# Modelling of Extremely High Rainfall in Limpopo Province of South Africa

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The r-Largest Order Statistics

#### 2.2. Peaks-over-Threshold Approach

#### 2.2.1. The Generalised Pareto Distribution

#### 2.2.2. Threshold Selection and Declustering

#### 2.3. Parameter Estimation

#### 2.3.1. The Delta Method

#### 2.3.2. The Profile Likelihood Method

#### 2.4. Forecast Combination

#### 2.4.1. Combining Estimated Return Levels and Prediction Intervals

#### 2.4.2. Evaluation of Prediction Intervals

#### 2.5. Data and Study Area

#### 2.5.1. Description of the Study Area

#### 2.5.2. Rainfall

#### 2.5.3. El Niño Southern Oscillation Indices

#### 2.5.4. Indian Ocean Dipole

## 3. Results

#### 3.1. Exploratory Data Analysis

#### 3.2. GEVD${}_{r}$ Results

#### 3.3. GPD Results

#### 3.4. Model Comparisons

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CRPS | Continuous Rank Probability Score |

DSS | Directory of open access journals |

EVT | Extreme Valu Theory |

GEVD | Generalised Extreme Value Distribution |

GPD | Generalised Pareto Distribution |

LogS | Log Score |

MK | Mann-Kendall |

MLE | Maximum Likelihood Estimation |

POT | Peaks Over Threshold |

PINAW | Prediction Interval Normalised Average Width |

PIW | Prediction Interval Width |

SAWS | South African Weather Services |

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**Figure 4.**Return level plot using the profile likelihood and delta methods for determining best value of $r=8$.

**Figure 5.**Threshold stability plots.

**Top panel:**Stability plot for the extremal index.

**Middle panel:**Stability plot of the scale parameter.

**Bottom panel:**Stability plot of the shape parameter.

**Figure 7.**Cluster maxima plots.

**Top left panel:**Scatter plot of cluster maxima data.

**Top right panel:**Histogram of cluster maxima data.

**Bottom left panel:**Box plot of cluster maxima data.

Station Name | Latitude | Longitude | Altitude (m) | Data Availability |
---|---|---|---|---|

Thabazimbi | −24.6170 | 27.4000 | 1026 | 1960–2020 |

Min | Q1 | Q2 | Mean | Q3 | Max | Std. | Skew | Kurt | |
---|---|---|---|---|---|---|---|---|---|

MR | 0 | 0 | 23 | 48.3 | 77.1 | 326.8 | 60.15 | 1.65 | 3.11 |

AMR | 21.8 | 134.3 | 159.7 | 172 | 202 | 326.8 | 66.48 | 0.658 | 0.223 |

$\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\xi}}$ | |
---|---|---|---|

GEVD${}_{r=1}$ | 143.6 (9.92) | 64 (6.91) | −0.202 (0.0812) |

GEVD${}_{r=8}$ | 137.6 (10.38) | 66.3 (7.31) | −0.103 (0.0892) |

GPD | 60.9 (1.21) | −0.0678 (0.142) |

Model | Min | Q1 | Q2 | Mean | Q3 | Max | Skew | Kurt | StDev |
---|---|---|---|---|---|---|---|---|---|

GEVD${}_{r=1}$ (delta) | 40 | 50 | 56 | 54.33 | 60 | 64 | −0.46 | −1.33 | 8.08 |

GEVD${}_{r=1}$ (profile) | 22 | 27 | 30 | 28.89 | 32 | 34 | −0.33 | −1.51 | 4.14 |

GEVD${}_{r=8}$ (delta) | 24 | 27 | 29 | 28.33 | 30 | 31 | −0.49 | −1.43 | 2.55 |

GEVD${}_{r=8}$ (profile) | 34 | 44 | 51 | 49.33 | 56 | 59 | −0.46 | −1.36 | 8.59 |

GPD (MLE) | 2 | 21 | 32 | 27.6 | 38 | 42 | −0.63 | −1.19 | 13.8 |

Mean | 24 | 34 | 40 | 37.78 | 43 | 46 | −0.57 | −1.24 | 7.51 |

Median | 24 | 27 | 32 | 32.33 | 38 | 42 | 0.13 | −1.80 | 6.73 |

PINC(%) | Model | PINAW(%) | Average PIW |
---|---|---|---|

95 | GEVD${}_{r=1}$ (delta) | 2.26 | 54.33 |

95 | GEVD${}_{r=1}$ (profile) | 2.41 | 28.89 |

95 | GEVD${}_{r=8}$ (delta) | 1.97 | 28.33 |

95 | GEVD${}_{r=8}$ (profile) | 4.05 | 49.33 |

95 | GPD (MLE) | 0.07 | 27.56 |

95 | Mean | 0.17 | 108.78 |

95 | Median | 0.22 | 126.11 |

Delta Method | Profile Likelihood Method | ||
---|---|---|---|

Return Period (Years) | (L95,RL,U95) | (L95,RL,U95) | Exceedance Probability |

10 | (272,284,296) | (269,284,303) | 0.100 |

15 | (294,307,319) | (289,307,329) | 0.067 |

20 | (309,322,336) | (303,322,347) | 0.050 |

25 | (320,334,348) | (313,334,361) | 0.040 |

30 | (329,343,358) | (321,343,372) | 0.033 |

35 | (336,351,366) | (327,351,381) | 0.029 |

40 | (343,358,373) | (333,358,389) | 0.025 |

45 | (348,363,379) | (338,363,396) | 0.022 |

50 | (353,368,384) | (342,368,401) | 0.020 |

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**MDPI and ACS Style**

Sikhwari, T.; Nethengwe, N.; Sigauke, C.; Chikoore, H.
Modelling of Extremely High Rainfall in Limpopo Province of South Africa. *Climate* **2022**, *10*, 33.
https://doi.org/10.3390/cli10030033

**AMA Style**

Sikhwari T, Nethengwe N, Sigauke C, Chikoore H.
Modelling of Extremely High Rainfall in Limpopo Province of South Africa. *Climate*. 2022; 10(3):33.
https://doi.org/10.3390/cli10030033

**Chicago/Turabian Style**

Sikhwari, Thendo, Nthaduleni Nethengwe, Caston Sigauke, and Hector Chikoore.
2022. "Modelling of Extremely High Rainfall in Limpopo Province of South Africa" *Climate* 10, no. 3: 33.
https://doi.org/10.3390/cli10030033