A Regional Methodology for Deriving Flood Frequency Curves (FFC) in Partially Gauged Catchments with Uncertain Knowledge of Soil Moisture Conditions

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1 A Regonal Methodology for Dervng Flood Frequency Curves (FFC) n Partally Gauged Catchments wth Uncertan Knowledge of Sol Mosture Condtons G. Aronca a A. Candela b a Dpartmento d Costruzon e Tecnologe Avanzate, Unverstà d Messna, Messna, Italy, aronca@ngegnera.unme.t b Dpartmento d Ingegnera Idraulca e Applcazon Ambental, Unverstà d Palermo, Palermo, Italy Abstract: In ths paper a Monte Carlo procedure for dervng frequency dstrbutons of extreme dscharges startng from a smplfed descrpton of ranfall and surface runoff processes and usng regonal data s presented. The procedure s based on two modules: a stochastc ranfall generator module and a catchment response module. In the ranfall generator module the ranfall storm,.e. the maxmum ranfall depth for a fxed duraton, s assumed to follow the Two Components Extreme Value (TCEV) dstrbuton whose parameters have been estmated at regonal scale for Scly. The catchment response has been modelled by usng the Sol Conservaton Servce Curve Number for the total-effectve ranfall transformaton and the classcal ratonal formula for the flood routng. Ths method allows ncorporaton of nformaton on sol type, land use, sol cover condton and Antecedent Sol Mosture (AMC). Furthermore, to take n account for the spatal varaton of CN wthn the catchment, a sem-dstrbuted approach of the ranfall-runoff model was mplemented. Fnally, the Generalsed Lkelhood Uncertanty Estmaton (GLUE) procedure has been used to explore the estmaton of the uncertan knowledge of AMC affectng dervaton of FFC. Keywords: Flood; frequency analyss; Extreme events; Monte Carlo smulaton, Uncertanty, PUB. INTRODUCTION For engneerng applcatons the estmaton of the peak dscharge at the basn outlet for a gven return tme s mportant especally because the plannng and desgn of water resource projects and floodplan management depend on the frequency and magntude of peak dscharges. In gauged basns, ths s possble through a statstcal analyss of data or the calbraton of a ranfall-runoff model of varyng degrees of complexty. In ungauged basns the modellng of flood formaton process must be bult up usng smplfed descrptons of the hydrologcal processes charactersed by a reduced number of robust parameters n order to ensure a reduced uncertanty n model predctons. Eagleson [97] developed the dea of dervng flood statstcs from a smplfed schematsaton of storm and basn characterstcs proposng the so-called Dervaton Dstrbuton Technque. The approach allows consderaton of the knowledge about the hydrologcal processes generatng streamflow,.e. developng the chan of events n the runoff formaton process that lead to a certan frequency of streamflow. Streamflow varables are related to precptaton data (whch have longer record hstory, spatally more dense and more unform), antecedent mosture condtons n the dranage basn and the basn response to a precptaton nput. [Hebson and Wood 98; Daz-Granados et al.,984; Gottschalk and Wengartner, 998; Iacobells and Forentno, ; De Mchele and Salvador, ]. The Derved Dstrbuton Approach can be used to derve, analytcally, or numercally, the cumulatve dstrbuton functon of the flood runoff.as the analytcal methods to derve the probablty dstrbuton of peak streamflow showed mathematcal complexty,.e. dffcultes n parameter estmaton, avalablty of long seres of hystorcal data some authors [among others Loukas, ; Blöschl and Svapalan, 997; Muzk, ; Rahman et al., ] adopted Monte Carlo smulaton approach to determne the flood probablty dstrbuton. Ths

2 methodology nvolves random samplng from contnuous dstrbutons of nput varables to obtan the flood hydrographs. The procedure s mathematcally smple, despte havng a heavy computatonal demand. Therefore, from the consderaton of practcal applcablty and flexblty, the Monte Carlo smulaton technque appears to be the most promsng method to determne derved flood frequency dstrbutons, and thus has been adopted n ths study.. MONTE CARLO PROCEDURE FOR DERIVING FFC Ths secton descrbes the Monte Carlo procedure used n the present study to derve the flood frequency curves. The procedure s based on two modules: a stochastc ranfall generator module and a catchment response module. It was the ntenton of the authors to develop a method as smple as possble requrng lmted data and suted for ungauged or partally gauged catchments. In the next paragraphs the ranfall and the catchment modules wll be presented.. The stochastc ranfall generator module In the ranfall generator module the storm h s assumed to follow the Two Components Extreme Value (TCEV) dstrbuton [Ross et al. 984]. Ths dstrbuton has been adopted by the Italan Natonal Research Group for the Preventon of Hydro-Geologcal Dsasters for the analyss of extreme ranfall n Italy. The man advantage of ths dstrbuton les n usng two components to model the observed hydrologcal varable. Ths capablty s sgnfcant n Medterranean catchments where maxmum ranfalls and floods are often due to storms wth dfferent meteorologcal characterstcs [Ross and Vllan, 99]. The Cumulatve Dstrbuton Functon (CDF) of the TCEV dstrbuton wrtten usng a dmensonless varable h equal to the rato between the hydrologcal varable h and the mean value µ of the dstrbuton s the followng: F( h' * Λ ( Λ ) [ ( expα ) ) = exp Λ α * Θ exp * Θ h' h' () In order to apply ths dstrbuton to ungauged catchments Cannarozzo et al., [995] estmated the fve parameters α, Λ, Λ *, Θ *, µ at regonal scale for Scly. In ther work the authors proposed a dvson of the Scly nto three hydrologcally homogeneous sub-regons for whch the parameters α, Λ, µ have been estmated usng Maxmum Lkelhood (ML) method usng the annual maxmum ranfall wth,3,6, and 4 hours duraton recorded at rangauges located n each sub-regon. In comparson, the two parameters Λ *, Θ * have been estmated usng the data recorded n the entre regon. Furthermore, these fve parameters were observed to be dependent on the ranfall duraton and smple relatonshps were proposed for ther estmaton (Cannarozzo et al., 995): for the entre regon: * Θ = t Λ =.75 t.3 dependng on the sub-regons: Λ = p t α = r t s q dependng on rangauge ste: () (3) n µ = a t (4) Agan, to allow a smple estmaton of the parameter µ n ungauged or partally gauged stes two maps reportng contour lnes wth constant a or n values were also produced.. The catchment response module For modellng the catchment response a parsmonous hydrologcal approach was preferred over more complex models because of ts smplcty and ablty to approxmate catchment runoff behavour charactersed by a fast hydrologcal response, a small area and nadequate streamflow data. Agan, n the model we consder the ranfall duraton as constant and unformly dstrbuted n space over the catchment. Such an hypothess s physcally accepted for small-medum sze catchments, whch are more usually ungauged or partally gauged. The classcal ratonal formula [Dooge, 957], although an old concept, remans a practcal tool n engneerng hydrology. From the ratonal formula, the peak runoff Q peak, caused by an effectve unform ranfall h e (t c ) for a duraton t c whch leads, for a gven return perod, to the maxmum peak dscharge (crtcal duraton) can be calculated from:

3 ( tc ) A he Q peak = (5) tc where A s the catchment area. The SCS-CN method, adopted by USDA Sol Conservaton Servce [986], s used to transform the ranfall depth h to effectve ranfall h e. Ths method allows us to ncorporate nformaton on sol type, land use, sol cover condton and antecedent sol mosture (AMC). The total depth of effectve ranfall h e can be expressed n terms of the ranfall depth h as: ( h. S ) h >. S he = ( h +.8 S ) (6) h. S where S s the maxmum sol potental retenton gven by the followng expresson: S = 54 (7) CN Now, to take n account for the spatal varaton of CN wthn the catchment, a sem-dstrbuted probablstc approach was mplemented for the modellng of the runoff producton from (6). Usng GIS, a classfed map of the Curve Number, reportng the number of pxels n each class of CN, can be easly obtaned gven the land use and the sol type. In ths perspectve the effectve ranfall can be computed by applyng (6) n a way to nclude the nformaton from ths type of map. Followng these premses, equaton (6) can be rewrtten n the form: ( h. S ) h >. S h = ( h +.8 S ) (8) e, h. S and, consequently, the equaton (5): ( t ) N he, c Q peak = a (9) t = c where h e, s the runoff produced the -th CN class, S s the maxmum sol potental retenton n -th CN class, a s the area of the catchment charactersed by a partcular value of CN wth N A = a = the GIS map. and N s the number of CN classes n The SCS-CN method assumes that the CN values vary wth the degree of saturaton of the sol before the start of the storm. Partcularly, the sol condtons are descrbed through the defnton of three AMC classes (I, II and III) dependng on the total 5-days antecedent ranfall. To take nto account the catchment pror-to-storm condtons and to relax the classcal hypothess of sofrequency between ranfall nput and peak dscharges, the AMC has been treated as a random varable wth a dscrete probablty dstrbuton: λ λ λ3 3 = = Prob = Prob = Prob λ = [ AMC = I] [ AMC = II] [ AMC = III] () where {λ, λ, λ 3 } are the probabltes of occurrence of the three dfferent mosture condtons of the catchment [De Mchele and Salvador, ].Fnally, the dstrbuton of peak flood condtoned by the AMC dstrbuton s thus gven by: 3 ( t ) N he,,j c Q peak = λ a j () t = j= c 3. IMPLEMENTATION OF MONTE CARLO PROCEDURE The procedure above outlned was appled for the dervaton of the FFC n a relatvely small catchment located n the north-western part of Scly, Italy (Fgure ): Oreto Rver (catchment area = 75.6 km ) km Flow gauge Tyrrenan Sea Fgure. Locaton of Oreto catchment. In ths catchment the heavy storm ranfall s the only flood producng factor and the presence of streamflow gaugng staton wth a long recordng perod (N=57 yrs) at outflow allowed dervaton of the AMC probablty dstrbuton usng daly data (Fgure ). The TCEV parameters were

4 evaluated usng ()-(4) wth the parameters p, q, r, s, specfed for the catchment sub-regon and the parameters a, n obtaned by the so-a, so-n contour lnes [Cannarozzo et al., 995] (Table ). The crtcal storm duraton t c was assumed to be equal to the concentraton tme of the catchment. For ts evaluaton we used the smple relatonshp t c =.46 A / v (t c n hours, A n square klometres and v = - m s - ) deduced by Agnese and D Asaro [99] by applyng the Geomorphologcal Instantaneous Unt Hydrograph theory to several Sclan catchments. The CN spatal dstrbuton map has been produced n a GIS envronment wth a m grd resoluton (Fgure 3). T m m.4 = () NG +. where NG s the number of smulated peaks and m s the rank of the m-th peak flood value arranged n ncreasng order of magntude λ AMCI AMCII AMCIII Antecedent Mosture Condton Fgure. AMC dscrete probablty dstrbuton from hstorcal data Table. Parameter values for the TCEV dstrbuton Parameters Values p 4.55 q.49 r 3.58 s.34 a 6. n.37 t c (hours).7 Monte Carlo runs were performed to obtaned synthetc FFC wth a return tme range from to 5 years. The mplementaton can be descrbed by the followng steps: (a) values of total ranfall depth h for the storm duraton t c were randomly drawn from TCEV dstrbuton by solvng (); (b) for the three AMC condtons (I, II, III) effectve ranfall values have been calculated usng (8) for each CN class; (c) the fnal values of the peak flood dscharges condtoned by the AMC dstrbuton were obtaned usng (); (d) the return tme for each generated peak flood value has been computed from the plottng poston formula: Fgure 3. CN spatal dstrbuton map The resultng Flood Frequency Curve s reported n Fgure 4 and compared wth the observed annual maxmum peak flood values plotted usng the same plottng poston. The agreement s farly good and ths shows that the Monte Carlo smulaton technque can reproduce observed flood frequency curves wth reasonable accuracy over a wde range of return tme usng a smple and parsmonous approach. 4. UNCERTAINTY ANALYSIS USING GLUE As many authors ponted out [Wood, 976, Muzk et. al.,, De Mchele and Salvador, ] the sol mosture condton of the basn could be the most mportant factor nfluencng the estmaton of flood frequency dstrbuton and hence, the uncertanty of the dstrbuton. The Generalsed Lkelhood Uncertanty Estmaton (GLUE) procedure [Beven and Bnley, 99] s used here to explore the estmaton of the uncertan knowledge of AMC affectng the predctons of the ranfall-runoff model. GLUE s a Monte Carlo smulaton-based approach developed as an attempt to recognse more explctly the underlyng uncertantes of models smulatng envronmental processes. numbers of λ values of (correspondng to dfferent AMC condtons) were generated, each value beng drawn wthn range from to wth the condton λ =. 3 =

5 Flood quantle (m 3 /s) Observed floods Syntethc floods Return tme (years) Fgure 4. Flood Frequency Curve for Oreto catchment compared wth observed values Smulatons were performed for each parameter set θ ={λ, λ, λ 3 } for comparson wth the measured flood peak values at catchment outlet. Due to the dfferent sample sze for the generated and the observed values the comparson was carred out wth the generated dscharge values charactersed by the same return tme of the hstorcal values (a lnear nterpolaton was used to extract the value of the dscharge from the generated sample). Each smulaton was evaluated usng a performance ndex n the form of the classcal Nash and Sutclffe Effcency Crteron (97): where L(θ /Y) s the lkelhood measure for the - th model smulaton for parameter vector θ condtoned on a set of observatons Y, σ s the assocated error varance for the -th model and σ obs s the observed varance of the observed peak floods. Fgure 5 shows scatter plots for the lkelhood based on (3) for each of the AMC parameters sampled for the Flood Frequency Curve. Each dot represents one run of the model wth dfferent randomly chosen parameter values wthn the ranges. The generaton of the lkelhood surface nvolves a decson about the crteron for model rejecton and, partcularly, smulatons that acheve a lkelhood value less than zero are rejected as non-behavoural. The remanng are rescaled between to n order to calculate the cumulatve dstrbuton of the predctve varables from whch the chosen dscharge quantles, 5 and 95%, have been calculated to represent the model uncertanty n the model predctons (Fgure 6). As mght be expected these plot show how the model response s hghly senstve to varaton of antecedent mosture condton n the catchment. / Y) = ( σ / σobs ) obs L ( θ σ < σ (3) L L L λ λ λ 3 Fgure 5. Scatter plots llustratng the dstrbuton of lkelhood weghted qualtatve parameter values It worth to pont out how the hgher values of rescale lkelhood were found for hgh value of λ 3 (AMC III ). In the lmtaton of a sngle applcaton ths s a typcal behavour of Medterranean catchments n whch extreme events occur when the sol s close to saturaton and hence the sol mosture s the man factor affectng the flood formaton process. Fgure 6 llustrates the 9% lkelhood weghted uncertanty flood frequency bounds derved from the behavoural smulatons parameter sets, In addton to above stated the estmates of the hgher return perod floods are subject to the greatest amounts of uncertanty. Flood quantle (m 3 /s) 6 4 Observed floods Syntethc floods 5% quantles 95% quantles Return tme (years)

6 Fgure 6. 9% uncertanty bounds derved from annual maxmum peaks of behavoural parameter set wth -year smulaton length 5. CONCLUSIONS Ths paper presents a Monte Carlo procedure for dervng frequency dstrbutons of extreme dscharges startng from a smplfed descrpton of ranfall and surface runoff processes. The procedure s talored for small-medum sze ungauged or partally gauged catchments It focuses on a parsmonous ranfall-runoff modellng approach and a stochastc ranfall generator both fed by data at regonal scale, as CN maps and spatal nterpolaton of maxmum ranfall depth over the study area. The applcaton of ths procedure to a Medterranean catchment showed how Monte Carlo smulaton technque can reproduce the observed flood frequency curves wth reasonable accuracy over a wde range of return tme (Fgure 4) usng a smple and parsmonous approach and lmted data nput. In addton to ths we put the emphass on the mportance of the sol mosture condtons for flood formaton process and the uncertanty n ther knowledge as common n ungauged catchments. The applcaton of GLUE procedure for explorng ths uncertanty showed the mportance of the pror-to-storm condtons to derve the FFC and the senstvty of the ranfallrunoff model (despte, ts smplcty) to sol mosture varatons. 6. REFERENCES Agnese, C. and M. D Asaro, On the dervaton of the IUH from the geomorphology of lnk structured channel networks. Excerpta, 5, 99; Beven K.J. and A. Bnley, The future of dstrbuted models: model calbraton and uncertanty predcton, Hydrologcal Processes, 6, 79-98, 99; Bloschl, G. and M. Svapalan, Process control on flood frequency.. Runoff generaton, storm propertes and return. Centre for Water Research Envronmental Dynamcs Report, ED 59 MS, Department of Cvl Engneerng, The Unversty of Western Australa, 997; Cannarozzo, M., D'Asaro, F. and V. Ferro, Regonal ranfall and flood frequency analyss for Scly usng the two component extreme value dstrbuton. Journal of Hydrologcal Scences, 4(), 9-4, 995; De Mchele, C. and G. Salvador, On the derved flood frequency dstrbuton: analytcal formulaton and the nfluence of antecedent sol mosture condton. Journal of Hydrology, 6, 45-58, ; Daz-Granados, M. A., J. B. Valdes and R. L. Bras, A physcally based flood frequency dstrbuton, Water Resources Research, (7), 995, 984; Dooge, J.C.I., The ratonal method for estmatng flood peaks. Engneerng, 84, 3 33, , 957; Eagleson, P.S., Dynamcs of flood frequency, Water Resources Research, 8 (4), , 97; Gottschalk, L. and R. Wengartner, Dstrbuton of peak flow derved from a dstrbuton of ranfall volume and runoff coeffcent, and a unt hydrograph, Journal of Hydrology, 8, 48 6, 998; Hebson, C. and E. F. Wood, A derved flood frequency dstrbuton usng Horton order rato, Water Resources Research, 8(5), 59 58, 98; Iacobells, V. and M. Forentno, Derved dstrbuton of floods based on the concept of partal area coverage wth a clmatc appeal, Water Resources Research, 36(), , ; Loukas, A., Flood frequency estmaton by a derved dstrbuton procedure, Journal of Hydrology, 55, 69-89, ; Muzk I., A frst-order analyss of the clmate change effect on flood frequences n a subalpne watershed by means of a hydrologcal ranfall-runoff model. Journal of Hydrology, 67, 65-73, ; Nash J.E., and J.V. Sutclffe, Rver flow forecastng through conceptual models. Journal of Hydrology,, 8-9, 97; Rahaman, A., P.E. Wenmann,, T.M.T. Hoang and E.M. Laurenson, Monte Carlo smulaton of flood frequency curves from ranfall. Journal of Hydrology, 56, 96-, ; Ross, F., M. Forentno and P. Versace, Two components extreme value dstrbuton for flood frequency analyss. Water Resources Research, (7), , 984; Ross, F. and P. Vllan, Regonal methods for flood estmaton, Proceedngs of NATO-ASI Conference on Copng wth floods, Erce, Italy, 99. US Department of Agrcolture, Sol Conservaton Servce, Natonal Engneerng Handbook, Hydrology, Sec.4, Washngton DC, 986