## 1. Introduction

[2] In hydrologic engineering, several design and modeling problems are tackled by using a so-called event-based approach. For example, in flood risk assessment, the floodplain corresponding to a given return period *T* is obtained by driving flow routing models with design hydrographs whose shape synthesizes the temporal evolution of the observed flood events and the peak *X _{p}* assumes the value corresponding to a prescribed probability of exceedance or return period resulting from a univariate frequency analysis [e.g.,

*Di Baldassarre et al*., 2010;

*Grimaldi et al*., 2012a;

*Serinaldi and Grimaldi*, 2011]. When the hydrographs do not result from the simulation of rainfall series and a subsequent continuous rainfall-runoff transformation [

*Grimaldi et al*., 2012], they are defined from design hyetographs that synthesize the temporal evolution of rainfall storms and are often characterized by a peak value resulting from a univariate frequency analysis. Both hyetographs and hydrographs are complex objects that are characterized by several properties, such as

*X*, volume

_{p}*V*, duration

*D*, and average intensity

*I*, which can be of interest in practical applications. These properties are commonly treated as random variables owing to the inherent variability of their values and the complexity of the rainfall and runoff processes. In practical applications, this study is often limited to a univariate frequency analysis of

*X*or

_{p}*I*summarized by intensity-duration-frequency curves [

*Chow et al*., 1988] for rainfall and flow-duration-frequency curves for discharge [e.g.,

*Meunier*, 2001]. However, as , and

*I*can all be of interest [

*Salvadori and De Michele*, 2006;

*Karmakar and Simonovic*, 2008, 2009;

*Serinaldi and Grimaldi*, 2011;

*Vandenberghe et al*., 2012], more refined multivariate techniques have been proposed in the literature. In more detail, as these variables can exhibit significant values of indices of association such as the Pearson product moment correlation coefficient , Kendall rank correlation coefficient , or Spearman correlation , they have been deemed suitable to be modeled by joint distributions.

[3] The first attempts relied on the use of the meta-Gaussian framework under the hypothesis that the transformation of the marginal distributions into Gaussian can guarantee that the joint distribution is multivariate Gaussian. As is well known, this hypothesis is hardly ever fulfilled by real-world data; however, the difficulty of splitting the analysis and modeling of the marginal distributions and joint behavior (as well as computing limitations) limited the applications in that early stage. Since the late 1990s, a series of papers by Yue and coworkers [*Yue et al*., 1999; *Yue*, 2000a-2000c, 2001a, 2001b, 2002] has revitalized this research area by showing the application of a set of suitable bivariate non-Gaussian distributions to analyze hyetograph and hydrograph properties. However, the literature on the topic has actually grown fast after the introduction of copulas in geosciences by the seminal paper of *De Michele and Salvadori* [2003]. The up-to-date list of references provided by the International Commission of Statistics in Hydrology of the International Association of Hydrological Sciences acknowledges this research activity (http://www.stahy.org/Activities/STAHYReferences/ReferencesonCopulaFunctiontopic/tabid/78/ Default.aspx).

[4] As copulas allow splitting the analyses of the marginal distribution and the so-called structure of dependence or copula, they provide a virtually infinite set of multivariate distributions with arbitrary marginals and dependence structure that fall outside the field of the meta-Gaussian and metaelliptical multivariate distributions. However, the increased ease of modeling and the simplified inference procedures as well as the availability of free statistical software has led to a focus on the inference procedures and applications overlooking to some extent a more thorough understanding of the variables at hand.

[5] In this study, we attempt to fill this gap. Instead of trying to find the best fitting copula that describes the hyetograph and hydrograph properties, we try to interpret the true nature of the dependence structures exhibited by , and *I* and their generating mechanism. The analysis is based on a large data set of rainfall and streamflow time series in order to support the generality of the results. We use some simple bootstrap techniques that can be easily implemented to repeat the analysis on other data sets without requiring any specific knowledge of the multivariate frequency analysis and copulas. These ad hoc bootstrap algorithms allow checking the working hypotheses by a nonparametric framework free from modeling errors and uncertainty. A large set of time series simulated from universal multifractal processes is also used to further support the analysis and conclusions.

[6] This study is organized as follows. In section 2, some basic definitions of dependence structure and copula-related concepts are briefly recalled in order to introduce the subject of this study. Section 3 introduces the data sets used in the analyses. Sections 4 and 5 present the analyses and the results referring to hyetographs and hydrographs, respectively. In these sections, we also introduce the bootstrap algorithms used to test the working hypotheses deduced from theoretical remarks and the preliminary inspection of the pairwise dependence structures of , and *I*. Without loss of generality, the discussion is focused on one time series of each data set, whereas the results for all time series are provided as supporting information. A discussion about the relationship between marginal distributions and dependence structure resulting from the hypothesized generating processes is provided in section 6 along with the analysis of the synthetic multifractal time series. Conclusions in section 7 close this study.