Developing regional flood frequency models for Victoria, Australia by using Index Flood (IF) approach

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1 18 th World IMACS / MODSIM Congress, Carns, Australa July Developng regonal flood frequency models for Vctora, Australa by usng Index Flood (IF) Masan, M. N 1. and Hewa, G. A 1 1 Unversty of South Australa, Mawson Lakes, South Australa Emal:masmy008@students.unsa.edu.au Abstract: The Index Flood (IF) method s a wdely used procedure n regonal flood frequency analyses (RFFA) to make flood predctons for ungauged catchments. The IF method assumes that a group s hydrologcally homogeneous and ths homogenety s exploted to produce quantles whch have been shown to be more relable than at-ste estmates (Cunnane 1988). The IF method has been wdely appled n a number of studes and found to be a relable f the groups are hydrologcally homogenous. Hence, the RFFA by the IF method s often accomplshed n three steps: delneaton of homogeneous groups, determnaton of regonal growth curves, and the development of regonal predcton models for the IF. The selecton of an approprate regonal frequency dstrbuton s of prme mportance n RFFA and has been nvestgated for many decades, but hydrologsts stll debate the relatve merts of dfferent dstrbutons. For nstance, Engneers Australa (1987) recommends Log Pearson Type III (LPIII) ftted by the Method of Moments (MOM), LPIII/MOM for Australan catchments. However, many studes (Vogel et al. 1993; Wang 1997; Rahman et al. 1999) have found that the Generalzed Extreme Value (GEV) dstrbuton ftted by L moments (GEV/L) method yelds more relable flood quantle estmates than LPIII/MOM. Although drect statstcal analyses of at-ste and regonal floods have been conducted, no regonal growth curves have been derved for general use n Vctora, Australa. The am of ths study was to dentfy the most approprate regonal flood frequency dstrbuton for Vctora and to conduct RFFA by usng the IF method. Hence the study reported n ths paper conssts of: (1) assessng the sutablty of the GEV ftted by L2 moments (GEV/L2) for RFFA for Vctora, Australa; (2) delneaton of homogenous groups; (3) development of regonal growth curves for the defned groups; (4) development of regonal predcton models for the IF and; (5) makng predctons for ungauged (test) catchments. 72 catchments were selected and assgned nto groups usng cluster analyss; homogenety of each group was also assessed. Regonal growth curves for the defned groups were developed and RFFA was carred out by usng the Index Flood (IF) method. A total of 72 unregulated catchments were selected and the extracted Annual Maxmum seres (AM) of each staton was modelled by LPIII/MOM, LN/MOM, GEV/L and GEV/L2 methods. The results ndcated GEV/L2 as the most approprate method for at-ste flood frequency analyses n Vctora as the quantles made by the GEV/L2 exhbted smaller bas and mean square errors than those of the other models. Sutablty of GEV/L and GEV/L2 for RFFA was assessed by constructng correspondng moment rato dagrams. These show that L2 moment ratos of the 72 catchments le symmetrcally around the theoretcal GEV curve whle the majorty of the L moment ratos are scattered below the theoretcal GEV curve ndcatng that GEV/L2 s therefore more approprate than GEV/L for RFFA n Vctora. Keywords: Fgure 1. LH moment rato dagrams: (a) L moment rato dagram and (b) L2 moment rato dagram Regonal flood frequency analyss, Vctora, GEV, L2 moments, ndex flood 3407

2 Masan and Hewa, Developng regonal flood frequency models for Vctora, Australa by usng ndex-flood 1. INTRODUCTION Szng of mnor hydraulc structures, such as culverts and brdges, s based on desgn flood quantles (Q T ) of medum to hgh return perods (T). If the length of the avalable data seres s shorter than the T of nterest, or when the ste of nterest s ungauged (no flow data avalable) obtanng a satsfactory estmate of Q T s dffcult. Regonal flood Frequency analyss (RFFA) s one of the es that can be used n such stuatons. The Index Flood (IF) method s a wdely used procedure n RFFA: t requres that any group of catchments selected for analyss s hydrologcally homogeneous. Ths homogenety s exploted to produce quantles whch have been shown to be more relable than at-ste estmates (Cunnane 1988). The method has been wdely appled n a number of studes and found to be a relable (Potter & Lettenmaer 1990; Kjeldsen et al. 2002; Portela & Das 2005; Saf 2008). The RFFA s often accomplshed n three steps: delneaton of homogeneous groups, determnaton of approprate regonal frequency dstrbuton, and development of regonal predcton models. The selecton of an approprate regonal frequency dstrbuton s of prme mportance n flood frequency analyss, and has been nvestgated for many decades but hydrologsts stll debate the relatve merts of dfferent dstrbutons. In Australa, for example, the LPIII/MOM has been recommended by the Australan Ranfall and Runoff (ARR) (Engneers Australa, 1987). However, many studes (Vogel et al. 1993; Wang 1997; Rahman et al. 1999) have found that the GEV method yelds more relable flood quantle estmates than LPIII. Ths paper presents the work undertaken to: () nvestgate the sutablty of GEV ftted by two orders of LH moments; η = 0 (L moments) and η = 2 (L2 moments) for both at-ste and regonal flood frequency analyses; () delneate homogenous groups; () derve regonal growth curves and; (v) develop of regonal predcton model for the selected IF quantle (Q 2.33 ). Fnally, the relablty of the ungauged catchment predcton s assessed. 2. STUDY AREA AND DATA 72 Vctoran catchments were selected for ths study by consderng the followng crtera: (1) catchment has over 10 years flow data, (2) catchment was not regulated and, (3) catchment was not affected by more than 10% urbanzaton. The geographcal dstrbutons of the selected catchments n Vctora are shown n Fgure 2. Fgure 2. Map of Vctora and the locatons of the study catchments Based on the publshed lterature and the data avalablty, a total of 8 catchment characterstcs were dentfed as sutable for the regonal model development. The selected attrbutes are tabulated n Table

3 Masan and Hewa, Developng regonal flood frequency models for Vctora, Australa by usng ndex-flood Table 1: The selected catchment characterstcs Varable Defnton unts Maxmum Mnmum Mean AREA Sze of the catchment km ELEV Elevaton at the catchment centrod mahd FOREST Fracton of area covered by forest LENGTH Man stream length km SLOPE Slope of the central 75% of the manstream length SFREQ Total number of stream junctons dvded by km catchment area SHAPE Catchment permeter dvded by area m RAIN Mean annual ranfall mm METHODOLOGY RFFA by the IF method was accomplshed n the followng steps: Step 1 - At-ste flood frequency analyss and selecton of the most approprate flood frequency model- Quantles of LPIII/MOM and LN/MOM at sx selected return perods (T = 2, 5, 10, 20, 50 and 100 years) were computed usng the HYDSTRA program, whle those of the GEV/L and GEV/L2 were computed accordng to Wang (1997). The relatve performance of each method was assessed usng an emprcal suggested by Rahman et al. (1999). In ths method, the devaton (D) between the estmated quantles (E G ) and the correspondng value from the best ft curve (E D ) was calculated usng Equaton 1. ( E - E ) D G G D = (1) E A small value of D corresponds to a small standard error of estmates. These devatons were then used to evaluate the performance of each dstrbuton n descrbng the observed data. The relablty of the GEV/L2 estmates over GEV/L estmates was assessed through Monte Carlo smulaton. Bas (Equaton 2) and MSE (Equaton 3) of the GEV flood quantles of the two methods were computed usng 1000 smulated samples of the same sze as thosee observed. Relatve performance of the GEV/L2 compared to GEV/L was assessed usng the Effcency (φ ) measure gven n Equaton 4. Fnally, the sutablty of the GEV/L2 and GEV/L methods were assessed by constructng and L2 moment rato dagram and L moment rato dagram (Fgure 1). () θˆ = Ε[ θ ˆ - θ] () θ = Ε ( θ ˆ 2 θ) = 2 ˆ - Bas() θˆ + VAR() θˆ Bas (2) MSE ( MSE) L ( MSE) L2 φ = (4) Step 2 - Delneaton of homogeneous groups and testng of homogenety. As noted above, the IF estmates may not be relable f the group of catchments s not homogeneous. Hoskng and Walls (1997) recommended of use methods that rely on ste characterstcs only when dentfyng homogeneous groups, and consequently use at-ste characterstcs to ndependently test homogenety. Hoskng and Walls (1997) further recommended usng Ward s method, whch s a herarchcal clusterng method based on mnmzng the Eucldean dstance n ste characterstcs space wthn each cluster. The 72 catchments were therefore subject to cluster analyss by Ward s method and homogenety assessment of the delneated groups by the method proposed by Hoskng and Walls (1997). Ths s a statstcal test based on L moment ratos for testng the heterogenety of the proposed groups. The test compares the between-ste varaton n sample L moment ratos wth the expected varaton for a homogeneous group. The method fts four parameter Kappa dstrbutons to regonal average L moment ratos. The estmated Kappa dstrbuton uses to generate 500 homogeneous groups wth populaton parameters equal to regonal average (3) 3409

4 sample L moment ratos. The propertes of the smulated homogeneous groups were compared to the sample L moment ratos as H ( V μ ) V V = (5) σ where μv s the mean of smulated V values and σv s the standard devaton of smulated V values. For the sample and smulated groups V s calculated as R ( t t ) N n = 1 V = (6) N n = 1 Where N s the number of stes, n s the record length at ste, t s the sample L moment rato at ste and t R s the correspondng regonal average sample L moment rato. In the study, three measures of H were computed: H 1 (based on L coeffcent of varaton (L Cv)), H 2 (based on L Cv and L coeffcent of skewness (L Cs)) and H 3 (based L Cs and L kurtoss (L Kurt)). If H < 2, the group can be regarded as acceptable homogeneous, f 1 H < 2 t s possble homogeneous, and f H 2 t s defntely homogeneous (Hoskng and Walls 1997). Step 3 - Development of regonal dmensonless flood frequency curves. Havng dentfed the homogeneous groups and the most approprate flood frequency dstrbuton, the next task s to conduct at-ste flood frequency analyses. For each of the defned group, average flood quantle, ( Q T ) ave at T=2, 5, 10, 20, 50, 100 and T=2.33 were then computed. The regonal dmensonless flood quantles were estmated by dvdng ( T Q The regonal growth curve for each of the ) ave Q by the average ndex flood, ( ) ave homogeneous groups was obtaned by plottng the dmensonless average quantles ( QT ) ( ) ave Q2.33 ave aganst T. Step 4 - Development of regonal predcton models for Q A regresson relatonshp between Q 2.33 and the catchment characterstcs was developed by Ordnary Least Squares (OLS) regresson usng stepwse method for each of the defned groups. As the stepwse method maxmzes the predctve ablty by selectng only the varable that mproves the adjusted R 2, the modellng was started wth all selected catchment characterstcs (Table 1). The performance of the developed models was assessed usng the coeffcent of determnaton (R 2 ): ths coeffcent was adjusted to accommodate degree of freedom (adjusted R 2 ) and Standard Error of Estmate (SES). The modellng was carred out by leavng one catchment asde at a tme as a test catchment and the modellng was repeated N tmes, where N s the number of catchments n the group. The averages of the parameters were taken as the parameters of the regonal model. Step 5 - Model evaluaton. Model evaluaton was conducted by comparng the predctons aganst the at-ste quantles as well as the upper and the lower bounds at the 95% confdence ntervals. 4. RESULT AND DISCUSSION The results of the at-ste flood frequency analyss showed that the GEV/L2 was the most approprate method for the majorty of selected catchments. It was also found that LPIII/MOM and GEV/L gave comparable results for the majorty of catchments, whle GEV/L2 descrbed the data better than any the other methods. The results of the devaton D (Equaton 1) ndcated that D of the GEV/L2 was smaller than that of the LN, LPIII, and GEV/L models for 24 out of 36 cases. The average devaton for the GEV/L2 was 7.46% compared to 37.65% and 13.84% and 9.67% for LN, LPIII, and GEV/L respectvely. Ths ndcated that the use of GEV/L2 estmates would result n smaller SEE. Ths was further confrmed by the results of Monte Carlo smulaton studes. Monte Carlo results for staton are presented n Fgure 3 to llustrate ths. 3410

5 Fgure 3. Results of the Monte Carlo Smulaton Saton It can be seen from Fgure 3 that GEV/L2 estmates show smaller bas compared to GEV/L estmates at all selected T values. The effcency of the GEV/L2 s greater than 1 for medum to hgher T values, ndcatng that GEV/L2 outperforms the GEV/L at medum to hgh Ts. Ths s a common observaton for majorty of catchments. Sutablty of GEV/L2 and GEV/L methods for regonal flood frequency analyses gauged from the moment rato dagrams presented n Fgure 1. It s clear from Fgure 1 that a greater proporton of the catchments n the L moment rato dagram le below the theoretcal GEV curve whle those n the L2 moment rato dagram are evenly scattered above and below the GEV curve. Ponts n the L moments rato dagram are also more scattered than those n the L2 moments rato dagram. These observatons ndcate that the GEV/L2 method s capable of descrbng the regonal data better than GEV/L. Hence, GEV/L2 s selected as the most approprate method for flood frequency analyses n Vctora. The catchment characterstcs of AREA, ELEV, FOREST, LENGTH, SLOPE, SFREQ, SHAPE and RAIN were used n the cluster analyss. Of the 4 ntally ndentfed groups, three were found to be non-homogeneous (H>2). These four groups were further refned by movng a ste or two from one group to another as proposed n Hoskng and Walls (1997). As a result, 5 homogeneous groups were delneated. The H statstcs of these fve groups are presented n Table 2. Table 2. The heterogenety measure of the 5 groups Group Number of stes H 1 H 2 H Fgure 4 shows the regonal growth curves derved for the fve homogeneous groups. It s clear that group 3 has the steepest flood frequency curve ndcatng greater varablty of floods n ths area compared to the other groups. Group 5 has the flattest frequency curve ndcatng least varablty. Havng developed regonal growth curves, the next step was to develop regonal predcton models for Q 2.33 for the defned groups. 3411

6 Fgure 4. Dmensonless regonal flood frequency curves (Q T /Q 2.33 ) for the delneated groups Table 3 shows the best ftted models for the fve group and ther performances. It s observed from Table 3 that for four out of the fve models, only the AREA was sgnfcant. The model effcences are not so attractve for two out of the fve groups and need further attenton n terms of dentfyng sgnfcant catchment characterstcs other than AREA and dentfyng the most sutable transformaton to mnmze the unexplaned varance. Also t s hoped that f the Generalzed Least Square (GLS) regresson s performed, model effcences can be mproved. Table 3: Model for the fve groups Group Model R 2 Adj. R 2 SEE as a % of ( Q ) Ave ( AREA) ( AREA) Q ( LENGTH) ( SFREQ) ( RAIN) ( AREA) = ( AREA) The number of tme model development was carred out corresponds to the number of catchments n the group by takng one catchment at a tme as a test catchment. Hence, the parameters shown n Table 3 are averaged values. Fgure 5 compares the predcted Q 2.33 aganst the 95% confdence ntervals for group 1. Of the 17 test runs undertaken, 14 predctons are wthn the 95% confdence nterval and only 3 are at the upper boundary ndcatng that model predctons are acceptable. However, there were cases n groups 2 and 4 whch gave predcted values outsde the confdence ntervals: Ths s expected as the model performances of these two groups were not satsfactory. When predctng Q T for a test catchment, the correspondng dmensonless value needs to be multpled by the Q 2.33 of the group. 3412

7 Fgure 5. Comparng the predcted Q 2.33 aganst the 95% confdence ntervals of the group 1 5. CONCLUSIONS Ths study has shown the GEV/L2 model as the most approprate methodology for flood frequency analyses n Vctora, Australa. Regonal predcton models for Q 2.33 (ndex flood) were developed and regonal growth curves for fve homogenous groups were derved. The developed predcton models gven n Table 2 and the regonal growth curves presented n Fgure 4 wll be valuable tools for makng quck predctons of flood quantles for ungauged statons. It s beleved that regonal models for groups 2 and 4 can be mproved by selectng more approprate catchment attrbutes and dentfyng the most approprate transformatons as well as by usng the GLS regresson for the model development. REFERENCES Cunnane, C (1988), Methods and merts of regonal flood frequency analyss, Journal of Hydrology, 100, Grngoten, I.I (1963), A plottng rule for extreme probablty paper, Journal of Geophscal Research, 68(3), Har, J., Black, W., Babn, B., Anderson, R. and Tatham, R (2006), Multvarate Data Analyss, Pearson Prentce Hall, Pearson Prentce Hall, Upper Saddle Rver, New Jersey. Hoskng, J. and Walls, J. (1997), Regonal Frequency Analyss: An Approach Based on L-moments, Cambrdge Unversty Press, UK. Engneers Australa (1987), Australan Ranfall and Runoff: A gude to flood estmaton, Plgrm, HD (ed.), The Insttute of Engneers Australa, Canberra. Kjeldsen, T.R., Smthers, J.C. and Schulze, R.E (2002), Regonal flood frequency analyss n the kwazulunatal provnce, South Afrca, usng the ndex-flood method, Journal of Hydrology, 255, Potter, K and Lettenmaer, D. (1990), A comparson of the regonal flood frequency estmaton methods usng a resamplng method, Water Resources Research, 26(3), Portela, M. and Das, A. (2005), Applcaton of The Index-Flood Method to the regonalzaton of flood peak dscharges on the Portugal manland, n Rver Basn Management, eds Breba, C.A and AntonesdoCarmo, J.S. WIT Press, UK, Rahman, A., Wenmann, P.E. and Men, R.G. (1999), At-ste frequency analyss: LP3-product moment, GEV-L moment and GEV-LH moment procedures compared. Water 99 Jont Congress, Saf, B. (2008), Applcaton of ndex procedures to flood frequency analyss n Turkey, Journal of the Amercan Water Resources Assocaton, 44(1), Vogel, R.M., Thomas Jr, W.O. and McMahon, T.A. (1993), Flood-flow frequency model selecton n Southwestern Unted States, Journal of Water Resources Plannng and Management, 119( 3), Wang, Q. J. (1997), LH moments for statstcal analyss of extreme events, Water Resources Research, 33(12),