, Volume 4, Issue 1, pp 5–22 | Cite as

Scaling Properties of Flood Peaks

  • Julia E. Morrison
  • James A. Smith


The scaling behavior of flood peak distributions is examined using a statistical model of the spatio-temporal distribution of rainfall and a hydrological model that describes the transformation of rainfall to discharge within a drainage network. Of particular interest is the empirical observation made by a number of investigators that the coefficient of variation (CV) of annual flood peaks for a region increases with drainage area up to drainage areas of approximately 100 km2, and decreases with drainage area for larger drainage basins. This observation is neither consistent with simple scaling models, in which the coefficient of variation does not vary with drainage area, nor multiscaling models, in which the coefficient of variation decreases monotonically with drainage area. Model analyses illustrate that knowledge of the spatial and temporal organization of the rainfall, together with the details of the network structure of the drainage basin, is sufficient information with which to explain the observed behavior of sample CV. The interaction between the temporal variability of rainfall, relative to basin size, and the network structure is shown to be of particular importance.

flood peaks scaling coefficient of variation simulations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bloschl, G., “Scaling issues in snow hydrology,” Hydrol. Proc. 13, 2149–2175, (1999).Google Scholar
  2. [2]
    Bloschl, G. and Sivapalan, M., “Process controls on regional flood frequency: Coefficient of variation and basin scale,” Water Resour. Res. 33(12), 2967–2980, (1997).Google Scholar
  3. [3]
    Caracena, F. and Fritch, J.M., “Focusing mechanisms in the Texas Hill Country flash floods of 1978,” Mon. Wea. Rev. 111, 2319–2332, (1983).Google Scholar
  4. [4]
    Costa, J.E., “Hydraulics and basin morphometry of the largest flash floods in the conterminous United States,” J. Hydrol. 93, 313–338, (1987).Google Scholar
  5. [5]
    Cox, D.R. and Isham, V.S., “Stochastic spatial-temporal models for rain,” In: Stochastic Methods in Hydrology (O.E. Barnoff-Nielsen, V.K. Gupta, V. Perez-Abreu, and E. Waymire, eds), World Scientific, New York, (1998).Google Scholar
  6. [6]
    Gupta, V.K., Castro, S., and Over, T.M., “On scaling exponents of spatial peak flows from rainfall and river network geometry,” J. Hydrol. 187, 81–104, (1996).Google Scholar
  7. [7]
    Gupta, V.K. and Dawdy, D.R., “Physical interpretations of regional variations in the scaling exponents of flood quantiles,” Hydrological. Proc. 9, 347–361, (1995).Google Scholar
  8. [8]
    Gupta, V.K., Messa, O.J., and Dawdy, D.R., “Multiscaling theory of flood peaks: Regional quantile analysis,” Water Resour. Res. 30(12), 3405–3421, (1994).Google Scholar
  9. [9]
    Gupta, V.K. and Waymire, E., “Multiscaling properties of spatial rainfall and river flow distributions,” J. Geophys. Res. 95(D3), 1999–2009, (1990).Google Scholar
  10. [10]
    Gupta, V.K. and Waymire, E., “Spatial variability and scale invariance in hydrologic regionalization,” In: Scale Dependence and Scale Invariance in Hydrology, Cambridge University Press, 1998.Google Scholar
  11. [11]
    Harris, D., Menabde, M., Seed, A., and Austin, G., “Multifractal characterization of rain fields with a strong orographic influence,” J. Geophys. Res. 101(D21), 26405–26414, (1996).Google Scholar
  12. [12]
    Hosking, J.R.M. and Wallis, J.R., “The theory of probability weighted moments,” Research report RC20349, IBM Research, Yorktown Heights, NY, (1996).Google Scholar
  13. [13]
    Marani, A., Rigon, R., and Rinaldo, A., “A note on fractal channel structure,” Water Resour. Res. 27, 3041–3049, (1991).Google Scholar
  14. [14]
    Morrison, J.E. and Smith, J.A., “Stochastic modeling of flood peaks using generalized extreme value distribution,” submitted to Water Resour. Res. Google Scholar
  15. [15]
    Northrop, F., “A clustered spatial-temporal model of rainfall,” Proc. Roy. Soc. Series A 454, 1875–1888, (1998).Google Scholar
  16. [16]
    Ogden, F.L. and Saghafian, B., “Green and Ampt infiltration with redistribution,” J. of Irrigat. and Drainage Eng — ASCE 123(5), 386–393, (1997).Google Scholar
  17. [17]
    Onof, C., Northrop, P., Wheater, H.S., and Isham, V., “Spatiotemporal storm structure and scaling property analysis for modeling,” J. Geophys. Res. 101(D21), 26415–26425, (1996).Google Scholar
  18. [18]
    Over, T.M. and Gupta, V.K., “A space-time theory of mesoscale rainfall using random cascades,” J. Geophys. Res. 101(D21), 26319–26331, (1996).Google Scholar
  19. [19]
    Perica, S. and Foufoula-Georgiou, E., “Model for multiscale disaggregation of spatial rainfall based on coupling meteorological and scaling description,” J. Geophys. Res. 101(D21), 26347–26361, (1996).Google Scholar
  20. [20]
    Robinson, J.S. and Sivapalan, M., “Catchment scale runoff generation model by aggregation and similarity analysis,” In: Scale Issues in Hydrologic Modeling (J. Kalma and M. Sivapalan, eds), Wiley, New York, 311–330, (1995).Google Scholar
  21. [21]
    Robinson, J.S. and Sivapalan, M., “An investigation into the physical causes of scaling and heterogeneity of regional flood frequency,” Water Resour. Res. 33(5), 1045–1059, (1997).Google Scholar
  22. [22]
    Robinson, J.S., Sivapalan, M., and Snell, J., “On the relative roles of hillslope processes, channel routing, and network geomorphology in the hydrologic response of natural catchments,” Water Resour. Res. 31, 3089–3101, (1995).Google Scholar
  23. [23]
    Rodriguez-Iturbe, I., Cox, D.R., and Isham, V., “Some models for rainfall based on stochastic point processes,” Proc. R. Soc. London, Ser. A 410, 269–288, (1987).Google Scholar
  24. [24]
    Rodriguez-Iturbe, I. and Rinaldo, A., Fractal River Basins: Chance and Self-Organization, Cambridge University Press, 1998.Google Scholar
  25. [25]
    Smith, J.A., “Representation of basin scale in flood peak distributions,” Water Resour. Res. 28(11), 2993–2999, (1992).Google Scholar
  26. [26]
    Smith, J.A., Baeck, M.L., Morrison, J.E., and Sturdevant-Rees, P., “Catastrophic rainfall and flooding in Texas,” J. of Hydrometeor. 1, 5–25, (2000).Google Scholar
  27. [27]
    Smith, J.A., Baeck, M.L., Morrison, J.E., Sturdevant-Rees, P., Turner-Gillespie, D., and Bates, P., “The regional hydrology of extreme floods in an urban environment,” submitted to J. of Hydrometeor. Google Scholar
  28. [28]
    Smith, J.A., Baeck, M.L., Steiner, M., and Miller, A., “Catastrophic rainfall from an upslope thunderstorm in the central Appalachians: The Rapidan storm of June 27, 1995,” Water Resour. Res. 32(10), 3099–3113, (1996).Google Scholar
  29. [29]
    Smith, J.A. and Karr, A.F., “A statistical model of extreme storm rainfall,” Journal of Geophysical Research 95(D3), 2083–2092, (1990).Google Scholar
  30. [30]
    Waymire, E., Gupta, V.K., and Rodriguez-Iturbe, I., “A spectral theory of rainfall intensity at the meso-β scale,” Water Resour. Res. 20(10), 1453–1465, (1984).Google Scholar
  31. [31]
    Wood, E.F., Sivapalan, M., Beven, K., and Band, L., “Effects of spatial variability and scale with implications to hydrologic modeling,” J. Hydrol. 102, 29–47, (1988).Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Julia E. Morrison
    • 1
  • James A. Smith
    • 2
  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrinceton
  2. 2.Department of Civil and Environmental EngineeringPrinceton UniversityPrinceton

Personalised recommendations