## 1. Introduction and Literature Review

[2] Extreme events, such as floods, storms and droughts, have serious economic, environmental and social consequences. It is hence of high importance to develop the appropriate models for the prediction of such events both at gauged and ungauged sites. Local and regional frequency analysis (FA) procedures are commonly used tools for the analysis of extreme hydrological events. The objective of regional frequency analysis (RFA) is to transfer information from gauged sites to an ungauged target site within a homogeneous region.

[3] Generally, hydrological events are characterized by several correlated variables. For instance, floods are described through their volume, peak and duration [*Ashkar*, 1980; *Yue et al.*, 1999; *Ouarda et al.*, 2000; *Yue*, 2001; *Shiau*, 2003; *De Michele et al.*, 2005; *Zhang and Singh*, 2006; *Chebana and Ouarda*, 2009]. These studies have pointed out the importance of jointly considering all these variables. Depending on data sources and the number of variables that characterize the event, frequency analysis can be divided into four classes: univariate-local, univariate-regional, multivariate-local and multivariate-regional. The first two classes have been extensively studied [see, e.g., *Stedinger and Tasker*, 1986; *Burn*, 1990; *Hosking and Wallis*, 1993; *Durrans and Tomic*, 1996; *Nguyen and Pandey*, 1996; *Alila*, 1999, 2000; *Ouarda et al.*, 2001, 2006; *Chebana and Ouarda*, 2008]. Recently, increasing attention has been given to multivariate-local FA by, e.g., *Yue et al.* [1999], *Yue* [2001], *Shiau* [2003], *De Michele et al.* [2005], *Zhang and Singh* [2006], and *Chebana and Ouarda* [2009]. However, much less attention is given to multivariate-regional FA. In this category, we find few references such as *Ouarda et al.* [2000] and *Chebana and Ouarda* [2007].

[4] Justifications for adopting the multivariate framework to treat extreme events were discussed in several references. In bivariate FA, *Yue et al.* [1999] concluded that single-variable hydrological FA can only provide limited assessment of extreme events. A better understanding of the probabilistic characteristics of such events requires the study of their joint distribution. It was also outlined by *Shiau* [2003] that multivariate FA requires considerably more data and more sophisticated mathematical analysis. Univariate FA can be useful when only one random variable is significant for design purposes or when the two random variables are less dependent. However, a separate analysis of random variables cannot reveal the significant relationship between them if the correlation is an important information in the design criteria. Therefore, it is of importance to jointly consider all the random variables that characterize the hydrological event.

[5] Three main elements are treated in multivariate-local FA literature: (1) explaining the usefulness and importance of considering the multivariate framework, (2) modeling extreme events by fitting the appropriate copula and marginal distributions, and estimating the corresponding parameters, and (3) defining bivariate return periods. However, despite the importance of quantiles in FA, the literature on multivariate-local FA did not specifically address the estimation of multivariate quantiles. Recently, *Chebana and Ouarda* [2009] introduced the notion of multivariate quantile in hydrological FA.

[6] Regional FA is generally composed by two main steps: regional delineation and extreme quantile estimation [see, e.g., *Groupe de recherche en hydrologie statistique* (*GREHYS*), 1996a]. In the multivariate context, the delineation step was treated by *Chebana and Ouarda* [2007] where multivariate discordancy and homogeneity statistical tests were proposed. In univariate-regional FA different quantile estimation methods were proposed in the literature, such as the index flood method and regressive models [see *GREHYS*, 1996a, 1996b]. As a natural continuation of the study by *Chebana and Ouarda* [2007] and in order to present a complete multivariate-regional FA framework, an estimation procedure in the multivariate context is presented in this paper. The present procedure is an extension of the index flood model to the multivariate context.

[7] The multivariate index flood model is based on two main concepts: multivariate quantile curves and the notion of copulas. The univariate index flood model aims to obtain an estimation of quantiles at ungauged sites using data (and hence quantiles) from sites within a specified region. The objective of FA is the quantile estimation which can be obtained through the cumulative distribution function or the density function. The multivariate quantile version adopted in this paper is a curve composed of combinations of the variables corresponding to the same risk. Copulas are employed in order to model the dependence between the variables describing the event.

[8] The paper is organized as follows. In section 2, we present some background elements required for the development of the methodology: the index flood model and multivariate quantile curves. In section 3, we present the multivariate index flood model. In section 4 a simulation study is carried out to evaluate the performance of the proposed model with an adaptation of the procedure to flood events. Results and discussions are reported in section 5, and the conclusions are presented in section 6.