Thesis title: A study on probabilistic approaches to describe rainfall extreme events
Rainfall is the main component of the hydrological cycle. It is vital for both human society and ecosystems, as it provides most of the fresh water on Earth, and it is also the primary input for several hydrological applications.
Despite its countless benefits, rainfall can also produce dramatic impacts. It is, indeed, the main responsible of many hazards, such as droughts and floods, that every year cause loss of human lives, destruction of built up areas, infrastructures and crops, and billion euros of damages all over the world. The impacts of these catastrophic events are expected to increase in the near future, due to the growth of population, uncontrolled urbanization, unsustainable land management policies and global warming.
For these reasons, understanding, modelling and predicting rainfall, especially its extreme events, is paramount for both scientists and politicians. Indeed, several countries are investing many economic resources to develop and apply mitigation strategy plans to reduce the devastating impacts of such events. The design of such mitigation strategies requires reliable predictions of rainfall values. And, since deterministic long-term rainfall predictions are not feasible, using a probabilistic approach is a necessity and its based on observed data of historical events. Therefore, the correct description of the statistical behaviour of extreme events is crucial to avoid failures in mitigation plans.
Given accurate rainfall data, the study of extreme events represents the first step for most of hydrological studies. The extreme value analysis basically consists on fitting probability distribution functions to observed data and extrapolating the tails of the distribution related to low exceedance probabilities. The focus on the upper tail, which controls the magnitude and frequency of extremes, allows giving a quantitative information on how extreme are the extremes, to retrieve the loading force for most of hydraulic structures and infrastructures. The main issue in this methodology lies in the way in which probabilities are calculated and in the level of accuracy associated with such probabilities. Since in practice the true probability distribution of the phenomenon (i.e., the parent distribution) is unknown, the question that naturally arises is: which distribution better describes the tail? The question is crucial, since a naive choice may lead to serious under- or over-estimation of rainfall amounts for assigned return periods. Additionally, selecting the best method to estimate the distribution's parameters may help to reduce the bias of rainfall estimates, and thus to obtain risk estimates of satisfactory accuracy.
To answer this question, in this thesis, we explore the behaviour of extremes in a highly vulnerable territory, that is Italy, where the rainfall data accessibility is limited due to strict regulation and bureaucracy. We employ two rainfall datasets (i.e., a gauged-base and a satellite-based) to highlight the effects of bad practice in estimating extremes and their repercussions in the definition and management of flood risk and droughts characteristics. Specifically, we focus our attention on two extreme events, extreme rainfall and droughts, investigating the reliability of the most commonly used distributions to model them, i.e., Gumbel and Gamma, respectively.
The results for the extreme rainfall case indicate that, for both datasets, the use of heavy-tailed distributions, thanks to their ability in detecting extremes, are the best choice to overcome possible under-estimation due to the use of light-tailed distributions. The bias in rainfall quantiles emerged mainly for high return periods, while for low return periods (up to 40 years) are negligible. We also observed a considerable difference of the rainfall quantiles for high return periods when comparing two common sample selection methods. We then explore the reliability of one of the most commonly used distribution to describe the extremes, that is the Generalized Extreme Value (GEV) distribution. Our findings reveal that, due to mathematical artifact in its formulation and to sample variation, the shape parameter of the GEV should be restricted only to positive values (we name this variant GEV+) to avoid that the GEV may converge to an upper bounded distribution (i.e., the reverse Weibull). We also test the performance of a non-extreme value distribution, that is the Burr type XII (BrXII), which provides promising results in describing extremes. Finally, our results on the drought case, reveal that the Gamma distribution is not always the best model to describe monthly rainfall time series. Thus, using the best fitted probability distribution provides different results in terms of frequency and characteristics of drought events.