Maximum entropy probability distributions and statistical - stochastic modelling of rainfall

S.M. Papalexiou, Maximum entropy probability distributions and statistical - stochastic modelling of rainfall, PhD thesis, 188 pages, Department of Water Resources and Environmental Engineering – National Technical University of Athens, Athens, June 2013.



Three main issues are examined: (a) the potential to use a theoretical principle, namely the principle of maximum entropy, as a basis for formulating and selecting probabilistic distributions suitable for geophysical variables and more specifically for rainfall, (b) the probabilistic-statistical analysis of daily rainfall and of extreme daily rainfall on a global scale, and (c) the stochastic structure of daily rainfall at fine temporal scales. The main goal of this research is to formulate simple yet fundamental and of wide interest questions, mainly regarding the statistical-stochastic nature of rainfall, and try to provide answers not only of theoretical but mostly of practical value. Regarding the principle of maximum entropy the emphasis is given on formulating and logically justifying simple constraints to be used along with the maximization of the classical definition of entropy, i.e., the Boltzmann-Gibbs-Shannon entropy, that will lead suitable probability distributions for rainfall, or more generally, for geophysical processes. Regarding the statistical analysis of daily rainfall, three different aspects are examined. First, the seasonal variation of daily rainfall is investigated focusing on the properties of its marginal distribution. A massive empirical analysis is performed of more than 170 000 monthly daily rainfall records from more than 14 000 stations from all over the globe aiming to answer two major questions: (a) which statistical characteristics of daily rainfall vary the most over the months and how much, and (b) whether or not there is a relatively simple probability model that can describe the nonzero daily rainfall at every month and every area of the world. Second, the distribution tail of daily rainfall is studied, i.e., the distribution’s part that describes the extreme events. More than 15 000 daily rainfall records are analysed in order to test the performance of four common distribution tails that correspond to the Pareto, the Weibull, the Lognormal and the Gamma distributions aiming to find out which of them better describes the behaviour of extreme events. Third, the annual maximum daily rainfall is analysed. The annual maxima time series from more than 15 000 stations from all over the world are extracted and examined in order to answer one of the most basic questions in statistical hydrology, i.e., which one of the three Extreme Value distributions better describes the annual maximum daily rainfall. Finally, regarding the stochastic properties of rainfall at fine temporal scales, a unique dataset, comprising measurements of seven storm events at a temporal resolution of 5-10 seconds, is studied. The question raised and attempted to be answered is if it is possible for a single and simple stochastic model to generate a plethora of temporal rainfall patterns, as well as to detect the major characteristics of such a model.

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