# Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Data

#### 2.1. Estimation of IDF Curves

#### 2.1.1. Using the Duration-Dependent GEV

`R`-package

`IDF`[22]. Equation (3) is used to get intensities for arbitrary durations $(d\ge 1)$.

#### 2.1.2. Using a Max-Stable Process

`R`package

`SpatialExtremes`[35].

`R`language [38]. The data and code is available as supplementary information.

#### 2.2. Verification and Model Comparison

#### 2.3. Data

#### 2.3.1. Synthetic Data

`R`-package

`SpatialExtremes`. For the marginal d-GEV distribution, we used a set of parameters characteristic of those d-GEV distributions fitted from the stations in the observational dataset. Then, to fulfill the constraints of annual rainfall maxima averaged over durations ${d}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,k$, we transformed the initial simulated data from having unit-Fréchet margins to GEV margins that follow the constraints given by [5,20]

#### 2.3.2. Observations

## 3. Results

#### 3.1. Case Study

#### 3.1.1. Structure of the Extremal Dependence

#### 3.1.2. Estimation of IDF Curves

#### 3.1.3. Performance Averaged over All Durations

#### 3.1.4. Performance for Individual Durations

#### 3.2. Simulation Study

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IDF | Intensity-Duration-Frequency (curve) |

GEV | Generalized Extreme Value distribution |

d-GEV | Duration-dependent GEV |

rd-GEV | Reduced d-GEV-based approach for modeling IDF curves |

MS-GEV | Max-stable-based approach for modeling IDF curves |

QS | Quantile Score |

QSS | Quantile Skill Score |

QSI | Quantile Skill Index |

## Appendix A. Inference from the Brown-Resnick Max-Stable Process

## Appendix B. Comparison of 100-year Return Level between Euclidean and Log-Distance for MS-GEV Approach

**Figure A1.**Comparison of the 100-year return level intensity for the MS-GEV approach using the euclidean distance ${h}_{e}$ and the log-distance ${h}_{l}$.

## Appendix C. QQ-Plots for Selected Stations and Durations

**Figure A2.**QQ plots for model checking of the marginal distributions for three stations: Bever (

**top row**), Leverkusen (

**middle row**), and Neumuehle (

**bottom row**). The duration used for the accumulation of the rainfall maxima is indicated in each plot.

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**Figure 1.**

**Left**: (Lower panel) Distribution of the annual rainfall maxima at a 1-h accumulation duration plotted against time. Each boxplot shows the distribution of the pooled maxima from the six stations used for the case study in the Wupper catchment region. (Upper panel) Time series of the annual rainfall maxima, showing the values of each station as a different color.

**Right**: Map of the Wupper catchment (dashed line) showing the location of all 6 stations; the lower-right corner shows the location of the catchment within Germany.

**Figure 2.**Nonparametric (dots = ${\widehat{\theta}}_{emp}$) and parametric (solid line = ${\theta}_{\mathrm{BR}}$) estimates for the pairwise extremal coefficient $\theta $ using the euclidean distance ${h}_{e}$. The estimated nonparametric mean of $\theta $ for each duration lag bin is shown as black dots. Each color represents the lower duration ${d}_{i}$ used for each duration pair.

**Figure 3.**Nonparametric (dots = ${\widehat{\theta}}_{emp}$) and parametric (solid line = ${\theta}_{\mathrm{BR}}$) estimates for the pairwise extremal coefficient $\theta $ using the logarithmic distance ${h}_{l}$. For ease of interpretation, the values of the log-distance ${h}_{l}$ in the x-axis were transformed to the duration ratio ${d}_{j}/{d}_{i}$. The estimated nonparametric mean of $\theta $ for each duration ratio bin is shown as black dots. Each color represents the lower duration ${d}_{i}$ used for each duration pair. Notice the difference in the variability of the point clouds when compared to those of Figure 2.

**Figure 4.**Comparison of IDF curves for the MS-GEV (

**solid line**) and rd-GEV approach (

**dashed line**) for all stations. Different colors represent different return periods; from bottom to top: (5, 10, 20, 40, 100) years.

**Figure 5.**Quantile Skill Index comparing the MS-GEV versus the rd-GEV approach for all stations in the Wuppertal catchment. Positive values favor the MS-GEV approach.

**Figure 6.**Quantile Skill Index conditioned on duration comparing the MS-GEV (using log-distance ${h}_{l}$) versus the rd-GEV approach for all stations in the Wuppertal catchment. Positive values favor the MS-GEV approach for different durations.

**Figure 7.**Quantile Skill Index calculated from QS averaged for all durations $d=(1,2,..,120)\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$, comparing the MS-GEV versus the rd-GEV approach as a function of simulated data’s dependence parameter. Positive values of the QSI favor the MS-GEV approach. Each boxplot represents the distribution from the results of 1000 simulations.

**Figure 8.**Quantile Skill Index, as in Figure 7, but showing the quantile score index (QSI) as a function of return period and duration.

**Table 1.**Parameter estimates from the MS-GEV approach for stations in the Wupper catchment using durations $d=(1,3,\dots ,119,120)\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$.

Station | $\mathit{\alpha}$ | $\mathit{\rho}$ | ${\mathit{\mu}}_{0}$ | ${\mathit{\sigma}}_{0}$ | ${\mathit{\xi}}_{0}$ | c |
---|---|---|---|---|---|---|

Bever | 1.42 | 2.09 | 13.32 | 2.84 | 0.03 | −0.58 |

Buchenhofen | 1.39 | 2.22 | 13.38 | 3.03 | 0.02 | −0.63 |

Leverkusen | 1.32 | 2.19 | 10.74 | 2.34 | 0.05 | −0.64 |

Lindscheid | 1.54 | 2.44 | 13.69 | 3.23 | 0.06 | −0.64 |

Neumuehle | 1.54 | 1.84 | 13.52 | 2.74 | 0.04 | −0.60 |

Schwelm | 1.53 | 1.74 | 13.07 | 2.77 | 0.05 | −0.62 |

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**MDPI and ACS Style**

Jurado, O.E.; Ulrich, J.; Scheibel, M.; Rust, H.W.
Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves. *Water* **2020**, *12*, 3314.
https://doi.org/10.3390/w12123314

**AMA Style**

Jurado OE, Ulrich J, Scheibel M, Rust HW.
Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves. *Water*. 2020; 12(12):3314.
https://doi.org/10.3390/w12123314

**Chicago/Turabian Style**

Jurado, Oscar E., Jana Ulrich, Marc Scheibel, and Henning W. Rust.
2020. "Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves" *Water* 12, no. 12: 3314.
https://doi.org/10.3390/w12123314