# A Global Map for Selecting Stationary and Nonstationary Methods to Estimate Extreme Floods

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Comparison of the Annual Maximun Discharge Generated with Multiple GCM Runoff Datasets

#### 3.2. Reference Map for Extreme Flood Estimation Using Stationary and Nonstationary Methods

#### 3.3. Application of Referenced Map for Estimating Flood Magnitudes Using Stationary and Nonstationary Methods

## 4. Discussion

#### 4.1. GCMs Unveiling Spatial Suitability: Reference Map for Methodological Selection

#### 4.2. Implications of Nonstationarity on Extreme Flood Estimation

#### 4.3. Navigating Uncertainty: GCMs and Model Selection

#### 4.4. Limitation and Future Research

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Mean annual maximum discharge during 1850–2015 using CaMa-Flood model driven by the other four GCM-output runoff products.

**Figure A2.**Slope of the annual maximum discharge during 1850–2015 using CaMa-Flood model driven by the other four GCM-output runoff products.

**Figure A3.**Geological Variation and GCM Differences in Best Distribution for Extreme Flood Estimation. Constant means the stationary method, while changing mu, changing sigma, and changing both are the nonstationary approaches.

**Figure A4.**Best distribution model for extreme flood estimation using CaMa-Flood with different GCM-output runoff datasets. The best model is selected in terms of BIC.

## References

- Slater, L.J.; Anderson, B.; Buechel, M.; Dadson, S.; Han, S.; Harrigan, S.; Kelder, T.; Kowal, K.; Lees, T.; Matthews, T.; et al. Nonstationary weather and water extremes: A review of methods for their detection, attribution, and management. Hydrol. Earth Syst. Sci. Discuss.
**2021**, 25, 3897–3935. [Google Scholar] [CrossRef] - UNDRR. Economic Losses, Poverty & Disasters: 1998–2017. 2018. Available online: https://www.undrr.org/publication/economic-losses-poverty-disasters-1998-2017 (accessed on 4 August 2023).
- Bouchard, J.P.; Pretorius, T.B.; Kramers-Olen, A.L.; Padmanabhanunni, A.; Stiegler, N. Global warming and psychotraumatology of natural disasters: The case of the deadly rains and floods of April 2022 in South Africa. Ann. Médico-Psychol. Rev. Psychiatr.
**2023**, 181, 234–239. [Google Scholar] [CrossRef] - Hirabayashi, Y.; Mahendran, R.; Koirala, S.; Konoshima, L.; Yamazaki, D.; Watanabe, S.; Kim, H.; Kanae, S. Global flood risk under climate change. Nat. Clim. Chang.
**2013**, 3, 816–821. [Google Scholar] [CrossRef] - Hoegh-Guldberg, O.; Jacob, D.; Taylor, M.; Bindi, M.; Brown, S.; Camilloni, I. Impacts of 1.5 °C of global warming on natural and human systems. In Global Warming of 1.5 °C; An IPCC Special Report; IPCC: Geneva, Switzerland, 2018. [Google Scholar]
- Blum, A.G.; Ferraro, P.J.; Archfield, S.A.; Ryberg, K.R. Causal Effect of Impervious Cover on Annual Flood Magnitude for the United States. Geophys. Res. Lett.
**2020**, 47, e2019GL086480. [Google Scholar] [CrossRef] - Vogel, R.M.; Yaindl, C.; Walter, M. Nonstationarity: Flood Magnification and Recurrence Reduction Factors in the United States1. JAWRA J. Am. Water Resour. Assoc.
**2011**, 47, 464–474. [Google Scholar] [CrossRef] - Yan, L.; Xiong, L.; Ruan, G.; Zhang, M.; Xu, C.-Y. Design flood estimation with varying record lengths in Norway under stationarity and nonstationarity scenarios. Hydrol. Res.
**2021**, 52, 1596–1614. [Google Scholar] [CrossRef] - Yan, L.; Xiong, L.; Guo, S.; Xu, C.-Y.; Xia, J.; Du, T. Comparison of four nonstationary hydrologic design methods for changing environment. J. Hydrol.
**2017**, 551, 132–150. [Google Scholar] [CrossRef] - Salas, J.D.; Obeysekera, J.; Vogel, R.M. Techniques for assessing water infrastructure for nonstationary extreme events: A review. Hydrol. Sci. J.
**2018**, 63, 325–352. [Google Scholar] [CrossRef] - Berghuijs, W.R.; Aalbers, E.E.; Larsen, J.R.; Trancoso, R.; Woods, R.A. Recent changes in extreme floods across multiple continents. Environ. Res. Lett.
**2017**, 12, 114035. [Google Scholar] [CrossRef] - Archfield, S.A.; Hirsch, R.M.; Viglione, A.; Blöschl, G. Fragmented patterns of flood change across the United States. Geophys. Res. Lett.
**2016**, 43, 10232–10239. [Google Scholar] [CrossRef] - Eastoe, E.F. Nonstationarity in peaks-over-threshold river flows: A regional random effects model. Environmetrics
**2019**, 30, e2560. [Google Scholar] [CrossRef] - Hecht, J.S.; Vogel, R.M. Updating urban design floods for changes in central tendency and variability using regression. Adv. Water Resour.
**2020**, 136, 103484. [Google Scholar] [CrossRef] - Steirou, E.; Gerlitz, L.; Apel, H.; Sun, X.; Merz, B. Climate influences on flood probabilities across Europe. Hydrol. Earth Syst. Sci.
**2019**, 23, 1305–1322. [Google Scholar] [CrossRef] - Prosdocimi, I.; Kjeldsen, T.R.; Miller, J.D. Detection and attribution of urbanization effect on flood extremes using nonstationary flood-frequency models. Water Resour. Res.
**2015**, 51, 4244–4262. [Google Scholar] [CrossRef] - Slater, L.; Villarini, G.; Archfield, S.; Faulkner, D.; Lamb, R.; Khouakhi, A.; Yin, J. Global changes in 20-year, 50-year, and 100-year river floods. Geophys. Res. Lett.
**2021**, 48, e2020GL091824. [Google Scholar] [CrossRef] - Rigby, R.; Stasinopoulos, M.; Heller, G.; De Bastiani, F. Distributions for Modelling Location, Scale and Shape: Using GAMLSS in R; CRC Press: Boca Raton, FL, USA, 2019. [Google Scholar]
- Villarini, G.; Smith, J.A.; Serinaldi, F.; Bales, J.; Bates, P.D.; Krajewski, W.F. Flood frequency analysis for nonstationary annual peak records in an urban drainage basin. Adv. Water Resour.
**2009**, 32, 1255–1266. [Google Scholar] [CrossRef] - López, J.; Francés, F. Non-stationary flood frequency analysis in continental Spanish rivers, using climate and reservoir indices as external covariates. Hydrol. Earth Syst. Sci.
**2013**, 17, 3189–3203. [Google Scholar] [CrossRef] - Faulkner, D.; Warren, S.; Spencer, P.; Sharkey, P. Can we still predict the future from the past? Implementing non-stationary flood frequency analysis in the UK. J. Flood Risk Manag.
**2020**, 13, e12582. [Google Scholar] [CrossRef] - Towe, R.; Tawn, J.; Eastoe, E.; Lamb, R. Modelling the Clustering of Extreme Events for Short-Term Risk Assessment. J. Agric. Biol. Environ. Stat.
**2020**, 25, 32–53. [Google Scholar] [CrossRef] - Zhang, T.; Wang, Y.; Wang, B.; Tan, S.; Feng, P. Nonstationary Flood Frequency Analysis Using Univariate and Bivariate Time-Varying Models Based on GAMLSS. Water
**2018**, 10, 819. [Google Scholar] [CrossRef] - Rigby, R.A.; Stasinopoulos, M.D. Mean and Dispersion Additive Models. In Statistical Theory and Computational Aspects of Smoothing; Physica-Verlag HD: Heidelberg, Germany, 1996; pp. 215–230. [Google Scholar]
- Villarini, G.; Strong, A. Roles of climate and agricultural practices in discharge changes in an agricultural watershed in Iowa. Agric. Ecosyst. Environ.
**2014**, 188, 204–211. [Google Scholar] [CrossRef] - Giuntoli, I.; Villarini, G.; Prudhomme, C.; Hannah, D.M. Uncertainties in projected runoff over the conterminous United States. Clim. Chang.
**2018**, 150, 149–162. [Google Scholar] [CrossRef] - Yamazaki, D.; Kanae, S.; Kim, H.; Oki, T. A physically based description of floodplain inundation dynamics in a global river routing model. Water Resour. Res.
**2011**, 47, W04501. [Google Scholar] [CrossRef] - Kimura, Y.; Hirabayashi, Y.; Kita, Y.; Zhou, X.; Yamazaki, D. Methodology for constructing a flood-hazard map for a future climate. Hydrol. Earth Syst. Sci.
**2023**, 27, 1627–1644. [Google Scholar] [CrossRef] - Markovic, R.D. Probability Functions of the Best Fit to Distributions of Annual Precipitation and Runoff Hydrology. Doctoral Dissertation, Colorado State University, Fort Collins, CO, USA, 1965. [Google Scholar]
- Vogel, R.M.; Wilson, I. Probability Distribution of Annual Maximum, Mean, and Minimum Streamflows in the United States. J. Hydrol. Eng.
**1996**, 1, 69–76. [Google Scholar] [CrossRef] - Stasinopoulos, M.; Rigby, B.; Akantziliotou, C. Instructions on How to Use the Gamlss Package in R Second Edition. 2008. Available online: https://www.researchgate.net/publication/228429663_Instructions_on_how_to_use_the_gamlss_package_in_R_Second_Edition (accessed on 4 August 2023).
- Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Nelson, D.B. Stationarity and persistence in the GARCH (1, 1) model. Econom. Theory
**1990**, 6, 318–334. [Google Scholar] [CrossRef] - Shumway, R.; Stoffer, D. Time Series Analysis and Its Applications with R Examples; Springer: New York, NY, USA, 2011; Volume 9. [Google Scholar]
- Chen, H.-L.; Rao, A.R. Testing hydrologic time series for stationarity. J. Hydrol. Eng.
**2002**, 7, 129–136. [Google Scholar] [CrossRef] - Buuren, S.v.; Fredriks, M. Worm plot: A simple diagnostic device for modelling growth reference curves. Stat. Med.
**2001**, 20, 1259–1277. [Google Scholar] [CrossRef]

**Figure 1.**Mean annual maximum discharge during 1850–2015 using CaMa-Flood model driven by eight GCM-output runoff products. For the remaining results, please see the Figure A1.

**Figure 2.**Slope of the annual maximum discharge during 1850–2015 using CaMa-Flood model driven by eight GCM-output runoff products. For the remaining results, please see the Figure A2.

**Figure 3.**Slope of the annual maximum discharge in seven discharge groups among two combinations of GCMs. The discharge data are categorized into seven groups, representing the ranges [0, 1000, 5000, 10,000, 20,000, 30,000, 40,000, 80,000] m

^{3}/s. Each combination of GCMs is used to fit curves among their respective simulations, and the slope of the simulated discharge is calculated within each discharge group.

**Figure 4.**Spatial Variation and GCM Differences in Best Distribution for Extreme Flood Estimation. Constant means the stationary method, while changing mu, changing sigma, and changing both are the nonstationary approaches. For the remaining results, please see the Figure A3.

**Figure 5.**Best distribution model for extreme flood estimation using CaMa-Flood with different GCM-output runoff datasets. The best model is selected in terms of BIC. For the remaining results, please see the Figure A4.

**Figure 6.**Reference map for extreme flood estimation using stationary and nonstationary methods after merging results from all GCMs.

**Figure 7.**One hundred-year flood magnitudes (m

^{3}/s) estimated by stationary method based on the annual maximum discharge simulated by CaMa-Flood with different GCM-output runoff datasets.

**Figure 8.**Examples to illustrate the difference of 100-year flood magnitudes (m

^{3}/s) estimated by stationary and nonstationary methods (

**a**,

**c**,

**e**). Note that nonstationary conditions have three cases: changing mu, changing sigma, and changing both. We selected three representative grids from the reference map, each identified as necessitating nonstationary methodologies, encompassing three pivotal scenarios—changing μ (

**b**), altering σ (

**d**), and simultaneous variations in both (

**f**).

Changing Mean (μ) | Changing Scale (σ) | |
---|---|---|

Stationary | - | - |

Nonstationary | Y | - |

- | Y | |

Y | Y |

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**MDPI and ACS Style**

Li, Z.; Yan, Z.; Tang, L.
A Global Map for Selecting Stationary and Nonstationary Methods to Estimate Extreme Floods. *Water* **2023**, *15*, 3835.
https://doi.org/10.3390/w15213835

**AMA Style**

Li Z, Yan Z, Tang L.
A Global Map for Selecting Stationary and Nonstationary Methods to Estimate Extreme Floods. *Water*. 2023; 15(21):3835.
https://doi.org/10.3390/w15213835

**Chicago/Turabian Style**

Li, Zhenzhen, Zhongyue Yan, and Li Tang.
2023. "A Global Map for Selecting Stationary and Nonstationary Methods to Estimate Extreme Floods" *Water* 15, no. 21: 3835.
https://doi.org/10.3390/w15213835