In the framework of the regional analysis of hydrological extreme events, a probabilistic mixture model, given by a convex combination of two Gumbel distributions, is proposed for rainfall extremes modelling. The papers concerns with statistical methodology for parameter estimation. With this aim, theoretical expressions of the L-moments of the mixture model are here defined. The expressions have been tested on the time series of the annual maximum height of daily rainfall recorded in Tuscany (central Italy)

Caporali, Enrica

Cavigli, Elisabetta

Moro, Antonio

Tartaglia, Vincenzo

Advances in Geosciences

English

International audienceIn the framework of the regional analysis of hydrological extreme events, a probabilistic mixture model, given by a convex combination of two Gumbel distributions, is proposed for rainfall extremes modelling. The papers concerns with statistical methodology for parameter estimation. With this aim, theoretical expressions of the L-moments of the mixture model are here defined. The expressions have been tested on the time series of the annual maximum height of daily rainfall recorded in Tuscany (central Italy)

Tartaglia, V.

Caporali, E.

Cavigli, E.

Moro, A.

HAL-INSU

L-moments based assessment of a mixture model forfrequency analysis of rainfall extremesV. Tartaglia, E. Caporali, E. Cavigli, A. MoroTo cite this version:V. Tartaglia, E. Caporali, E. Cavigli, A. Moro. L-moments based assessment of a mixture modelfor frequency analysis of rainfall extremes. Advances in Geosciences, European GeosciencesUnion, 2006, 2, pp.331-334. <hal-00296911>HAL Id: hal-00296911https://hal.archives-ouvertes.fr/hal-00296911Submitted on 23 Jan 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Advances in Geosciences, 2, 331–334, 2006SRef-ID: 1680-7359/adgeo/2006-2-331European Geosciences Union© 2006 Author(s). This work is licensedunder a Creative Commons License.Advances inGeosciencesL-moments based assessment of a mixture model for frequencyanalysis of rainfall extremesV. Tartaglia, E. Caporali, E. Cavigli, and A. MoroDipartimento di Ingegneria Civile, Universita` degli Studi di Firenze, ItalyReceived: 2 December 2004 – Revised: 2 October 2005 – Accepted: 3 October 2005 – Published: 23 January 2006Abstract. In the framework of the regional analysis of hy-drological extreme events, a probabilistic mixture model,given by a convex combination of two Gumbel distributions,is proposed for rainfall extremes modelling. The papers con-cerns with statistical methodology for parameter estimation.With this aim, theoretical expressions of the L-moments ofthe mixture model are here defined. The expressions havebeen tested on the time series of the annual maximum heightof daily rainfall recorded in Tuscany (central Italy).1 IntroductionIn hydrology the most frequent use of statistics has been thatof frequency analysis (Haan, 1994). This applies in particu-lar when the magnitude of an event with a given return periodor the rarity of a hydrological event, has to be estimated. Of-ten the common use of the asymptotic extreme value type I,or Gumbel, distribution does not represent an acceptable fit-ting of the model to the observations of rare frequency, i.e.it is not able to model the upper tail of the hydrological ex-treme events. This limit has been overcome, for example,by the Two Components Extreme Value (TCEV) distribution(Rossi et al., 1984). Observing the presence of outliers data,the TCEV shows, in fact, a better fitting to the data sam-ple skewness (Matalas et al., 1975), but it needs of a largersample in order to obtain a robust estimation of the param-eters. For this reason such kind of distributions are oftenused on regional basis. The assumption that a region is aset of gauging sites whose hydrologic events frequency be-haviour is homogeneous, in some quantifiable manner (Cun-nane, 1988), allows to estimate hydrologic variables quin-tiles even in ungauged sites. In Italy a regionalized procedurehave been developed in the VaPI (Valutazione delle Piene inItalia) project, based on the TCEV distribution (Rossi et al.,1992). In the framework of the regional analysis of rainfallCorrespondence to: E. Caporali(enrica@dicea.unifi.it)events in Tuscany (central Italy), a statistical methodologyfor parameter estimation of a probabilistic mixture model ishere developed. The use of a probabilistic mixture model inthe regional analysis of rainfall extreme events, comes fromthe hypothesis that rainfall phenomenon arises from two in-dependent populations (Becchi et al., 2000; Caporali et al.,2002). The first population describes the basic componentscorresponding to the more frequent events of low magnitude,controlled by the local morpho-climatic conditions. The sec-ond population characterizes the outlier component of moreintense and rare events, which is here considered due to theaggregation at the large scale of meteorological phenomena.As the rare component is due to the large scale state of en-ergy and to the morpho-climatic conditions, it is assumedthat any regionalization procedure can be reduced to the useof parameters of the basic or ordinary component.The contribution of the basic and the outlier componentsis presented as a mixture model, i.e. a convex combinationof two independent populations having Gumbel distribution.Linearity of “sum” – operator leaves separated the contribu-tions of the two populations, and it allows to evaluate sepa-rately their statistical parameters (Becchi et al., 2000).Being very suitable for frequency analysis of extreme val-ues, the L-moments method is proposed for parameter es-timation (Hosking and Wallis, 1993). The L-moments arein fact expectations of linear combinations of order statis-tics (Hosking, 1990) and are more robust to the data outliersand virtually unbiased for small samples. Moreover the L-moments have the very important advantage, over the con-ventional moments, of being less affected from the effects ofsampling variability being linear functions of the data.In the following the probabilistic model and the parame-ter estimation method are described. A procedure to find anestimation of the outlier percentage is also investigated.332 V. Tartaglia et al.: L-moments assessment for frequency analysis of rainfall extremes2 Model development and parameter estimationThe probability model is defined by the following convexcombination or mixture of two Gumbel distributions.FX(x) = (1 − α) · exp[−31 exp(− xθ1)]+ α · exp[−32 exp(− xθ2)](1)with : 31= exp(−µ1θ1)and 32= exp(−µ2θ2)(2)where µ1, µ2 are the modal values and θ1, θ2 are propor-tional to the standard deviations of the two Gumbel distribu-tions appearing in the mixture. The indexes 1 and 2 indicaterespectively the basic and the outlier component. Using the αproportion, the model underlines the contribution of the twoindependent components to the rainfall extreme values.The fitting of the mixture model to the time series of rain-fall extreme values is evaluated comparing the L-moments ofthe at-site data (Hosking, 1990) with their theoretical expres-sions.The theoretical expressions of the L-moments are definedas linear combinations of the Probability Weighted Moments(PWM) (Greenwood et al., 1979). The first four momentsare:λ1 = β0;λ2 = 2β1 − β0,λ3 = 6β2 − 6β1 + β0;λ4 = 20β3 − 30β2 + 12β1 + β0 (3)where:βi =+∞∫−∞x · FX(x)idF (x) =+∞∫−∞x · FX(x)i · f (x)dx (4)is the i-th PWM given by:β0 = (1 − α) · θ1 · ln λ1 + α · θ2 · ln λ2 + γ [(1 − α) · θ1 + α · θ2] (5)β1 = (1 − α)2 · I (1, 2, 1, 0)+ (1 − α)α · I (1, 2, 0, 1)+ (1 − α) α · I (2, 1, 0, 1)+ α2 · I (2, 1, 1, 0) (6)β2 = (1 − α)3 · I (1, 2, 2, 0)+ 2(1 − α)2α · I (1, 2, 1, 1)+(1 − α)α2 · I (1, 2, 0, 2)+ (1 − α)2α · I (2, 1, 0, 2)+2(1 − α)α2 · I (2, 1, 1, 1)+ α3 · I (2, 1, 2, 0) (7)β3 = (1 − α)4 · I (1, 2, 3, 0)+ 3(1 − α)3α · I (1, 2, 2, 1)+3(1 − α)2α2 · I (1, 2, 1, 2)+ (1 − α)α3 · I (1, 2, 0, 3)+(1 − α)3α · I (2, 1, 0, 3)+ 3(1 − α)2α2 · I (2, 1, 1, 2)+3(1 − α)α3 · I (2, 1, 2, 1)+ α4 · I (2, 1, 3, 0) (8)where:I (i, j, r, s) =+∞∫−∞x · F ri (x) · F sj (x) · fidx (9)if s=0 is expressed by:I (i, j, r, s) = θi · [ln λi + γ + ln(r + 1)]r + 1 (10)if s>0 and sλj[(r+1)λi ]θi /θj<1 is expressed by:I (i, j, r, s) = θir + 1 ln ((r + 1) · λi) ·[ ∞∑j=0(−1)jλjj ! · 0(jθ+ 1)]++− θir + 1 ∞∑j=0(−1)jλjj ! ·−γ − 0(jθ+ 1)jθ+ 1 (11)here θ= θjθi, λ= sλj[(r+1)λi ]1/θ and γ is the Euler’s constant;if s>0 and sλj[(r+1)λi ]θi /θj≥1 is expressed by:I (i, j, r, s) = λr + 1 ·[ln (s · λi) · θj · 1λ[1 −∞∑j=0(−1)jλjj ! · 0(jθ+ 1)]]+− λr + 1 ·[θ2jθi·∞∑j=0(−1)jλjj ! ·[−γ − θj+10(j+1θ)]](12)here θ= θiθjand λ= (r+1)λi(sλj )1/θ and γ is the Euler’s constant.3 Application and preliminary conclusionThe analysis of the capability of the mixture model to repro-duce the L-moments of the sample values has been carriedout using L-kurtosis (τ4=λ4/λ2) and L-skewness (τ3=λ3/λ2),evaluated from Eqs. (3) and (5)–(8).The study has been developed using time series, more than30 years long, of annual maximum of daily rainfall height,over a territory of about 22 000 km2 in Tuscany. In particular201 time series, from 1916 to 1996, having a low spatial cor-relation, have been extracted from a set of 584 rain gauges.Recently the database has been updated to the year 2000, and307 time series have been extracted from a set of 880 raingauges.The analysis has been worked out on both the data sets andthe L-moments, i.e. the L-kurtosis and the L-skewness, of thesample have been compared with the corresponding theoret-ical ones (Fig. 1). The comparison shows that the mixturemodel is able to reproduce the L-moments of the sample se-ries on a wide range of values.4 Future developments4.1 Characterization of the outliers percentageWith the aim to develop a regionalization procedure for theparameters, based on the identification of homogeneous ar-eas with respect to the basic component of the rainfall phe-nomena (Gabriele and Arnell, 1991), the spatial distributionof the outliers, in the studied region, has been investigated.V. Tartaglia et al.: L-moments assessment for frequency analysis of rainfall extremes 333 α=1E-06 α=0.3 Data set      up to 2000 Fig. 1. Comparison of the sample L-skewness versus L-kurtosis (·)with the corresponding theoretical L-moments of the mixture modelgiven the α proportion of rare events.Particularly, the α proportion has been estimated and its spa-tial distribution has been analyzed. As a first approximation,the α proportion has been evaluated as the ratio of the num-ber of outliers on the total number of events recorded in a raingauge station. In future its relationships with territory char-acteristics (i.e. local elevation, slopes, exposure etc.) will bealso investigated.A box-and-whisker plot (Tukey, 1977) has been used tofind the outliers of each site. The whisker extends to thefarthest points that are considered as frequent events. Out-liers’ values are defined as values that are larger than the up-per quartile plus 1.5 times the interquartile range. Each siteis classified with the number of outliers per total number ofdata. The ratio of the outliers with the total number of data isthe α proportion of the specific measurement site.4.2 Outliers interpolation gridTo represent the particular trend of the outliers in the studiedregion, an inverse distance weighted interpolation has beenused to predict the value for any unmeasured site. A 2×2 kmgrid has been generated (Fig. 2) assuming that each measuredpoint has a local influence that decreases with the distanceand data sets that are close are more similar than those thatare farther apart.The relationship between sample data and associated valuewith each cell is:p(i, j) =n∑k=1pk · 1Dνk−(i,j)n∑k=11Dνk−(i,j)(13)where p(i,j) is the value of the quantity associated with the ijcell; pk is the value of the quantity in the site k; Dk−(i,j) is range. Each site is classified with the number of outliers per total number of data. The ratio of the outliers with the total number of data is the α proportion of the specific measurement site.  4.2 Outliers interpolation grid To represent the particular trend of the outliers in the studied region, an inverse distance weighted interpolation has been used to predict the value for any unmeasured site. A 2x2 km grid has been generated (figure 2) assuming that each measured point has a local influence that decreases with the distance and data sets that are close are more similar than those that are farther apart.   Figure 2. Spatial distribution of outliers’ percentage α in the region (UTM coordinates [m]).  The relationship between sample data and associated value with each cell is: ∑∑=−=−⋅= nk jiknk jikkDDpjip1 ),(1 ),(11),(νν         (13) where p(i,j) is the value of the quantity associated with the ij cell; pk is the value of the quantity in the site k; Dk-(i,j) is the distance between the site k and the ij cell center; ν is the exponent.  Fig. 2. Spatial distribution of outliers’ percentage α in the region(UTM coordinates [m]).the distance between the site k and the ij cell center; ν is theexponent.To allocate values on the grid, distance between the ij celland the measurement sites has been weighted. The decreas-ing influence of the rain gauge site is controlled by the expo-nent ν. After several tests, an interested value for the expo-nent seems to be ν= 2. However, it still has to be proved thatit is the optimal value, according to the minimization of theroot mean square prediction error.Acknowledgements. Authors would like to thank I. Becchi and A.Petrucci for suggestions and helpful discussions. This researchhas received support from the “Group for Prevention from Hydro-geological Disasters” (GNDCI) of the Italian National ResearchCouncil.Edited by: L. FerrarisReviewed by: anonymous refereesReferencesBecchi, I., Caporali, E., Moro, A., and Sorbi, A.: Misture di dis-tribuzione di valori estremi di precipitazione, XXVII Convegnodi Idraulica e Costruzioni Idrauliche, II, 147–150, 2000.Beran, M., Hosking, J. R. M., and Arnel, N.: Comment on “Two-Component Extreme Value Distribution for Flood FrequencyAnalysis” by Fabio Rossi, Mauro Fiorentino, and Pasquale Ver-sace, Water Resour. Res., 22, 263–266, 1986..Caporali, E., Petrucci, A., and Tartaglia, V.: A mixture model ofrainfall extreme values, 3rd Plinius Conference on MediterraneanStorms, edited by: Deidda, R., Mugnai, A., and Siccardi, F.,GNDCI publ. n. 2560, 245–248, 2002.Cunnane, C.: Methods and merits of regional flood frequency anal-ysis, J. Hydrol., 100(1–4), 269–290, 1988.Gabriele, S. and Arnell, N. W.: A hierarchical approach to regionalflood frequency analysis, Water Resour. Res., 27, 1281–1289,1991.Greenwood, J. A., Landwehr, J. M., Matalas, N. C., and Wallis,J. R.: Probability weighted moments: definition and relation to334 V. Tartaglia et al.: L-moments assessment for frequency analysis of rainfall extremesparameters of several distributions expressable in inverse form,Water Resour. Res., 15, 1049–1054, 1979.Haan, C. T.: Statistical Methods in hydrology, The Iowa State Uni-versity Press/Ames., 1994.Hosking, J. R. M.: L-moments: Analysis and Estimation of Distri-butions using Linear Combinations of Order Statistics, Journalof Royal Statistical Society, Series B, 52, 105–124, 1990.Hosking, J. R. M. and Wallis, J. R.: Some statistics useful in re-gional frequency analysis, Water Resour. Res., 29, 271–281,1993.Matalas, N. C., Slack, R., and Wallis, J. R.: Regional skew in searchof a parent, Water Resour. Res., 11(6), 815–826, 1975.Rossi, F., Fiorentino, M., and Versace, P.: Two-Component ExtremeValue Distribution for Flood Frequency Analysis, Water Resour.Res., 20(7), 847–856, 1984.Rossi, F. and Villani, P.: A project for regional analysis of flood inItaly, Proceedings of the NATO ADVANCED Study institute onCoping with Floods, Erice, Italy, 193–217, 1992.Tukey, J. W.: Exploratory data analysis, Addison-Wesley, Reading,Mass., 1977.

L-moments based assessment of a mixture model for frequency analysis of rainfall extremes

V. Tartaglia

E. Caporali

E. Cavigli

A. Moro

Directory of Open Access Journals

Florence Research

Advances in Geosciences, 2, 331–334, 2006SRef-ID: 1680-7359/adgeo/2006-2-331European Geosciences Union© 2006 Author(s). This work is licensedunder a Creative Commons License.Advances inGeosciencesL-moments based assessment of a mixture model for frequencyanalysis of rainfall extremesV. Tartaglia, E. Caporali, E. Cavigli, and A. MoroDipartimento di Ingegneria Civile, Universita` degli Studi di Firenze, ItalyReceived: 2 December 2004 – Revised: 2 October 2005 – Accepted: 3 October 2005 – Published: 23 January 2006Abstract. In the framework of the regional analysis of hy-drological extreme events, a probabilistic mixture model,given by a convex combination of two Gumbel distributions,is proposed for rainfall extremes modelling. The papers con-cerns with statistical methodology for parameter estimation.With this aim, theoretical expressions of the L-moments ofthe mixture model are here defined. The expressions havebeen tested on the time series of the annual maximum heightof daily rainfall recorded in Tuscany (central Italy).1 IntroductionIn hydrology the most frequent use of statistics has been thatof frequency analysis (Haan, 1994). This applies in particu-lar when the magnitude of an event with a given return periodor the rarity of a hydrological event, has to be estimated. Of-ten the common use of the asymptotic extreme value type I,or Gumbel, distribution does not represent an acceptable fit-ting of the model to the observations of rare frequency, i.e.it is not able to model the upper tail of the hydrological ex-treme events. This limit has been overcome, for example,by the Two Components Extreme Value (TCEV) distribution(Rossi et al., 1984). Observing the presence of outliers data,the TCEV shows, in fact, a better fitting to the data sam-ple skewness (Matalas et al., 1975), but it needs of a largersample in order to obtain a robust estimation of the param-eters. For this reason such kind of distributions are oftenused on regional basis. The assumption that a region is aset of gauging sites whose hydrologic events frequency be-haviour is homogeneous, in some quantifiable manner (Cun-nane, 1988), allows to estimate hydrologic variables quin-tiles even in ungauged sites. In Italy a regionalized procedurehave been developed in the VaPI (Valutazione delle Piene inItalia) project, based on the TCEV distribution (Rossi et al.,1992). In the framework of the regional analysis of rainfallCorrespondence to: E. Caporali(enrica@dicea.unifi.it)events in Tuscany (central Italy), a statistical methodologyfor parameter estimation of a probabilistic mixture model ishere developed. The use of a probabilistic mixture model inthe regional analysis of rainfall extreme events, comes fromthe hypothesis that rainfall phenomenon arises from two in-dependent populations (Becchi et al., 2000; Caporali et al.,2002). The first population describes the basic componentscorresponding to the more frequent events of low magnitude,controlled by the local morpho-climatic conditions. The sec-ond population characterizes the outlier component of moreintense and rare events, which is here considered due to theaggregation at the large scale of meteorological phenomena.As the rare component is due to the large scale state of en-ergy and to the morpho-climatic conditions, it is assumedthat any regionalization procedure can be reduced to the useof parameters of the basic or ordinary component.The contribution of the basic and the outlier componentsis presented as a mixture model, i.e. a convex combinationof two independent populations having Gumbel distribution.Linearity of “sum” – operator leaves separated the contribu-tions of the two populations, and it allows to evaluate sepa-rately their statistical parameters (Becchi et al., 2000).Being very suitable for frequency analysis of extreme val-ues, the L-moments method is proposed for parameter es-timation (Hosking and Wallis, 1993). The L-moments arein fact expectations of linear combinations of order statis-tics (Hosking, 1990) and are more robust to the data outliersand virtually unbiased for small samples. Moreover the L-moments have the very important advantage, over the con-ventional moments, of being less affected from the effects ofsampling variability being linear functions of the data.In the following the probabilistic model and the parame-ter estimation method are described. A procedure to find anestimation of the outlier percentage is also investigated.332 V. Tartaglia et al.: L-moments assessment for frequency analysis of rainfall extremes2 Model development and parameter estimationThe probability model is defined by the following convexcombination or mixture of two Gumbel distributions.FX(x) = (1 − α) · exp[−31 exp(− xθ1)]+ α · exp[−32 exp(− xθ2)](1)with : 31= exp(−µ1θ1)and 32= exp(−µ2θ2)(2)where µ1, µ2 are the modal values and θ1, θ2 are propor-tional to the standard deviations of the two Gumbel distribu-tions appearing in the mixture. The indexes 1 and 2 indicaterespectively the basic and the outlier component. Using the αproportion, the model underlines the contribution of the twoindependent components to the rainfall extreme values.The fitting of the mixture model to the time series of rain-fall extreme values is evaluated comparing the L-moments ofthe at-site data (Hosking, 1990) with their theoretical expres-sions.The theoretical expressions of the L-moments are definedas linear combinations of the Probability Weighted Moments(PWM) (Greenwood et al., 1979). The first four momentsare:λ1 = β0;λ2 = 2β1 − β0,λ3 = 6β2 − 6β1 + β0;λ4 = 20β3 − 30β2 + 12β1 + β0 (3)where:βi =+∞∫−∞x · FX(x)idF (x) =+∞∫−∞x · FX(x)i · f (x)dx (4)is the i-th PWM given by:β0 = (1 − α) · θ1 · ln λ1 + α · θ2 · ln λ2 + γ [(1 − α) · θ1 + α · θ2] (5)β1 = (1 − α)2 · I (1, 2, 1, 0)+ (1 − α)α · I (1, 2, 0, 1)+ (1 − α) α · I (2, 1, 0, 1)+ α2 · I (2, 1, 1, 0) (6)β2 = (1 − α)3 · I (1, 2, 2, 0)+ 2(1 − α)2α · I (1, 2, 1, 1)+(1 − α)α2 · I (1, 2, 0, 2)+ (1 − α)2α · I (2, 1, 0, 2)+2(1 − α)α2 · I (2, 1, 1, 1)+ α3 · I (2, 1, 2, 0) (7)β3 = (1 − α)4 · I (1, 2, 3, 0)+ 3(1 − α)3α · I (1, 2, 2, 1)+3(1 − α)2α2 · I (1, 2, 1, 2)+ (1 − α)α3 · I (1, 2, 0, 3)+(1 − α)3α · I (2, 1, 0, 3)+ 3(1 − α)2α2 · I (2, 1, 1, 2)+3(1 − α)α3 · I (2, 1, 2, 1)+ α4 · I (2, 1, 3, 0) (8)where:I (i, j, r, s) =+∞∫−∞x · F ri (x) · F sj (x) · fidx (9)if s=0 is expressed by:I (i, j, r, s) = θi · [ln λi + γ + ln(r + 1)]r + 1 (10)if s>0 and sλj[(r+1)λi ]θi /θj<1 is expressed by:I (i, j, r, s) = θir + 1 ln ((r + 1) · λi) ·[ ∞∑j=0(−1)jλjj ! · 0(jθ+ 1)]++− θir + 1 ∞∑j=0(−1)jλjj ! ·−γ − 0(jθ+ 1)jθ+ 1 (11)here θ= θjθi, λ= sλj[(r+1)λi ]1/θ and γ is the Euler’s constant;if s>0 and sλj[(r+1)λi ]θi /θj≥1 is expressed by:I (i, j, r, s) = λr + 1 ·[ln (s · λi) · θj · 1λ[1 −∞∑j=0(−1)jλjj ! · 0(jθ+ 1)]]+− λr + 1 ·[θ2jθi·∞∑j=0(−1)jλjj ! ·[−γ − θj+10(j+1θ)]](12)here θ= θiθjand λ= (r+1)λi(sλj )1/θ and γ is the Euler’s constant.3 Application and preliminary conclusionThe analysis of the capability of the mixture model to repro-duce the L-moments of the sample values has been carriedout using L-kurtosis (τ4=λ4/λ2) and L-skewness (τ3=λ3/λ2),evaluated from Eqs. (3) and (5)–(8).The study has been developed using time series, more than30 years long, of annual maximum of daily rainfall height,over a territory of about 22 000 km2 in Tuscany. In particular201 time series, from 1916 to 1996, having a low spatial cor-relation, have been extracted from a set of 584 rain gauges.Recently the database has been updated to the year 2000, and307 time series have been extracted from a set of 880 raingauges.The analysis has been worked out on both the data sets andthe L-moments, i.e. the L-kurtosis and the L-skewness, of thesample have been compared with the corresponding theoret-ical ones (Fig. 1). The comparison shows that the mixturemodel is able to reproduce the L-moments of the sample se-ries on a wide range of values.4 Future developments4.1 Characterization of the outliers percentageWith the aim to develop a regionalization procedure for theparameters, based on the identification of homogeneous ar-eas with respect to the basic component of the rainfall phe-nomena (Gabriele and Arnell, 1991), the spatial distributionof the outliers, in the studied region, has been investigated.V. Tartaglia et al.: L-moments assessment for frequency analysis of rainfall extremes 333 α=1E-06 α=0.3 Data set      up to 2000 Fig. 1. Comparison of the sample L-skewness versus L-kurtosis (·)with the corresponding theoretical L-moments of the mixture modelgiven the α proportion of rare events.Particularly, the α proportion has been estimated and its spa-tial distribution has been analyzed. As a first approximation,the α proportion has been evaluated as the ratio of the num-ber of outliers on the total number of events recorded in a raingauge station. In future its relationships with territory char-acteristics (i.e. local elevation, slopes, exposure etc.) will bealso investigated.A box-and-whisker plot (Tukey, 1977) has been used tofind the outliers of each site. The whisker extends to thefarthest points that are considered as frequent events. Out-liers’ values are defined as values that are larger than the up-per quartile plus 1.5 times the interquartile range. Each siteis classified with the number of outliers per total number ofdata. The ratio of the outliers with the total number of data isthe α proportion of the specific measurement site.4.2 Outliers interpolation gridTo represent the particular trend of the outliers in the studiedregion, an inverse distance weighted interpolation has beenused to predict the value for any unmeasured site. A 2×2 kmgrid has been generated (Fig. 2) assuming that each measuredpoint has a local influence that decreases with the distanceand data sets that are close are more similar than those thatare farther apart.The relationship between sample data and associated valuewith each cell is:p(i, j) =n∑k=1pk · 1Dνk−(i,j)n∑k=11Dνk−(i,j)(13)where p(i,j) is the value of the quantity associated with the ijcell; pk is the value of the quantity in the site k; Dk−(i,j) is range. Each site is classified with the number of outliers per total number of data. The ratio of the outliers with the total number of data is the α proportion of the specific measurement site.  4.2 Outliers interpolation grid To represent the particular trend of the outliers in the studied region, an inverse distance weighted interpolation has been used to predict the value for any unmeasured site. A 2x2 km grid has been generated (figure 2) assuming that each measured point has a local influence that decreases with the distance and data sets that are close are more similar than those that are farther apart.   Figure 2. Spatial distribution of outliers’ percentage α in the region (UTM coordinates [m]).  The relationship between sample data and associated value with each cell is: ∑∑=−=−⋅= nk jiknk jikkDDpjip1 ),(1 ),(11),(νν         (13) where p(i,j) is the value of the quantity associated with the ij cell; pk is the value of the quantity in the site k; Dk-(i,j) is the distance between the site k and the ij cell center; ν is the exponent.  Fig. 2. Spatial distribution of outliers’ percentage α in the region(UTM coordinates [m]).the distance between the site k and the ij cell center; ν is theexponent.To allocate values on the grid, distance between the ij celland the measurement sites has been weighted. The decreas-ing influence of the rain gauge site is controlled by the expo-nent ν. After several tests, an interested value for the expo-nent seems to be ν= 2. However, it still has to be proved thatit is the optimal value, according to the minimization of theroot mean square prediction error.Acknowledgements. Authors would like to thank I. Becchi and A.Petrucci for suggestions and helpful discussions. This researchhas received support from the “Group for Prevention from Hydro-geological Disasters” (GNDCI) of the Italian National ResearchCouncil.Edited by: L. FerrarisReviewed by: anonymous refereesReferencesBecchi, I., Caporali, E., Moro, A., and Sorbi, A.: Misture di dis-tribuzione di valori estremi di precipitazione, XXVII Convegnodi Idraulica e Costruzioni Idrauliche, II, 147–150, 2000.Beran, M., Hosking, J. R. M., and Arnel, N.: Comment on “Two-Component Extreme Value Distribution for Flood FrequencyAnalysis” by Fabio Rossi, Mauro Fiorentino, and Pasquale Ver-sace, Water Resour. Res., 22, 263–266, 1986..Caporali, E., Petrucci, A., and Tartaglia, V.: A mixture model ofrainfall extreme values, 3rd Plinius Conference on MediterraneanStorms, edited by: Deidda, R., Mugnai, A., and Siccardi, F.,GNDCI publ. n. 2560, 245–248, 2002.Cunnane, C.: Methods and merits of regional flood frequency anal-ysis, J. Hydrol., 100(1–4), 269–290, 1988.Gabriele, S. and Arnell, N. W.: A hierarchical approach to regionalflood frequency analysis, Water Resour. Res., 27, 1281–1289,1991.Greenwood, J. A., Landwehr, J. M., Matalas, N. C., and Wallis,J. R.: Probability weighted moments: definition and relation to334 V. Tartaglia et al.: L-moments assessment for frequency analysis of rainfall extremesparameters of several distributions expressable in inverse form,Water Resour. Res., 15, 1049–1054, 1979.Haan, C. T.: Statistical Methods in hydrology, The Iowa State Uni-versity Press/Ames., 1994.Hosking, J. R. M.: L-moments: Analysis and Estimation of Distri-butions using Linear Combinations of Order Statistics, Journalof Royal Statistical Society, Series B, 52, 105–124, 1990.Hosking, J. R. M. and Wallis, J. R.: Some statistics useful in re-gional frequency analysis, Water Resour. Res., 29, 271–281,1993.Matalas, N. C., Slack, R., and Wallis, J. R.: Regional skew in searchof a parent, Water Resour. Res., 11(6), 815–826, 1975.Rossi, F., Fiorentino, M., and Versace, P.: Two-Component ExtremeValue Distribution for Flood Frequency Analysis, Water Resour.Res., 20(7), 847–856, 1984.Rossi, F. and Villani, P.: A project for regional analysis of flood inItaly, Proceedings of the NATO ADVANCED Study institute onCoping with Floods, Erice, Italy, 193–217, 1992.Tukey, J. W.: Exploratory data analysis, Addison-Wesley, Reading,Mass., 1977.

L-MOMENTS BASED ASSESSMENT OF A MIXTURE MODEL FOR FREQUENCY ANALYSIS OF RAINFALL EXTREMES.

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L-MOMENTS BASED ASSESSMENT OF A MIXTURE MODEL FOR FREQUENCY ANALYSIS OF RAINFALL EXTREMES.

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