# Smooth Spatial Modeling of Extreme Mediterranean Precipitation

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

#### 2.1. Rainfall Station Data

^{2}(see the leftmost panel of Figure 2). The maximum altitude is located in the mountainous region known as Djebel Abderrahmane and then decreases toward the coast. Daily precipitation data are available from 1919 to 2014 in 18 stations, with a percentage of missing data that varies between $19.29\%$ and $67.5\%$. Average annual rainfall totals over the rainy season range from $377.3$ to $583.2$ mm. Both daily rainfall data sets in Tunisia were collected from the Tunisian General Directorate of Water Resources.

#### 2.2. CHIRPS Data Set

#### 2.3. Inter-Covariate Correlation Analysis

## 3. Statistical Methods

#### 3.1. Extreme Value Theory

#### 3.2. Smooth Spatial Modeling for Extremes

#### 3.2.1. Generalized Linear Models

#### 3.2.2. Artificial Neural Networks

- Shuffle the data set randomly.
- Split the data set into k = 10 folds.
- For each fold:
- –
- Define that fold as the validation data set.
- –
- Define the remaining folds as the training data set.
- –
- Fit the model on the training set and evaluate on the validation set.

- The error is calculated as the average of the error over all validation sets. The optimal number of hidden units corresponds to the minimum of errors.

## 4. Results

#### 4.1. Pointwise GEV Parameters Estimation

#### 4.2. Spatial GEV Parameters Estimation

#### 4.3. Model Evaluation

^{2}is $6.57$, i.e., 1 station per 152 km

^{2}. For Merguellil, the number of stations per 1000 km

^{2}is $3.6$, i.e., 1 station per 277 km

^{2}. To have approximately the same density, we selected 100 stations instead of 183. The selection of covariates in this case for the ANN model is 4 hidden units and the covariates (x,y,z). We even attempted to take the same number of stations, i.e., 26 stations for the French Mediterranean region. In this case, the result of the cross-validation for the ANN model is to use (x,y,z,chirps) as covariates and 2 hidden units. From these experiments, we can deduce that there is a close relationship between model complexity and input size, which explains the difference between the number of hidden units in Table 2 (four for the French Mediterranean and one for each Tunisian site).

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Estimation of the GEV Parameters Using ANN on the Tunisian Sites

**Figure A1.**Estimates of the GEV parameters $(\mu ,\sigma ,\xi )$ and the 100-year return level over the Merguellil region. The spatial estimates are resulted from an ANN using (y,chirps) as covariates with 1 hidden unit and the point estimations are obtained by the L-moments method.

**Figure A2.**Estimates of the GEV parameters $(\mu ,\sigma ,\xi )$ and the 100−year return level over the Cap Bon region. The spatial estimates are resulted from an ANN using (y,z) as covariates with 1 hidden unit, and the point estimations are obtained by the L−moments method.

#### Appendix A.2. 10-Fold Cross-Validation Results

**Figure A3.**Result of the 10-fold cross-validation with the ANN model for the French Mediterranean site.

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**Figure 2.**DEMs for the three study sites. The selected stations are represented with average annual rainfall totals over rainy season represented by the blue scale. The stations numbered in red will serve as test stations. The sky blue area represents the Mediterranean Sea and the blue lines delimit the catchment areas.

**Figure 3.**Interpolated average seasonal precipitation totals computed from the CHIRPS data constituting the CHIRPS spatial covariate. The gauging stations are depicted as black points, and contour lines from DEM are represented by the gray lines.

**Figure 5.**Box−plots of estimated L-moments parameters of GEV distribution $(\mu ,\sigma ,\xi )$ and 100-year return level.

**Figure 6.**Return levels at the selected test stations for the French Mediterranean region. The 95% bootstrap confidence bands of the return levels obtained from the fitted models: the ANN in gray and the GLM in blue. The dots represent the empirical return levels.

**Figure 7.**Return levels at the selected test stations for the Merguellil region, showing 95% bootstrap confidence bands of the return levels obtained from the fitted models: the ANN in gray and the GLM in blue. The dots represent the empirical return levels.

**Figure 8.**Return levels at the selected test stations for the Cap Bon region, showing 95% bootstrap confidence bands of the return levels obtained from the fitted models: the ANN in gray and the GLM in blue. The dots represent the empirical return levels.

**Figure 9.**Estimates of the GEV parameters $(\mu ,\sigma ,\xi )$ and the 100-year return level over the French Mediterranean region. Spatial estimates result from an ANN using (x,y,z,chirps) as covariates with four hidden units, and the point estimations are obtained by the L−moments method.

Covariate | French Mediterranean | Merguellil | Lebna |
---|---|---|---|

x | 4341.18 | 488.39 | 364.70 |

y | 4382.88 | 488.36 | 367.07 |

z | 4341.94 | 488.46 | 366.97 |

chirps | 4326.7 | 488.56 | 365.08 |

(x,y) | 4305.67 | 487.34 | 361.42 |

(x,z) | 4301.3 | 487.21 | 364.51 |

(y,z) | 4340.35 | 488.63 | 363.98 |

(x,chirps) | 4296.11 | 486.64 | 364.42 |

(y,chirps) | 4313.43 | 487.00 | 364.65 |

(z,chirps) | 4314.11 | 487.50 | 363.04 |

(x,y,z) | 4275.84 | 486.52 | 360.75 |

(x,y,chirps) | 4295.64 | 486.93 | 360.91 |

(y,z,chirps) | 4301.79 | 487.46 | 361.89 |

(x,z,chirps) | 4277.24 | 486.60 | 362.39 |

(x,y,z,chirps) | 4274.04 | 486.55 | 361.25 |

**Table 2.**The selected covariates for the spatial model (GLM and ANN) of the GEV parameters over the three sites. The number of hidden units concerns only the ANN model.

Site | GLM Covariate | ANN Covariate | Number of Hidden Units |
---|---|---|---|

French Mediterranean | (x,y,z,chirps) | (x,y,z,chirps) | 4 |

Merguellil | (x,y,z) | (y,chirps) | 1 |

Lebna | (x,y,z) | (y,z) | 1 |

Site | Negative Log-Likelihood | Kolmogorov–Smirnov |
---|---|---|

French Mediterranean | 72.67% | 74.35% |

Merguellil | 61.53% | 65.38% |

Lebna | 61.1% | 61.1% |

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**MDPI and ACS Style**

Hammami, H.; Carreau, J.; Neppel, L.; Elasmi, S.; Feki, H.
Smooth Spatial Modeling of Extreme Mediterranean Precipitation. *Water* **2022**, *14*, 3782.
https://doi.org/10.3390/w14223782

**AMA Style**

Hammami H, Carreau J, Neppel L, Elasmi S, Feki H.
Smooth Spatial Modeling of Extreme Mediterranean Precipitation. *Water*. 2022; 14(22):3782.
https://doi.org/10.3390/w14223782

**Chicago/Turabian Style**

Hammami, Hela, Julie Carreau, Luc Neppel, Sadok Elasmi, and Haifa Feki.
2022. "Smooth Spatial Modeling of Extreme Mediterranean Precipitation" *Water* 14, no. 22: 3782.
https://doi.org/10.3390/w14223782