This article analyzes six probability distributions from the Generalized Pareto family, with three, four and five parameters, with the main purpose of identifying other distributions from this family with applicability in flood frequency analysis compared to the distribution already used in the literature from this family such as Generalized Pareto Type II and Wakeby. This analysis is part of a larger and more complex research carried out in the Faculty of Hydrotechnics regarding the elaboration of a norm for flood frequency analysis using the linear moments method. In Romania, the standard method of parameter estimation is the method of ordinary moments, thus the transition from this method to the method of linear moments is desired. All the necessary elements for the distribution use are presented, such as the probability density functions, the complementary cumulative distribution functions, the quantile functions, and the exact and approximate relations for estimating parameters, for both methods of parameter estimation. All these elements are necessary for a proper transition between the two methods, especially since the use of the method of ordinary moments is done by choosing the skewness of the observed data depending on the origin of the maximum flows. A flood frequency analysis case study, using annual maximum and annual exceedance series, was carried out for the Prigor River to numerically present the analyzed distributions. The performance of this distribution is evaluated using a linear moments diagram.

The study of extreme phenomena in hydrology generally involves frequency analysis and a time series analysis. In this article we provide enough mathematics to enable hydrology researchers to apply a wide range of probability distributions in frequency analyses of hydrological drought. The article presents a hydrological drought frequency analysis methodology for the determination of minimum annual flows, annual drought durations and annual deficit volumes for exceedance probabilities common in water management. Eight statistical distributions from different families and with different numbers of parameters are used for the frequency analysis. The method of ordinary moments and the method of linear moments are used to estimate the parameters of the distributions. All the mathematical characteristics necessary for the application of the eight analyzed distributions, for the method of ordinary moments and the method of linear moments, are presented. The performance of the analyzed distributions is evaluated using relative mean error and relative absolute error. For the frequency analysis of the annual minimum flows, only distributions that have a lower bound close to the annual minimum value should be used, a defining characteristic having the asymptotic distributions at this value. A case study of hydrological drought frequency analysis is presented for the Prigor River. We believe that the use of software without the knowledge of the mathematics behind it is not beneficial for researchers in the field of technical hydrology; thus, the dissemination of mathematical methods and models is necessary. All the research was conducted within the Faculty of Hydrotechnics.

This article presents six probability distributions from the gamma family with three parameters for the flood frequency analysis in hydrology. The choice of the gamma family of statistical distributions was driven by its frequent use in hydrology. In the Faculty of Hydrotechnics, the improvement of the estimation of maximum flows, including the methodological bases for the realization of a regionalization study with the linear moments method with the corrected parameters, was researched and is an element of novelty. The linear moments method performs better than the method of ordinary moments because it avoids the choice of skewness depending on the origin of the flows, and is the method practiced in Romania. The L-moments method conforms to the current trend for estimating the parameters of statistical distributions. Observed data from hydrometric stations are of relatively short length, so the statistical parameters that characterize them are of a sample that requires correction. The correction of the statistical parameters is proposed using the method of least squares based on the inverse functions of the statistical distributions expressed with the frequency factor for L-moments. All the necessary elements for their use are presented, such as quantile functions, the exact and approximate relations for estimating parameters, and frequency factors. A flood frequency analysis case study was carried out for the Ialomita river to verify the proposed methodology. The performance of this distributions is evaluated using Kling–Gupta and Nash–Sutcliff coefficients.

Accurately determining the maximum designed water discharges of dams is extremely important, considering the economic costs of carrying out these types of hydrotechnical works and the possible disastrous consequences resulting from their incorrect design. This article describes and applies probability distributions used in hydrology, with some recommended by Romanian legislation standard NP 129-2011. The methods for estimating the parameters presented in this article, as well as the establishment of directions for correlating the normative with international regulations, resulting from the research on many rivers with different characteristics, conducted within the Faculty of Hydrotechnics, were completed with specialized computer applications for applying the normative. In this article, two case studies reflecting this research are presented. The verification of the proposed recommendations, on rivers with hydrographic basins with different physiographic characteristics, confirmed the opportunity to implement rigorous and simple criteria. The presentation of the quantile form of some distributions (especially Pearson III) and of the expressions of moments (central and raw) of high order, as well as the presentation of the frequency factors of each analyzed distribution necessary to calculate the confidence interval, constitute novelties, thus facilitating the ease of use of these distributions.

Estimating the parameters of probability distributions generally involves solving a system of nonlinear equations or a nonlinear equation, being a technical difficulty in their usual application in hydrology. The choice of probability distributions for the calculation of extreme values in hydrology is, in most cases, made according to the ease of calculation of the estimated parameters and the explicit form of the inverse probability function. This article presents improved approximations and, in some cases, new approximations for the estimation with the method of ordinary moments and the method of linear moments, which are useful for the direct calculation of the parameters, because the errors in the approximate estimation are similar to the use of iterative numerical methods. Thirteen probability distributions of two and three parameters frequently used in hydrology are presented, for which parameter estimation was laborious. Thus, the approximate estimation of the parameters by the two methods is simple but also precise and easily applicable by hydrology researchers. The new and improved approximate forms presented in this article are the result of the research conducted within the Faculty of Hydrotechnics to update the Romanian normative standards in the hydrotechnical field.

This study examines all of the equations necessary to derive the parameters for seven probability distributions of three parameters typically used in flood frequency research, namely the Pearson III (PE3), the generalized extreme value (GEV), the Weibull (W3), the log-normal (LN3), the generalized Pareto Type II (PG), the Rayleigh (RY) and the log-logistic (LL3) distributions, using the higher-order linear moments method (LH-moments). The analysis represents the expansion of previous research whose results were presented in previous materials, and is part of hydrological research aimed at developing a standard for calculating maximum flows based on L-moments and LH-moments. The given methods for calculating the parameters of the examined distributions are used to calculate the maximum flows on Romania’s Prigor River. For both methods, the criterion for selecting the most suitable distribution is represented by the diagram of the L-skewness–L-kurtosis and LH-skewness–LH-kurtosis. The results for Prigor River show that the PG distribution is the best model for the L-moments method, the theoretical values of the statistical indicators being 0.399 and 0.221. The RY distribution is the best model for the LH-moments technique, with values of 0.398 and 0.192 for the two statistical indicators.