Journal of Hydrologic Engineering

This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.

The use of copulas in hydrology is quite recent, and many important results still are to be discovered and/or derived. In this work we present some recent advances in hydrological modeling that exploit copulas. These include: (1) the calculation of conditional probabilities and their use in bivariate simulation; (2) the calculation of the level curves of joint distributions; (3) the calculation of the return periods of bivariate events, both in the conditional and unconditional cases; (4) the definition and calculation of the secondary return period; (5) a trivariate model for the temporal structure of the sequence of storms; (6) the statistical calculation of the storm depth; and (7) the calculation of the convolution variance. Several applications to hydrological data are shown. The models presented here may have important implications in many areas of water resources and hydrologic systems.

This paper discusses the bivariate modeling of extreme tails of correlated hydrological random variables. We take a copula approach and model the dependence structure independently of the marginal distributions. We apply results from the classical extreme value theory to choose marginal distributions for excesses of high thresholds and consider six copula families to capture the dependence structure of these excesses. While copulas can differ somewhat in the degree of association that they provide, differences in which part of the distribution this association is more pronounced can be substantial. We discuss certain pertinent properties of copulas and give some insight to assist the practitioner in their selection. We examine the effects of model misspecification and the impact of the chosen method of estimation, targeting the estimated quantities frequently used by hydrologists. A simulation study shows not only the dangers of improper copula selection, but also the possible benefits of using a bivariate approach to estimate univariate quantities. We apply the methodology to the study of low-flow events and analyze two Canadian hydrometric data sets.

This paper highlights the importance of taking into account the tail dependence in the context of bivariate frequency analysis based on copulas. Three nonparametric estimators of the tail-dependence coefficient are compared by simulations with seven families of copulas. We choose the two estimators most adapted to a bivariate frequency analysis of the annual maximum flows and the corresponding flow hydrograph volumes of the Loire River (France). In this example, the bivariate return period and the conditional density of the volume given that the flow exceeds a given threshold are computed. The results show, as can be expected, that out of the seven copula families tested, five overestimate the return periods of correlated extreme events. These results bring to the forefront the importance of taking into account the tail dependence in order to estimate the risk adequately.

The dependence between temporal sampling error in satellite-derived rainfall estimates and rainfall rate is of scientific and practical interest. We explore the use of copulas to construct the needed joint distribution between the sampling error and the corresponding rainfall rate. Our approach is to first estimate the marginal distribution functions in a parametric way, and then use these with a number of copula functions in search of the one most appropriate. We use maximum likelihood to estimate the parameters of the copulas. We select the best-fitted parametric copula function as the one that gives the largest likelihood. Our findings have important implications for the interpretation and propagation studies of remote sensing precipitation uncertainties.

Joint distributions of rainfall intensity, duration, and depth or those of rainfall intensity and duration, rainfall depth and duration, and rainfall intensity and depth are important in hydrologic design and floodplain management. Considering the dependence among rainfall intensity, depth, and duration, multivariate rainfall frequency distributions have been derived using one of three fundamental assumptions. Either the rainfall intensity, duration, and depth have been assumed independent, or they each have the same type of marginal probability distribution or they have been assumed to have the normal distribution or have been transformed to have the normal distribution. In reality, however, rainfall intensity, duration, and depth are dependent, do not follow, in general, the normal distribution, and do not have the same type of marginal distributions. This study aims at deriving trivariate rainfall frequency distributions using the Gumbel–Hougaard copula which does not assume the rainfall variables to be independent or normal or have the same type of marginal distributions. The trivariate distribution is then employed to determine joint conditional return periods, and is tested using rainfall data from the Amite River Basin in Louisiana.

In multivariate frequency analysis, when the number of variables increases, different mutual structures of dependence among the analyzed quantities are usually observed. To correctly model this behavior, a very flexible joint distribution function is needed. A quite simple approach to build such distributions is based on the copula function. Precisely, using the so-called fully nested or asymmetric Archimedean copulas, it is possible not only to focus attention on the structures of dependence overlooking the margins—a property common to all copulas—but also to analyze more complex asymmetric structures of dependence. The aim of this paper is to describe the inference procedure to carry out a trivariate frequency analysis via asymmetric Archimedean copulas. The writers highlight the differences between the symmetric Archimedean copulas and asymmetric ones, show the inference procedure, and carefully describe some goodness- of-fit tests proposed in the literature to choose the best fitting model when one uses the fully nested Archimedean copulas. Finally, the methodology is applied to observed hydrological data, and results are shown and commented on.

Using the Gumbel–Hougaard copula, trivariate distributions of flood peak, volume, and duration were derived, and then conditional return periods were obtained. The derived distributions were tested using flood data from the Amite River Basin in Louisiana. A major advantage of the copula method is that marginal distributions of individual variables can be of any form and the variables can be correlated.