UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE CAMINOS, CANALES Y PUERTOS FLOOD FREQUENCY ANALYSIS BY A BIVARIATE MODEL BASED ON COPULAS TESIS DOCTORAL Ana Isabel Requena Rodríguez Ingeniero de Caminos, Canales y Puertos 2015 DEPARTAMENTO DE INGENIERÍA CIVIL: HIDRÁULICA, ENERGÍA Y MEDIO AMBIENTE ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE CAMINOS, CANALES Y PUERTOS FLOOD FREQUENCY ANALYSIS BY A BIVARIATE MODEL BASED ON COPULAS Ana Isabel Requena Rodríguez Ingeniero de Caminos, Canales y Puertos Director: Luis Mediero Orduña Doctor Ingeniero de Caminos, Canales y Puertos 2015 TESIS DOCTORAL FLOOD FREQUENCY ANALYSIS BY A BIVARIATE MODEL BASED ON COPULAS Autor: Ana Isabel Requena Rodríguez Ingeniero de Caminos, Canales y Puertos Director: Luis Mediero Orduña Doctor Ingeniero de Caminos, Canales y Puertos TRIBUNAL CALIFICADOR Presidente: Secretario: Vocal: Vocal: Vocal: Realizado el acto de defensa y lectura de la tesis doctoral el día de de 2015, en Madrid Calificación: EL PRESIDENTE LOS VOCALES EL SECRETARIO To my mother Acknowledgements I would like to express my gratitude to my thesis director Dr Luis Mediero for encouraging and guiding me during the doctorate, as well as for sharing his knowledge and enthusiasm for travelling and collaborating with people around the world I would also like to thank Professor Luis Garrote for his continuous support, together with the Department of Civil Engineering: Hydraulics, Energy and Environment My sincere thanks to Professor Fateh Chebana for hosting me and offering me the opportunity to work under his guidance at the Institute National de la Reserche Scientifique in Canada, which resulted in the research included in Chapter I would also like to thank Dr Thomas Kjeldsen from the University of Bath and Dr Ilaria Prosdocimi from the Centre for Ecology & Hydrology in the United Kingdom, for the nice collaboration done together resulting in Chapter I would also like to express my gratitude to Andrew Selby for always answering kindly my questions about English grammar and style I thank my colleagues for turning congresses not only about working but also about having a good time with very nice people, which motivates me to continue researching I specially thank the Fundación Carlos González Cruz of the E.T.S.I de Caminos, Canales y Puertos in the Universidad Politécnica de Madrid for its financial support, as the present doctoral thesis would not have been possible without its grant My gratitude also to the financial support given by the Cost Office grant ES0901: European procedures for flood frequency estimation Finally, I would like to thank my family and friends for their unconditional support and for being always there vii Hence, a multivariate analysis on flood variables should rence of a flood and the return period of a flood in terms of these assumptions could not be satisfied by the dependence structure of flood variables Copula models can avoid these difficulties A copula is a function that connects univari variate marginal distributions can be defined independently efit as the study of joint return periods is essential to flood other fields such as finance, they have been only recently applied to model hydrological events such as floods, storms copula families are the most used in modelling flood vari studied variables, which is very important in flood frequency But selection of the copula model that best fits the ob in order not to underestimate the flood risk, as the upper tail study the bivariate flood frequency analysis of peak and vol regarding multivariate flood frequency analysis using copu presented a methodology to classify floods regarding the hy ter level reached at the dam by each flood was estimated, In this paper, a bivariate flood frequency analysis was car currence of floods and a return period defined in terms of to assess the influence of the routing process on them The currence were estimated from the fitted copula In addition, the fitted copula was used to generate a large set of synthetic peak-volume flood pairs in the catchment Synthetic investigate how the flood variables control the routed return are briefly described Thirdly, the procedure to generate syn Identification of the copula that best fits the observations is (ii) parameter estimation method; (iii) goodness-of-fit tests; that best fits the data can be performed through different methods A first group consists of rank-based meth ification of the traditional maximum likelihood method, in IFM methods are non-robust against misspecification of the can be affected if such models not fit adequately Con cumstances in which a goodness-of-fit test based on it is not goodness-of-fit, formal tests are needed to quantify it Sev procedures These procedures are classified in three groups: value associated to the goodness-of-fit test The aim of a goodness-of-fit test is selecting the copula A first idea of the behaviour of the copulas can be drawn can be elaborated by fitting the marginal distributions of the 2.1.3 Goodness-of-fit tests goodness-of-fit test utilised in the present paper The statis Once the goodness of fit of the copula to observed data is focus on the frequency analysis of extreme flood events for the capacity to link extreme flood peaks to extreme volumes This measure can be quantified by the upper tail dependence coefficient, the upper tail coefficient obtained from each copula should be compared to the coefficient estimated from the observed dence coefficient is also obtained in order to be compared quantitatively to the upper tail dependence coefficient of each also works well when this hypothesis is not fulfilled, except dependence in the extremes are identified Because of a good upper tail dependence fit does not mean a good whole data fit, the assessment of the tail dependence is developed at this Different joint return periods estimated by the fitted copula have been developed for the case of a bivariate flood fre inequality is always fulfilled: sociated with the primary return period, as it can be defined tion is completed, the level curves of the fitted copula will be Synthetic hydrographs were estimated from flood peak should be defined in terms of acceptable risk to the structure In addition, the influence of reservoir volume and spill rence of floods regarding the theoretical joint return periods Team, 2012) Specifically, the use of the R package copula 889 m, being the flooded area at the spillway crest height of The dam is an earthfill embankment with a mum annual flood peak ( ) and flood volume ( event with the annual maximum flood peak As this work is based on the flood frequency curve of maximum annual nual flood volumes were assumed to be linked to hydro The marginal distributions for both variables were fitted priate for fitting the annual maximum flood peaks (Jim ity, the Gumbel distribution was considered to fit the data Smirnov test was applied to the flood peak and volume data first step consists of studying the dependence between the The set of copulas considered is classified into three bel copulas belong to the first class, while Galambos, H of-fit criterion, small values are always desirable The results tau method are drawn The figure shows that extreme value Scatter plot of 100 000 values generated from the copulas fitted by the inversion of Kendall’s tau method and the observed data ula that best fits the data that extreme value copulas are slightly worse in terms of fit goodness-of-fit test based on the can lead to significant differences in results It can be seen method is the most appropriate in terms of fitting the bulk , considering the copulas fitted by the inversion of of the studied copulas The coefficients were esti null result, as they have by definition an upper tail depen per tail dependence coefficient of the observed data obtained per tail dependence coefficient of each considered copula As er-von Mises goodness-of-fit test ( Upper tail dependence coefficient of the considered A comparison between a sample generated from the fitted Also, contours of the fitted copula that represent the events estimated from the fitted cific events with a copula value of 0.9 and 0.99 The results fulfil Eq (10) is caused by flood control properties in a reservoir: the higher higher the attenuation of the flood peak In this case, the hydrographs that have more influence on the risk to the dam, or smaller the attenuation of the flood peak In this case, the hydrographs that have more influence on the risk to the dam are characterised by a high flood peak and by the marginal return period of flood peaks (the curves are As available flood control volume and spillway crest significantly with the reservoir volume, showing a severe re comparing it to the probability of occurrence of a flood For that purpose a bivariate flood frequency analysis of flood ula model fitted to observed data In addition, a set of synthetic flood hydrographs was generated from the fitted Gum return period is independent of reservoir and spillway config a large sample through the reservoir has benefits from a con of dam overtopping because this return period is specific for sider the secondary return period, flood hydrographs should It was appreciated that as the available flood control volume pendent on flood peak (the slope of the return period curves played by flood control volume and spillway crest length on flood control volume the dominant variable is hydrograph that represent the probability of occurrence of a flood event structure) provides valuable information about flood control provided by the flood event-based return periods, taking into account the dam characteristics that exert a significant influ The authors are also grateful for the financial contribution made by the COST Office grant ES0901 “European procedures for flood Chowdhary, H., Escobar, L A., and Singh, V P.: Identification of suitable copulas for bivariate frequency analysis of flood peak and flood volume data, Hydrol Res., 42, 193–216, 2011 dependence coefficient: Properties and pitfalls, Insur Math for goodness-of-fit testing in semiparametric models, Ann I H emillard, B.: Goodness-of-fit Proce emillard, B., and Beaudoin, D.: Goodness-of-fit tests flood frequency analysis, Adv Water Resour., 29, 1155–1167, and Garrote, J.: Map of maximum flows of intercommunity ity analysis of hydrological loads for the design of flood con Alvarez, A., and Garrote, L.: Design flood hydrographs from the relationship between flood peak and volume, upper tail dependence coefficient, and 2-copulas with application to rainfall fields, Stoch Environ Res Risk A, 22, 671–688, floods using copulas, J Am Water Resour As., 42, 1549–1564, ` Zhang, L and Singh, V P.: Bivariate flood frequency analysis using Zhang, L and Singh, V P.: Trivariate flood frequency analysis using ... (local) flood frequency analysis; (iii) regional flood frequency analysis; (iv) non-stationary flood frequency analysis; and (v) synthetic data for flood frequency analysis Univariate flood frequency. .. llevar a cabo un análisis de frecuencia de avenidas regional bivariado para proporcionar cuantiles en estaciones sin aforar o para mejorar la estimación de dichos cuantiles en estaciones aforadas;... 2.6 Application of copulas in local flood frequency analysis 41 Regional flood frequency analysis 45 3.1 Univariate regional approach 47 3.2 Multivariate regional approach