Regional flood frequency analysis based on L-moment approach (case study Halil-River basin)

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1 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 273 Regioal flood frequecy aalysis based o L-momet approach (case study Halil-River basi) Rahama M. B., Rostami R., 2 -Assistat Prof. Irrigatio Dept., Shahid Bahoar Uiversity, Kerma, Ira. Phoe: mbr@mail.uk.ac.ir 2- Master of sciece, Irrigatio Dept., Shahid Bahoar Uiversity, Kerma, Ira. Phoe: rami_ak23@yahoo.com. Abstract Flood estimatio with certai frequecy is oe of the fudametal factors for desig of hydraulic structures, flood plai, river coastal stablig, basi maagemet, etc. Accurate estimatio of flood frequecy discharge icreases safety of the structures. L-momet approach was used for flood frequecy aalysis i Halil-River basi. For idetifyig homogeeous regios, the Ward hierarchical cluster method was used. Site data were used for idepedet testig of the cluster of the statio for homogeeity. The Halil-River basi was divided ito two regios (regio A ad B). I these regios parameters of the regioal frequecy distributio were evaluated by L-momet ratios. The L-momet diagram, goodess of fit test, ad plottig positio methods were used for the selectio of appropriate distributios. I Halil-River basi, Geeralized Pareto distributio for regio A, Geeralized extreme values, Pearso type III, Logormal, Geeralized Logistic, ad Geeralized Pareto for regio B, were selected as appropriate distributios. The relative Root Mea Square Error (rrmse) betwee observed ad estimated data i all statios was calculated. The results show a good agreemet betwee observed ad estimated data. Keywords: Halil-River, homogeeity, L-Momet, regioal frequecy Itroductio A importat practical applicatio of hydrology is the estimatio of extreme evets, especially because the plaig ad desig of water resource projects ad flood-plai maagemet deped o the frequecy ad magitude of peak discharges. Iformatio o flood magitudes ad their frequecies is eeded for desig of hydraulic structures such as dams, spillways, road ad railway bridges, culverts, urba draiage systems, flood plai zoig, ecoomic evaluatio of flood protectio projects etc. (Kumar et al. 23). Regioal flood frequecy aalysis is usually applied whe o local data are available at a site of iterest or the data are isufficiet for a reliable estimatio of flood quatiles for the required retur period. Regioal flood frequecy aalysis has three major compoets, amely, delieatio of homogeeous regio, determiatio of appropriate probability desity fuctio (or frequecy curves) of the observed data, ad the developmet of a regioal flood frequecy model (i.e., a relatioship betwee flows of differet retur periods, basi characteristics, ad climatic data). The study icludes idetificatio of homogeeous regios based o cluster aalysis of site characteristics, idetificatio of suitable regioal frequecy distributio ad developmet of regioal flood frequecy models. L-momets Recetly, Hoskig (99) has defied L-momets, which are aalogous to covetioal momets, ad ca be expressed i terms of liear combiatios of order statistics. Basically, L-momets are liear fuctios of probability-weighted momets (PWMs) (Sakarasubramaia ad Sriivasa 999). Procedures based o PWM ad L-momet are equivalet, but L-momet is more coveiet because it is directly iterpretable as measure of the scale ad of the shape of probability distributios. L-momets are robust to outliers ad

2 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 27 virtually ubiased for small samples, makig them suitable for flood frequecy aalysis (Adamowski 2). Similar to covetioal momets, the purpose of L-momets ad probability-weighted momets is to summaries theoretical distributio ad observed samples. As metioed i Schulze ad Smithers, 22, paper Greewood summarizes the theory of PWM ad defied them as (Schulze ad Smithers, 22): r β r = E{ X[ FX ( x)] } () where β r is the rth order PWM ad F X (x) is the cumulative distributio fuctio (cdf) of X. Ubiased sample estimators (b i ) of the first four PWMs are give as: β = m = X j j β = [ ] X ( j) ( ) 2 ( j)( j 2) β 2 = [ ] X ( j) ( )( 2) j= j= j= 3 ( j)( j )( j 2) 3 = [ ] ( )( 2)( 3) j = β X (2) ( j) where x (j) represets the raked Aual Maximum Series (AMS) with x () beig the highest value ad x () the lowest value, respectively. The first four L-momet are give as follow: λ = β λ2 = 2β β λ 3 = 6β2 6β + β λ = 2β3 3β2 + 2β β (3) Ubiased sample estimators of the first four L-momets are obtaied by substitutig the PWM sample estimators from Eq. (2) ito Eq. (3). The first L-momet λ is equal to the mea value of X. Fially, the L-momet ratios are calculated as: λ3 λ λ2 L γ = τ 3 = L k = τ = L C V = τ 2 = () λ2 λ2 λ Sample estimates of L-momet ratios are obtaied by substitutig the L-momets i Eq. () with sample L-momets (Hoskig ad Wallis 997). Idex flood The T-year evet X T is defied as the evet exceeded o average oce every T years. Whe the aual maximum floods are distributed accordig to a specified frequecy distributio with (sfd), the T-year evet ca be calculated as (Cadma et. al. 23): X T =F - (-/T) (5) Regioal frequecy aalysis methods, such as the idex flood method, iclude iformatio from early statios exhibitig similar statistical behavior as at the site uder cosideratio i order to obtai more reliable estimates (Schulze ad Smithers, 22). Regioal methods ca also be used to obtai estimates at ugauged sites, which is importat i regio such as Halil- River basi, where the flow gaugig etwork desity is relatively low. Cosider a homogeeous regio with sites, each site i havig sample size i ad observed AMS x ij, j=,, i. The AMS from a homogeeous regio are idetically distributed except for a sitespecific scalig factor, viz., the idex flood. At each site the AMS is ormalized usig the idex flood as: q ij = Q ij / µ i (6) Where µ i is the Mea Aual Flood (MAF) at site i, which is ofte used as the idex flood. The sample L-momet ratios are estimated at each site ad the regioal record legth weighted average L-momet ratios are calculated as: ) ) R ( i) λr = iλr / i (7) i= i= (i) where ) λ r is the rth order sample L-momet ratio at site i, ad ) λ r R is the rth order regioal average sample L-momet ratio. The parameters of a regioal frequecy distributio ca be estimated usig the method of L-momet ratios, as show, for example, by Hoskig ad Wallis (997). Fially, the T-year evet at site i ca be estimated as:

3 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 275 ) ) ) Q T q where ) µ i is the MAF at site i, ad q ) T, i = µ i T (8) is the regioal growth curve. The regioal growth curve is the (-/T)-quatile of the regioal distributio of the ormalized AMS as defied through Eq. 8 (Hoskig ad Wallis 997). Idetificatio of homogeeous regios I Halil-River basi six hydrometric sites, which have sufficiet legth record of data ad are importat for frequecy aalysis, were selected. For idetificatio of homogeeous regios, Hoskig ad Wallis recommeded usig Ward s method, which is a hierarchical clusterig method based o miimizig the Euclidea distace i site characteristics space withi each cluster. The site characteristics selected i this study for each site icluded: latitude (LAT) ad logitude (LO) of the flow gaugig weir, Mea Aual Flood (MAF), statio area (AREA), altitude (ALT) ad desig storm itesity (ID). Table shows the site characters for six statios i Halil-River basi. Usig this method Halil-River basi divided to two regios (A ad B). After idetificatio of homogeeous regios, usig Hoskig s method discordacy measure (Di) of the sites was determiate i each regio. Table 2 shows the L- momet ratios ad discordacy measure for regio A ad B statios. Fig. shows locatio of gaugig sites ad homogeous regios i Halil-River basi. Heterogeeity test Hoskig ad Wallis (997) proposed a statistical test based o L-momet ratios for testig the heterogeeity of the proposed regios. The test compares the betwee-site variatio i sample L-CV with the expected variatio for a homogeeous regio. The method fits a four parameters kappa distributio to the regioal average L-momet ratios. The estimated kappa distributio is used to geerate 5 homogeeous regios with populatio parameters equal to the regioal average sample L-momet ratios. The properties of the simulated homogeeous regio are compared to the sample L-momet ratios as H = ( V µ ) / σ (9) V V where µ V is the mea of simulated V values, ad σ V is the stadard deviatio of simulated V values. For the sample ad simulated regios, respectively, V is calculated as: V ( i) R 2 i ( t t ) / i= i= = { } () i 2 where is the umber of sites, i is the record legth at site i, t (i) is the sample L-CV at site I, ad t R is the regioal average sample L-Cv. If H<, the regio ca be regarded as acceptable homogeeous, H<2 is possible homogeeous, ad H 2 is defiitely heterogeeous (Hoskig ad Wallis, 997). Table 3 shows the heterogeeity measure for idetified regios i Halil-River basi. Goodess-of-fit test The goodess-of-fit test described by Hoskig ad Wallis (997) is based o a compariso betwee sample L-kurtosis ad populatio L-kurtosis for differet distributios. The test statistic is termed Z DIST ad give as follows: DIST DIST R Z = ( τ t + B ) / σ () where DIST refer to a cadidate distributio. τ DIST is the populatio L-kurtosis of selected R distributio, t is the regioal average sample L-kurtosis, ad σ is the stadard deviatio of regioal average sample L-kurtosis. A four-parameter kappa distributio is fitted to the regioal average sample L-momet ratios. The kappa distributio was used to simulate 5

4 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 276 regios similar to the observed regios. From these simulated regios B ad σ are estimated. Declare the fit to be adequate if Z DIST is sufficietly close to zero, a reasoable DIST critererio for selectio of suitable beig Z. 6 (Hoskig ad Wallis 997). The test described above was applied to the four homogeeous regios. For each regio the data were tested agaist the Geeral Logistic (GLO), Geeral Pareto (GPA), Geeral Extreme Value (GEV), Geeral ormal (GO) ad Pearso Type 3 (PE3) distributio. Table shows the results. Regioal flood frequecy distributio Several methods are available for selectig appropriate regioal distributios. I this study the regioal frequecy distributios were selected based o the results of L-momet ratios as described by Hoskig ad Wallis (997). Additioal probability plots (plottig positio) were used to verify that the selected regioal distributios provided a satisfactory descriptio of the observed AMS. L-momet ratio diagrams A L-momet ratio diagram of L-kurtosis versus L-skewess compares sample estimates of the dimesioless ratios with their populatio couterparts for rages of statistical distributios iclude GLO, GEV, GO, PE3 ad GPA. L-momet diagrams are useful for discerig groupig of sites with similar flood frequecy behavior, ad idetifyig the statistical distributio likely to adequately describe this behavior. Fig.2 shows the L-momet ratio diagram for homogeeous regios i Halil-River basi (A ad B). As the sample L- momets, are ubiased, the sample poits should be distributed above ad below the theoretical lie of a suitable distributio (Hoskig ad Wallis 997). From the above L- momet diagrams, it appears that the GPA distributio for regio A ad the GPA, GLO, GEV, PE3 ad GO for regio B are appropriate. Plottig Positio As poited out by Hoskig et al. (985), compariso of differet regioal frequecy distributios agaist observed data caot be used to discrimiate betwee differet distributios, as the observed data represets oly oe of a ifiite umber of realizatios of the true uderlyig populatio (Schulze ad Smithers, 22). However, the probability plots may reveal tedecies such as systematic regioal bias i the estimatio of the extreme evets. To assess how well the proposed regioal frequecy distributio fit to the observed AMS, the calculated X T -T relatioships for Koarueyeh statio i regio B are show i Fig.3. The empirical exceedace probability for the ordered observatios x (i) were calculated usig the media probability plottig positio as described by Hoskig, ad show bellow: P[ X > x( i ) ] = i.35/ (2) From above three methods, goodess-of-fit test, L-momet ratio diagram ad plottig positio, the GPA distributio for regio A ad the GO, GEV, GLO, GPA ad PE3 distributios for regio B were selected as regioal frequecy distributios. Quitiles estimatio After the regioal distributios selected, usig these distributios the quitiles with differet oexceedace probability estimated for regios A ad B i Halil-River basi. Table 5 shows the estimated value usig GPA distributio i regios A ad B. The accuracy of estimated values (regioal ad at-site estimatios) was determiate usig relative Root Mea Squire Error (rrmse). Fig. shows the rrmse i Koarueyeh statio. From these charts the rrmse values i high retur period are low. This idicates that both at-site ad regioal estimatio procedure i high retur period give accurate results.

5 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 277 Regioal Model Differet methods are available to obtai regioal models with hydrologic ad basis parameters which ca estimate AMS values i ugaged regios. I this study we obtaied the regioal models for regios A ad B i Halil-River basi usig multiple regressio ad stepwise method. I these models Q P, A, L ad I d are average AMS, basis area, latitude ad desig storm itesity respectively. Coclusio I this study usig site characteristic ad Ward s method, hierarchical clusterig method based o miimizig the Euclidea distace i site characteristics space withi each cluster, the Halil-River basi divided ito two acceptably homogeeous regios. The heterogeeity measures based o H were -.67 ad.88 for regios A ad B respectively. The idetificatio of suitable regioal distributio for each of two regios was based o the L- momet diagram, a goodess-of-fit test ad evaluated usig probability plots. The GPA distributio for regio A ad PE3, GO, GLO, GPA ad GEV distributios for regio B were suitable ad selected. The rrmse values betwee computed ad observed data were obtaied. These values i high retur period were low ad idicate that both at-site ad regioal estimatio procedure i high retur period give accurate results. Regioal models for homogeeous regios was obtaied usig the multiple regressio ad stepwise method ad with catchmets ad hydrologic characteristics. Refereces Adamowski k., (2), Regioal aalysis of aual maximum ad partial duratio flood data by oparametric ad L-Momet method, Joural of Hydrology 229: Cadma D., Zaidma M. D., Keller V., ad Yog A. R., (23), Flow-duratio-frequecy behavior of British rivers based o aual miima data, Joural of Hydrology 277: Hoskig J. R. M., (99), L-momet aalysis ad estimatio of distributios usig liear combiatios of order statistics, Joural of the Royal Statistical Society, series B 52: 5-2. Hoskig J. R. M., ad Wallis J. R. (997), Regioal frequecy aalysis: A approach based o L- Momet, Cambridge Uiversity Press, Lodo, UK. Hoskig J. R. M., Wallis J. R., ad Wood E. F., (985), Estimatio of the geeralized extreme value distributio by the method of probability weighted momet, Techometrics 27: Kumar R., Chatterjee C., Paigrihy., Patwary B. C., ad Sigh R. D., (23), Developmet of regioal flood formulae usig L-momets for gauged ad ugauged catchmets of orth Brahmaputra river system, IE (I) Joural 8: Lee S.H, ad Meag S. J., (23), Frequecy Aalysis of extreme raifall usig L-momet, Irrigatio ad Draiage 52: Sakarasubramaia A., ad Sriivasa K., (999), Ivestigatio ad compariso of samplig properties of L-Momets ad Covetioal Momets, Joural of Hydrology 28: 3-3. Schulze R. E., ad Smithers J. C., (22), Regioal flood frequecy aalysis i KwaZulu-atal provice, South Africa, usig the Idex-Flood method, Joural of Hydrology 255:9-2. Tables Table. Site characteristic for Halil-River statios Statio ame LAT LO ALT (m) MAF (m 3 /s) AREA (km 2 ) ID(mm/hr) Soltai Meyda

6 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 278 Pol baft Heja Koarueyeh Cheshm Aroos Table 2. L-momet ratios ad discordacy measure for regio A ad B statios Table 3. Regio A Statio ame Record legth (year) L-Cv L-Skew L-Kurt Di Soltai Meyda Pol baft Heja Regio B Koarueyeh Cheshm Aroos Heterogeeity measure for idetified regios i Halil-River basi Sl. o. Heterogeeity measures Values Regio A Regio B. Heterogeeity measure H (a) Observed stadard deviatio of group L-CV (b) Simulated mea of stadard deviatio of group L-CV (c) Simulated stadard deviatio of stadard deviatio of group L-CV (d) Stadardized test value H Heterogeeity measure H (2) (a) Observed average of L-CV/L-Skewess distace (b) Simulated mea of average L-CV/L-Skewess distace (c) Simulated stadard deviatio of average L-CV/L-Skewess distace (d) Stadardized test value H (2) Heterogeeity measure H (3) (a) Observed average of L-Skewess/L-Kurtosis distace (b) Simulated mea of average L-Skewess/L-Kurtosis distace (c) Simulated stadard deviatio of average L-Skewess/L-Kurtosis distace (d) Stadardized test value H (3) Table. Test statistic Z DIST of regioal distributios Distributio Regio A Regio B GLO GEV GO PE GPA.23.6 Table 5. Regioal parameters for the various distributios for regio B Parameters of the distributio GEV ξ=.55 α=.26 k=-.2 GLO ξ=.63 α=.355 k=-.7 GO ξ=.596 α=.65 k=-.9 PE3 µ= α=.78 k=2.87 WAK ξ=-.93 α= β= γ=.65 δ=.279 Table 6. Regioal models for Halil-River basis. Regio Regioal Model R 2 A Q P =.236A +.53EL ID Table 7. Estimated B Q( p) =.9A discharge (m 3 /s)

Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 279 from GPA distributio i regio A ad B oexceedece probability Statio ame.5.8.9.95.96.98.99.995.998.

7 Proceedigs of The Fourth Iteratioal Ira & Russia Coferece 279 from GPA distributio i regio A ad B oexceedece probability Statio ame Soltai Meyda Pol baft Heja Koarueyeh Cheshm Aroos Figures Soltai Polbaft Regio Regio.2 GPA PE3 GEV Regio B GLO L3 Regio A Cheshm Aroos L-kurtosis Jiroft dam Koaroeyeh Heja Meyda L-skewess Fig.. Homogeeous regios ad Halil-River statios Fig. 2. L-momet diagrams for regios A ad B Discharge (CMS) Observed data At-site GPA distributio Regioal GPA distributio Retur period (year) Discharge (CMS) Observed data At-site GEV distributio Regioal GEV distributio Retur period (year) Fig. 3. Probability plot for Koarueyeh statio i regio B rrmse At-site GPA distributio Regioal GPA distributio Retur period (year) At-site GEV distributio.2 Regioal GEV distributio Retur period (year) Fig.. rrmse betwee computed ad observed data for Koarueyeh statio rrmse

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.