An uncertain journey around the tails of multivariate hydrological distributions


  • Francesco Serinaldi

    Corresponding author
    1. School of Civil Engineering and Geosciences, Newcastle University, Newcastle Upon Tyne, UK
    2. Willis Research Network, London, UK
    • Corresponding author: F. Serinaldi, School of Civil Engineering and Geosciences, Newcastle University, Newcastle Upon Tyne NE1 7RU, UK. (

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  • The copyright line for this article was changed on 9 October 2014 after original online publication.


[1] Moving from univariate to multivariate frequency analysis, this study extends the Klemeš' critique of the widespread belief that the increasingly refined mathematical structures of probability functions increase the accuracy and credibility of the extrapolated upper tails of the fitted distribution models. In particular, we discuss key aspects of multivariate frequency analysis applied to hydrological data such as the selection of multivariate design events (i.e., appropriate subsets or scenarios of multiplets that exhibit the same joint probability to be used in design applications) and the assessment of the corresponding uncertainty. Since these problems are often overlooked or treated separately, and sometimes confused, we attempt to clarify properties, advantages, shortcomings, and reliability of results of frequency analysis. We suggest a selection method of multivariate design events with prescribed joint probability based on simple Monte Carlo simulations that accounts for the uncertainty affecting the inference results and the multivariate extreme quantiles. It is also shown that the exploration of the p-level probability regions of a joint distribution returns a set of events that is a subset of the p-level scenarios resulting from an appropriate assessment of the sampling uncertainty, thus tending to overlook more extreme and potentially dangerous events with the same (uncertain) joint probability. Moreover, a quantitative assessment of the uncertainty of multivariate quantiles is provided by introducing the concept of joint confidence intervals. From an operational point of view, the simulated event sets describing the distribution of the multivariate p-level quantiles can be used to perform multivariate risk analysis under sampling uncertainty. As an example of the practical implications of this study, we analyze two case studies already presented in the literature.