# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## 2 Methodology
The annual deficit volumes corresponding to the three thresholds are obtained from the time series by daily summation of the volumes lower than the chosen threshold. The annual drought durations corresponding to the three thresholds are obtained from the time series by summing the days with flows lower than the chosen threshold. These annual data represent the statistical data for frequency analysis. In the frequency analysis of low flow, when the observed values are lower than 100 values, a correction of the skewness coefficient (\(C_{s}\)) is necessary to estimate the parameters with MOM [11; 18; 19; 20; 21]. In Romania, for the low flow frequency analysis, the skewness coefficient is established according to the perennial/ephemeral character of the river. The skewness coefficient of the flows is established based on the coefficient of variation of the minimum annual flows by multiplying with a coefficient \(C_{v}\), depending on the perennial/ephemeral character of the river, chosen as follows [22; 23; 24]:  
* 3 for rivers with strong supply from the aquifer;
* 2 for rivers that never dry up and with a significant supply from the aquifer in periods of low water;
* 1.5 for rivers with a deficient supply from the aquifer and with a strong trend of decreasing flows;  
Figure 3: Methodological approach. 0-1 in the case of rivers that dry up (small rivers, without water supply). This method for the calculation of the corrected skewness is an outdated method, needing an update by aligning with modern norms and methodologies. A solution is the use of the L-moments method for estimating the parameters of statistical distributions, knowing that this estimation method is more stable and less influenced by the small lengths of the observed data [15; 16; 17; 18; 19; 25; 26; 27]. To estimate the parameters with MOM, for some three-parameter distributions, the L-moments method can be used to obtain an equivalent skewness. This involves the calculation of the shape parameter for each distribution using the L-moments method, and this will substitute the shape parameter from the MOM, obtaining the skewness, for the \(C_{s}/C_{v}\) ratio. This method preserves the mean and coefficient of variation of the observed data. This skewness estimation method is better than the arbitrary choice, many times, of the flow genesis function. The use of MOM with skewness determined from L-moments is useful for a period of transition from MOM to L-moments in Romania, because the interpretation of the \(C_{s}/C_{v}\) ratio is well known. For the application of this method, the equivalence relations of the \(C_{s}/C_{v}\) ratio according to \(\tau_{3}\) are presented in Table 13 from Appendix C.  
For frequency analysis of the annual volume deficit and duration, due to the fact that the data can contain zero values, it is necessary to use a conditional probability model [3; 11; 28]. The same model is recommended for application in the statistical analysis of annual low flows when the observed data contains zero values (rivers that dry up). This model assumes a correction of the cumulative function and the quantile of the analyzed distribution, as follows:  
\[F(x)^{*}=(1-p_{0})\cdot F(x) \tag{3}\]  
\[x(p)^{*}=\left\{ \begin{array}{ll}0,&p>F(0)^{*}\\ x\Big{(}\frac{p}{1-p_{0}}\Big{)}\end{array} \right. \tag{4}\]  
where \(p_{0}\) is the empirical probability from which the values are zero. The next section presents the theoretical distributions analyzed in the research of the Faculty of Hydrotechnics regarding the regionalization studies of the low flows in Romania. The inverse forms of the distributions (quantile functions) can be expressed both for MOM and for L-moments with the frequency factor. The frequency factors of the analyzed distributions are presented in Appendix A.