Superstatistical distribution of daily precipitation extremes: A worldwide assessment

Abstract

Maximum annual daily precipitation is a fundamental hydrologic variable that does not attain asymptotic conditions. Thus the classical extreme value theory (i.e., the Fisher-Tippett’s theorem) does not apply and the recurrent use of the Generalized Extreme Value distribution (GEV) to estimate precipitation quantiles for structural-design purposes could be inappropriate. In order to address this issue, we first determine the exact distribution of maximum annual daily precipitation starting from a Markov chain and in a closed analytical form under the hypothesis of stochastic independence. As a second step, we formulate a superstatistics conjecture of daily precipitation, meaning that we assume that the parameters of this exact distribution vary from a year to another according to probability distributions, which is supported by empirical evidence. We test this conjecture using the world GHCN database to perform a worldwide assessment of this superstatistical distribution of daily precipitation extremes. The performances of the superstatistical distribution and the GEV are tested against data using the Kolmogorov-Smirnov statistic. By considering the issue of model’s extrapolation, that is, the evaluation of the estimated model against data not used in calibration, we show that the superstatistical distribution provides more robust estimations than the GEV, which tends to underestimate (7–13%) the quantile associated to the largest cumulative frequency. The superstatistical distribution, on the other hand, tends to overestimate (10–14%) this quantile, which is a safer option for hydraulic design. The parameters of the proposed superstatistical distribution are made available for all 20,561 worldwide sites considered in this work.

On the Exact Distribution of Correlated Extremes in Hydrology

“…In Figures we may observe that for all daily rainfall data sets the magnitudes of extreme events taken from excesses of a low threshold (the 5th percentile of the nonzero sample) can be considered independent and identically distributed, and this is consistent with the results shown in the literature using different approaches (see, e.g., Marani & Ignaccolo, ; Zorzetto et al, ; De Michele & Avanzi, ). In addition, we may notice that the classical model of POT analyses assuming Poisson occurrences (see equation ) seems to be appropriate to study rainfall extremes.…”

Advances in Deriving the Exact Distribution of Maximum Annual Daily Precipitation

“…First, Wilson and Toumi [25] provided some physical argumentations in favor of the probability distribution of daily non-zero precipitation being stretched exponential, also known as Weibull distribution. This seminal result is coherent with later findings by [10,17,22,[26][27][28] about both the distribution of the daily amount and the distribution of daily values above a given threshold. Also, Porporato et al [29] obtained a distribution of daily non-zero precipitation with a stretched exponential tail by compounding an exponential distribution with a parameter having a Gamma distribution.…”

“…A (non-exhaustive) list of papers published in the period 2000-2015 that exclusively used the GEV, or its three asymptotic laws, to describe the behavior of maximum annual daily precipitation is reported in [10]. Some studies have also investigated which of these three asymptotic laws is the most appropriate to represent the statistical variability of maximum annual daily precipitation [8,11].…”

“…The violation of the asymptotic theory also raises doubts about the extrapolation capability of the fitted GEV model to values (quantiles) associated to high values of probability or return period [10,17]. From this standpoint, an increasing number of studies have recently considered a pool of distributions to model maximum precipitation, including several families of asymptotic-and standard-type (see Table S2 of [10]). Results show that the asymptotic laws often do not yield the best fit according to goodness-of-fit tests and/or statistical indicators.…”

Spatial Dependency in Nonstationary GEV Modelling of Extreme Precipitation over Great Britain

“…The -values, which indicates the goodness of fitness, are all very close to 1.0, which indicate a failure on rejecting the null hypothesis, i.e., the AMDR follows the GEV distribution at 5% significance level. It should be noted that the high p-values cannot be used to confirm that the AMDR follows the GEV 180 distribution, however, we follow other researchers here to use them to indicate that the AMDR is highly likely to follow the GEV distribution (De Michele and Avanzi, 2018;Hasan et al, 2012;Machiwal and Jha, 2008;Martin, 2013). Meanwhile, the Diff is applied to identify the best performance and uncertainty on nonstationary-based assumption.…”