# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## 4 Results  
The frequency analysis of hydrological drought consisted in determining the minimum flows, annual volumes and durations for exceedance probabilities used in water management [39]. Eight probability distributions, with three and five parameters, were used. The distribution parameters were estimated for MOM and L-moments for the analysis of minimum annual flows, and for L-moments for the analysis of annual volume deficits and annual drought durations. For the analysis of minimum flows with MOM, the skewness coefficient was chosen, depending on the ephemeral/perennial character of the river. In this case, the multiplication coefficient 3, applied to the coefficient of variation of the data, was used, resulting in a skewness of 1.351, different from 1.053 of the observed values. Figures 6-8 show the frequency curves using the annual minimum flow (minimum 1-day flow), minimum 7-day flow, and minimum 30-day flow, respectively. For plotting positions, the Landwehr formula was used [18]. Figure 6: The probability distribution curves for minimum 1-day low flows. Figure 7: The probability distribution curves for minimum 7-day low flows. If the presented graphs are for three time series, the data are presented in the tables only for annual minimums, being specific to the run-of-river facilities. A frequency analysis was performed with distributions of three, four and five parameters for the two estimation methods, and the distributions that have a lower bound close to the minimum recorded value were selected. In general, it is considered that with a larger number of parameters, anything can be fit. In the case of parameter estimation with L-moments (the method recommended in this article, being much less influenced by the length of the observed data), it is necessary to estimate the first four linear moments, an aspect fulfilled properly only by distributions of at least four parameters. In the case of three-parameter distributions, they properly calibrate only L-skewness (L\({}_{3}\)/L\({}_{2}\)), and L-kurtosis (L\({}_{4}\)/L\({}_{2}\)) is the natural one of the distributions, not being able to calibrate according to the fourth order moment. It should be mentioned, however, that not every distribution of at least four parameters gives good results; they must be used carefully, depending on the nature of extreme events (low flow or flood). In the analysis of extreme events, the most used are the Wakeby and lambda distributions, these having the advantage of the fact that the left hand is less influenced by the right hand, fulfilling the so-called \"separation effect\" described by  
Figure 8: The probability distribution curves for minimum 30-day low flows. Matalas. Their advantage in the frequency analysis of low flows is represented by their asymptotic character in the field of high probabilities, with a lower bound close to the minimum value of the data sets. The supplementary file graphically presents the results obtained for other four-parameter distributions (four-parameter kappa, four-parameter Kumaraswamy, four-parameter Burr, four-parameter Johnson SU). However, these cannot be applied for the analysis of the minimum extreme values, because for high probabilities of exceeding (right hand), they have a decreasing trend, being characterized by a small lower bound (even zero or negative); this is an unrealistic aspect for perennial rivers, generating unrealistic values in this field of probabilities. From the case study, it is observed that among all the distributions analyzed, the ones that meet the conditions mentioned above are Wilson-Hilferty, Weibull, Wakeby and lambda. The Pearson III distribution was presented comparatively, because this is the parent distribution in Romania in the frequency analysis of extreme events. For the L-moments method, this gives good results. Tables 7 and 8 present the result values of quantiles for some of the most common exceedance probabilities for 1-day low flow analysis.