Skip to main content
SpringerLink
Log in

Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study

  • Original Article
  • Published:
Extremes Aims and scope Submit manuscript

Abstract

Estimation of flood and drought frequencies is important for reservoir design and management, river pollution, ecology and drinking water supply. Through an example based on daily streamflow observations, we introduce a stepwise procedure for estimating quantiles of the hydrological extremes floods and droughts. We fit the generalised extreme value (GEV) distribution by the method of block maxima and the generalised Pareto (GP) distribution by applying the peak over threshold method. Maximum likelihood, penalized maximum likelihood and probability weighted moments are used for parameter estimation. We incorporate trends and seasonal variation in the models instead of splitting the data, and investigate how the observed number of extreme events, the chosen statistical model, and the parameter estimation method effect parameter estimates and quantiles. We find that a seasonal variation should be included in the GEV distribution fitting for floods using block sizes less than one year. When modelling droughts, block sizes of one year or less are not recommended as significant model bias becomes visible. We conclude that the different characteristics of floods and droughts influence the choices made in the extreme value modelling within a common inferential strategy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aune, B., Air temperature normals, normal period 1961–90, Climate Report No. 02/93, the Norwegian Meteorological Institute, DNMI, Norway, 1993.

  • Bobée, B. and Rasmussen, P.F., “Recent advances in flood frequency analysis,” Rev. Geophys. 33, 1111–1116, (1995).

    Google Scholar 

  • Buishand, T.A., and Demaré, G.R., “Estimation of the annual maximum distribution from samples of maxima in separate season,” Stoch. Hydrol. Hydraul. 4, 89–103, (1990).

    Google Scholar 

  • Clausen, B. and Pearson, C.P., “Regional frequency analysis of annual maximum streamflow drought,” J. Hydrol. 173, 11–130, (1995).

    Google Scholar 

  • Coles, S.G., An Introduction to Statistical Modelling of Extreme Values, Springer, London, 2001.

    Google Scholar 

  • Coles, S.G. and Casson, E., “Extreme value modelling of hurricane wind speed,” Struct. Saf. 20, 283–296, (1998).

    Google Scholar 

  • Coles, S.G. and Dixon, M.J., “Likelihood-based inference for extreme value models,” Extremes 2(1), 5–23, (1999).

    MATH  Google Scholar 

  • Coles, S., Pericchi, L.R. and Sisson S., “A fully probabilistic approach to extreme rainfall modelling,” J. Hydrol. 273, 35–50, (2003).

    Google Scholar 

  • Cramér, H. and Leadbetter, M.R., Stationary and Related Stochastic Processes, John Wiley and Sons, New York, 1967.

    MATH  Google Scholar 

  • Cunnane, C., “A note on the Poisson assumption in partial duration series model,” Water Resour. Res. 15(2), 489–494, (1979).

    Google Scholar 

  • Davison, A.C. and Hinkley, D.V., Bootstrap Methods and Their Applications, Cambridge University Press, 1997.

  • Davison, A.C. and Smith, R.L., “Models for exceedances over high thresholds,” J. R. Stat. Soc., Ser. B 52(3), 393–443, (1990).

    MATH  MathSciNet  Google Scholar 

  • Dupuis, D.J., “Exceedances over high threshold: A guide to threshold selection,” Extremes 1(3), 251–261, (1998).

    MathSciNet  Google Scholar 

  • Efron, B. and Tibshirani, R.J., An Introduction to the Bootstrap, Monographs on Statistics and Applied Probability, Chapmann & Hall, New York, 57, 1998.

    Google Scholar 

  • Embrects, P., Klüppenberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1999.

    Google Scholar 

  • FEH, Flood Estimation Handbook, Institute of Hydrology, Wallingford, UK, 1999.

    Google Scholar 

  • Fisher, R.A. and Tippet, L.H.C., “Limiting forms of the frequency distributions of the largest or smallest member of a sample,” Proc. Camb. Phil. Soc. 24, 180–190, (1928).

    Article  MATH  Google Scholar 

  • Frigessi, A., Haug, O. and Rue, H., “A dynamic mixture model for unsupervised tail estimation without threshold selection,” Extremes 5(3), 219ndash;235, (2002).

    MathSciNet  Google Scholar 

  • Førland, E., Precipitation Normals, Normal Period 1961–90. Climate Report No. 39/93, the Norwegian Meteorological Institute, DNMI, Norway, 1993.

  • Førland, E., Roald, L.A., Tveito, O.E., and Hanssen-Bauer, I., Past and Future Variations in Climate and Runoff in Norway. Climate Report No. 19/00, The Norwegian Meteorological Institute, DNMI, Norway, 2000.

  • GREHYS, “Presentation and review of some methods for regional flood frequency analysis,” J. Hydrol. 186, 63–84, (1996).

    Google Scholar 

  • Heo, J.-H., Salas, J.D. and Boes, C.D., “Regional flood frequency analysis based on a Weibull model: Part 2. Simulations and applications,” J. Hydrol. 242, 171–182, (2001).

    Google Scholar 

  • Hisdal, H., Stahl, K., Tallaksen, L.M. and Demuth, S., “Have streamflow droughts in Europe become more severe or frequent?” Int. J. Climatol. 21, 317–333, (2001).

    Google Scholar 

  • Hosking, J.R.M. and Wallis, J.R. Regional Frequency Analysis, Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  • Hosking, J.R.M., Wallis, J.R. and Wood, E.F., “Estimation of the generalized extreme- value distribution by the method of probability-weighted moments,” Technometrics 27(3), 251–261, (1985).

    MathSciNet  Google Scholar 

  • Kjeldsen, T.R., Lundorf, A. and Rosbjerg, D., “Use of a two-component exponential distribution in partial duration modelling of hydrological droughts in Zimbabwean rivers,” Hydrol. Sci. J. 45(2), 285–298, (2000).

    Article  Google Scholar 

  • Klemeš, V., “Tall tales about tails of hydrological distributions. I,” J. Hydrol. Eng. 5(3), 227–231, (2000a).

    Google Scholar 

  • Klemeš, V., “Tall tales about tails of hydrological distributions. II,” J. Hydrol. Eng. 5(3), 232–239, (2000b).

    Google Scholar 

  • Lang, M., Ouarda, T.B.M.J. and Bobée, B., “Towards operational guidelines for over-threshold modeling,” J. Hydrol. 225(3–4), 103–117, (1999).

    Google Scholar 

  • Leadbetter, M.R. and Rootzén, H., “Extremal theory for stochastic processes,” Ann. Probab. 16(2), 431–478, (1988).

    MATH  MathSciNet  Google Scholar 

  • Leadbetter, M.R., Lindgren, G. Rootzén, H., Extremes and Related Properties of Random Sequences and Processes, Springer, New York, 1983.

    MATH  Google Scholar 

  • Madsen, H. and Rosbjerg, D., “A Regional Bayesian Method for Estimation of Extreme Streamflow Droughts,” in Statistical and Bayesian Methods in Hydrological Sciences (E. Parent, B. Bobeé, P. Hubert and J. Miquel, eds), Studies and Reports in Hydrology, UNESCO, Paris, France: 327–340, 1998.

    Google Scholar 

  • Madsen, H., Pearson, C.P., Rasmussen, P.F. and Rosbjerg, D., “Comparison of annual maximum series and partial duration series methods for modelling extreme hydrological events 1. At-site modelling,” Water Resour. Res. 33(4), 747–758, (1997a).

    Google Scholar 

  • Madsen, H., Pearson, C.P. and Rosbjerg, D., “Comparison of annual maximum series and partial duration series methods for modelling extreme hydrologic events 2. Regional modelling,” Water Resour. Res. 33(4), 759–769, (1997b).

    Google Scholar 

  • Martins, E.S., and Stedinger, J.R., “Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data,” Water Resour. Res. 36(3), 737–744, (2000).

    Google Scholar 

  • Martins, E.S., and Stedinger, J.R., “Generalized maximum likelihood Pareto-Poisson flood risk analysis for partial duration series,” Water Resour. Res. 37(10), 2551–2557, (2001).

    Google Scholar 

  • Pettersson, L.-E., Fjelstad, K., Engen, T., Hydrometriske stasjoner for sanntidsdata. Vannføringsstatistikk og feltbeskrivelser. (Real time hydrometric stations. Flow statistics and catchment descriptions. In Norwegian), Publication No. 19, NVE, Oslo, Norway, 1996.

  • Pickands, J., “Statistical inference using extreme order statistics,” Ann. Stat. 3(1), 119–131, (1975).

    MATH  MathSciNet  Google Scholar 

  • Rasmussen, P. and Rosbjerg, D., “Prediction uncertainty in seasonal partial duration series.” Water Resour. Res. 27(11), 2875–2883, (1991).

    Google Scholar 

  • Rice, S.O., “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–156, (1945).

    MATH  MathSciNet  Google Scholar 

  • Rosbjerg, D., Madsen, H. and Rasmussen, P.F., “Prediction in partial duration series with generalized pareto distributed exceedances,” Water Resour. Res. 28(11), 3001–3010, (1992).

    Google Scholar 

  • Rossi, F., Fiorentino, M. and Versace, P., “Two-component extreme value distribution for flood frequency analysis,” Water Resour. Res. 20(7), 847–856, (1984).

    Google Scholar 

  • Rootzén, H. and Tajvidi, N., “Extreme value statistics and wind storm losses: A case study,” Scand. Actuar. J. 1, 70–94, (1997).

    Google Scholar 

  • Smith, R.L., “Maximum likelihood estimation in a class of nonregular cases,” Biometrika 72, 69–90, (1985).

    Google Scholar 

  • Smith, R.L., “Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone,” Stat. Sci. 4(4), 367–377, (1989).

    MATH  Google Scholar 

  • Stedinger, J.R., Vogel, R.M. and Foufoula-Georgiou, E., “Frequency Analysis of Extreme Events,” in Handbook of Hydrology (D.R. Maidment, ed), McGraw-Hill, New York, 18.41, 1993.

    Google Scholar 

  • Strupczewski, W.G., Singh, V.P. and Feluch, W., “Non-stationarity approach to at-site flood frequency modelling I. Maximum likelihood estimation,” J. Hydrol. 248, 123–142, (2001).

    Google Scholar 

  • Tallaksen, L.M., Streamflow drought frequency analysis, Contribution to report from Exploratory Workshop on Drought and Drought Mitigation in Europe, 1–3 March 1999, JCRR-Ispra (VA), Italy (to be published in the “Natural and Technological Hazards” series of Kluwer Academic Publishers), 2000.

  • Tallaksen, L.M. and Hisdal, H. “Regional analysis of extreme streamflow drought duration and deficit volume. In FRIEND’97 - Regional Hydrology: Concepts and Models for Sustainable Water Management,” IAHS 246, 141–150, (1997).

    Google Scholar 

  • Tallaksen, L. M., Madsen, H. and Clausen, B., “On the definition and modelling of streamflow drought duration and deficit volume,” Hydrol. Sci. J. 42(1), 15–33, (1997).

    Google Scholar 

  • Tate, E.L. and Freeman, S.N., “Three modelling approaches for seasonal streamflow droughts in southern Africa: The use of Censored data,” Hydrol. Sci. J. 45(1), 27–42, (2000).

    Google Scholar 

  • Tawn, J.A.,“Estimating probabilities of extreme sea-levels,” App. Stat. 41, 77–93, (1992).

    MATH  Google Scholar 

  • Todorovic, P. and Rouselle, J., “Some problems of flood analysis,” Water Resour. Res. 7(5), 1144–1150, (1971).

    Google Scholar 

  • USWRC, Guidelines for determining flood flow frequency. United States Water Resources Council, Bull. 17, Hydrl. Comm. Washington, DC, 1976.

  • Vogel, R.M., McHahon, T.A. and Chiew, F.H.S., “Flood-flow frequency model selection in Australia,” J. Hydrol. 146, 421–449, (1993).

    Google Scholar 

  • Waylen, P. and Woo, M.-K., “Prediction of annual floods generated by mixed processes,” Water Resour. Res. 18(4), 1283–1286, (1982).

    Google Scholar 

  • Woo, M.-K. and Tarhule, A., “Streamflow droughts of northern Nigerian rivers,” Hydrol. Sci. J. 39(1), 19–34, (1994).

    Google Scholar 

  • Yevjevich, V., An Objective Approach to Definition and Investigations of Continental Hydrologic Droughts, Hydrology Papers, Colorado State University, Fort Collins, 23, 1967.

    Google Scholar 

  • Zaidman, M.D., Keller, V. Young, A.R. Cadman, D., “Flow-duration-frequency behaviour of British rivers based on annual minimum data,” J. Hydrol. 277, 195–213, (2003).

    Google Scholar 

  • Zelenhasic, E. and Salvai, A., “A method of streamflow drought analysis.” Water Resour. Res. 23(1), 156–168, (1987).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cemagref, Lyon, France

    Kolbjørn Engeland

  2. Department of Geosciences, University of Oslo, and Norwegian Water Resources and Energy Directorate, Oslo, Norway

    Hege Hisdal

  3. Section of Medical Statistics, University of Oslo and Norwegian Computing Center, Oslo, Norway

    Arnoldo Frigessi

Authors
  1. Kolbjørn Engeland
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hege Hisdal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arnoldo Frigessi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kolbjørn Engeland.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Engeland, K., Hisdal, H. & Frigessi, A. Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study. Extremes 7, 5–30 (2004). https://doi.org/10.1007/s10687-004-4727-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-004-4727-5

Key words

Search

Navigation