablty of Gumbel dstrbuton for modellng the seres of AMDR. Fgure2. Plots of observed and estmated 1-day extreme ranfall wth confdence lmts Estmaton of Peak Flood Dscharge The requrement of the study s to estmate the PFD for Srsa and ts trbutares. ether Srsa nor ts trbutares are gauged and thus needs to be estmated by ndrect estmatons from ranfall and catchment characterstcs. From an observaton of catchment sze from the Toposheets and Google Earth of the regon of these trbutares t was found that these are small Internatonal Journal of Emergng Engneerng Research and Technology V5 I

4 catchments that respond quckly to ranfall. From the catchment characterstcs, the tme lag (t L ) and tme of concentraton (t c ) were estmated for each of these catchments. As the present study s concerned wth peak flood estmaton the desgn storm duratons for each of the catchment were estmated from t L. From the analyss of catchment characterstcs, the computed desgn storm duratons for dfferent catchments vary from 1-hour (hr) to 6-hr. The desgn storm duraton s 1-hr for Bagbana Khad, Manpura Khad, Ratta man rver, Sandhol Khad and Pal Mahadev Khad; 2-hr for Khera Khad and Srsa RD 28300; 3-hr for Chkn Khad; 4-hr for Srsa RD 18000; 5-hr for Srsa RD 10000; and 6-hr for Srsa RD 5000 and Srsa RD 0. It s to state that, the shorter duraton ranfall (say, 1-hr) were not avalable for the study area. Based on the desgn storm duratons, the approprate storms (2-hr, 3-hr, 4-hr, 5-hr and 6-hr) were estmated adoptng the procedures as detaled n CWC report [14] and used for computng the desgn storm of the respectve catchments from the estmated 1-day extreme ranfall (100-year return perod) by usng sutable converson factors (Fgure 3), as gven n Central Water Commsson Report ttled Flood estmaton report for Western Hmalayas Zone - 7. The computed values of dstrbuted ranfall are presented n Table 3. Ths value was used as nput (ranfall ntensty) for estmaton of PFD for the catchments of Srsa rver and ts trbutares. The catchment areas are n the range of 7.40 km 2 to km 2, the largest of the catchment beng Srsa rver at RD 0 and the smallest catchment beng Sandhol Khad at RD 500. As mentoned these trbutares are ungauged and hence the PFD for these catchments are computed by usng ratonal formula, whch s gven below: q = * C I A (6) where, q s PFD (m 3 /s), C s runoff coeffcent, I s ranfall ntensty (mm/hour) and A s catchment area (km 2 ). Fgure3. Converson factor for computaton of dstrbuted ranfall for shorter duraton Table3. Dstrbuted ranfall for short duratons Return perod Dstrbuted ranfall (mm) for short duratons based on desgn storm (year) 1-hr 2-hr 3-hr 4-hr 5-hr 6-hr By consderng topography and general land use of the catchments, the value of C s consdered as 0.55 for estmaton of flood dscharge. The computed PFD for dfferent return perods for trbutares of Srsa rver are presented n Table 4. It s to state that the desgn dscharge [15] may be approprately chosen wth a partcular return perod. EVA of ranfall and hydrologc consderaton adoptng the desgn storms estmated for the catchments wth approprate return perod may be used for flood protecton works. Table4. Peak flood dscharge for dfferent return perods for trbutares of Srsa rver ame of the Trbutary Peak flood dscharge (10 3 m 3 /s) for dfferent return perods Srsa rver at RD Srsa rver at RD Srsa rer at RD Srsa rver at RD Srsa rver at RD Chkn Khad at RD Khera Khad at RD Bagbana Khad at RD Manpura Khad at RD Ratta Man Rver at RD Sandhol Khad at RD Pal Mahadev at Rd Internatonal Journal of Emergng Engneerng Research and Technology V5 I5 2017

5 COCLUSIOS The paper descrbed brefly the study carred out for statstcal analyss of ranfall data adoptng Gumbel dstrbuton and estmaton of PFD for trbutares of rver Srsa. From the results of the data analyss, the followng conclusons were drawn from the study: ) The KS test result confrmed the sutablty of Gumbel dstrbuton (usng MLM) for modellng the data seres of AMDR. ) From Fgure 2, t was observed that about 75 percent of the observed AMDR data are wthn the confdence lmts of the estmated 1-day extreme ranfall. ) The 1-day extreme ranfall was used to compute the values of the dstrbuted ranfall adoptng CWC gudelnes descrbed n the Flood estmaton report for Western Hmalayas-Zone 7. v) By usng the ratonal formula, the PFD for dfferent return perods for dfferent trbutares of Srsa rver was estmated. v) The study suggested that the PFD, as gven n Table 4, could be consdered for desgn of flood protecton works. However, by consderng the data length made avalable for the study, t was cautoned to use the PFD for return perods beyond 50-year because of uncertanty n the estmated values. ACKOWLEDGEMETS The authors are grateful to the Drector, Central Water and Power Research Staton (CWPRS), Pune, for provdng research facltes to carry out the study. The authors are thankful to Dr. R.G. Patl, Scentst-D, CWPRS, Pune, for supply of ranfall data used n the study. REFERECES [1] Sngh, R.D., Mshra, S.K., and Chowdhary, H. 2001, Regonal flow duraton models for 1200 ungauged Hmalayan watersheds for plannng mcro-hydro projects, ASCE Journal of Hydrologc Engneerng, 6(4): [2] Casas, M.C., Rodrguez, R., Prohom, M., Gazquez, A., and Redano, A., 2011, Estmaton of the probable maxmum precptaton n Barcelona (Span), Journal of Clmatology, 31(9): [3] Deka. S., and Borah. M, 2009, Dstrbuton of annual maxmum ranfall seres of orth East Inda, European Water Publcatons, 27/28: [4] Sharma, M.A., and Sngh, J.B., 2010, Use of probablty dstrbuton n ranfall analyss, ew York Scence Journal, [5] Mujere,., 2011, Flood frequency analyss usng the Gumbel dstrbuton, Journal of Computer Scence and Engneerng, 3(7): [6] Baratt, E., Montanar, A., Castellarn, A., Salnas, J.L., Vglone, A., and Bezz, A., 2012, Estmatng the flood frequency dstrbuton at seasonal and annual tme scales, Hydrologcal Earth System Scence, 16(12): [7] Esteves, L.S., 2013, Consequences to flood management of usng dfferent probablty dstrbutons to estmate extreme ranfall, Journal of Envronmental Management, 115(1): [8] Olumde, B.A., Sadu, M., and Oluwasesan, A., 2013, Evaluaton of best ft probablty dstrbuton models for the predcton of ranfall and runoff volume (Case study: Tagwa Dam, Mnna-gera), Journal of Engneerng and Technology, 3(2): [9] Vvekanandan,., 2015, Assessng the adequacy of parameter estmaton methods of the Gumbel dstrbuton for modellng the seasonal and annual ranfall, Journal of Management Scence and Engneerng Management, 10(4): [10] Rasel, M.M., and Hossan, S.M., 2015, Development of ranfall ntensty duraton frequency equatons and curves for seven dvsons n Bangladesh, Internatonal Journal of Scentfc and Engneerng Research, 6(5): [11] Gumbel, E.J., 1960, Statstc of Extremes, Columba Unv. Press, ew York. [12] Charles Anns, P.E., 2009, Goodness-of-Ft tests for statstcal dstrbutons, [ engneerng.com/ goodness. html]. [13] Zhang, J., 2002, Powerful goodness-of-ft tests based on the lkelhood rato, Journal of Royal Statstcal Socety, 64(2): [14] CWC, 1994, Flood estmaton report for Western Hmalayas-Zone 7, Central Water Commsson (CWC) Desgn Offce Report o.: WH/22/1994, ew Delh. [15] IS: 12094, 2000, Indan Standard on Gudelnes Internatonal Journal of Emergng Engneerng Research and Technology V5 I

6 for plannng and desgn of rver embankments (levees), ew Delh. AUTHOR S BIOGRAPHY. Vvekanandan post graduated n mathematcs from Madura Kamaraj Unversty n He obtaned Master of Engneerng n hydrology from Unversty of Roorkee n He also obtaned Master of Phlosophy n mathematcs from Bharathar Unversty n 2006 and MBA (Human Resources) from Manonmanam Sundaranar Unversty n He s assocated n conductng R&D studes related to the felds vz., analyss of hydrologcal and hydrometeorologcal data usng probablstc methods, ranfall-runoff modellng usng soft computng, water resources plannng and management, water qualty modellng, etc. Dr. C. Ramesh post graduated n physcs from Sr Venkateshwara Unversty n He receved Ph. D. n Cvl Engneerng from Indan Insttute of Technology, Bombay n From March 2013 to tll date, he s workng as Scentst-C n CWPRS. He s nvolved n montorng of progress of varous R&D projects and assocated wth preparaton of techncal reports and documents for the studes related to flood forecastng, ranfall-runoff modellng, water resources management, etc. Ctaton:. Vvekanandan and C. Ramesh, "Statstcal Analyss of Ranfall Data for Estmaton of Peak Flood Dscharge Usng Ratonal Formula", Internatonal Journal of Emergng Engneerng Research and Technology, vol. 5, no. 5, pp. 1-6, Copyrght: Vvekanandan, et al. 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 I7 July