The use of geometric and gamma-related distributions for frequency analysis of water deficit

Abstract

This paper presents an approach to perform statistical frequency analysis of water deficit duration and severity using respectively the geometric and exponential distributions. Monthly mean water discharges are compared to a given threshold and classified in two mutually exclusive ways. This leads to a two state random variable such that: a success represents the absence of a water deficit event (mean monthly discharge exceeds threshold), and a failure, a water deficit event (mean monthly discharge is below threshold). If we suppose that this random variable gives rise to a Markov process of order 1, then the duration of a water deficit event X (consecutive months in deficit) will have a geometric distribution. In turn, the summation of discharges in deficit will give the severity of a water deficit event which can be represented by a one-parameter exponential distribution. The threshold or base level is taken as a percentile of the observed mean discharges of a given month. This base level, which varies from month to month, can be viewed as the limit of an acceptable deficit (or energetic failure) associated to a given empirical probability of being in deficit. The second step of the approach is to estimate the value of the parameter for each distribution using the maximum likelihood method. Expressions for the estimator of a given percentile, $$\hat x_q $$ , as well as its variance are deduced. Finally, the presented models are applied to observed data.