# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## 2 Methodology
### Parameter Estimation
#### 2.2.4 Wilson-Hilferty (WH3)  
The equations needed to estimate the parameters with MOM have the following expressions [20]:  
\[\mu=\gamma+\frac{\beta}{\Gamma(\alpha)}\cdot\Gamma\left(\alpha+\frac{1}{3}\right) \tag{53}\]  
\[\sigma^{2}=\frac{\beta^{2}}{\Gamma(\alpha)}\cdot\left[\Gamma\left(\alpha+ \frac{2}{3}\right)-\frac{1}{\Gamma(\alpha)}\cdot\Gamma\left(\alpha+\frac{1}{ 3}\right)^{2}\right] \tag{54}\]  
\[C_{s}=\frac{\alpha-\frac{3}{\Gamma(\alpha)^{2}}\cdot\Gamma\left(\alpha+\frac{ 2}{3}\right)\cdot\Gamma\left(\alpha+\frac{1}{3}\right)+\frac{2}{\Gamma(\alpha )^{3}}\cdot\Gamma\left(\alpha+\frac{1}{3}\right)^{3}}{\sqrt{\frac{\Gamma \left(\alpha+\frac{2}{3}\right)}{\Gamma\left(\alpha\right)}-\frac{\Gamma \left(\alpha+\frac{1}{3}\right)^{2}}{\Gamma\left(\alpha\right)^{2}}\cdot\left( \Gamma\left(\alpha+\frac{2}{3}\right)-\frac{\Gamma\left(\alpha+\frac{1}{3} \right)^{2}}{\Gamma\left(\alpha\right)}\right)}} \tag{55}\]  
The shape parameter can be obtained approximately, depending on the skewness coefficient, using the following exponential function [20]:  
\[\alpha=\exp\left(\begin{array}{l}-1.6047146-1.2117058\cdot\ln(C_{s})-2.46279 86\cdot 10^{-1}\cdot\ln(C_{s})^{2}-\\ 3.0754515\cdot 10^{-2}\cdot\ln(C_{s})^{3}+1.3529125\cdot 10^{-2}\cdot\ln(C_{s})^{ 4}+\\ 5.4495596\cdot 10^{-3}\cdot\ln(C_{s})^{5}+6.031030\cdot 10^{-7}\cdot\ln(C_{s})^{ 6}-\\ 3.5860178\cdot 10^{-4}\cdot\ln(C_{s})^{7}-7.3564689\cdot 10^{-5}\cdot\ln(C_{s})^{ 8}-4.7318329\cdot 10^{-6}\cdot\ln(C_{s})^{9}\end{array}\right) \tag{56}\]  
\[\beta=\frac{\sigma}{\sqrt{\frac{\Gamma\left(\alpha+\frac{2}{3}\right)}{\Gamma \left(\alpha\right)}-\frac{\Gamma\left(\alpha+\frac{1}{3}\right)^{2}}{\Gamma \left(\alpha\right)^{2}}}} \tag{57}\]  
\[\gamma=\mu-\frac{\beta\cdot\Gamma\left(\alpha+\frac{1}{3}\right)}{\Gamma \left(\alpha\right)} \tag{58}\]  
The parameter estimation with the L-moments method is performed numerically (definite integrals) based on the equations using the quantile of the function.  
An approximate form can be adopted based on the parameter estimation, depending on L-skewness (\(\tau_{3}\)), as follows [20]:  
\[\alpha=\exp\left(\begin{array}{l}-4.16202506-3.261604018\cdot\ln(\tau_{3})- 1.783702334\cdot\ln(\tau_{3})^{2}-\\ 0.770946644\cdot\ln(\tau_{3})^{3}-0.221698815\cdot\ln(\tau_{3})^{4}-0.041191426 \cdot\ln(\tau_{3})^{5}-\\ 0.004665295\cdot\ln(\tau_{3})^{6}-0.000287712\cdot\ln(\tau_{3})^{7}-0.00000720 7\cdot\ln(\tau_{3})^{8}\end{array}\right) \tag{59}\]\[\text{if}\;1/3<\tau_{3}\leq 2/3\text{:}\] \[\alpha=\exp\left(\begin{array}{l}-5.264027693-9.134993593\cdot\ln( \tau_{3})-\\ 15.845477811\cdot\ln(\tau_{3})^{2}-19.874003352\cdot\ln(\tau_{3})^{3}-\\ 15.495267042\cdot\ln(\tau_{3})^{4}-6.752325319\cdot\ln(\tau_{3})^{5}-1.255615645 \cdot\ln(\tau_{3})^{6}\end{array}\right) \tag{60}\]  
\[\text{if}\;2/3<\tau_{3}<1\text{:}\]  
\[\alpha=\exp\left(\begin{array}{l}-7.712526023-93.06660109\cdot\ln(\tau_{3})- 1.519099681\cdot 10^{3}\cdot\ln(\tau_{3})^{2}-\\ 1.602587644\cdot 10^{3}\cdot\ln(\tau_{3})^{4}-1.041173897\cdot 10^{5}\cdot\ln( \tau_{3})^{4}-\\ 4.152482998\cdot 10^{5}\cdot\ln(\tau_{3})^{5}-9.887282\cdot 10^{5}\cdot\ln( \tau_{3})^{6}-\\ 1.287749298\cdot 10^{6}\cdot\ln(\tau_{3})^{7}-7.050767642\cdot 10^{5}\cdot\ln( \tau_{3})^{8}\end{array}\right) \tag{61}\]  
\[\beta=\frac{L_{2}}{\frac{\Gamma\left(\alpha+\frac{1}{2}\right)}{\Gamma\left( \alpha\right)}-2\cdot z} \tag{62}\]  
\[\gamma=L_{1}-\frac{\beta\cdot\Gamma\left(\alpha+\frac{1}{3}\right)}{\Gamma \left(\alpha\right)} \tag{63}\]  
where \(z=\int\limits_{0}^{1}qgamma(1-p,\alpha)^{1/3}\cdot p\cdot dp\), which can be approximated with the following equation:  
\[z=\exp\left(\begin{array}{l}-1.037385169+0.592727202\cdot\ln(\alpha)-0.1074 94558\cdot\ln(\alpha)^{2}+\\ 0.027616773\cdot\ln(\alpha)^{3}-0.002977204\cdot\ln(\alpha)^{4}-\\ 0.000546413\cdot\ln(\alpha)^{5}+0.00123125\cdot\ln(\alpha)^{6}+\\ 0.000420922\cdot\ln(\alpha)^{7}+0.000052295\cdot\ln(\alpha)^{8}+0.000002315 \cdot\ln(\alpha)^{9}\end{array}\right) \tag{64}\]