G. Salvadori

On the derived flood frequency distribution: analytical formulation and the influence of antecedent soil moisture condition

Abstract In this paper we present an analytical formulation of the derived distribution of peak flood and maximum annual peak flood, starting from a simplified description of rainfall and surface runoff processes, and we show how such a distribution is useful in practical applications. The assumptions on rainfall dynamics include the hypotheses that the maximum storm depth has a Generalized Pareto distribution, and that the temporal variability of rainfall depth in a storm can be described via power–law relationships. The SCS-CN model is used to describe the soil response, and a lumped model is adopted to transform the rainfall excess into peak flood; in particular, we analyse the influence of antecedent soil moisture condition on the flood frequency distribution. We then calculate the analytical expressions of the derived distributions of peak flood and maximum annual peak flood. Finally, practical case studies are presented and discussed.

A multivariate copula‐based framework for dealing with hazard scenarios and failure probabilities

This paper is of methodological nature, and deals with the foundations of Risk Assessment. Several international guidelines have recently recommended to select appropriate/relevant Hazard Scenarios in order to tame the consequences of (extreme) natural phenomena. In particular, the scenarios should be multivariate, i.e., they should take into account the fact that several variables, generally not independent, may be of interest. In this work, it is shown how a Hazard Scenario can be identified in terms of (i) a specific geometry and (ii) a suitable probability level. Several scenarios, as well as a Structural approach, are presented, and due comparisons are carried out. In addition, it is shown how the Hazard Scenario approach illustrated here is well suited to cope with the notion of Failure Probability, a tool traditionally used for design and risk assessment in engineering practice. All the results outlined throughout the work are based on the Copula Theory, which turns out to be a fundamental theoretical apparatus for doing multivariate risk assessment: formulas for the calculation of the probability of Hazard Scenarios in the general multidimensional case ( d≥2) are derived, and worthy analytical relationships among the probabilities of occurrence of Hazard Scenarios are presented. In addition, the Extreme Value and Archimedean special cases are dealt with, relationships between dependence ordering and scenario levels are studied, and a counter-example concerning Tail Dependence is shown. Suitable indications for the practical application of the techniques outlined in the work are given, and two case studies illustrate the procedures discussed in the paper.

From generalized Pareto to extreme values law: Scaling properties and derived features

Given the fact that, assuming a generalized Pareto distribution for a process, it is possible to derive an asymptotic generalized extreme values law for the corresponding maxima, in this paper we consider the theoretical relations linking the parameters of such distributions. In addition, temporal scaling properties are shown to hold for both laws when considering proper power-law forms for both the position and the scale parameters; also shown is the relation between the scaling exponents of the distributions of interest, how the scaling properties of one distribution yield those of the other, and how the scaling features may be used to estimate the parameters of the distributions at different temporal scales. Finally, an application to rainfall is given.

Multivariate Extreme Value Methods

Multivariate extremes occur in several hydrologic and water resources problems. Despite their practical relevance, the real-life decision making as well as the number of designs based on an explicit treatment of multivariate variables is yet limited as compared to univariate analysis. A first problem arising when working in a multidimensional context is the lack of a “natural” definition of extreme values: essentially, this is due to the fact that different concepts of multivariate order and failure regions are possible. Also, in modeling multivariate extremes, central is the issue of dependence between the variables involved: again, several approaches are possible. A further practical problem is represented by the construction of multivariate Extreme Value models suitable for applications: the task is indeed difficult from a mathematical point of view. In addition, the calculation of multivariate Return Periods, quantiles, and design events, which represent quantities of utmost interest in applications, is rather tricky. In this Chapter we show how the use of Copulas may help in dealing with (and, possibly, solving) these problems.

Multifractal analysis of wide dynamic range neuron discharge profiles in normal rats and in rats with sciatic nerve constriction

Using normal rats and rats with a chronic constriction injury of the sciatic nerve (injured, model of Bennett-Xie) we investigated the possibility of classifying, by statistical tools, the temporal sequences of neuronal discharges during different noxious and non-noxious stimuli. An analysis was made of both the distribution of the inter-spike intervals and the temporal density of spike trains, the latter being studied within the framework of stochastic universal multifractals, to allow the identification of different random processes involved in the discharge distributions through the Lévy index alpha. The statistical analysis shows that the parametrization based on the Lévy index seems able to discriminate between different noxious stimuli (mechanical pinching and thermal), both in normal and injured animals. Furthermore, comparing normal and injured animals, although the spontaneous basal and non-noxious stimuli (brushing) evoked activities presented different frequencies, these seem to have the same multifractal structure, while the corresponding statistics of the inter-spike intervals are quite different. This information might be relevant to the understanding of a code of neuronal firing and to the modelling of temporal patterns in acute and chronic noxious signals.

Quantification of the environmental structural risk with spoiling ties: is randomization worthwhile?

Many recent works show that copulas turn out to be useful in a variety of different applications, especially in environmental sciences. Here the variables of interest are usually continuous, being times, lengths, weights, and so on. Unfortunately, the corresponding observations may suffer from (instrumental) adjustments and truncations, and eventually may show several repeated values (i.e., ties). In turn, on the one hand, a tricky issue of identifiability of the model arises, and, on the other hand, the assessment of the risk may be adversely affected. A possible remedy is to adopt suitable randomization procedures: here three different strategies are outlined. The goal of the work is to carry out a simulation study in order to evaluate the effects of the randomization of multivariate observations when ties are present. In particular, it is investigated whether, how, and to what extent, the randomization may change the estimation of the structural risk: for this purpose, a coastal engineering example will be used, as archetypical of a broad class of models and problems in engineering applications. Practical advices and warnings about the use of randomization techniques are hence given.

Multifractal objective analysis of seveso ground pollution

We analyse the Dioxin (TCDD) pollution of the Seveso (Milan, Italy) territory, seeking to statistically parametrize it in terms of Universal Multifractals. The data set contains the measurements collected from 1976 up to 1981. We apply the Double Trace Moment (DTM) technique in order to estimate both α (the degree of multifractality) and C 1 (the codimension of the mean field) and (with the help of spectral analysis) we also calculate H (the degree of non‐conservation of the process). We then discuss the effects introduced by statistical undersampling and network sparseness and provide a way to statistically correct for these effects. We conclude that the ground distribution of Dioxin shows clear multifractal features and can be classified as an unconditionally hard universal multifractal process.

Conditional risk based on multivariate hazard scenarios

We present a novel methodology to compute conditional risk measures when the conditioning event depends on a number of random variables. Specifically, given a random vector

A distributional multivariate approach for assessing performance of climate-hydrology models

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