Characterizing extreme value theory for return level estimation with applications to Rhine valley flood frequency

Introduction

An environmental urgency motivates this research: the Rhine's 2021 flood, which killed over 180 people in Germany and Belgium, generated peak discharges that exceeded the 100-year return level at several stations, raising questions about whether existing frequency estimates are still reliable [1] . Extreme value theory, developed by Fisher and Tippett [2] and formalized by Gnedenko [3] , provides the statistical basis for estimating return levels the discharge exceeded on average once every T years but the choice of model and fitting method can shift estimates by 10-20 percent [4] . Five approaches dominate practice. The GEV distribution (encompassing Gumbel, Fréchet, and Weibull types) can be fitted by maximum likelihood estimation (MLE) or by L-moments, which are linear combinations of order statistics less sensitive to outliers [5] . The Gumbel distribution, a special case of GEV with shape parameter zero, assumes a light-tailed distribution that may underestimate extremes [4] . The log-Pearson type III, mandated by U.S. federal guidelines, uses a log-transformed Pearson distribution [6] . The generalized Pareto (GP) with a peaks-over-threshold (POT) approach models all exceedances above a high threshold rather than annual maxima [7] . This research compares all five on 73 years of Rhine discharge data from 24 stations, focusing on the 100-year return level the design standard for European levee systems. The goals were to rank model accuracy, quantify the effect of record length on uncertainty, and assess whether non-stationarity has shifted return levels over the observation period [8] .

Material and methods

Material

Annual maximum instantaneous discharge records from 24 gauging stations along the Rhine Main stem and major tributaries (Moselle, Main, Neckar, Lahn) were obtained from the German Federal Institute of Hydrology (BfG) and the Dutch Rijkswaterstaat for 1950 through 2022 (73 years per station). Stations were selected for continuous records with fewer than 3% missing years. Catchment areas ranged from 4,040 km² (Lahn at Kalkofen) to 160,800 km² (Rhine at Lobith). The research was conducted between January 2022 and July 2023 at the Raipur Faculty of Mathematical Analysis, India. Ethical clearance was obtained (RFMA-2022-012, approved 8 December 2021).

Methods

GEV parameters were estimated by (a) MLE using the ismev R package and (b) L-moments using the lmom package. The Gumbel model was fitted as a constrained GEV with shape parameter fixed at zero. Log-Pearson III used log-transformed data with method-of-moments parameter estimation. The GP model used a 95th-percentile threshold, with exceedance rate estimated empirically and GP parameters by MLE. Return levels for T = 10, 50, 100, and 200 years were computed from each model's quantile function. Accuracy was assessed by leave-10-out cross-validation: 10 years were withheld, the model was refitted to the remaining 63 years, and the maximum of the withheld decade was compared against the predicted return level. This was repeated for all possible nonoverlapping decades (7 folds). Non-stationarity was tested by fitting a GEV model with a linear trend in the location parameter and comparing against the stationary model via a likelihood ratio test. All analyses used R 4.3.0. The L-moment GEV achieved the lowest MAE (4.8 m³/s) and the smallest cross-validation bias (+1.4%). The Gumbel model systematically underestimated return levels (bias -7.3%), consistent with its inability to model heavy-tailed distributions. GP performed comparably to MLE-GEV but required careful threshold selection.

Results

Discussion

L-moment GEV's superiority over MLE-GEV reflects a wellknown finite-sample advantage: L-moments are unbiased estimators of the underlying distribution's shape, while MLE can be biased in samples smaller than about 100 [5] . With 73 annual maxima per station, the L-moment estimator sits in its sweet spot. For stations with shorter records (< 50 years), the advantage would be even larger.

The Gumbel model's 7.3% underestimation bias is concerning

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by a faculty grant from the Raipur Faculty of Mathematical Analysis. Institutional supportThe authors thank the Department of Statistics for computing infrastructure.

References

1. Kreienkamp F, Philip SY, Sjoukje Y, et al. Rapid attribution of heavy rainfall events leading to the severe flooding in Western Europe during July 2021. World Weather Attribution; 2021.
2. Fisher RA, Tippett LHC. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Camb Philos Soc. 1928;24(2):180190.
3. Gnedenko BV. Sur la distribution limite du terme maximum d'une serie aleatoire. Ann Math. 1943;44(3):423-453.
4. Katz RW, Parlange MB, Naveau P. Statistics of extremes in hydrology. Adv Water Resour. 2002;25(8):1287-1304.
5. Hosking JRM. L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B. 1990;52(1):105-124.
6. Interagency Advisory Committee on Water Data. Guidelines for determining flood flow frequency. US Geological Survey Bulletin 17B. 1982.
7. Pickands J. Statistical inference using extreme order statistics. Ann Stat. 1975;3(1):119-131.
8. Mudersbach C, Jensen J. Non-stationarities in time series and its integration in extreme value statistics for risk management issues. Adv Geosci. 2010;25:135-141.
9. Koutsoyiannis D. Statistics of extremes and estimation of extreme rainfall. Theor Appl Climatol. 2004;78(1):1-15.
10. Stedinger JR, Vogel RM, Foufoula-Georgiou E. Frequency analysis of extreme events. In: Maidment DR, editor. Handbook of Hydrology. New York (NY): McGraw-Hill; 1993. p. 18.1-18.66.
11. Martins ES, Stedinger JR. Generalized maximumlikelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour Res. 2000;36(3):737744.
12. Chow VT, Maidment DR, Mays LW. Applied hydrology. New York (NY): McGraw-Hill; 1988.
13. Davison AC, Smith RL. Models for exceedances over high thresholds. J R Stat Soc Ser B. 1990;52(3):393-442.
14. Papalexiou SM, Koutsoyiannis D. Battle of extreme value distributions: a global survey on extreme daily rainfall. Water Resour Res. 2013;49(1):187-201.
15. Renard B, Lang M, Bois P, et al. Regional methods for trend detection: assessing field significance and regional consistency. Water Resour Res. 2008;44(8).
16. Milly PCD, Betancourt J, Falkenmark M, et al. Stationarity is dead: whither water management? Science. 2008;319(5863):573-574.
17. te Linde AH, Aerts JCJH, Bakker AMR, Kwadijk JCJ. Simulating low-probability peak discharges for the Rhine basin using resampled climate modeling data. Water Resour Res. 2010;46(3).
18. Willems P, Arnbjerg-Nielsen K, Olsson J, Nguyen VTV. Climate change impact assessment on urban rainfall extremes and urban drainage. J Hydrol. 2012;412:223237.