WIREs Water
WIREs Water / All Issues / Vol 5 Issue 4 (July 2018)
Flood frequency analysis: The Bayesian choice
Eric Gaume
DOI: 10.1002/wat2.1290
After an introduction to the traditional flood frequency analysis methods, this article discusses their limits and the risks associated with their thoughtless use: overconfidence in the estimated values of flood quantiles or return periods and systematic underestimation of risks. The article then presents and illustrates the added value of modern Bayesian flood frequency inference procedures that are statistically consistent, numerically accurate, and now computationally affordable. The implementation of such methods shows that estimated flood frequencies, based on observed samples of limited size, are generally affected by large uncertainties. This acknowledgement should be an incentive for increasing the size of the analyzed samples through a more systematic use of historic information as well as regional approaches in flood frequency analyses. It also clearly points out that the margin of errors should be considered when using inference results for design or risk assessment purposes. Several pieces of software are now available to conduct Bayesian flood frequency analyses relatively straightforwardly. There is no remaining obstacle to the implementation of these modern approaches in operational hydrological studies. This article is categorized under: Science of Water > Methods Science of Water > Water Extremes
Plot of measured stages versus corresponding estimated discharges at the Montcel gauge on the Morge river (France) according to the available dataset (left panel). View of the Morge river in the vicinity of the gauged cross section (right panel). The stage–discharge relation has obviously been modified in 2003 with few changes in its lower part but tremendous evolutions of its upper, extrapolated part
(Left) Distribution of exceedance probabilities of the 100‐year estimated quantiles for 2,000 samples of 30 values drawn from a GEV(10, 10, −0.2) distribution: red vertical line (theoretical value: 1%), black vertical line (sample average). (Right) Illustration of the "predictive" distribution in red along with parent distribution (bold line), maximum likelihood distribution (continuous line), 90% confidence interval (dotted lines), sample plotting positions (dots)
Inference results based on a sample of 30 values (left) and on the same sample complemented with historic information over 100 years (right). The two largest historic peak discharges could be estimated with a large uncertainty range (distance between and ) indicated by the dotted lines; 90% credibility limits (black dotted line), maximum likelihood distribution (continuous line), sample plotting positions (dots), GEV(10, 10, −0.2) parent distribution (bold line), historic records (vertical dotted lines). Note that the plotting positions have been recomputed in the right panel to account for the historic records
(Left) Comparison between the Bayesian–MCMC 90% credibility limits (black dotted line) and the 90% bootstrap confidence interval (gray dotted line) for a sample of 30 values drawn from a GEV(10, 10, −0.2) distribution. Sample plotting positions (dots), parent distribution (bold line), and maximum likelihood distribution (line). (Right) Uniformity test for the confidence intervals computed with the Bayesian MCMC procedure
Impact of sampling variability. Inference results based on a sample of 30 values (left): moments (continuous line), L‐moments (dashed line), maximum likelihood (dotted line), sample plotting positions (dots), parent GEV (10, 10, −0.2) distribution (bold line). Scattering of the plotting positions for 100 samples of 30 values drawn from the parent distribution (right)
Boxplots of the 100‐year quantile estimation relative errors for the three standard inference methods. A total of 1,000 series of N values drawn from a parent GEV (10, 10, −0.2) distribution. N = 30 (left) and N = 100 (right)
Illustration of the two classic flood peak discharge sampling strategies applied to a 10‐year series of discharges measured in Paris on the Seine river: annual block maxima (left) and peaks over threshold (right). Red dots (sampled values), dotted vertical lines (separation between hydrological years), red horizontal line (selected threshold)
Coles,, S. (2001). An introduction to statistical modeling of extreme values. London, England: Springer.
Cunnane,, C. (1978). Unbiased plotting position. Journal of Hydrology, 37(3–4), 205–222.
Fisher,, R., & Tippett,, L. (1928). Limiting forms of frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24, 180–190.
Fréchet,, M. (1927). Sur la loi de probabilité de l`écart maximum. Annales de la Société Polonaise de Mathématiques, 6, 93–126.
Gaume,, E. (2006). On the asymptotic behavior of flood peak distributions. Hydrology and Earth System Sciences, 10(2), 233–243.
Gnedenko,, B. (1943). Sur la distribution limite du terme maximum d`une série aléatoire. Annales de Mathématiques, 44, 423–453.
Halbert,, K., Nguyen,, C., Payrastre,, O., & Gaume,, E. (2016). Reducing uncertainty in flood frequency analyses: A comparison of local and regional approaches involving information on extreme historical floods. Journal of Hydrology, 541, 90–98. https://doi.org/10.1016/j.jhydrol.2016.01.017
Hershfield,, D., & Kohler,, M. (1960). An empirical appraisal of the gumbel extreme value procedure. Journal of Geophysical Research, 65(6), 1737–1746.
Hosking,, J. (1990). L‐moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society: Series B, 52, 105–124.
Hosking,, J., & Wallis,, J. (1997). Regional frequency analysis: An approach based on l‐moments. Cambridge, England: Cambridge University Press. ISBN 0‐521‐43045‐3.
International Commission on Large Dams. (2014). Integrated flood risk management, bulletin 156. Author.
Klemes,, V. (1988). The improbable probabilities of extreme floods and droughts. In O. Starosolszky, & O. Melder, (Eds.), Hydrology of disasters (pp. 43–51). London, England: James %26 James.
Kuczera,, G. (1999). Comprehensive at‐site flood frequency analysis using Monte Carlo bayesian inference. Water Resources Research, 35(5), 1551–1557. https://doi.org/10.1029/1999WR900012
Laio,, F., Di Baldassarre,, G., & Montanari,, A. (2009). Model selection techniques for the frequency analysis of hydrological extremes. Water Resources Research, 45(7). https://doi.org/10.1029/2007WR006666
N`Guyen,, C., Payrastre,, O., & Gaume,, E. (2014). Regional flood frequency analyses involving extraordinary flood events at ungauged sites: Further developments and validations. Journal of Hydrology, 508, 385–396. https://doi.org/10.1016/j.jhydrol.2013.09.058
Payrastre,, O., Gaume,, E., & Andrieu,, H. (2011). Usefulness of historical information for flood frequency analyses: A case study. Water Resources Research, 47, W08511. https://doi.org/10.1029/2010WR009812
Robert,, C. (2007). The Bayesian choice: From decision‐theoretic foundations to computational implementation (2nd ed.). Paris: Springer. ISBN 978‐0‐387‐71598‐8.
Robert,, C., & Casella,, G. (2004). Monte carlo statistical methods (2nd ed.). New York: Springer. ISBN 0‐387‐21239‐6.
Rogger,, M., Viglione,, A., Derx,, J., & Bloeschl,, G. (2013). Quantifying effects of catchments storage thresholds on step changes in the flood frequency curve. Water Resources Research, 49, 69466958. https://doi.org/10.1002/2013WR020553
Stedinger,, J. (1983). Confidence intervals for design events. Journal of Hydraulic Engineering, 109(1), 13–27.
Stedinger,, J., & Cohn,, T. (1986). Flood frequency analysis with historical and paleoood information. Water Resources Research, 22(5), 785–793.
Tanner,, M. (1996). Tools for statistical inference—Methods for the extrapolation of posterior distributions and likelihood functions (3rd ed.). New York, NY: Springer ISBN 0‐387‐94688‐8.
Viglione,, A., Merz,, R., Salinas,, J., & Blöschl,, G. (2013). Flood frequency hydrology: 3. A Bayesian analysis. Water Resources Research, 49. https://doi.org/10.1029/2011WR010782
Volpi,, E., Fiori,, A., Grimaldi,, S., Lombardo,, F., & Koutsoyiannis,, D. (2015). One hundred years of return period: Strengths and limitations. Water Resources Research, 51, 85708585.
Understanding and Managing Floods: an Interdisciplinary Challenge
Science of Water > Water Extremes
Science of Water > Methods
Gaume Eric. Flood frequency analysis: The Bayesian choice. WIREs Water 2018, 5: null. doi: 10.1002/wat2.1290