Daily precipitation with stretched-exponential tails could explain the statistics of observed annual maxima

The parent distribution of daily precipitation is usually not known, and the exceedance probability of extremes is described using a Generalized Extreme Value distribution (GEV) fitting the annual maxima. However, knowing the parent distribution would allow us to use ordinary statistics to describe extremes, with two advantages: (i) a decreased parameter estimation uncertainty; (ii) the possibility to establish direct relations between ordinary and extreme events. Recent studies suggest that daily precipitation could have Weibull tails, meaning that the probability of exceeding large values decrease as a stretched exponential. Here, we exploit a global dataset of long and quality-controlled continuous rain gauge records (~8,000 stations, ≥50 complete years) to investigate this question.

We find that the observed annual maxima are likely samples from Weibull tails in ~88% of the stations worldwide. On average, ~36% of the wet days belong to these tails. We find a strong climatic dependence in their definition, with smaller portions of data in the Weibull tails in central Europe, US east coast and southern Australia. We then generate synthetic records with the same characteristics (yearly number of wet days, Weibull tails with the same shape parameter) and increasing lengths (10-110 years); we estimate the corresponding GEV shape parameters and contrast them with the ones obtained from very long annual maxima records (~15,000 stations, 40-163 years; Papalexiou and Koutsoyiannis, 2013, https://doi.org/10.1029/2012WR012557). We show that parent distributions with Weibull tails well explain the properties of the observed GEV shape parameters. These GEV tails (type-III, Frechet) are heavier than the limiting GEV for Weibull parent distributions (type-I, Gumbel); this implies a pre-asymptotic behavior: the average yearly number of wet days (globally, n=100±50) is not large enough to fulfill the asymptotic assumption (n~∞) of extreme value theory. Contrasting our results with generalized Pareto tails, as predicted by extreme value theory for high threshold exceedances, we find that the two models are equivalent within the observational uncertainties; the Weibull model, however, describes a portion of data which is, on average, 7 times larger.