# Evaluation of Statistical PMP Considering RCP Climate Change Scenarios in Republic of Korea

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Datasets

#### 2.1. Study Area

#### 2.2. Datasets

## 3. Methodology

#### 3.1. Hershfield Method

#### 3.2. Hershfield’s Nomograph

#### 3.3. Frequency Factor Method

#### 3.4. Hydrometeorological Method

#### 3.5. Statistical Measures

## 4. Application and Results

#### 4.1. Statistical Probable Maximum Precipitation for Historical Period

#### 4.2. PMP of Modified Hershfield’s Nomograph for Future Period

#### 4.3. Comparision of SPMP by Each Method for Future Period

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Study area and selected 615 sites including 62 sites (red dots show sites with rainfall data spanning more than 40 years) in this study.

**Figure 2.**Nomograph of ${K}_{M}$ values based on mean annual maximum series and rainfall duration (Hershfield [28]).

**Figure 3.**Nomograph of ${K}_{M}$ values based on mean annual maximum 24 h precipitation data from 615 sites. Hershfield’s nomograph (Case 3) is indicated in the blue dashed line, and the modified Hershfield’s nomograph (Case 4) is indicated in the red dashed line. The dots represent the frequency factor corresponding to mean annual maximum rainfall.

**Figure 4.**Statistical and hydrometeorological probable maximum precipitation based on observed data up to 2020. (

**a**) SPMP (Case 1, ${K}_{M}=15$); (

**b**) SPMP (Case 3, Hershfield’s Nomograph); (

**c**) SPMP (Case 4, Modified Nomograph); (

**d**) HPMP (2020).

**Figure 5.**Nomograph of frequency factor ${\mathit{K}}_{\mathit{M}}$ for RCP 4.5 and RCP 8.5 scenarios for future periods. (

**a**) RCP 4.5; (

**b**) RCP 8.5.

**Figure 6.**SPMPs for Case 4 (modified Hershfield’s nomograph) for RCP scenarios and future periods 2040, 2070, and 2100. (

**a**) RCP 4.5 (2040); (

**b**) RCP 8.5 (2040); (

**c**) RCP 4.5 (2070); (

**d**) RCP 8.5 (2070); (

**e**) RCP 4.5 (2100); (

**f**) RCP 8.5 (2100).

**Figure 7.**Statistical and hydrometeorological probable maximum precipitation using RCP 4.5 (2100). (

**a**) RCP 4.5 (Hershfield’s ${K}_{M}=15$); (

**b**) RCP 4.5 (Hershfield’s Nomograph); (

**c**) RCP 4.5 (Modified Nomograph); (

**d**) RCP 4.5 (HPMP).

**Figure 8.**Statistical and hydrometeorological probable maximum precipitation using RCP 8.5 (2100). (

**a**) RCP 8.5 (Hershfield’s ${K}_{M}=15$); (

**b**) RCP 8.5 (Hershfield’s Nomograph); (

**c**) RCP 8.5 (Modified Nomograph); (

**d**) RCP 8.5 (HPMP).

Climate Scenario | Period (Year) |
---|---|

RCP 4.5 and 8.5 | Obs. start year~2040 |

Obs. start year~2070 | |

Obs. start year~2100 |

**Table 2.**Statistical measures and formulas for evaluating the results of SPMPs compared to the result of HPMP, where $n$ is number of observation sites, ${x}_{i}$ is HPMP value, and ${\widehat{x}}_{i}$ is SPMP value.

Statistical Measure | Formula |
---|---|

Mean Absolute Error (MAE) | $\mathrm{M}\mathrm{A}\mathrm{E}=\frac{\sum _{i=1}^{n}\left|{x}_{i}-{\widehat{x}}_{i}\right|}{n}$ |

Mean Absolute Percentage Error (MAPE) | $\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\frac{\sum _{i=1}^{n}\left(\frac{\left|{x}_{i}-{\widehat{x}}_{i}\right|}{{x}_{i}}\right)}{n}\times 100\%$ |

Root Mean Square Error (RMSE) | $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}{\left({x}_{i}-{\widehat{x}}_{i}\right)}^{2}}$ |

Case | Statistical Methods |
---|---|

Case 1 | $\mathrm{Hershfield}\u2019\mathrm{s}\text{}\mathrm{fixed}\text{}\mathrm{frequency}\text{}\mathrm{factor}\text{}({K}_{M}=15$) |

Case 2 | $\mathrm{Hershfield}\u2019\mathrm{s}\text{}\mathrm{method}\text{}\mathrm{with}\text{}\mathrm{varying}\text{}{K}_{M}$ for each site |

Case 3 | Hershfield’s original nomograph |

Case 4 | Modified Hershfield’s nomograph |

Case 5 | Chow’s frequency factor method (T = 60,000) |

**Table 4.**MAE, MAPE, RMSE, SPMP/HPMP ratio, ${K}_{M}$, and SPMPs compared to HPMP (2020) for 615 sites.

Estimation of PMP | SPMPs | HPMP (2020) | |||||
---|---|---|---|---|---|---|---|

Case 1 Hershfield’s ${\mathit{K}}_{\mathit{M}}=15$ | Case 2 $\mathbf{Varying}\text{}{\mathit{K}}_{\mathit{M}}$ for Each Site | Case 3 Hershfield’s Nomograph | Case 4 Modified Nomograph | Case 5 Chow’s T = 60,000 | |||

PMP | Max. | 2083 | 1420 | 1608 | 1719 | 2966 | 1396 |

Mean | 1035 | 357 | 909 | 955 | 1316 | 967 | |

Min. | 395 | 141 | 403 | 411 | 437 | 598 | |

${K}_{M}$ | Max. | 15.0 | 10.4 | 15.4 | 15.8 | 35.1 | - |

Mean | 15.0 | 3.4 | 13.1 | 13.8 | 19.8 | - | |

Min. | 15.0 | 1.3 | 9.7 | 11.0 | 9.7 | - | |

Evaluation | |||||||

MAE | 184 | 613 | 150 | 149 | 372 | - | |

MAPE | 19 | 63 | 16 | 15 | 39 | - | |

RMSE | 249 | 634 | 194 | 189 | 487 | - | |

SPMP/HPMP Ratio | 1.071 | 0.369 | 0.941 | 0.988 | 1.361 | - |

**Table 5.**Statistical information of SPMP (calculated using modified Hershfield’s nomograph, Case 4) based on RCP scenarios for future periods.

Scenario (Period) | RCP 4.5 | RCP 8.5 | |||||
---|---|---|---|---|---|---|---|

2040 | 2070 | 2100 | 2040 | 2070 | 2100 | ||

${K}_{M}$ | Max | 16.5 | 17.0 | 25.5 | 16.6 | 17.0 | 25.0 |

Mean | 12.7 | 12.9 | 17.7 | 12.4 | 12.8 | 17.3 | |

Min | 9.1 | 10.0 | 16.5 | 8.8 | 9.9 | 16.1 | |

PMP | Max | 1573 | 1904 | 2665 | 1810 | 1831 | 2614 |

Mean | 633 | 763 | 1079 | 788 | 779 | 1101 | |

Min | 377 | 444 | 594 | 393 | 432 | 622 |

**Table 6.**MAE, MAPE, RMSE, SPMP/HPMP ratio, ${K}_{M}$, and SPMPs compared with HPMP (2100) for 62 sites.

Estimation of PMP | SPMPs | HPMP (2100) | ||||||
---|---|---|---|---|---|---|---|---|

Case 1 $\mathbf{Hershfield}\u2019\mathbf{s}\phantom{\rule{0ex}{0ex}}{\mathit{K}}_{\mathit{M}}=15$ | Case 2 $\mathbf{Varying}\text{}{\mathit{K}}_{\mathit{M}}$ for Each Site | Case 3 Hershfield’s Nomograph | Case 4 Modified Nomograph | Case 5 Chow’s T = 60,000 | ||||

RCP 4.5 scenario | SPMP | Max. | 1823 | 1600 | 1643 | 2665 | 2795 | 1655 |

Mean | 967 | 480 | 948 | 1118 | 1361 | 1273 | ||

Min. | 514 | 172 | 535 | 599 | 611 | 856 | ||

${K}_{M}$ | Max. | 15.0 | 12.6 | 18.9 | 25.5 | 27.4 | - | |

Mean | 15.0 | 4.9 | 12.6 | 17.7 | 18.5 | - | ||

Min. | 15.0 | 1.4 | 7.3 | 16.5 | 9.9 | - | ||

Evaluation | ||||||||

MAE | 372 | 798 | 367 | 327 | 407 | - | ||

MAPE | 29 | 62 | 28 | 25 | 32 | - | ||

RMSE | 431 | 841 | 430 | 408 | 522 | - | ||

SPMP/HPMP Ratio | 0.760 | 0.377 | 0.745 | 0.878 | 1.069 | - | ||

RCP 8.5 scenario | SPMP | Max. | 1968 | 1017 | 1828 | 2316 | 3634 | 2136 |

Mean | 1003 | 477 | 983 | 1137 | 1933 | 1566 | ||

Min. | 563 | 212 | 567 | 626 | 912 | 1019 | ||

${K}_{M}$ | Max. | 15.0 | 14.0 | 15.8 | 25.0 | 31.2 | - | |

Mean | 15.0 | 5.9 | 14.8 | 17.3 | 24.0 | - | ||

Min. | 15.0 | 3.4 | 13.6 | 16.1 | 17.7 | - | ||

Evaluation | ||||||||

MAE | 594 | 1089 | 605 | 512 | 514 | - | ||

MAPE | 37 | 69 | 38 | 32 | 34 | - | ||

RMSE | 646 | 1122 | 656 | 565 | 744 | - | ||

SPMP/HPMP Ratio | 0.641 | 0.304 | 0.628 | 0.726 | 1.234 | - |

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## Share and Cite

**MDPI and ACS Style**

Seo, M.; Kim, S.; Kim, H.; Kim, H.; Shin, J.-Y.; Heo, J.-H.
Evaluation of Statistical PMP Considering RCP Climate Change Scenarios in Republic of Korea. *Water* **2023**, *15*, 1756.
https://doi.org/10.3390/w15091756

**AMA Style**

Seo M, Kim S, Kim H, Kim H, Shin J-Y, Heo J-H.
Evaluation of Statistical PMP Considering RCP Climate Change Scenarios in Republic of Korea. *Water*. 2023; 15(9):1756.
https://doi.org/10.3390/w15091756

**Chicago/Turabian Style**

Seo, Miru, Sunghun Kim, Heechul Kim, Hanbeen Kim, Ju-Young Shin, and Jun-Haeng Heo.
2023. "Evaluation of Statistical PMP Considering RCP Climate Change Scenarios in Republic of Korea" *Water* 15, no. 9: 1756.
https://doi.org/10.3390/w15091756