# Estimating IDF Curves Consistently over Durations with Spatial Covariates

^{1}

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## Abstract

**:**

## 1. Introduction

- Under which conditions is the spatial d-GEV approach an improvement compared to the separate application of the GEV for each duration and station?
- Does the spatial d-GEV approach provide reliable estimates at ungauged sites?

## 2. Methods

#### 2.1. Data

^{2}. As the area extends from the Cologne-Bonn lowlands in the west to the Bergisches Land in the east, different altitudes are well represented by the stations and a great variability in topographic shapes is covered.

#### 2.2. d-GEV as a Model for Annual Maxima for Different Durations

#### 2.2.1. Station-Wise Model for a Range of Durations (d-GEV)

#### 2.2.2. Adding Spatial Covariates

`IDF`for the

`R`environment [32,33]. Typically the choice of a link function ensures parameters to be positive or within a predefined range. Here, we implemented intervals for the parameters directly into the optimizer and thus used the identity ${l}^{\varphi}\left(\varphi \right)=\varphi $ as link function for all parameters.

`poly`from the package

`stats`in the

`R`environment [33]. We also add interactions resulting from the products of the respective terms. This yields the following model for each d-GEV parameter

#### 2.3. Model Selection

#### 2.4. Verification

#### 2.5. Confidence Intervals

## 3. Results

#### 3.1. Model Performance

#### 3.1.1. Overall Performance

#### 3.1.2. Dependence on Time Series Length

#### 3.1.3. Ungauged Sites

#### 3.2. Quantile Estimation and Uncertainty

## 4. Discussion

## 5. Summary

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BHM | Bayesian Hierarchical Model |

DWD | German Weather Service (Deutscher Wetterdienst) |

EVT | Extreme Value Theory |

GEV | Generalized Extreme Value distribution |

d-GEV | duration-dependent GEV |

IDF | Intensity-Duration-Frequency (curve) |

QS | Quantile Score |

QSS | Quantile Skill Score |

QSI | Quantile Skill Index |

VGLM | Vector generalized linear model |

VGAM | Vector generalized additive model |

## Appendix A. Overview of Verification Variations

- the overall performance
- the dependence of the model performance on the length of the time series used for training the model
- the model performance at ungauged sites.

**Table A1.**Cross-validation sets used for training and validating the spatial d-GEV and the reference model (GEV). N represents the complete length of the time series and varies for different durations and stations, while ${n}_{\mathrm{train}}$ is a fixed number of years for each time series.

Overall Performance | Dependence on Time Series Length | Ungauged Sites | |||||
---|---|---|---|---|---|---|---|

Training | Validation | Training | Validation | Training | Validation | ||

spatial d-GEV | station | ($N-3$) years | 3 years | ${n}_{\mathrm{train}}$ years | 3 years | - | 3 years |

remaining stations | all data | - | all data | - | all data | - | |

GEV | station | ($N-3$) years | 3 years | ${n}_{\mathrm{train}}$ years | 3 years | ${n}_{\mathrm{train}}$ years | 3 years |

remaining stations | - | - | - | - | - | - |

## Appendix B. Coverage of Confidence Intervals

**Figure A1.**Coverage of Confidence Intervals, obtained by the bootstrap method (left column) and the delta method (right column). The coverage was calculated by re-sampling from a known d-GEV distribution 1000 times. Different colors indicate different sample sizes, which correspond to the length of the time series in years in this context. Different rows represent different non-exceedance probabilities p, whereby the confidence intervals were examined for the corresponding quantile estimates.

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**Figure 1.**Study area containing 92 gauge stations with different measurement periods. The black line borders the Wupper catchment. Gauges marked white are those used as example stations (Schwelm (square) and Solingen-Hohenscheid (diamond)). The altitude is coded along a grey scale and stems from http://www.diva-gis.org/gdata, river shapes come from https://www.openstreetmap.org.

**Figure 2.**Final model selection result. For each d-GEV parameter, the added covariates are shown as colored boxes, according to the order of their selection.

**Figure 3.**Average Quantile Skill Index ${\overline{\mathrm{QSI}}}_{d}\left(p\right)$ for different durations d and probabilities p. Upper panel: station-wise d-GEV. Lower panel: spatial d-GEV. Positive values (red) indicate an improvement compared to the quantile estimates obtained by modeling each station and duration separately.

**Figure 4.**Dependence of the spatial d-GEV model performance on length of the training time series. Upper panel: Average Quantile Skill Index ${\overline{\mathrm{QSI}}}_{d}\left(p\right)$ for different durations d and probabilities p as seen in Figure 3 but rotated. Different columns represent ${\overline{\mathrm{QSI}}}_{d}\left(p\right)$ for different numbers of years in the training set ${n}_{\mathrm{train}}$. Lower panel: Boxplot of the Quantile Skill Index ${\mathrm{QSI}}_{s,d}\left(p\right)$ with probability $p=0.99$ dependent on the length of the training time series.

**Figure 5.**Quantile Skill Index ${\mathrm{QSI}}_{s,d}\left(p\right)$ for the spatial d-GEV approach with probability $p=0.99$ for durations $d=1\phantom{\rule{0.166667em}{0ex}}$h (upper panel) and $d=24\phantom{\rule{0.166667em}{0ex}}$h (lower panel) at all stations, for the number of training years ${n}_{\mathrm{train}}=\left(\right)open="\{"\; close="\}">10,15,20,25$ (different columns). Colored dots indicate superiority of the spatial d-GEV (red) or inferiority (blue), while gray circles show stations without an estimate.

**Figure 6.**Average Quantile Skill Index ${\overline{\mathrm{QSI}}}_{d}\left(p\right)$ at ungauged sites for different durations d and probabilities p, similar to Figure 4. For the spatial d-GEV the data at the respective stations are omitted for fitting, while the reference model uses a different number of years ${n}_{\mathrm{train}}$ in the training set (different columns.)

**Figure 7.**Point-wise return level maps for durations $d\in \{5,30,120\}\phantom{\rule{0.166667em}{0ex}}$min (different columns) and probabilities $p=0.99$ (upper panel) and $p=0.95$ (lower panel) corresponding to return periods of 100 years and 20 years, respectively. The colors are provided as a general reference between the plots.

**Figure 8.**IDF-curve estimate for Solingen-Hohenscheid (marked with white square in Figure 1) (lower panel) obtained by using the spatial d-GEV (black dashed lines) and their $95\%$ confidence intervals (colored areas). Observations are shown as boxplots, where the width of the box is proportional to the number of data points available at a certain duration. The upper panel shows the corresponding QSI values at that station following the presentation of Figure 3.

**Figure 9.**Bootstrapped $95\%$ confidence intervals for IDF-estimates at the example stations Solingen-Hohenscheid and Schwelm (marked in Figure 1), using the station-wise d-GEV (left column) and the spatial d-GEV approach (right column).

Provider | Number of Stations | Measuring Interval | Device | Length of Time Series |
---|---|---|---|---|

DWD | 69 | 1 day | Hellmann | 9–121 years |

DWD | 17 | 1 min | Pluvio | 5–14 years |

WV | 6 | 1 hour | Pluvio | 38 years |

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**MDPI and ACS Style**

Ulrich, J.; Jurado, O.E.; Peter, M.; Scheibel, M.; Rust, H.W.
Estimating IDF Curves Consistently over Durations with Spatial Covariates. *Water* **2020**, *12*, 3119.
https://doi.org/10.3390/w12113119

**AMA Style**

Ulrich J, Jurado OE, Peter M, Scheibel M, Rust HW.
Estimating IDF Curves Consistently over Durations with Spatial Covariates. *Water*. 2020; 12(11):3119.
https://doi.org/10.3390/w12113119

**Chicago/Turabian Style**

Ulrich, Jana, Oscar E. Jurado, Madlen Peter, Marc Scheibel, and Henning W. Rust.
2020. "Estimating IDF Curves Consistently over Durations with Spatial Covariates" *Water* 12, no. 11: 3119.
https://doi.org/10.3390/w12113119