# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## 1 Introduction
Water deficit analysis was performed for calendar-year periods, as the use of a hydrological-year period is relevant to the water balance period but interrupts drought periods. Seasonal analysis of drought is not appropriate, because the freezing period leads to hydrological droughs smaller in magnitude and less important in water management, because there is not much water consumption, mainly only hydropower being critical. Through the annual summation of drought periods, water deficit analysis proves useful in water management decisions. Probability distributions with three and five parameters are presented in this article, for the first time, in the statistical analysis of hydrological drought. In other materials [7; 8; 9; 10], the Pearson III, Pearson V, log-normal, GEV, Frechet, Weibull, LogLogistic, etc., distributions are recommended for the frequency analysis of hydrological drought, especially of the minimum annual flows, all of which are three-parameter distributions. Pearson III, Pearson V, log-normal, and Weibull distributions are used in this article. In addition, the three-parameter distributions, Wilson-Hilferty and fatigue lifetime, and the five-parameter Wakeby and lambda distributions are used. Based on [7; 8; 9; 10], in the case of perennial rivers, only statistical distributions that have a positive lower bound should be used for frequency analysis, all being three-parameter distributions and applied with the MOM or maximum likelihood parameter estimation method. Thus, it is required that the frequency analysis be performed using the L-moments parameter estimation method, which is much more stable, and that distributions of at least four parameters be used. It is recommended to use some distributions that still have a lower bound that is not only positive, but also very close to the minimum value recorded. The methods for estimating the parameters of the presented distributions are MOM and L-moments [15]. The use of some software in hydrology limits the user's freedom of choice, but, above all, it distances the user from understanding the phenomenon. The elimination of these inconveniences requires the detailed presentation of the distributions used and the equations for estimating the parameters of the distributions for the two methods. Other novel elements are also presented, such as the presentation of some distributions which have received very little attention in hydrology, the expressions of the quantiles of some distributions, parameter approximation relations, and frequency factors for estimating the quantile with MOM and L-moments (Appendix A). The estimation of the parameters leads to the solution of some systems of nonlinear equations, often having the integral operator, so it was necessary to obtain approximation relations of the parameters to some probability distributions. For this, approximation tests were carried out, based on [16], with polynomial, exponential or rational functions. In some cases, the variable function by which the parameter is approximated is a logarithm. The relative approximation errors of the parameters are below \(10^{-3}\). For some distributions, no approximation relations were calculated because the skewness depends on two parameters, resulting in a function of two variables with poor results. For these, numerical methods were used to solve systems of equations. Thus, all these novelty elements for these distributions presented in Table 1 will help hydrology researchers to use these distributions easily. The fatigue lifetime and Wilson-Hilferty distributions are used for the first time in the frequency analysis of hydrological drought. In the research carried out in the Faculty of Hydrotechnics, other distributions were also analyzed (four-parameter Johnson SU, Kappa, Burr and Kumaraswamy distributions) but the results were unsatisfactory.