# Intensity–Duration–Frequency Curves for Dependent Datasets

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Classical Models and Vine Copula Approach

- Rainfall intensity sample definition;
- Determination and estimation of the marginal distributions and, for the sampled series, estimation of the copula parameters and construction of the joint distribution;
- Estimation of the quantiles corresponding to specified return periods;
- Adjustment of an IDF relationship to the estimated quantiles;

#### 2.1. GEV-Distribution-Based Rainfall Quantiles

#### 2.2. Vine-Copula-Based Rainfall Quantiles

**Definition**

**1**

**.**

- 1.
- $V=({T}_{1},\cdots ,{T}_{m})$.
- 2.
- ${T}_{1}$ is a tree of nodes ${N}_{1}=\{1,\cdots ,n\}$ and a set of edges denoted by ${E}_{1}$.
- 3.
- For $i=2,\cdots ,m$, ${T}_{i}$ is a tree of nodes ${N}_{i}\subset {N}_{1}\cup {E}_{1}\cup {E}_{2}\cup \cdots \cup {E}_{i-1}$ and a set of edges denoted by ${E}_{i}$.

**Definition**

**2**

**.**

#### 2.3. IDF Curve Formulation

## 3. Study Area and Dataset

## 4. Results and Discussion

#### 4.1. Quantile Estimation and Comparison

#### 4.2. IDF Curve Estimation and Comparison

## 5. IDF Projections for Global Climate Models (GCMs)

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Directed acyclic graph for a vine copula model with six durations showing the hierarchical representation by bivariate copulas. In tree 1, the first node (F1) represents the distribution of one of the four subhourly rainfall intensities (${d}_{1}$, ${d}_{2}$, ${d}_{3}$ or ${d}_{4}$), and the rest of the nodes $(F2,\cdots ,F6)$ are the distributions of the five hourly rainfall intensities $({d}_{5},\cdots ,{d}_{9})$. The edges of tree k, $k=1,\cdots ,4$ become nodes in trees k + 1, i.e., bivariate copulas for tree 2 and conditional bivariate copulas for the rest.

**Figure 3.**Monthly average of total precipitation records for the Moncton location for the 1899–2010 period.

**Figure 5.**Comparison between the quantiles obtained from the typical GEV model and quantile points from the copula method.

**Figure 6.**Comparison between the IDF curves obtained from the typical IDF empirical formula and IDF points from the copula method.

**Figure 7.**Spatial interpolation of the IDF curves for the RCP2.6 annual maximum series given by the proposed model for the Moncton location.

Station | ID | Lat. | Long. | Alt (m) | Years | Start | End |
---|---|---|---|---|---|---|---|

Moncton Int A | NB- 103201 | 46 7${}^{\prime}$ N | 64 41${}^{\prime}$ W | 70 | 67 | 1946 | 2016 |

Tree | Edge | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{4}$ |
---|---|---|---|---|---|

1 | 2,1 | Sur BB8 (6, 0.57) | N (0.7) | N (0.76) | N (0.88) |

3,2 | Gumbel (2.57) | Gumbel (2.57) | Gumbel (2.57) | Gumbel (2.57) | |

4,3 | N (0.76) | N (0.76) | N (0.76) | N (0.76) | |

5,4 | N (0.90) | N (0.90) | N (0.90) | N (0.90) | |

6,5 | N (0.92) | N (0.92) | N (0.92) | N (0.92) | |

2 | 3,1;2 | I (0) | I (0) | I (0) | I (0) |

4,2;3 | r270 BB7 ($-1.14,-0.66$) | r270 BB7 ($-1.14,-0.66$) | r270 BB7 ($-1.14,-0.66$) | r270 BB7 ($-1.14,-0.66$) | |

5,3;4 | r90 Joe ($-1.47$) | r90 Joe ($-1.47$) | r90 Joe ($-1.47$) | r90 Joe ($-1.47$) | |

6,4;5 | Frank ($-2.45$) | Frank ($-2.45$) | Frank ($-2.45$) | Frank ($-2.45$) | |

3 | 4,1;3,2 | I (0) | I (0) | I (0) | I (0) |

5,2;4,3 | Frank (1.5) | Frank (1.5) | Frank (1.5) | Frank (1.5) | |

6,3;5,4 | I (0) | I (0) | I (0) | I (0) | |

4 | 5,1;4,3,2 | I (0) | I (0) | I (0) | I (0) |

6,2;5,4,3 | I (0) | I (0) | I (0) | I (0) | |

5 | 6,1;5,4,3,2 | Frank ($-2.15$) | r90 Clayton ($-0.25$) | r90 Clayton ($-0.24$) | Frank ($-1.38$) |

Intensities over the Different Durations | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T (Years) | ${\mathit{d}}_{1}=5$ min | ${\mathit{d}}_{2}=10$ min | ${\mathit{d}}_{3}=15$ min | ${\mathit{d}}_{4}=30$ min | ||||||||

${\mathit{i}}_{\mathit{GEV}}^{\mathit{T}}$ | ${\mathit{i}}_{\mathit{cop}}^{\mathit{T}}$ | $\mathsf{\Delta}$ | ${\mathit{i}}_{\mathit{GEV}}^{\mathit{T}}$ | ${\mathit{i}}_{\mathit{cop}}^{\mathit{T}}$ | $\mathsf{\Delta}$ | ${\mathit{i}}_{\mathit{GEV}}^{\mathit{T}}$ | ${\mathit{i}}_{\mathit{cop}}^{\mathit{T}}$ | $\mathsf{\Delta}$ | ${\mathit{i}}_{\mathit{GEV}}^{\mathit{T}}$ | ${\mathit{i}}_{\mathit{cop}}^{\mathit{T}}$ | $\mathsf{\Delta}$ | |

2 | 74.6 | 76.6 | 2.7 | 52.5 | 51.8 | $-1.4$ | 40.7 | 40.2 | $-1.2$ | 27.1 | 27.2 | 0.3 |

5 | 106.2 | 115.9 | 8.4 | 74.5 | 82.6 | 9.8 | 57.3 | 63.8 | 10.2 | 37.2 | 40.7 | 8.7 |

10 | 128.5 | 140.2 | 8.3 | 90.1 | 106.8 | 15.7 | 69.3 | 82.9 | 16.5 | 44.4 | 51.7 | 14.1 |

25 | 158.4 | 168.2 | 5.8 | 110.8 | 141.6 | 21.7 | 85.7 | 111.6 | 23.2 | 54.2 | 68.2 | 20.6 |

50 | 181.7 | 191.4 | 5.0 | 127.1 | 172.1 | 26.1 | 98.9 | 137.4 | 28.0 | 61.9 | 82.5 | 25.0 |

100 | 206.0 | 215.2 | 4.3 | 144.1 | 203.6 | 29.2 | 112.9 | 165.0 | 31.6 | 70.1 | 98.5 | 28.9 |

Intensities over the Different Durations | |||||
---|---|---|---|---|---|

T (Years) | ${\mathit{d}}_{5}=1$ h | ${\mathit{d}}_{6}=2$ h | ${\mathit{d}}_{7}=6$ h | ${\mathit{d}}_{8}=12$ h | ${\mathit{d}}_{9}=24$ h |

2 | 18.0 | 12.2 | 6.6 | 4.1 | 2.4 |

5 | 24.4 | 16.2 | 8.6 | 5.4 | 3.3 |

10 | 29.6 | 19.5 | 9.9 | 6.4 | 3.9 |

25 | 37.6 | 24.5 | 11.6 | 7.6 | 4.7 |

50 | 44.7 | 29.1 | 12.8 | 8.5 | 5.3 |

100 | 52.9 | 34.4 | 14.0 | 9.4 | 5.9 |

**Table 5.**Rainfall intensities (mm/h) for historical data and future IDF curves for Moncton station and three RCP emission scenarios (RCP2.6; RCP4.5; RCP8.5) with different durations and return periods (2; 10; 100 years).

Duration | RCP2.6 | RCP4.5 | RCP8.5 | Observations | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

2 Years | 10 Years | 100 Years | 2 Years | 10 Years | 100 Years | 2 Years | 10 Years | 100 Years | 2 Years | 10 Years | 100 Years | |

5 min | 119 | 178.8 | 260 | 129.4 | 198.7 | 293.2 | 126.4 | 188.6 | 273.1 | 71.3 | 113.1 | 155.3 |

10 min | 72.2 | 108.6 | 157.8 | 78.6 | 120.6 | 178 | 76.7 | 114.5 | 165.8 | 50.2 | 79.2 | 108.4 |

15 min | 53.9 | 81.1 | 117.9 | 58.7 | 90.1 | 132.9 | 57.3 | 85.5 | 123.8 | 40.5 | 62.7 | 85.1 |

30 min | 32.8 | 49.2 | 71.6 | 35.6 | 54.7 | 80.7 | 34.8 | 51.9 | 75.2 | 27 | 40.7 | 54.5 |

1 h | 19.9 | 29.9 | 43.4 | 21.6 | 33.2 | 49 | 21.1 | 31.5 | 45.6 | 17.9 | 25.5 | 33.2 |

2 h | 12.1 | 18.1 | 26.4 | 13.1 | 20.2 | 29.7 | 12.8 | 19.1 | 27.7 | 12.1 | 16.8 | 21.5 |

6 h | 5.5 | 8.2 | 12 | 6 | 9.1 | 13.5 | 5.8 | 8.7 | 12.6 | 6.5 | 9.6 | 12.7 |

12 h | 3.3 | 5 | 7.3 | 3.6 | 5.5 | 8.2 | 3.5 | 5.3 | 7.6 | 4.1 | 6 | 8 |

24 h | 2 | 3 | 4.4 | 2.2 | 3.4 | 5 | 2.1 | 3.2 | 4.6 | 2.4 | 3.6 | 4.8 |

**Table 6.**Relative changes in the rainfall intensity (in %) between historical and future periods (RCP2.6; RCP4.5; RCP8.5) of the IDF curves for Moncton with different durations and return periods (2; 10 and 100 years).

Duration | RCP2.6 | RCP4.5 | RCP8.5 | ||||||
---|---|---|---|---|---|---|---|---|---|

2 Years | 10 Years | 100 Years | 2 Years | 10 Years | 100 Years | 2 Years | 10 Years | 100 Years | |

5 min | 67 | 58 | 67 | 82 | 76 | 89 | 77 | 67 | 76 |

10 min | 44 | 37 | 46 | 56 | 52 | 64 | 53 | 45 | 53 |

15 min | 33 | 29 | 39 | 45 | 44 | 56 | 41 | 36 | 46 |

30 min | 21 | 21 | 31 | 32 | 34 | 48 | 29 | 28 | 38 |

1 h | 11 | 17 | 31 | 21 | 30 | 48 | 18 | 24 | 38 |

2 h | 0 | 8 | 23 | 8 | 20 | 38 | 6 | 14 | 29 |

6 h | −16 | −14 | −6 | −9 | −5 | 6 | −11 | −10 | −1 |

12 h | −18 | −17 | −10 | −11 | −8 | 2 | −13 | −13 | −5 |

24 h | −16 | −16 | −9 | −9 | −7 | 3 | −11 | −11 | −4 |

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**MDPI and ACS Style**

El Hannoun, W.; Boukili Makhoukhi, A.; Zoglat, A.; El Adlouni, S.-E.
Intensity–Duration–Frequency Curves for Dependent Datasets. *Water* **2023**, *15*, 2641.
https://doi.org/10.3390/w15142641

**AMA Style**

El Hannoun W, Boukili Makhoukhi A, Zoglat A, El Adlouni S-E.
Intensity–Duration–Frequency Curves for Dependent Datasets. *Water*. 2023; 15(14):2641.
https://doi.org/10.3390/w15142641

**Chicago/Turabian Style**

El Hannoun, Wafaa, Anas Boukili Makhoukhi, Abdelhak Zoglat, and Salah-Eddine El Adlouni.
2023. "Intensity–Duration–Frequency Curves for Dependent Datasets" *Water* 15, no. 14: 2641.
https://doi.org/10.3390/w15142641