# Evaluation of Various Generalized Pareto Probability Distributions for Flood Frequency Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

^{−3}and 10

^{−4}.

#### 2.1. Probability Distributions

#### 2.2. Parameter Estimation

#### 2.2.1. Generalized Pareto Type IV (PGIV4)

#### 2.2.2. Generalized Pareto Type IV (PGIV3)

#### 2.2.3. Generalized Pareto Type III (PGIII)

#### 2.2.4. Generalized Pareto Type II (PGII)

#### 2.2.5. Generalized Pareto Type I (PGI)

#### 2.2.6. The Five-Parameter Wakeby Distribution (WK5)

## 3. Case Study

^{2}and the average altitude is 729 m. The river has a length of 33 km, with an average slope of 22‰ and a sinuosity coefficient of 1.83.

## 4. Results

## 5. Discussions

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MOM | the method of ordinary moments |

L-moments | the method of linear moments |

$\mu $ | expected value; arithmetic mean |

$\sigma $ | standard deviation |

${C}_{v}$ | coefficient of variation |

${C}_{s}$ | coefficient of skewness; skewness |

${C}_{k}$ | coefficient of kurtosis; kurtosis |

${L}_{1},{L}_{2},{L}_{3}$ | linear moments |

${\tau}_{2},L{C}_{v}$ | coefficient of variation based on the L-moments method |

${\tau}_{3},L{C}_{s}$ | coefficient of skewness based on the L-moments method |

${\tau}_{4},L{C}_{k}$ | coefficient of kurtosis based on the L-moments method |

Distr. | Distributions |

RME | relative mean error |

RAE | relative absolute error |

x_{i} | observed values |

## Appendix A. The Variation of the L-Kurtosis—L-Skewness

## Appendix B. The Frequency Factors for the Analyzed Distributions

Distribution | $\mathbf{Frequency}\mathbf{Factor},{\mathit{K}}_{\mathit{p}}\left(\mathit{p}\right)$ | |
---|---|---|

Quantile Function (Inverse Function) | ||

Method of Ordinary Moments (MOM) | L-Moments | |

$\mathrm{x}\left(\mathrm{p}\right)=\mathsf{\mu}+\mathsf{\sigma}\cdot {\mathrm{K}}_{\mathrm{p}}\left(\mathrm{p}\right)$ | $\mathrm{x}\left(\mathrm{p}\right)={\mathrm{L}}_{1}+{\mathrm{L}}_{2}\cdot {\mathrm{K}}_{\mathrm{p}}\left(\mathrm{p}\right)$ | |

PGIV4 | $\frac{\Gamma \left(\lambda \right)\cdot \left({\mathrm{p}}^{-\frac{1}{\lambda}}-1\right)-\Gamma \left(\mathsf{\alpha}+1\right)\cdot \Gamma \left(\lambda -\mathsf{\alpha}\right)}{\sqrt{\Gamma \left(2\cdot \mathsf{\alpha}+1\right)\cdot \Gamma \left(\lambda -2\cdot \mathsf{\alpha}\right)\cdot \Gamma \left(\lambda \right)-\Gamma {\left(\mathsf{\alpha}+1\right)}^{2}\cdot \Gamma {\left(\lambda -\mathsf{\alpha}\right)}^{2}}}$ | $\frac{{\left({\mathrm{p}}^{-\frac{1}{\lambda}}-1\right)}^{\mathsf{\alpha}}-\frac{\Gamma \left(\mathsf{\alpha}+1\right)\cdot \Gamma \left(\lambda -\mathsf{\alpha}\right)}{\Gamma \left(\lambda \right)}}{\Gamma \left(\mathsf{\alpha}+1\right)\cdot \left(\frac{\Gamma \left(\lambda -\mathsf{\alpha}\right)}{\Gamma \left(\lambda \right)}-\frac{\Gamma \left(2\cdot \lambda -\mathsf{\alpha}\right)}{\Gamma \left(2\cdot \lambda \right)}\right)}$ |

PGIV3 | $\frac{\Gamma \left(\lambda \right)\cdot \Gamma {\left({\mathrm{p}}^{-\frac{1}{\lambda}}-1\right)}^{\mathsf{\alpha}}-\Gamma \left(\mathsf{\alpha}+1\right)\cdot \Gamma \left(\lambda -\mathsf{\alpha}\right)}{\sqrt{\Gamma \left(2\cdot \mathsf{\alpha}+1\right)\cdot \Gamma \left(\lambda -2\cdot \mathsf{\alpha}\right)\cdot \Gamma \left(\lambda \right)-\Gamma {\left(\mathsf{\alpha}+1\right)}^{2}\cdot \Gamma {\left(\lambda -\mathsf{\alpha}\right)}^{2}}}$ | $\frac{\Gamma {\left({\mathrm{p}}^{-\frac{1}{\lambda}}-1\right)}^{\mathsf{\alpha}}-\frac{\Gamma \left(\mathsf{\alpha}+1\right)\cdot \Gamma \left(\lambda -\mathsf{\alpha}\right)}{\Gamma \left(\lambda \right)}}{\Gamma \left(\mathsf{\alpha}+1\right)\cdot \left(\frac{\Gamma \left(\lambda -\mathsf{\alpha}\right)}{\Gamma \left(\lambda \right)}-\frac{\Gamma \left(2\cdot \lambda -\mathsf{\alpha}\right)}{\Gamma \left(2\cdot \lambda \right)}\right)}$ |

PGIII | $\frac{{\left(\frac{1}{\mathrm{p}}-1\right)}^{\frac{1}{\mathsf{\alpha}}}-\frac{\pi}{\mathsf{\alpha}\cdot \mathrm{sin}\left(\frac{\pi}{\mathsf{\alpha}}\right)}}{\sqrt{\frac{2\cdot \pi}{\mathsf{\alpha}\cdot \mathrm{sin}\left(\frac{2\cdot \pi}{\mathsf{\alpha}}\right)}-\frac{{\pi}^{2}}{{\mathsf{\alpha}}^{2}\cdot \mathrm{sin}{\left(\frac{\pi}{\mathsf{\alpha}}\right)}^{2}}}}$ | $\frac{{\left(\frac{1-\mathrm{p}}{\mathrm{p}}\right)}^{\frac{1}{\mathsf{\alpha}}}\cdot {\mathsf{\alpha}}^{2}\cdot \mathrm{sin}\left(\frac{\pi}{\mathsf{\alpha}}\right)}{\pi}-\mathsf{\alpha}$ |

PGII | $\frac{\sqrt{1-2\cdot \mathsf{\alpha}}}{\mathsf{\alpha}}\cdot \left(1-\left(1+\mathsf{\alpha}\right)\cdot {\mathrm{p}}^{\mathsf{\alpha}}\right)$ | $1+\frac{2}{\mathsf{\alpha}}\cdot \left(1-{\mathrm{p}}^{\mathsf{\alpha}}\right)-\left(\mathsf{\alpha}+3\right)\cdot {\mathrm{p}}^{\mathsf{\alpha}}$ |

PGI | $\frac{\mathsf{\alpha}}{\left|\mathsf{\alpha}\right|}\cdot \frac{\left(\mathsf{\alpha}-1\right)\cdot \sqrt{\mathsf{\alpha}-2}\cdot \left({\left(1-\mathrm{p}\right)}^{-\frac{1}{\mathsf{\alpha}}}-\frac{\mathsf{\alpha}}{\mathsf{\alpha}-1}\right)}{\sqrt{\mathsf{\alpha}}}$ | $\frac{\mathsf{\alpha}}{\left|\mathsf{\alpha}\right|}\cdot \frac{\left(\mathsf{\alpha}-1\right)\cdot \left(2\cdot \mathsf{\alpha}-1\right)\cdot \left({\left(1-\mathrm{p}\right)}^{-\frac{1}{\mathsf{\alpha}}}-\frac{\mathsf{\alpha}}{\mathsf{\alpha}-1}\right)}{\mathsf{\alpha}}$ |

WK5 | $\frac{\begin{array}{l}\frac{\mathsf{\alpha}}{\mathsf{\beta}}\cdot \left(1-{\mathrm{p}}^{\mathsf{\beta}}\right)-\frac{\mathsf{\gamma}}{\mathsf{\delta}}\cdot \left(1-{\mathrm{p}}^{-\mathsf{\delta}}\right)-\\ \frac{\mathsf{\alpha}}{\mathsf{\beta}+1}+\frac{\mathsf{\gamma}}{\mathsf{\delta}-1}\end{array}}{\sqrt{\begin{array}{l}\frac{{\mathsf{\alpha}}^{2}}{{\left(\mathsf{\beta}+1\right)}^{2}\cdot \left(2\cdot \mathsf{\beta}+1\right)}-\frac{2\cdot \mathsf{\alpha}\cdot \mathsf{\gamma}}{\left(\mathsf{\beta}+1\right)\cdot \left(\mathsf{\beta}+1-\mathsf{\delta}\right)\cdot \left(\mathsf{\delta}-1\right)}\\ -\frac{{\mathsf{\gamma}}^{2}}{{\left(\mathsf{\delta}-1\right)}^{2}\cdot \left(2\cdot \mathsf{\delta}-1\right)}\end{array}}}$ | $\frac{\frac{\mathsf{\alpha}}{\mathsf{\beta}}\cdot \left(1-{\mathrm{p}}^{\mathsf{\beta}}\right)-\frac{\mathsf{\gamma}}{\mathsf{\delta}}\cdot \left(1-{\mathrm{p}}^{-\mathsf{\delta}}\right)-\frac{\mathsf{\alpha}}{\mathsf{\beta}+1}-\frac{\mathsf{\gamma}}{1-\mathsf{\delta}}}{\frac{\mathsf{\alpha}}{\left(\mathsf{\beta}+1\right)\cdot \left(\mathsf{\beta}+2\right)}+\frac{\mathsf{\gamma}}{\left(2-\mathsf{\delta}\right)\cdot \left(1-\mathsf{\delta}\right)}}$ |

## Appendix C. Estimation of the Frequency Factor for the PGIII Distribution

P [%] | a | b | c | d | e | f | g | h |
---|---|---|---|---|---|---|---|---|

0.01 | 5.111737 | 2.313409 | 1.34999 | −0.776028 | 0.182704 | −0.0232091 | 1.5563 × 10^{−3} | −4.330 × 10^{−5} |

0.1 | 3.808941 | 1.377403 | 345331 | −0.318752 | 0.0903167 | −0.0130843 | 9.7470 × 10^{−4} | −2.960 × 10^{−5} |

0.5 | 2.912620 | 0.812083 | 0.0170251 | −0.114635 | 0.0394326 | −6.29710 × 10^{−3} | 4.9970 × 10^{−4} | −1.590 × 10^{−5} |

1 | 2.527259 | 0.600436 | −0.0535487 | −0.0565675 | 0.0233015 | 3.98010 × 10^{−3} | 3.2800 × 10^{−4} | 1.070 × 10^{−5} |

2 | 2.140031 | 0.411098 | −0.0911407 | −0.0147256 | 0.0107835 | −2.10400 × 10^{−3} | 1.8480 × 10^{−4} | −6.200 × 10^{−6} |

3 | 1.911447 | 0.311395 | −0.100372 | 2.88960 × 10^{−3} | 5.08580 × 10^{−3} | −1.21600 × 10^{−3} | 1.1520 × 10^{−4} | −4.100 × 10^{−6} |

5 | 1.619324 | 0.197967 | −0.100797 | 0.0185338 | −4.64200 × 10^{−4} | −3.15600 × 10^{−4} | 4.3000 × 10^{−5} | −1.700 × 10^{−6} |

10 | 1.208950 | 0.066749 | −0.0847851 | 0.0291844 | −5.24780 × 10^{−3} | 5.25900 × 10^{−4} | −2.7400 × 10^{−5} | 6.000 × 10^{−7} |

20 | 0.763538 | −0.0362722 | −0.0531111 | 0.0286703 | −6.99220 × 10^{−3} | 9.32800 × 10^{−4} | −6.5700 × 10^{−5} | 1.900 × 10^{−6} |

40 | 0.224364 | −0.103222 | −6.80020 × 10^{−3} | 0.0162353 | −5.34190 × 10^{−3} | 8.40400 × 10^{−4} | −6.63000 × 10^{−5} | 2.100 × 10^{−6} |

50 | 0.001255 | −0.111838 | 0.0119116 | 8.67100 × 10^{−3} | −3.76100 × 10^{−3} | 6.53200 × 10^{−4} | −5.44000 × 10^{−5} | 1.8000 × 10^{−6} |

80 | −0.762791 | −0.0547579 | 0.0593627 | −0.0207334 | 3.89270 × 10^{−3} | −4.16000 × 10^{−4} | 2.37000 × 10^{−5} | −6.000 × 10^{−7} |

90 | −1.210684 | 0.0415455 | 0.0668030 | −0.036012 | 8.85020 × 10^{−3} | −1.19170 × 10^{−3} | 8.48000 × 10^{−5} | −2.500 × 10^{−6} |

P [%] | a | b | c | d | e | f | g | h |
---|---|---|---|---|---|---|---|---|

0.01 | 9.0865 | −9.0870 | −5.6824 | 14.136 | −19.532 | 15.682 | −6.8872 | 1.2851 |

0.1 | 6.8300 | −6.8367 | −4.3709 | 8.5137 | −9.3692 | 6.1000 | −2.2178 | 0.35120 |

0.5 | 5.3161 | −5.3440 | −3.2817 | 4.4099 | −2.2009 | −1.0766 | 1.7928 | 0.61653 |

1 | 4.6040 | −4.6529 | −2.9168 | 3.6177 | −1.7699 | −0.69899 | 1.2546 | −0.43830 |

2 | 3.8961 | −3.9826 | −2.5218 | 2.8156 | −1.2339 | −0.62320 | 0.99972 | −0.35026 |

3 | 3.4784 | −3.5994 | −2.2840 | 2.4127 | −1.0762 | −0.39996 | 0.73125 | −0.26304 |

5 | 2.9457 | −3.1313 | −1.9655 | 1.9096 | −0.81060 | −0.29906 | 0.55319 | −0.20219 |

10 | 2.1976 | −2.5343 | −1.4975 | 1.3114 | −0.53308 | −0.15403 | 0.33379 | −0.12411 |

20 | 1.3864 | −2.0186 | −0.95952 | 0.81735 | −0.27803 | −0.074878 | 0.18663 | −0.059390 |

40 | 0.40582 | −0.40582 | −0.25879 | 0.38534 | 0.16327 | −0.042751 | −0.062078 | 0.088377 |

50 | 1.934 × 10^{−4} | −1.6511 | 0.033144 | 0.37384 | 0.19857 | 0.011770 | −0.031091 | 0.064863 |

80 | −1.3869 | −0.52845 | −0.095705 | 1.2009 | −0.42838 | 0.047547 | 0.40683 | −0.21635 |

90 | −2.1977 | 1.0582 | −0.12265 | 0.70118 | −0.41310 | −0.11862 | 0.17678 | −0.084297 |

## Appendix D. Estimation of the Frequency Factor for the PGII Distribution

P [%] | a | b | c | d | e | f | g | h |
---|---|---|---|---|---|---|---|---|

0.01 | 1.932014 | 0.061904 | 2.87112 | −0.795586 | 0.0495418 | 1.03747 × 10^{−2} | −1.7871 × 10^{−3} | 7.910 × 10^{−5} |

0.1 | 1.809609 | 0.670411 | 2.12699 | −1.15732 | 0.284537 | −3.77979 × 10^{−2} | 2.6299 × 10^{−3} | −7.530 × 10^{−5} |

0.5 | 1.704417 | 1.213220 | 0.762347 | −0.651934 | 0.198072 | −3.05909 × 10^{−2} | 2.3988 × 10^{−3} | −7.580 × 10^{−5} |

1 | 1.664419 | 1.314780 | 0.178438 | −0.355564 | 0.125632 | −2.09008 × 10^{−2} | 1.7176 × 10^{−3} | −5.610 × 10^{−5} |

2 | 1.622183 | 1.265729 | −0.290832 | −0.0764258 | 0.0500117 | −9.92700 × 10^{−3} | 8.9160 × 10^{−4} | −3.080 × 10^{−5} |

3 | 1.590286 | 1.151049 | −0.479305 | 0.0583412 | 0.0103575 | −3.85500 × 10^{−3} | 4.1630 × 10^{−4} | −1.570 × 10^{−5} |

5 | 1.530888 | 0.909001 | −0.598849 | 0.179061 | −0.0288591 | 2.47620 × 10^{−3} | −9.6700 × 10^{−5} | 9.000 × 10^{−7} |

10 | 1.377562 | 0.424384 | −0.522494 | 0.228357 | −0.0536328 | 7.14020 × 10^{−3} | −5.0690 × 10^{−4} | 1.490 × 10^{−5} |

20 | 1.047703 | −0.1346753 | −0.197916 | 0.136073 | −0.0393659 | 5.98720 × 10^{−3} | −4.6780 × 10^{−4} | 1.480 × 10^{−5} |

40 | 0.355375 | −0.4660863 | 0.182363 | −0.0347085 | 2.20900 × 10^{−3} | 2.60000 × 10^{−4} | −4.8700 × 10^{−5} | 2.100 × 10^{−6} |

50 | 0.0051002 | −0.428129 | 0.242055 | −0.0758534 | 0.0142094 | −1.57970 × 10^{−3} | 9.61000 × 10^{−5} | −2.5000 × 10^{−6} |

80 | −1.042946 | 0.1095022 | 0.0744175 | −0.0546187 | 0.0156946 | −2.36020 × 10^{−3} | 1.82600 × 10^{−4} | −5.700 × 10^{−6} |

90 | −1.389758 | 0.3823067 | −0.0662687 | −9.67300 × 10^{−3} | 6.70430 × 10^{−3} | −1.27020 × 10^{−3} | 1.09700 × 10^{−4} | −3.700 × 10^{−6} |

P [%] | a | b | c | d | e | f | g | h | i | j |
---|---|---|---|---|---|---|---|---|---|---|

0.01 | 2.8976 | 16.049 | −166.33 | 2107.2 | −11711 | 41260 | −84364 | 1.0671 × 10^{5} | −68465 | 14604 |

0.1 | 2.9921 | 7.9122 | 21.160 | 46.340 | 207.03 | −394.80 | 1467.1 | −2026.5 | 667.72 | − |

0.5 | 2.9751 | 7.0733 | 21.320 | 5.3452 | 97.924 | −99.393 | −95.564 | −59.326 | − | − |

1 | 2.9389 | 6.8741 | 14.223 | 10.895 | 33.199 | −116.24 | 47.113 | − | − | − |

2 | 2.8667 | 6.5598 | 3.8790 | 28.740 | −66.614 | 23.564 | − | − | − | − |

3 | 2.8214 | 5.3368 | 6.0252 | 1.9543 | −31.305 | 14.174 | − | − | − | − |

5 | 2.7060 | 3.9951 | 4.1471 | −13.220 | −2.6956 | 4.0738 | − | − | − | − |

10 | 2.4017 | 2.0082 | −1.6983 | −9.9252 | 7.8868 | −1.6725 | − | − | − | − |

20 | 1.7989 | −0.48676 | −4.2616 | 1.0072 | 1.7490 | −0.80770 | − | − | − | − |

40 | 0.60025 | −2.4049 | −0.31267 | 2.3727 | −1.6966 | 0.44147 | − | − | − | − |

50 | 3.8891 × 10^{−4} | −2.3323 | 1.3905 | 0.48092 | −0.81390 | 0.27475 | − | − | − | − |

80 | −1.8003 | 0.52569 | 1.0120 | −1.3741 | 0.86642 | −0.22985 | − | − | − | − |

90 | −2.4001 | 2.1286 | −0.96924 | 0.23252 | 0.041754 | −0.033573 | − | − | − | − |

## Appendix E. The Skewness and Kurtosis for the WK5 Distribution

## Appendix F. The Skewness and Kurtosis for the PGIV Distribution

## References

- Popovici, A. Dams for Water Accumulations; Technical Publishing House: Bucharest, Romania, 2002; Volume 2. [Google Scholar]
- Teodorescu, I.; Filotti, A.; Chiriac, V.; Ceausescu, V.; Florescu, A. Water Management; Ceres Publishing House: Bucharest, Romania, 1973. [Google Scholar]
- Hosking, J.R.M.; Wallis, J.R. Regional Frequency Analysis: An Approach Based on L-moments; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar] [CrossRef]
- Rao, A.R.; Hamed, K.H. Flood Frequency Analysis; CRC Press LLC: Boca Raton, FL, USA, 2000. [Google Scholar]
- Singh, V.P. Entropy-Based Parameter Estimation in Hydrology; Springer Science + Business Media: Dordrecht, The Netherlands, 1998; ISBN 978-90-481-5089-2/978-94-017-1431-0. [Google Scholar] [CrossRef]
- Gubareva, T.S.; Gartsman, B.I. Estimating Distribution Parameters of Extreme Hydrometeorological Characteristics by L-Moment Method. Water Resour.
**2010**, 37, 437–445. [Google Scholar] [CrossRef] - Greenwood, J.A.; Landwehr, J.M.; Matalas, N.C.; Wallis, J.R. Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form. Water Resour. Res.
**1979**, 15, 1049–1054. [Google Scholar] [CrossRef] - Houghton, J.C. Birth of a parent: The Wakeby distribution for modeling flood flows. Water Resour. Res.
**1978**, 14, 1105–1109. [Google Scholar] [CrossRef] - Crooks, G.E. Field Guide to Continuous Probability Distributions; Berkeley Institute for Theoretical Science: Berkeley, CA, USA, 2019. [Google Scholar]
- Yang, T.; Shao, Q.X.; Hao, Z.C.; Chen, X.; Zhang, Z.X.; Xu, C.Y.; Sun, L.M. Regional frequency analysis and spatio-temporal pattern characterization of rainfall extremes in the Pearl River Basin, China. J. Hydrol.
**2010**, 380, 386–405. [Google Scholar] [CrossRef] - Zakaria, Z.A.; Shabri, A. Regional frequency analysis of extreme rainfalls using partial L-moments method. Theor. Appl. Climatol.
**2013**, 113, 83–94. [Google Scholar] [CrossRef] - Zhou, C.R.; Chen, Y.F.; Huang, Q.; Gu, S.H. Higher moments method for generalized Pareto distribution in flood frequency analysis. IOP Conf. Ser. Earth Environ. Sci.
**2017**, 82, 012031. [Google Scholar] [CrossRef] - Martins, A.L.A.; Liska, G.R.; Beijo, L.A.; Menezes, F.S.D.; Cirillo, M.Â. Generalized Pareto distribution applied to the analysis of maximum rainfall events in Uruguaiana, RS, Brazil. SN Appl. Sci.
**2020**, 2, 1479. [Google Scholar] [CrossRef] - Ciupak, M.; Ozga-Zielinski, B.; Tokarczyk, T.; Adamowski, J. A Probabilistic Model for Maximum Rainfall Frequency Analysis. Water
**2021**, 13, 2688. [Google Scholar] [CrossRef] - Shao, Y.; Zhao, J.; Xu, J.; Fu, A.; Wu, J. Revision of Frequency Estimates of Extreme Precipitation Based on the Annual Maximum Series in the Jiangsu Province in China. Water
**2021**, 13, 1832. [Google Scholar] [CrossRef] - Ashkar, F.; Ouarda, T.B. On some methods of fitting the generalized Pareto distribution. J. Hydrol.
**1996**, 177, 117–141. [Google Scholar] [CrossRef] - Mohsen, S.; Zulkifli, Y.; Fadhilah, Y. Comparison of Distribution Models for Peak flow, Flood Volume and Flood Duration. Res. J. Appl. Sci. Eng. Technol.
**2013**, 6, 733–738. [Google Scholar] - Swetapadma, S.; Ojha, C.S.P. Technical Note: Flood frequency study using partial duration series coupled with entropy principle. Hydrol. Earth Syst. Sci. Discuss.
**2021**. preprint. [Google Scholar] [CrossRef] - Rahman, A.S.; Rahman, A.; Zaman, M.A.; Haddad, K.; Ahsan, A.; Imteaz, M. A study on selection of probability distributions for at-site flood frequency analysis in Australia. Nat. Hazards
**2013**, 69, 1803–1813. [Google Scholar] [CrossRef] - Drissia, T.K.; Jothiprakash, V.; Anitha, A.B. Flood Frequency Analysis Using L Moments: A Comparison between At-Site and Regional Approach. Water Resour. Manag.
**2019**, 33, 1013–1037. [Google Scholar] [CrossRef] - Hosking, J.R.M.; Wallis, J.R. Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics
**1987**, 29, 339–349. [Google Scholar] [CrossRef] - Ilinca, C.; Anghel, C.G. Flood-Frequency Analysis for Dams in Romania. Water
**2022**, 14, 2884. [Google Scholar] [CrossRef] - Anghel, C.G.; Ilinca, C. Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions. Appl. Sci.
**2023**, 13, 3055. [Google Scholar] [CrossRef] - Anghel, C.G.; Ilinca, C. Parameter Estimation for Some Probability Distributions Used in Hydrology. Appl. Sci.
**2022**, 12, 12588. [Google Scholar] [CrossRef] - Viglione, A.; Merz, R.; Salinas, J.L.; Blöschl, G. Flood frequency hydrology: 3. A Bayesian analysis. Water Resour. Res.
**2013**, 49, 675–692. [Google Scholar] [CrossRef] - Gaume, E. Flood frequency analysis: The Bayesian choice. WIREs Water.
**2018**, 5, e1290. [Google Scholar] [CrossRef] - Lang, M.; Ouarda, T.B.; Bobée, B. Towards operational guidelines for over-threshold modeling. J. Hydrol.
**1999**, 225, 103–117. [Google Scholar] [CrossRef] - Bulletin 17B Guidelines for Determining Flood Flow Frequency; U.S. Department of the Interior, U.S. Geological Survey: Reston, VA, USA, 1981.
- Bulletin 17C Guidelines for Determining Flood Flow Frequency; U.S. Department of the Interior, U.S. Geological Survey: Reston, VA, USA, 2017.
- STAS 4068/1-82; Maximum Water Discharges and Volumes, Determination of Maximum Water Discharges and Volumes of Watercourses. The Romanian Standardization Institute: Bucharest, Romania, 1982.
- Diacon, C.P. Serban Hydrological Syntheses and Regionalizations; Technical Publishing House: Bucharest, Romania, 1994. [Google Scholar]
- Mandru, R.; Ioanitoaia, H. Ameliorative Hydrology; Agro-Silvica Publishing House: Bucharest, Romania, 1962. [Google Scholar]
- Constantinescu, M.; Golstein, M.; Haram, V.; Solomon, S. Hydrology; Technical Publishing House: Bucharest, Romania, 1956. [Google Scholar]
- Ministry of Regional Development and Tourism. The Regulations Regarding the Establishment of Maximum Flows and Volumes for the Calculation of Hydrotechnical Retention Constructions; Indicative NP 129-2011; Ministry of Regional Development and Tourism: Bucharest, Romania, 2012.
- Murshed, M.S.; Park, B.J.; Jeong, B.Y.; Park, J.S. LH-Moments of Some Distributions Useful in Hydrology. In Communications for Statistical Applications and Methods; The Korean Statistical Society: Seoul, Republic of Korea, 2009. [Google Scholar] [CrossRef]
- Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill, Inc.: New York, NY, USA, 1988; ISBN 007-010810-2. [Google Scholar]
- Ministry of the Environment. The Romanian Water Classification Atlas, Part I—Morpho-Hydrographic Data on the Surface Hydrographic Network; Ministry of the Environment: Bucharest, Romania, 1992.
- Singh, K.; Singh, V.P. Parameter Estimation for Log-Pearson Type III Distribution by Pome. J. Hydraul. Eng.
**1988**, 1, 112–122. [Google Scholar] [CrossRef] - Shaikh, M.P.; Yadav, S.M.; Manekar, V.L. Assessment of the empirical methods for the development of the synthetic unit hydrograph: A case study of a semi-arid river basin. Water Pract. Technol.
**2021**, 17, 139–156. [Google Scholar] [CrossRef] - Gu, J.; Liu, S.; Zhou, Z.; Chalov, S.R.; Zhuang, Q. A Stacking Ensemble Learning Model for Monthly Rainfall Prediction in the Taihu Basin, China. Water
**2022**, 14, 492. [Google Scholar] [CrossRef]

New Elements | Distribution |
---|---|

Exact parameter estimation | PGIV4, PGI |

Approximate estimation of parameters | PGIII, PGII, PGI |

The frequency factor for MOM | PGIV4, PGIV3, PGIII, PGII, PGI, WK5 |

The frequency factor for L-moments | PGIV4, PGIV3, PGIII, PGII, PGI, WK5 |

Approximate estimation of the frequency factor | PGIII, PGII |

Raw and central moments * | PGIV, PGIII, PGII |

Distr. | $\mathit{f}\left(\mathit{x}\right)$ | $\mathit{F}\left(\mathit{x}\right)$ | $\mathit{x}\left(\mathit{p}\right)$ |
---|---|---|---|

PGIV4 | $\frac{\lambda \cdot {\left(\frac{x-\gamma}{\beta}\right)}^{\frac{1}{\alpha}-1}\cdot {\left({\left(\frac{x-\gamma}{\beta}\right)}^{\frac{1}{\alpha}}+1\right)}^{-\lambda -1}}{\alpha \cdot \beta}$ | ${\left({\left(\frac{x-\gamma}{\beta}\right)}^{\frac{1}{\alpha}}+1\right)}^{-\lambda}$ | $\gamma +\beta \cdot {\left(\frac{1}{{p}^{\frac{1}{\lambda}}}-1\right)}^{\alpha}$ |

PGIV3 | $\frac{\lambda \cdot {\left(\frac{x}{\beta}\right)}^{\frac{1}{\alpha}-1}\cdot {\left({\left(\frac{x}{\beta}\right)}^{\frac{1}{\alpha}}+1\right)}^{-\lambda -1}}{\alpha \cdot \beta}$ | $1-{\left({\left(\frac{x}{\beta}\right)}^{\frac{1}{\alpha}}+1\right)}^{-\lambda}$ | $\beta \cdot {\left(\frac{1}{{p}^{\frac{1}{\lambda}}}-1\right)}^{\alpha}$ |

PGIII | $\frac{\alpha \cdot {\left(\frac{x-\gamma}{\beta}\right)}^{\alpha -1}\cdot {\left({\left(\frac{x-\gamma}{\beta}\right)}^{\alpha}+1\right)}^{-2}}{\beta}$ | ${\left(1+{\left(\frac{x-\gamma}{\beta}\right)}^{\alpha}\right)}^{-1}$ | $\gamma +\beta \cdot {\left(\frac{1}{p}-1\right)}^{\frac{1}{\alpha}}$ |

PGII | $\frac{1}{\beta}\cdot {\left(1-\frac{\alpha}{\beta}\cdot \left(x-\gamma \right)\right)}^{\frac{1}{\alpha}-1}$ | ${\left(1-\frac{\alpha}{\beta}\cdot \left(x-\gamma \right)\right)}^{\frac{1}{\alpha}}$ | $\gamma +\frac{\beta}{\alpha}\cdot \left(1-{p}^{\alpha}\right)$ |

PGI | $\frac{\alpha}{\beta}\cdot {\left(\frac{x-\gamma}{\beta}\right)}^{-\alpha -1}$ | ${\left(\frac{x-\gamma}{\beta}\right)}^{-\alpha}$ | $\gamma +\beta \cdot {p}^{-\frac{1}{\alpha}}$ |

WK5 | No closed form | No closed form | $\xi +\frac{\alpha}{\beta}\cdot \left(1-{p}^{\beta}\right)-\frac{\gamma}{\delta}\cdot \left(1-{p}^{-\delta}\right)$ |

Length [km] | Average Stream Slope [‰] | Sinuosity Coefficient [-] | Average Altitude [m] | Watershed Area [km ^{2}] |
---|---|---|---|---|

33 | 22 | 1.83 | 713 | 153 |

AMS | ||||||||||||

1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | ||

Flow | [m^{3}/s] | 9.96 | 15 | 10.1 | 14.8 | 7.30 | 21.2 | 18.2 | 21.4 | 13.1 | 14.5 | 35 |

2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | ||

Flow | [m^{3}/s] | 19.9 | 22.1 | 11.8 | 80.3 | 88 | 51.6 | 72.2 | 16.2 | 42.6 | 28.5 | 12.8 |

2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | ||||

Flow | [m^{3}/s] | 31.2 | 24.1 | 52.2 | 21.1 | 18.9 | 6.40 | 24.9 | 15.1 | 36.6 |

AES | ||||||||||||

1995 | 1996 | 1997 | 2000 | 2001 | 2002 | 2004 | 2004 | 2004 | 2005 | 2005 | ||

Flow | [m^{3}/s] | 21.2 | 18.2 | 21.4 | 35 | 19.9 | 22.1 | 80.3 | 22.2 | 19.2 | 88 | 38.9 |

2005 | 2005 | 2006 | 2007 | 2007 | 2007 | 2008 | 2009 | 2009 | 2010 | 2012 | ||

Flow | [m^{3}/s] | 24 | 17.5 | 51.6 | 72.2 | 33.8 | 15.9 | 16.2 | 42.6 | 23.1 | 28.5 | 31.2 |

2012 | 2012 | 2013 | 2014 | 2015 | 2016 | 2016 | 2018 | 2020 | ||||

Flow | [m^{3}/s] | 27.3 | 18.7 | 24.1 | 52.2 | 21.1 | 18.9 | 16.8 | 24.9 | 36.6 |

Prigor River | Statistical Indicators | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{k}}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{4}$ | ${\mathit{\tau}}_{2}$ | ${\mathit{\tau}}_{3}$ | ${\mathit{\tau}}_{4}$ | |

[m^{3}/s] | [m^{3}/s] | [-] | [-] | [-] | [m^{3}/s] | [m^{3}/s] | [m^{3}/s] | [m^{3}/s] | [-] | [-] | [-] | |

AMS | 27.6 | 21.1 | 0.762 | 1.66 | 5.16 | 27.6 | 10.7 | 4.26 | 2.43 | 0.386 | 0.399 | 0.228 |

AES | 31.7 | 18.9 | 0.595 | 1.83 | 5.77 | 31.7 | 9.30 | 4.22 | 2.14 | 0.293 | 0.454 | 0.230 |

Distribution | Annual Maximum Series (AMS) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Exceedance Probabilities [%] | ||||||||||||||||

MOM | L-Moments | |||||||||||||||

0.01 | 0.1 | 0.5 | 1 | 2 | 3 | 5 | 80 | 0.01 | 0.1 | 0.5 | 1 | 2 | 3 | 5 | 80 | |

PE3 | 214 | 160 | 123 | 107 | 90.7 | 81.5 | 69.5 | 12.0 | 231 | 172 | 130 | 113 | 95.4 | 85.3 | 72.7 | 11.4 |

PGIV4 | 265 | 166 | 118 | 100 | 84.7 | 76.2 | 66.1 | 11.3 | 364 | 217 | 145 | 119 | 96.9 | 84.9 | 70.9 | 11.7 |

PGIV3 | 260 | 166 | 118 | 101 | 85.5 | 76.9 | 66.6 | 11.3 | 813 | 323 | 169 | 128 | 96.8 | 82.1 | 66.5 | 12.6 |

PGIII | 279 | 169 | 117 | 98.7 | 82.7 | 74.2 | 64.3 | 12.0 | 800 | 320 | 168 | 128 | 95.7 | 82.1 | 66.6 | 12.5 |

PGII | 225 | 162 | 122 | 106 | 89.6 | 80.4 | 69.1 | 11.8 | 329 | 207 | 142 | 118 | 96.8 | 85.1 | 71.4 | 11.7 |

PGI | 171 | 138 | 112 | 100 | 87.6 | 80 | 70.2 | 10.5 | 329 | 207 | 142 | 118 | 86.8 | 85.1 | 71.4 | 11.7 |

WK5 | 227 | 163 | 122 | 106 | 89.4 | 80.3 | 69.00 | 11.8 | 358 | 216 | 145 | 120 | 97.0 | 85.0 | 71.0 | 11.7 |

Distribution | Annual Exceedance Series (AES) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Exceedance Probabilities [%] | ||||||||||||||||

MOM | L-Moments | |||||||||||||||

0.01 | 0.1 | 0.5 | 1 | 2 | 3 | 5 | 80 | 0.01 | 0.1 | 0.5 | 1 | 2 | 3 | 5 | 80 | |

PE3 | 178 | 138 | 110 | 97.7 | 85.4 | 78.2 | 69.1 | 16.6 | 233 | 172 | 130 | 113 | 95.3 | 85.3 | 72.9 | 18.0 |

PGIV4 | 233 | 150 | 108 | 93.5 | 80.3 | 73.2 | 64.9 | 17.2 | 277 | 190 | 137 | 116 | 96.0 | 85.1 | 72.0 | 18.1 |

PGIV3 | 219 | 146 | 108 | 94.3 | 81.6 | 74.6 | 66.1 | 16.6 | 836 | 327 | 170 | 128 | 96.4 | 81.7 | 66.4 | 19.2 |

PGIII | 232 | 150 | 109 | 93.7 | 80.4 | 73.2 | 64.7 | 17.4 | 940 | 338 | 168 | 126 | 94.4 | 80.1 | 65.3 | 18.8 |

PGII | 170 | 136 | 110 | 98.0 | 86.0 | 78.9 | 69.7 | 16.6 | 452 | 240 | 150 | 121 | 96.4 | 83.9 | 69.9 | 18.3 |

PGI | 134 | 117 | 101 | 92.6 | 83.5 | 77.7 | 70.0 | 15.6 | 452 | 240 | 150 | 121 | 96.4 | 83.9 | 69.9 | 18.3 |

WK5 | 150 | 128 | 108 | 98.1 | 87.3 | 80.5 | 71.4 | 17.0 | 202 | 164 | 130 | 114 | 96.7 | 86.6 | 73.7 | 18.2 |

Distri. | AMS | AES | ||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MOM | L-Moments | MOM | L-Moments | |||||||||||||||||||||

$\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\lambda}$ | $\mathit{\delta}$ | $\mathit{\xi}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\lambda}$ | $\mathit{\delta}$ | $\mathit{\xi}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\lambda}$ | $\mathit{\delta}$ | $\mathit{\xi}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\lambda}$ | $\mathit{\delta}$ | $\mathit{\xi}$ | |

PE3 | 0.766 | 24.1 | 9.2 | - | - | - | 0.694 | 26.9 | 9.0 | - | - | - | 1.254 | 16.9 | 10.6 | - | - | - | 0.526 | 28.2 | 16.9 | - | - | - |

PGIV4 | 0.505 | 47.4 | −2.51 | 2.66 | - | - | 0.940 | 80.0 | 7.47 | 5.19 | - | - | 0.222 | 49.3 | −17 | 1.26 | - | - | 1.29 | 1649 | 16.2 | 42.9 | - | - |

PGIV3 | 0.58 | 52.7 | - | 3.27 | - | - | 0.37 | 20.3 | - | 0.92 | - | - | 0.431 | 46.7 | - | 2.55 | - | - | 0.15 | 19.6 | - | 0.36 | - | - |

PGIII | 5.23 | 52.9 | −28.5 | - | - | - | 2.51 | 20.3 | 0.90 | - | - | - | 6.075 | 57.1 | −28 | - | - | - | 2.2 | 14.2 | 11.2 | - | - | - |

PGII | −0.042 | 19.3 | 7.5 | - | - | - | −0.14 | 17.1 | 7.78 | - | - | - | 0.039 | 20.4 | 12.1 | - | - | - | −0.25 | 12.2 | 15.4 | - | - | - |

PGI | −15.4 | −367 | 373 | - | - | - | 7.14 | 122 | −114 | - | - | - | −6.735 | −166 | 176 | - | - | - | 4.023 | 49.2 | −33.8 | - | - | - |

WK5 | 3.317 | 4.014 | 19 | - | 0.047 | 7.057 | 2.82 | 2.09 | 16 | - | 0.17 | 7.56 | −20.5 | 1.82 | 27.5 | - | −0.133 | 14.7 | −508 | 0.28 | 515 | - | −0.25 | 16.33 |

Distr. | Statistical Measures | |||||||
---|---|---|---|---|---|---|---|---|

Methods of Parameters Estimation | AES Values | |||||||

MOM | L-Moments | |||||||

RME | RAE | RME | RAE | ${\tau}_{3}$ | ${\tau}_{4}$ | ${\tau}_{3}$ | ${\tau}_{4}$ | |

PE3 | 0.0238 | 0.0879 | 0.0224 | 0.0902 | 0.399 | 0.192 | 0.399 | 0.228 |

PGIV4 | 0.0352 | 0.1574 | 0.0184 | 0.0772 | 0.228 | |||

PGIV3 | 0.0300 | 0.1394 | 0.0183 | 0.0750 | 0.303 | |||

PGIII | 0.0533 | 0.2081 | 0.0181 | 0.0736 | 0.299 | |||

PGII | 0.0228 | 0.1047 | 0.0190 | 0.0787 | 0.221 | |||

PGI | 0.0470 | 0.2327 | 0.0646 | 0.2671 | 0.221 | |||

WK5 | 0.0184 | 0.0807 | 0.0185 | 0.0775 | 0.228 |

Distr. | Statistical Measures | |||||||
---|---|---|---|---|---|---|---|---|

Methods of Parameters Estimation | AES Values | |||||||

MOM | L-Moments | |||||||

RME | RAE | RME | RAE | ${\tau}_{3}$ | ${\tau}_{4}$ | ${\tau}_{3}$ | ${\tau}_{4}$ | |

PE3 | 0.0219 | 0.1063 | 0.0093 | 0.0405 | 0.454 | 0.220 | 0.454 | 0.230 |

PGIV4 | 0.0395 | 0.1736 | 0.0094 | 0.0376 | 0.230 | |||

PGIV3 | 0.0347 | 0.1556 | 0.0156 | 0.0680 | 0.348 | |||

PGIII | 0.0399 | 0.1747 | 0.0153 | 0.0656 | 0.338 | |||

PGII | 0.0195 | 0.0967 | 0.0108 | 0.0394 | 0.272 | |||

PGI | 0.0269 | 0.1313 | 0.0108 | 0.0394 | 0.272 | |||

WK5 | 0.0137 | 0.0689 | 0.0086 | 0.0359 | 0.230 |

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## Share and Cite

**MDPI and ACS Style**

Anghel, C.G.; Ilinca, C.
Evaluation of Various Generalized Pareto Probability Distributions for Flood Frequency Analysis. *Water* **2023**, *15*, 1557.
https://doi.org/10.3390/w15081557

**AMA Style**

Anghel CG, Ilinca C.
Evaluation of Various Generalized Pareto Probability Distributions for Flood Frequency Analysis. *Water*. 2023; 15(8):1557.
https://doi.org/10.3390/w15081557

**Chicago/Turabian Style**

Anghel, Cristian Gabriel, and Cornel Ilinca.
2023. "Evaluation of Various Generalized Pareto Probability Distributions for Flood Frequency Analysis" *Water* 15, no. 8: 1557.
https://doi.org/10.3390/w15081557