A Generalized Pareto intensity‐duration model of storm rainfall exploiting 2‐Copulas

Abstract

[1] Stochastic models of rainfall, usually based on Poisson arrivals of rectangular pulses, generally assume exponential marginal distributions for both storm duration and average rainfall intensity, and the statistical independence between these variables. However, the advent of stochastic multifractals made it clear that rainfall statistical properties are better characterized by heavy tailed Pareto-like distributions, and also the independence between duration and intensity turned out to be a nonrealistic assumption. In this paper an improved intensity-duration model is considered, which describes the dependence between these variables by means of a suitable 2-Copula, and introduces Generalized Pareto marginals for both the storm duration and the average storm intensity. Several theoretical results are derived: in particular, we show how the use of 2-Copulas allows reproducing not only the marginal variability of both storm average intensity and storm duration, but also their joint variability by describing their statistical dependence; in addition, we point out how the use of heavy tailed Generalized Pareto laws gives the possibility of modeling both the presence of extreme values and the scaling features of the rainfall process, and has interesting connections with the statistical structure of the process of rainfall maxima, which is naturally endowed with a Generalized Extreme Value law. Finally, a case study considering rainfall data is shown, which illustrates how the theoretical results derived in the paper are supported by the practical analysis.

Frequency analysis via copulas: Theoretical aspects and applications to hydrological events

“…As a matter of principle, other bivariate distributions (i.e., copulas other than the one cited) could provide valuable fits; note that such tests can easily be carried out using empirical copulas [Nelsen, 1999;De Michele and Salvadori, 2003;De Michele et al, 2004;Favre et al, 2004].…”

“…Here d = 1.294; such a value was estimated exploiting a relation between d and Pearson's linear correlation coefficient r. Although it is easy to calculate, we discourage the practitioners to use r as a means for estimating d; instead, we suggest using other quantities, such as Kendall's or Spearman's rank correlation coefficients. In fact, the latter ones always exist, whereas r depends upon the marginal laws F X , F Y and may not exist at all (e.g., when using suitable Pareto-like marginals (see De Michele and Salvadori [2003, and also section 4.2)).…”

“…In particular, the first case study shows how copulas may simplify the calculations when the marginals are finite mixtures of distributions, and the second one illustrates how different marginals (specifically, a Gumbel and a Gamma law) can be used to model the pair peak flow -peak volume. Indeed, Frank copulas may provide reasonable models for quite a few phenomena [see Hutchinson and Lai, 1990;Nelsen, 1999]; for some recent hydrological applications, see also De Michele and Salvadori [2003] and Salvadori and De Michele [2004].…”

“…In most situations the analysis of the return period involves univariate cases; unfortunately, this may lead to an over/underestimation of the risk associated with a given event (see, e.g., Raynal-Villasenor and Salas [1987] and De Michele et al [2004] for a practical example; see also section 3.1 for a thorough discussion). As a matter of fact, hydrological events are often characterized by the joint behavior of several random variables (RV), and these are usually nonindependent: to mention a few examples, consider the storm duration and intensity [De Michele and Salvadori, 2003;Salvadori and De Michele, 2004], the flood peak and volume , drought events characterized by duration-magnitude-intensity [see Kim et al, 2003, and references therein], and so on. Other significant works concerning the application of joint probability distributions to model extreme hydrological events are by Yue et al [1999], Yue [2000], and Yue et al [2001].…”

“…Incidentally, we observe that all the bivariate distributions present in literature can be described in a straightforward way in terms of suitable copulas (for an exhaustive list of models see, e.g., Hutchinson and Lai [1990] and Nelsen [1999, and references therein]). Concerning the problems of simulating copulas and fitting copulas to experimental data, see, e.g., many of the references in Nelsen [1999]; other references can be found in the bibliographies of De Michele and Salvadori [2003] and Favre et al [2004].…”