Multivariate Flood Frequency Analysis Using Copula with Parametric and Nonparametric Marginal Distribution Function

Many variables under study in hydrology are continuous and random; hence, it necessitates using probability and statistics sciences to study them. In a specific classification, variables are categorized to be either explanatory or response variable. An explanatory variable is defined as a variable which explains or predicts changes in the value of another variable, while a response variable is a variable whose prediction of its changes is desired under the effect of other variables. Due to the mentioned definition, a response variable may depend on one or more explanatory variables. Therefore, in the first case the response variable is intrinsically univariate and in the second case is intrinsically multivariate. If it is known that a hydrological response variable is intrinsically multivariate, multivariate statistical approaches must be employed, especially in the case of dependency among explanatory variables, because it has been widely warned that implementing univariate statistical approaches may result in over/under estimations. According to the fact that flood is an intrinsically multivariate event, it is essential to employ multivariate approaches to analyze it. The most important characteristics of flood are peak discharge, volume and duration.A traditional approach in multivariate analyses is to use classical multivariate distribution functions with parametric marginal distribution functions. However, both classical multivariate distribution functions and parametric distribution functions face substantial limitations. Among the limitations attributed to classical multivariate distribution functions, one may refer to the necessity of identifying marginal distribution functions and their parameters and equality of the kind of marginal distribution functions as the most important limitations. Also in the use of parametric distribution functions for marginal variables, an assumed distribution function is used to describe the distribution of data, while perhaps the assumed distribution function does not accurately describe the real distribution of data. The aim of this article is to establish joint distributions of different combinations of flood characteristics and corresponding return periods. Hence, firstly marginal distribution functions are chosen among parametric distribution functions and non-parametric distribution functions, which are not restricted to estimation of some parameters. Then joint analyses of flood variables are performed using copulas, which do not confront limitations of classical multivariate distributions. Finally, having found and appropriate copula for each combination of flood characteristics, joint return periods are calculated and contour plot of joint return periods are plotted. Joint return periods of flood characteristics can be used by water resources decision makers and engineers as a hydraulic design criterion and provide useful information for risk analysis. In this article, joint analyses of flood variables are performed using copulas, which do not confront limitations of classical multivariate distributions, such that marginal distribution functions are chosen among parametric distribution functions and non-parametric distribution functions, which are not restricted to estimation of some parameters.It should be mentioned that the R language has been utilized as the primary tool in order to perform calculations and draw diagrams.