Probabilistic characterization of the dates of occurrence of annual floods using the von Mises distribution

Abstract

The planning and management of a river´s water resources, and the preparation of non-structural plans for flood damage mitigation, depend on the relationship between the annual maximum flows and their date of occurrence. Such dates, as they occur all year long, can be treated as circular data, whose statistics of mean direction and dispersion or seasonality index define the two parameters of a von Mises distribution (dvM). Such distribution allows the probabilistic characterization of the dates of occurrence of annual floods; that is, it defines their probability distribution function. This study describes the dvM and its maximum likelihood parameter estimation method when the annual data or dates are unimodal and cover the entire year. When the annual dates are concentrated in a period of the year, the dvM is fitted with numerical optimization, via the Rosenbrock algorithm. Finally, when dates of occurrence are bimodal, they are represented by a mixture of three dvMs, which are fitted by means of restricted numerical optimization, using the complex algorithm. As a case study, the dates of occurrence of 777 annual floods registered in 21 hydrometric stations of Hydrological Region No. 10 (Sinaloa), Mexico were processed; detailing seven typical applications of the three types of dvM fittings. The conclusions ratify the dvM, as a probabilistic model of the dates of occurrence of annual floods, either unimodal or bimodal.