The flood probability distribution tail: how heavy is it?

  • Pietro Bernardara
  • Daniel Schertzer
  • Eric Sauquet
  • Ioulia Tchiguirinskaia
  • Michel Lang
Original Paper


This paper empirically investigates the asymptotic behaviour of the flood probability distribution and more precisely the possible occurrence of heavy tail distributions, generally predicted by multiplicative cascades. Since heavy tails considerably increase the frequency of extremes, they have many practical and societal consequences. A French database of 173 daily discharge time series is analyzed. These series correspond to various climatic and hydrological conditions, drainage areas ranging from 10 to 10km2, and are from 22 to 95 years long. The peaks-over-threshold method has been used with a set of semi-parametric estimators (Hill and Generalized Hill estimators), and parametric estimators (maximum likelihood and L-moments). We discuss the respective interest of the estimators and compare their respective estimates of the shape parameter of the probability distribution of the peaks. We emphasize the influence of the selected number of the highest observations that are used in the estimation procedure and in this respect the particular interest of the semi-parametric estimators. Nevertheless, the various estimators agree on the prevalence of heavy tails and we point out some links between their presence and hydrological and climatic conditions.


Asymptotic behaviour Flood frequency analysis Power law Heavy tails Cascades Multifractals 



The authors gratefully acknowledge the French Ministry of Ecology and Sustainable Development for providing data. The paper has been improved by helpful comments from the two anonymous reviewers.


  1. Anderson PL, Meerschaert M (1998) Modeling river flows with heavy tails. Water Resour Res 34(9):2271–2280CrossRefGoogle Scholar
  2. Bak P (1997) How nature works: the science of self-organized criticality. Oxford University Press OxfordGoogle Scholar
  3. Beirlant J, Teugels JL, Vynckiee P (1996) Practical analysis of extreme values. Leuven University Press LeuvenGoogle Scholar
  4. Bendjoudi H et al (2004) Prédétermination multifractales des précipitations et des crues. Rapport final projet RIO2, MEDD, ParisGoogle Scholar
  5. Barnett V (1975) Probability plotting methods and order statistics. Appl Stat 24:95–108CrossRefGoogle Scholar
  6. Chambers JM (1977) Computational methods for data analysis. Wiley, New YorkGoogle Scholar
  7. Coles S (2001) An introduction to statistical modeling of extreme values. LondonGoogle Scholar
  8. Csorgo M, Deheuvels P, Mason D (1985) Kernel estimate of the tail index of a distribution. Ann Stat 13:1050–1077CrossRefGoogle Scholar
  9. Cunnane C (1973) A particular comparison of annual maxima and partial duration series methods of flood frequency prediction. J Hydrol 18:257–271CrossRefGoogle Scholar
  10. Cunnane C (1988) Methods and merits of regional flood frequency analysis. J Hydrol 100:269–290CrossRefGoogle Scholar
  11. De Haan L (1970) On regular variation and its applications to weak convergence of sample extremes, vol.32 de CWI Tract. AmsterdamGoogle Scholar
  12. De Michele C, Salvadori G (2005) Some hydrological applications of small sample estimators of Generalized Pareto and extreme value distributions. J Hydrol 301(1–4):37–53CrossRefGoogle Scholar
  13. Douglas E, Barros AP (2003) Probable maximum precipitation estimation using multifractals: application in the Eastern United States. J Hydrometeorol 4(6):1012–1024CrossRefGoogle Scholar
  14. Fisher R, Tippett L (1928) On the estimation of the frequency distributions of the largest or smallest member of a sample. Proceedings of the Cambridge Philosophical Society 24:180–190Google Scholar
  15. Gnedenko B (1943) Sur la distribution limite du terme maximum d’une série aléatoire. Ann Math 44:423–453CrossRefGoogle Scholar
  16. Greenwood JA, Landwher JM, Matalas NC, Wallis JR (1979) Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resour Res 15(5):1049–1054CrossRefGoogle Scholar
  17. Gupta VK, Waymire E (1990) Multiscaling properties of spatial rainfall and river flow distributions. J Geophys Res 95(D3):1999–2009CrossRefGoogle Scholar
  18. Hall P (1982) On some simple estimate of an exponent of regular variation. J R Stat Soc 44:37–42Google Scholar
  19. Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3(5):1163–1174CrossRefGoogle Scholar
  20. Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B (Methodological) 52(1):102–124Google Scholar
  21. Hosking JRM, Wallis JR (1987) Parameter and quantile estimation for Generalized Pareto distribution. Technometrics 29(3):339–349CrossRefGoogle Scholar
  22. Hubert P, Tessier Y, Lovejoy S, Schertzer D, Schmitt F, Ladoy P, Carbonnel JP, Violette S, Desurosne I (1993) Multifractals and extreme rainfall events. Geophys Res Lett 20:931–934CrossRefGoogle Scholar
  23. Hubert P, Tchiguirinskaia I, Bendjoudi H, Schertzer D, Lovejoy S (2002) Multifractal modeling of the Blavet River discharges at Guerledan, Third celtic hydrology colloquium, Galway, IrelandGoogle Scholar
  24. Hubert P, Tchiguirinskaia I, Bendjoudi H, Schertzer D, Lovejoy S (2005) Multifractal modeling of flood. In: Jiri M, Gheorghe S, Gabor B (eds) Transboundary floods: reducing risks through flood management proceedings of the NATO advanced research workshop on transboundary floods: reducing risks and enhancing security through improved flood management planning, Baile Felix (Oradea), Romania, 4–8 May 2005 Series: Nato Science Series: IV: Earth and Environmental Sciences, vol 72 2006, X, 336 p., Softcover ISBN-10: 1-4020-4901-3 ISBN-13: 978-1-4020-4901-9Google Scholar
  25. Hurst HE (1951) Long-term storage capacity of reservoir. Trans Am Soc Civ Eng 116:770Google Scholar
  26. Javelle P, Ouarda TBMJ, Lang M, Bobée B, Galéa G, Grésillon JM (2002) Development of regional flood-duration-frequency curves based on the index-flood method. J Hydrol 258:249–259CrossRefGoogle Scholar
  27. Jenkinson AF (1955) The frequency distribution of the annual maximum (or minimum) value of meteorological elements. Q J R Meteorol Soc 81:158–171CrossRefGoogle Scholar
  28. Kottegoda NT, Rosso R (1997) Statistics, probability and reliability for civil and environmental engineers. McGraw-Hill New YorkGoogle Scholar
  29. Labat D, Mangin A, Ababou R (2002) Rainfall-runoff relations for karstic springs: multifractal analyses. J Hydrol 256(3–4):176–195CrossRefGoogle Scholar
  30. Lang M, Ouarda TBMJ, Bobée B (1999) Towards operational guidelines for over-threshold modeling. J Hydrol 225(3–4):103–117CrossRefGoogle Scholar
  31. Lovejoy S, Schertzer D (1990) Multifractals, universality classes, satellite and radar measurement of clouds and rain. J Geophys Resour 95:2021–2034CrossRefGoogle Scholar
  32. Lovejoy S, Schertzer D (1995) Multifractals and rain. In: Kunzewicz ZW (ed) New uncertainity concepts in hydrology and water resources. Cambridge University Press, Cambridge, pp 62–103Google Scholar
  33. Madsen H, Rosbjerg D (1997) The partial duration series method in regional index-flood modeling. Water Resour Res 33(4):737–746CrossRefGoogle Scholar
  34. Majone B, Bellin A, Borsato A (2004) Runoff generation in karst catchments: multifractal analysis. J Hydrol 294(1–3):176–195CrossRefGoogle Scholar
  35. Malamud BD, Turcotte DL (2006) The applicability of power-law frequency statistics to floods. J Hydrol Hydrofractals ‘03’ 322(1–4):168–180Google Scholar
  36. Mandelbrot BB, Wallis JR (1968) Noah, Joseph, and operational hydrology. Water Resour Res 4(5):909–918CrossRefGoogle Scholar
  37. Merz R, Bloschl G (2005) Flood frequency regionalisation–spatial proximity versus catchment attributes. J Hydrol 302(1–4):283–306CrossRefGoogle Scholar
  38. Moon YI, Lall U, Bosworth K (1993) A comparison of tail probability estimators for flood frequency analysis. J Hydrol 151(2–4):343–363CrossRefGoogle Scholar
  39. Naulet R, Lang M, Coeur D, Gigon C (2001) Collaboration between historians and hydrologists on the Ardèche River (France). First step: inventory of historical flood information. In: Paola Albini TG, Felix France (eds) Advances in natural and technological research: the use of historical data in natural hazard assessment. Kluwer, Dordrecht, pp 113–129Google Scholar
  40. Naulet R et al (2005) Flood frequency analysis on the Ardèche river using French documentary sources from the last two centuries. J Hydrol Palaeofloods, historical data and climate variability: applications in flood risk assessment 313(1–2):58–78Google Scholar
  41. Pandey G, Lovejoy S, Schertzer D (1998) Multifractal analysis of daily river flows including extremes for basins of five to two million square kilometres, one day to 75 years. J Hydrol 208(1–2):62–81CrossRefGoogle Scholar
  42. Pandey MD, Van Gelder PHAJM, Vrijling JK (2001) The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach. Struct Saf 23(2):179–192CrossRefGoogle Scholar
  43. Payrastre O, Gaume E, Andrieu H (2005) Use of historical data to assess the occurrence of floods in small watershed in the French Mediterranean area. Adv Geosci 2:313–320CrossRefGoogle Scholar
  44. Pickands J (1975) Statistical inference using extreme order statistics. Ann Stat 3(1):119–131CrossRefGoogle Scholar
  45. Renard B (2006) Détection et prise en compte d’éventuels impacts du changement climatique sur les extrêmes hydrologiques en France. Thèse INP Grenoble, Cemagref Lyon, 20 September, p 361Google Scholar
  46. Rodriguez-Iturbe I, Rinaldo A (1997) Fractal River basin, chance and self-organization. Cambridge University Press, New YorkGoogle Scholar
  47. Schertzer D, Lovejoy S (1987) Physical modeling and analysis of rain and clouds by anisotropic scaling of multiplicative processes. J Geophys Res D8(8):9693–9714CrossRefGoogle Scholar
  48. Schertzer D, Lovejoy S (1991) Non-linear variability in geophysics, scaling and fractals, Kluwer, Dordrecht-Boston, pp 318Google Scholar
  49. Schertzer D, Lovejoy S (1992) Hard and soft multifractal processes. Physica A 185:187–194CrossRefGoogle Scholar
  50. Schertzer D, Lovejoy S, Hubert P (2002) An introduction to stochastic multifractal fields. In: Ern A, Liu W (eds) ISFMA symposium on environmental science and engineering with related mathematical problems. High Education Press, Beijing, pp 106–179Google Scholar
  51. Stedinger JR, Cohn TA (1986) Flood frequency analysis with historical and paleoflood information. Water Resour Res 22(5):785–793CrossRefGoogle Scholar
  52. Tchiguirinskaia I, Schertzer D, Hubert P, Bendjoudi H, Lovejoy S (2004) Multiscaling geophysics and sustainable development. In: Tchiguirinskaia I, Hubert P (eds) Scales in hydrology and water management. IAHS, Paris, pp 113–136Google Scholar
  53. Tessier Y, Lovejoy S, Hubert P, Schertzer D, Pecknold S (1996) Multifractal analysis and modeling of rainfall and river flows and scaling, causal transfer functions. J Geophys Res Atmos 101(D21):26427–26440CrossRefGoogle Scholar
  54. Turcotte DL, Greene A (1993) Scale-invariant approach to flood frequency analysis. Stochastic Hydrology and Hydraulics 7:33–40CrossRefGoogle Scholar
  55. Wang QJ (1997) Using higher probability weighted moments for flood frequency analysis. J Hydrol 194(1–4):95–106CrossRefGoogle Scholar
  56. Willems P (1998) Hydrological applications of extreme value analysis. In: Wheater H, Kirby C (eds) Hydrology in a changing environment. Wiley, Chichester, pp 15–25Google Scholar
  57. Willems P (2000) Compound intensity/duration/frequency-relationships of extreme precipitation for two seasons and two storm types. J Hydrol 233(1–4):189–205CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Pietro Bernardara
    • 1
    • 2
  • Daniel Schertzer
    • 1
    • 3
  • Eric Sauquet
    • 2
  • Ioulia Tchiguirinskaia
    • 1
  • Michel Lang
    • 2
  1. 1.CEREVE, ENPCMarne-la-Vallée Cedex 2France
  2. 2.HHLY, CEMAGREFLyon Cedex 09France
  3. 3.CNRM, Météo FranceParisFrance

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