Abstract
As determining the probability of the exceedance of maximum precipitation over a specified duration is critical to hydrotechnical design, particularly in the context of climate change, a model was developed to perform a frequency analysis of maximum precipitation of a specified duration. The PMAXΤP model (Precipitation MAXimum Time (duration) Probability) harbors a pair of computational modules fulfilling different roles: (i) statistical analysis of precipitation series, and (ii) estimation of maximum precipitation for a specified duration and its probability of exceedance. The input data consist of homogeneous 30-element series of precipitation values for 16 different durations: 5, 10, 15, 30, 45, 60, 90, 120, 180, 360, 720, 1080, 1440, 2160, 2880, and 4320 min, obtained through Annual Maximum Precipitation (AMP) and Peaks-Over-Threshold (POT) approaches. The statistical analysis of the precipitation series includes: (i) detecting outliers using the Grubbs-Beck test; (ii) checking for the random variable’s independence using the Wald-Wolfowitz test and the Anderson serial correlation coefficient test; (iii) checking the random variable’s stationarity using nonparametric tests, e.g., the Kruskal-Wallis test and Spearman rank correlation coefficient test for trends of mean and variance; (iv) identifying the trend of the random variables using correlation and regression analysis, including an evaluation of the form of the trend function; and (v) checking for the internal correlation of the random variables using the Anderson autocorrelation coefficient test. To estimate maximum precipitations of a specified duration and with a specified probability of exceedance, three-parameter theoretical probability distributions were used: a shifted gamma distribution (Pearson type III), a log-normal distribution, a Weibull distribution (Fisher-Tippett type III), a log-gamma distribution, as well as a two-parameter Gumbel distribution. The best distribution was selected by: (i) maximum likelihood estimation of parameters; (ii) tests of the hypothesis of goodness of fit of the theoretical probability distribution function with the empirical distribution using Pearson’s χ2 test; (iii) selection of the best-fitting function within each type according to the criterion of minimum Kolmogorov distance; (iv) selection of the most credible probability distribution function from the set of various types of best-fitting functions according to the Akaike information criterion; and (v) verification of the most credible function using single-dimensional tests of goodness of fit: the Kolmogorov-Smirnov test, the Anderson-Darling test, the Liao-Shimokawa test, and Kuiper’s test. The PMAXTP model was tested on data from two meteorological stations in northern Poland (Chojnice and Bialystok) drawn from a digital database of high-resolution precipitation records for the period of 1986 to 2015, available for 100 stations in Poland (i.e., the Polish Atlas of Rainfall Intensities (PANDa)). Values of maximum precipitation with a specified probability of exceedance obtained from the PMAXTP model were compared with values obtained from the probabilistic Bogdanowicz-Stachý model. The comparative analysis was based on the standard error of fit, graphs of the density function for the probability of exceedance, and estimated quantile errors. The errors of fit were lower for the PMAXTP compared to the Bogdanowicz-Stachý model. For both stations, the smallest errors were obtained for the quantiles determined on the basis of maximum precipitation POT using PMAXTP.
Highlights
A frequency analysis of values of maximum precipitation of a specified duration and probability of exceedance is an essential part of engineering [1]
The purpose of this paper is to present the PMAXTP model for a frequency analysis of maximum precipitation with a specified duration and probability of exceedance, together with the results of testing the model against data from two meteorological stations located in northern Poland: Chojnice and Białystok
Based on a visual assessment of the measurement series and information contained in IMGW—PIB reports (Meteorological Yearbooks and Precipitation Yearbooks Report [51]), no significant factors were found that might have an impact on the genetic homogeneity of the series of maximum precipitation values observed in the years 1986–2015
Summary
A frequency analysis of values of maximum precipitation of a specified duration and probability of exceedance is an essential part of engineering [1]. Given the significant impact of maximum precipitation on various spheres of human activity (e.g., the economy, agriculture, industry, and the environment), such an analysis is widely applied, in the context of observed climate change [2,3]. A widely used tool in the statistical description of rare meteorological (climatic) events is the extreme value theorem (EVT). Two probability distributions are used when employing the EVT: the generalized extreme value distribution (GEV) and the generalized. Used methods for the estimation of the unknown parameters of theoretical probability distributions include: maximum likelihood, L-moments, and the Bayesian method [7,8,9]. Ragulina and Reitan [10] proposed a Bayesian hierarchical model approach to the selection of a GEV distribution, where
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