# Estimating IDF Curves Consistently over Durations with Spatial Covariates

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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.

## References

- Hattermann, F.F.; Kundzewicz, Z.W.; Huang, S.; Vetter, T.; Gerstengarbe, F.W.; Werner, P. Climatological drivers of changes in flood hazard in Germany. Acta Geophys.
**2013**, 61, 463–477. [Google Scholar] [CrossRef] - Seneviratne, S.; Nicholls, N.; Easterling, D.; Goodess, C.; Kanae, S.; Kossin, J.; Luo, Y.; Marengo, J.; McInnes, K.; Rahimi, M.; et al. Changes in climate extremes and their impacts on the natural physical environment. In Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation; Field, C.B., Barros, V., Stocker, T.F., Dahe, Q., Dokken, D.J., Ebi, K.L., Mastrandrea, M.D., Mach, K.J., Plattner, G.K., Allen, S.K., Eds.; Cambridge University Press: Cambridge, UK, 2012; pp. 109–230. [Google Scholar]
- Chow, V.T. Frequency analysis of hydrologic data with special application to rainfall intensities. Univ. Ill. Bull.
**1953**, 50, 86. [Google Scholar] - KOSTRA-DWD. Available online: https://www.dwd.de/DE/leistungen/kostra_dwd_rasterwerte/kostra_dwd_rasterwerte.html (accessed on 4 August 2020).
- Junghänel, T.; Ertelund, H.; Deutschländer, T. KOSTRA-DWD-2010R: Berichtzur Revisionderkoordinierten Starkregenregionalisierung und -auswertung des Deutschen Wetterdienstes in der Version 2010; Deutscher Wetterdienst, Abteilung Hydrometeorologie: Offenbach, Germany, 2017.
- Precipitation Frequency Data Server. Available online: https://hdsc.nws.noaa.gov/hdsc/pfds/ (accessed on 15 October 2020).
- Perica, S.; Pavlovic, S.; St. Laurent, M.; Trypaluk, C.; Unruh, D.; Wilhite, O. NOAA Atlas 14: Precipitation-Frequency Atlas of the United States, Volume 11 Version 2.0; U.S. Department of Commerce National Oceanic and Atmospheric Administration National Weather Service: Silver Spring, MD, USA, 2018.
- Hosking, J.R.M.; Wallis, J.R. Regional Frequency Analysis: An Approach Based on L-Moments; Cambridge University Press: Cambridge, UK, 1998; p. 208. [Google Scholar]
- Fukutome, S.; Schindler, A.; Capobianco, A. MeteoSwiss Extreme Value Analyses: User Manual and Documentation, 3rd ed.; Technical Report, 255; Federal Office of Meteorology and Climatology, MeteoSwiss: Zürich, Switzerland, 2018; 80p.
- MeteoSwiss Maps of Extreme Precipitation. Available online: https://www.meteoswiss.admin.ch/home/climate/swiss-climate-in-detail/extreme-value-analyses/maps-of-extreme-precipitation.html (accessed on 15 October 2020).
- Goudenhoofdt, E.; Delobbe, L.; Willems, P. Regional frequency analysis of extreme rainfall in Belgium based on radar estimates. Hydrol. Earth Syst. Sci.
**2017**, 21, 5385–5399. [Google Scholar] [CrossRef][Green Version] - Olsson, J.; Södling, J.; Berg, P.; Wern, L.; Eronn, A. Short-duration rainfall extremes in Sweden: A regional analysis. Hydrol. Res.
**2019**, 50, 945–960. [Google Scholar] [CrossRef] - Gaur, A.; Schardong, A.; Simonovic, S.P. Gridded Extreme Precipitation Intensity-Duration-Frequency Estimates for the Canadian Landmass. J. Hydrol. Eng.
**2020**, 25, 05020006. [Google Scholar] [CrossRef] - Coles, S. An Introduction to Statistical Modeling of Extreme Values; Springer: New York, NY, USA, 2001. [Google Scholar] [CrossRef]
- Koutsoyiannis, D.; Kozonis, D.; Manetas, A. A mathematical framework for studying rainfall Intensity-Duration-Frequency relationships. J. Hydrol.
**1998**, 206, 118–135. [Google Scholar] [CrossRef] - Ritschel, C.; Ulbrich, U.; Névir, P.; Rust, H.W. Precipitation extremes on multiple timescales—Bartlett-Lewis rectangular pulse model and Intensity-Duration-Frequency curves. Hydrol. Earth Syst. Sci.
**2017**, 21, 6501–6517. [Google Scholar] [CrossRef][Green Version] - Lehmann, E.; Phatak, A.; Soltyk, S.; Chia, J.; Lau, R.; Palmer, M. Bayesian hierarchical modelling of rainfall extremes. In Proceedings of the 20th International Congress on Modelling and Simulation, Adelaide, Australia, 1–6 December 2013; pp. 2806–2812. [Google Scholar]
- Van de Vyver, H.; Demarée, G.R. Construction of Intensity-Duration-Frequency (IDF) curves for precipitation at Lubumbashi, Congo, under the hypothesis of inadequate data. Hydrol. Sci. J. J. Sci. Hydrol.
**2010**, 55, 555–564. [Google Scholar] [CrossRef] - Stephenson, A.G.; Lehmann, E.A.; Phatak, A. A max-stable process model for rainfall extremes at different accumulation durations. Weather Clim. Extrem.
**2016**, 13, 44–53. [Google Scholar] [CrossRef][Green Version] - Blanchet, J.; Ceresetti, D.; Molinié, G.; Creutin, J.D. A regional GEV scale-invariant framework for Intensity-Duration-Frequency analysis. J. Hydrol.
**2016**, 540, 82–95. [Google Scholar] [CrossRef] - Davison, A.C.; Padoan, S.A.; Ribatet, M. Statistical modeling of spatial extremes. Stat. Sci.
**2012**, 27, 161–186. [Google Scholar] [CrossRef][Green Version] - Dyrrdal, A.V.; Lenkoski, A.; Thorarinsdottir, T.L.; Stordal, F. Bayesian hierarchical modeling of extreme hourly precipitation in Norway. Environmetrics
**2015**, 26, 89–106. [Google Scholar] [CrossRef][Green Version] - Fischer, M.; Rust, H.; Ulbrich, U. A spatial and seasonal climatology of extreme precipitation return-levels: A case study. Spat. Stat.
**2019**, 34. [Google Scholar] [CrossRef] - Van de Vyver, H. Spatial regression models for extreme precipitation in Belgium. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Yee, T.W.; Stephenson, A.G. Vector generalized linear and additive extreme value models. Extremes
**2007**, 10, 1–19. [Google Scholar] [CrossRef] - Mélèse, V.; Blanchet, J.; Molinié, G. Uncertainty estimation of Intensity-Duration-Frequency relationships: A regional analysis. J. Hydrol.
**2018**, 558, 579–591. [Google Scholar] [CrossRef][Green Version] - Bentzien, S.; Friederichs, P. Decomposition and graphical portrayal of the quantile score. Q. J. R. Meteorol. Soc.
**2014**, 140, 1924–1934. [Google Scholar] [CrossRef] - Wilks, D.S. Statistical Methods in the Atmospheric Sciences; Academic Press: Cambridge, MA, USA, 2011; Volume 100. [Google Scholar]
- Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill series in water resources and environmental engineering; Tata McGraw-Hill Education: New York, NY, USA, 1988. [Google Scholar]
- Singh, V. Elementary Hydrology; Prentice Hall: New Jersey, NJ, USA, 1992. [Google Scholar]
- García-Bartual, R.; Schneider, M. Estimating maximum expected short-duration rainfall intensities from extreme convective storms. Phys. Chem. Earth Part B
**2001**, 26, 675–681. [Google Scholar] [CrossRef] - Ulrich, J.; Ritschel, C. IDF: Estimation and Plotting of IDF Curves; R package version 2.0.0. 2019. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2020. [Google Scholar]
- Tyralis, H.; Langousis, A. Estimation of Intensity-Duration-Frequency curves using max-stable processes. Stoch. Environ. Res. Risk Assess.
**2019**, 33, 239–252. [Google Scholar] [CrossRef] - Jurado, O.E.; Ulrich, J.; Rust, H.W. Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves. 2020; Manuscript submitted for publication. [Google Scholar]
- Pasternack, A.; Grieger, J.; Rust, H.W.; Ulbrich, U. Recalibrating Decadal Climate Predictions—What is an adequate model for the drift? Geosci. Model Dev. Discuss.
**2020**, 2020, 1–38. [Google Scholar] [CrossRef] - Arlot, S.; Celisse, A. A survey of cross-validation procedures for model selection. Statist. Surv.
**2010**, 4, 40–79. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R.; Friedman, J. Elements of Statistical Learning, 2nd ed.; Stanford University: Stanford, CA, USA, 2009. [Google Scholar] [CrossRef]
- Davison, A.C.; Hinkley, D.V. Bootstrap Methods and Their Application; Number 1; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Van de Vyver, H. A multiscaling-based Intensity-Duration-Frequency model for extreme precipitation. Hydrol. Process.
**2018**, 32, 1635–1647. [Google Scholar] [CrossRef]

**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