# Predicting Future Flood Risks in the Face of Climate Change: A Frequency Analysis Perspective

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Probability Distributions

#### 2.2. Determination of Distribution Parameters

#### 2.2.1. Dagum Distribution (DG)

#### 2.2.2. Paralogistic Distribution (PR)

#### 2.2.3. Inverse Paralogistic (IPR)

#### 2.2.4. The Four Parameters Burr Distribution (BR4)

## 3. Case Studies

^{2}, representing about 4.6% of Romania’s surface [34].

## 4. Results and Discussions

#### 4.1. Estimated Parameters and Quantiles

^{3}/s (IPR distribution) and 1116 m

^{3}/s (BR4 distribution) using L-moments, between 1219 m

^{3}/s (DG distribution) and 1005 m

^{3}/s (BR4 distribution) using LH-moments. In the case of the Buhai river, the maximum flows vary between 1505 m

^{3}/s (IPR distribution) and 1176 m

^{3}/s (BR4 distribution) using L-moments, between 1257 m

^{3}/s (DG distribution) and 968 m

^{3}/s (BR4 distribution) using LH-moments. For the Miletin river, these maximum values vary between 863 m

^{3}/s (BR4 distribution) and 649 m

^{3}/s (PR distribution) using L-moments, between 1122 m

^{3}/s (BR4 distribution) and 799 m

^{3}/s (IPR distribution) using LH-moments. The results in the case of the Sitna river vary between 1388 m

^{3}/s (DG distribution) and 1049 m

^{3}/s (BR4 distribution) using L-moments, between 1379 m

^{3}/s (DG distribution) and 1083 m

^{3}/s (BR4 distribution) using LH-moments.

#### 4.2. Best-Fit Distribution Selection

_{0.01%}being characterized by a bias of less than 20%, a more than acceptable error regarding the rarity of this event.

#### 4.3. Confidence Intervals

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Observed Data for Jijia, Buhai, Miletin and Sitna Rivers

Jijia River | Buhai River | Miletin River | Sitna River | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Date | Flow | Date | Flow | Date | Flow | Date | Flow | Date | Flow | Date | Flow | Date | Flow | Date | Flow |

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

1961 | 12.1 | 1989 | 1.44 | 1981 | 25.4 | 2010 | 85 | 1981 | 60.4 | 2010 | 41.6 | 1961 | 35.3 | 1989 | 61.2 |

1962 | 35.4 | 1990 | 2.29 | 1982 | 7.31 | 2011 | 1.58 | 1982 | 50.4 | 2011 | 36.9 | 1962 | 47.6 | 1990 | 11.6 |

1963 | 15.8 | 1991 | 40.5 | 1983 | 5.68 | 2012 | 2.34 | 1983 | 64.5 | 2012 | 6.21 | 1963 | 58.7 | 1991 | 149 |

1964 | 5.75 | 1992 | 9.5 | 1984 | 37.6 | 2013 | 6.14 | 1984 | 55.4 | 2013 | 18.5 | 1964 | 5.27 | 1992 | 16.7 |

1965 | 49.1 | 1993 | 7.28 | 1985 | 22.4 | 2014 | 9.09 | 1985 | 204 | 2014 | 25 | 1965 | 290 | 1993 | 10.5 |

1966 | 10.8 | 1994 | 9.83 | 1986 | 2.75 | 2015 | 2.15 | 1986 | 9.02 | 2015 | 6.58 | 1966 | 26.9 | 1994 | 113 |

1967 | 9.6 | 1995 | 1.51 | 1987 | 4.4 | 2016 | 11.2 | 1987 | 2.68 | 2016 | 17.7 | 1967 | 28.2 | 1995 | 48 |

1968 | 3.27 | 1996 | 39.9 | 1988 | 11.2 | 2017 | 5.05 | 1988 | 104 | 2017 | 25.3 | 1968 | 11 | 1996 | 97 |

1969 | 170 | 1997 | 7.3 | 1989 | 1.8 | 1989 | 27.7 | 1969 | 176 | 1997 | 28.9 | ||||

1970 | 45.9 | 1998 | 59.2 | 1990 | 3.2 | 1990 | 6.81 | 1970 | 42.5 | 1998 | 56.8 | ||||

1971 | 49.1 | 1999 | 17.2 | 1991 | 12.9 | 1991 | 113 | 1971 | 105 | 1999 | 48.1 | ||||

1972 | 9.2 | 2000 | 16.4 | 1992 | 15.2 | 1992 | 34.4 | 1972 | 44.9 | 2000 | 34.4 | ||||

1973 | 36.6 | 2001 | 6.43 | 1993 | 6.86 | 1993 | 12.8 | 1973 | 84.5 | 2001 | 35.4 | ||||

1974 | 102 | 2002 | 32.2 | 1994 | 8.14 | 1994 | 42.1 | 1974 | 66.4 | 2002 | 72.5 | ||||

1975 | 16 | 2003 | 9.06 | 1995 | 9.6 | 1995 | 35.5 | 1975 | 82.7 | 2003 | 41.4 | ||||

1976 | 20.4 | 2004 | 3.02 | 1996 | 14.5 | 1996 | 70.8 | 1976 | 14.2 | 2004 | 16.2 | ||||

1977 | 57.5 | 2005 | 79.5 | 1997 | 2.87 | 1997 | 44.2 | 1977 | 51.2 | 2005 | 69.5 | ||||

1978 | 47 | 2006 | 90.6 | 1998 | 96 | 1998 | 70.1 | 1978 | 37.2 | 2006 | 55.2 | ||||

1979 | 127 | 2007 | 2.47 | 1999 | 6.68 | 1999 | 42.7 | 1979 | 100 | 2007 | 6.2 | ||||

1980 | 33.5 | 2008 | 54.38 | 2000 | 5.53 | 2000 | 39.8 | 1980 | 56.3 | 2008 | 41.8 | ||||

1981 | 56.7 | 2009 | 13.32 | 2001 | 4.96 | 2001 | 26.6 | 1981 | 36.5 | 2009 | 15.6 | ||||

1982 | 31.4 | 2010 | 190 | 2002 | 8.55 | 2002 | 47.9 | 1982 | 41 | 2010 | 23 | ||||

1983 | 14.8 | 2011 | 7.304 | 2003 | 1.02 | 2003 | 28.6 | 1983 | 12.2 | 2011 | 31.6 | ||||

1984 | 20.9 | 2012 | 4.5 | 2004 | 1.34 | 2004 | 8.73 | 1984 | 82.8 | 2012 | 4.65 | ||||

1985 | 54.2 | 2013 | 16.4 | 2005 | 25 | 2005 | 46.5 | 1985 | 125 | 2013 | 28.5 | ||||

1986 | 7.21 | 2014 | 17.82 | 2006 | 24.2 | 2006 | 39.56 | 1986 | 15.9 | 2014 | 30.4 | ||||

1987 | 1.34 | 2015 | 1.636 | 2007 | 0.77 | 2007 | 6.81 | 1987 | 5.74 | 2015 | 7.4 | ||||

1988 | 14.9 | 2016 | 25.5 | 2008 | 40.6 | 2008 | 68.6 | 1988 | 149 | 2016 | 48.74 | ||||

2017 | 8.306 | 2009 | 3.644 | 2009 | 32.8 | 2017 | 36.4 |

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**Figure 1.**The positioning of the rivers: Jijia, Buhai, Miletin and Sitna; and the positioning of the hydrometric stations: Dorohoi, Padureni, Sipote and Todireni.

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

Exact relationships for LH moments | DG, PR, IPR, BR4 |

Approximate relations for LH moments | PR, IPR |

Approximate relations for L-moments | PR |

LH moments diagrams and relationships | PR, IPR |

Exact frequency factors | DG, PR, IPR, BR4 |

Approximate frequency factors | PR, IPR |

Probability Distribution | $\mathbf{Quantile}\text{}\mathbf{Functions}\text{}\mathit{x}\left(\mathit{p}\right)$ |
---|---|

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

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

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

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

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

Jijia | 275 | 1.0 | 1.45 | 152 | 5757 |

Buhai | 18 | 10 | 1.17 | 279 | 134 |

Miletin | 90 | 3.0 | 1.24 | 166 | 675 |

Sitna | 78 | 2.0 | 1.4 | 166 | 943 |

River | MOM | L-Moments Method | LH-Moments Method | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

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

Jijia | 32.1 | 1.22 | 32.1 | 18.3 | 8.22 | 4.47 | 0.5703 | 0.4483 | 0.2436 | 50.5 | 19.9 | 8.46 | 4.59 | 0.3945 | 0.4247 | 0.2307 |

Buhai | 14.4 | 1.452 | 14.4 | 8.78 | 4.93 | 3.28 | 0.6102 | 0.5607 | 0.3731 | 23.2 | 10.3 | 5.47 | 3.40 | 0.4436 | 0.5319 | 0.3303 |

Miletin | 42.5 | 0.883 | 42.5 | 18.1 | 5.52 | 4.89 | 0.264 | 0.3041 | 0.2698 | 60.7 | 17.8 | 6.94 | 5.63 | 0.2924 | 0.3912 | 0.3175 |

Sitna | 53.9 | 0.931 | 53.9 | 24.3 | 8.53 | 5.96 | 0.4508 | 0.3513 | 0.2451 | 78.2 | 24.6 | 9.66 | 6.15 | 0.3149 | 0.3923 | 0.2496 |

River | Homogeneity | Outliers | Q_{max} for the Observed Data |
---|---|---|---|

von Newman | Grubb-Beck | ||

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

Jijia | 2.0712 | 548 | 190 |

Buhai | 2.3503 | 159 | 96 |

Miletin | 2.1681 | 353 | 204 |

Sitna | 2.4471 | 507 | 290 |

Parameter | Distribution | |||||||
---|---|---|---|---|---|---|---|---|

DG | PR | IPR | BR4 | DG | PR | IPR | BR4 | |

L-Moments | LH-Moments (First Level) | |||||||

Jijia River | ||||||||

$\alpha $ | 2.3353 | 1.5672 | 2.3426 | 0.1799 | 2.5609 | 1.6837 | 2.9441 | 0.1437 |

$\beta $ | 48.1 | 34.5 | 22.7 | 2.6613 | 59.0 | 43.8 | 35.0 | 2.8001 |

$\gamma $ | 0.3093 | −4.27 | −16.1 | 2.32 | 0.2334 | −9.86 | −35.1 | 3.95 |

$\lambda $ | - | - | - | 66.6 | - | - | - | 74.6 |

Bahna River | ||||||||

$\alpha $ | 1.798 | 1.3757 | 1.8029 | 0.2621 | 1.906 | 1.4172 | 1.9788 | 0.1528 |

$\beta $ | 12.1 | 10.3 | 6.57 | 1.9499 | 16.2 | 11.9 | 8.2 | 2.1129 |

$\gamma $ | 0.5647 | −0.298 | −2.56 | 1.34 | 0.3994 | −1.345 | −5.34 | 2.67 |

$\lambda $ | - | - | - | 20.7 | - | - | - | 30.1 |

Miletin River | ||||||||

$\alpha $ | 3.1175 | 1.9546 | 4.2066 | 1.0611 | 2.8317 | 1.7967 | 3.5184 | 0.761 |

$\beta $ | 57.5 | 58.9 | 56.7 | 3.2033 | 46.8 | 47.3 | 42.4 | 2.6568 |

$\gamma $ | 0.3811 | −4.56 | −51.8 | −16.34 | 0.5509 | 2.12 | −31.99 | 0.25 |

$\lambda $ | - | - | - | 48.7 | - | - | - | 38.8 |

Sitna River | ||||||||

$\alpha $ | 2.7591 | 1.8033 | 3.297 | 22.676 | 2.7679 | 1.7925 | 3.4943 | 5.3107 |

$\beta $ | 66.5 | 66.3 | 52.7 | 3.7366 | 67.0 | 65.2 | 58.2 | 3.5955 |

$\gamma $ | 0.4376 | −3.47 | −41.6 | −63.8 | 0.4323 | −2.83 | −49.5 | −55.3 |

$\lambda $ | - | - | - | 41.0 | - | - | - | 55.2 |

Distribution | Annual Exceedance Probabilities [%] | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

L-Moments Method | LH-Moments (First Level) | |||||||||||

0.01 | 0.1 | 0.5 | 1 | 40 | 80 | 0.01 | 0.1 | 0.5 | 1 | 40 | 80 | |

Jijia River | ||||||||||||

DG | 1503 | 560 | 280 | 207 | 26.0 | 5.20 | 1219 | 496 | 263 | 199 | 26.3 | 4.0 |

PR | 1460 | 566 | 287 | 213 | 25.5 | 6.14 | 1117 | 487 | 267 | 204 | 26.3 | 3.89 |

IPR | 1652 | 608 | 297 | 217 | 25.5 | 6.76 | 1118 | 492 | 270 | 206 | 26.4 | 3.86 |

BR4 | 1116 | 471 | 257 | 197 | 25.8 | 4.63 | 1005 | 443 | 250 | 195 | 25.1 | 5.32 |

Buhai River | ||||||||||||

DG | 1471 | 409 | 166 | 113 | 9.72 | 2.55 | 1257 | 375 | 161 | 111 | 9.84 | 1.98 |

PR | 1337 | 394 | 166 | 114 | 9.60 | 2.62 | 1160 | 366 | 162 | 113 | 9.74 | 2.06 |

IPR | 1505 | 418 | 169 | 114 | 9.65 | 2.81 | 1211 | 374 | 163 | 113 | 9.86 | 1.96 |

BR4 | 1176 | 362 | 158 | 111 | 9.60 | 2.23 | 968 | 327 | 153 | 110 | 8.95 | 2.87 |

Miletin River | ||||||||||||

DG | 810 | 387 | 230 | 184 | 41.2 | 14.9 | 981 | 435 | 246 | 192 | 40.3 | 17.0 |

PR | 649 | 349 | 223 | 182 | 40.1 | 15.4 | 820 | 399 | 239 | 191 | 39.8 | 17.5 |

IPR | 660 | 360 | 229 | 186 | 40.4 | 16.1 | 799 | 400 | 241 | 192 | 39.9 | 17.5 |

BR4 | 863 | 412 | 242 | 192 | 40.2 | 16.4 | 1122 | 471 | 257 | 198 | 39.7 | 18.6 |

Sitna River | ||||||||||||

DG | 1388 | 602 | 335 | 260 | 49.9 | 17.7 | 1379 | 600 | 335 | 260 | 49.9 | 17.6 |

PR | 1119 | 545 | 325 | 258 | 49.3 | 18.1 | 1140 | 550 | 327 | 259 | 49.2 | 18.3 |

IPR | 1195 | 574 | 336 | 264 | 49.0 | 19.1 | 1112 | 551 | 329 | 261 | 49.3 | 18.4 |

BR4 | 1049 | 537 | 326 | 260 | 49.1 | 18.7 | 1083 | 545 | 328 | 261 | 49.2 | 18.4 |

Distribution | L-Moments | Observed Data | LH-Moments | Observed Data | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

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

Jijia River | ||||||||||||

DG | 0.0319 | 0.1724 | 0.4483 | 0.2881 | 0.4483 | 0.2436 | 0.0415 | 0.2125 | 0.4247 | 0.2586 | 0.4247 | 0.2307 |

PR | 0.0773 | 0.2668 | 0.3106 | 0.1867 | 0.5742 | 0.2717 | ||||||

IPR | 0.119 | 0.3504 | 0.3327 | 0.2714 | 0.7564 | 0.2777 | ||||||

BR4 | 0.0287 | 0.1476 | 0.2436 | 0.0651 | 0.2426 | 0.2307 | ||||||

Buhai River | ||||||||||||

DG | 0.0327 | 0.1252 | 0.5607 | 0.4157 | 0.5607 | 0.3731 | 0.0478 | 0.1986 | 0.5319 | 0.3689 | 0.5319 | 0.3303 |

PR | 0.0341 | 0.1186 | 0.4115 | 0.0799 | 0.2493 | 0.3684 | ||||||

IPR | 0.061 | 0.166 | 0.4309 | 0.14 | 0.3625 | 0.375 | ||||||

BR4 | 0.0305 | 0.1312 | 0.3731 | 0.0899 | 0.2695 | 0.3303 | ||||||

Miletin River | ||||||||||||

DG | 0.0292 | 0.1275 | 0.3041 | 0.2143 | 0.3041 | 0.2698 | 0.0399 | 0.1424 | 0.3912 | 0.2596 | 0.3912 | 0.3175 |

PR | 0.0402 | 0.1544 | 0.2125 | 0.0552 | 0.1992 | 0.2457 | ||||||

IPR | 0.0619 | 0.1858 | 0.2300 | 0.0473 | 0.1706 | 0.2507 | ||||||

BR4 | 0.0635 | 0.1852 | 0.2698 | 0.0603 | 0.2095 | 0.3175 | ||||||

Sitna River | ||||||||||||

DG | 0.0174 | 0.0781 | 0.3513 | 0.2470 | 0.3513 | 0.2451 | 0.0175 | 0.0781 | 0.3923 | 0.257 | 0.3923 | 0.2496 |

PR | 0.0221 | 0.089 | 0.2406 | 0.0205 | 0.0872 | 0.2465 | ||||||

IPR | 0.0399 | 0.1172 | 0.2606 | 0.0482 | 0.1276 | 0.2516 | ||||||

BR4 | 0.0341 | 0.1084 | 0.2451 | 0.0435 | 0.1202 | 0.2496 |

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## Share and Cite

**MDPI and ACS Style**

Anghel, C.G.; Ilinca, C.
Predicting Future Flood Risks in the Face of Climate Change: A Frequency Analysis Perspective. *Water* **2023**, *15*, 3883.
https://doi.org/10.3390/w15223883

**AMA Style**

Anghel CG, Ilinca C.
Predicting Future Flood Risks in the Face of Climate Change: A Frequency Analysis Perspective. *Water*. 2023; 15(22):3883.
https://doi.org/10.3390/w15223883

**Chicago/Turabian Style**

Anghel, Cristian Gabriel, and Cornel Ilinca.
2023. "Predicting Future Flood Risks in the Face of Climate Change: A Frequency Analysis Perspective" *Water* 15, no. 22: 3883.
https://doi.org/10.3390/w15223883