Probabilistic analysis of coincident sums of precipitation at two measurement stations. Introduction to the method and an example

Wroclaw University of Environmental and Life Sciences, Department of Mathematics, ul.Grunwaldzka 53, 50-357 Wrocław, Poland

^* Corresponding author: gabriela.biel@upwr.edu.pl

Abstract

This article proposes the use of copula (copula function) for the purpose of two-dimensional analysis of the sums of precipitation as measured with a Hellman rain-gauge. The sums of precipitation are characterized by a two-dimensional random variable: the sum of uninterrupted sequence of rainfalls which were measured in Jelcz-Laskowice and the corresponding (coincident) sum of precipitation at the Botanical Garden in Wrocław. Several problems occur from the very start: debonding from time and lack of precipitation on one of stations. For the purpose of greater precision and correction it should be stated that in order to apply the two-dimensional copula functions we will use a random vector determining the sum of uninterrupted sequences of rainfalls at two simultaneous stations. In that way, this will not be a characteristics of the phenomenon, but rather the definition of two-dimensional random variable under analysis. Data for analysis has been derived from observational logs of the Institute of Meteorology and Water Management, branch in Wrocław. The results obtained in years 1980-2014 were subject to analysis. The aim of the work was to find the best two-dimensional probability distribution of a random variable (Opad_Jelcz, Opad_Ogród). The following were analysed from among the known copulas: the Archimedean copulas (the Gumbel copula, the Frank copula and the Clayton copula) and the Gaussian elliptical copula. The study of fitting of copulas to observed variables was carried out using the Spearmann's rank correlation coefficient and the best fitting was obtained for the Frank's copula.