Drought length properties for periodic-stochastic hydrological data

Extreme droughts may be characterized by their duration, severity (magnitude or intensity), spatial extent, and frequency or return period. Comparing the time series of water supply and water demand and analyzing droughts based on the theory of runs may determine these characteristics. This study is focused on drought analysis where the underlying water supply process is periodic stochastic, such as for monthly streamflows. The probability mass function (pmf) of drought length and associated low-order moments are derived assuming a periodic simple Markov chain. The derived pmf allows estimating the occurrence probability of droughts of a given length and its return period. The applicability of the drought formulations has been illustrated using a variety of water supply series such as monthly and weekly precipitation, monthly streamflows, the Palmer hydrologic drought index, and the standardized precipitation index. The results obtained confirm the validity of the analytical derivations for drought lengths and associated return periods. The overall conclusion of the study is that simple definitions of droughts enables one characterizing droughts using stochastic approaches and analytical derivations. They are particularly useful for drought analysis because the limited hydrologic records that are generally available do not allow observing many drought events of a particular duration and, in fact, extremely long droughts may not even be observable from the historical sample. This hinders the applicability of an inferential approach for finding the probability distributions of drought lengths and their associated return periods because it is either impractical or not feasible.