The Gumbel Mixed Model for Flood Frequency Analysis of Tarbela


  • S. Naz University of Karachi, Karachi
  • M. J. Iqbal University of Karachi, Karachi
  • S. M. Akhter University of Karachi, Karachi
  • I. Hussain University of Karachi, Karachi



Pakistan is considered a great natural reservoir of fresh water from the River Indus. When the river is in flood it causes immense damage. Flood frequency analysis provides guidance related to the behavior of anticipated flood flows using historical flow records. Flood frequency analysis usually focuses on the annual maximum peak discharge (flood peak) values, which is insufficient to solve many problems related to hydrological engineering design, management and planning. Therefore, the values of flood peak are not considered the only flood characteristic to assess the flood event. Flood durations and volumes are also associated with the flood frequency distributions as the characteristics of a flood event for authenticated results. The application of Gumbel mixed model on the recorded flood flows is proposed in this paper to analyze the joint probability distribution of flood volumes and peaks as well as flood durations and volumes, which are mutually correlated. Gumbel mixed model is a bivariate extreme value distribution with marginal distributions of two random variables by which the joint probability distributions, the conditional probability functions and the related return periods are obtained. Application of the suggested model on the observed data of Tarbela dam in Pakistan reveals that the model is appropriate for representing the joint probability distribution of flood volumes and peaks, and the joint probability distribution of flood durations and volumes. Hence, it is concluded that a bivariate probability distribution provides detailed information regarding future floods, whereas the univariate probability distribution is insufficient in providing extensive information regarding future flows.

Author Biographies

S. Naz, University of Karachi, Karachi

Department of Mathematics

M. J. Iqbal, University of Karachi, Karachi

Institute of Space and Planetary Astrophysics

S. M. Akhter, University of Karachi, Karachi

Department of Applied Physics,

I. Hussain, University of Karachi, Karachi

Department of Applied Physics


K, Smith, “Environmental hazard”, Rutledge, London, 1996.

Z. Kaczmarek, “The impact of climate variability on flood risk in Poland”, Risk Anal., vol. 23, no. 3 , pp. 559-566, 2003.

G. Khan, “Flood hazard assessment and mitigation along River Indus from Chashma Barrage to Sukkur Barrage using satellite image.” M. Phil thesis, Institute of Space and Planetary Astrophysics”, University of Karachi, 2007.

T. Tahirkaili and F. Nawaz, “Role of GIS and RS for flood hazard management in Pakistan: A case study of Jhelum, Pakistan”, MAPAsia, ( 003/disaster/index.htm , 2003.

B. Khan, M. J. Iqbal and M.A. Yousufzai, “Flood risk assessment of River Indus of Pakistan”, Arabian Journal of Geosciences, vol. 4, pp. 115-122, 2011.

F. Ashkar, “Partial duration series models for flood analysis”. Ph.D. thesis, Ecole Poly technique of Montreal, Montrea l, Canada, 1980.

F.N. Correia, “Multivariate partial duration series in flood risk analysis”, V.P. Singh (Ed.), Hydrologic Frequency Modeling, Reidel, Dordrecht, pp. 541–554, 1987.

E.J. Gumbel, “Statistics of extremes”, Columbia University Press, New York, 1958.¯

P. Todorovic, “Stochastic models of floods”, Water Resource Research, vol.4, no. 2 , pp. 345–356, 1978.

E. Castillo, “Extreme value theory in engineering”, 1st edn., Academic Press, New York, 1988.

W.E. Watt, K.W. Lathem, C.R. Neill, T.L. Richard and J. Rousselle, “Hydrology of floods in Canada: A Guide to planning and design”, National Research Council of Canada, 1989.

S. Yue, B.M.J Ouarda, B. Bobée, P. Legendre and P. Bruneau, “The Gumbel mixed model for flood frequency analysis”, Journal of Hydrology, vol. 226, pp. 88-100, 1999.

S. Yue, “The Gumbel mixed model applied to storm frequency analysis”, Water resources Management, vol. 14, no. 5, pp. 377-38, 2000.

E.J. Gumbel, “Multivariate extreme distributions”, Bulletin of the International Statistical Institute, vol. 39, no. 2, pp. 471–475, 1960.

J.T.D. Oliveria, “Bivariate extremes: extensions”, Bulletin of the International Statistical Institute, vol. 46 , no. 2, pp. 241-251, 1975.

J.T.D. Oliveria, ”Bivariate extremes: models and statistical decision”, Technical Report No. 14, Center for Stochastic

Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC, USA, 1982.

I.I. Gringorten, “A plotting rule for extreme probability”, Journal of Geophysical Research vol. 68, no. 3, pp. 813–814, 1963.

C. Cunnane, “Unbiased plotting positions: A Review”, Journal of Hydrology, vol. 37, no. 3/4, pp. 205-222, 1978.

S.L. Guo, “A discussion on unbiased plotting positions for the general extreme value distribution”, Journal of Hydrology, vol.121, no.1, pp. 33-44, 1990.

V. Fortin, Bernier, B. Bobée, Determination des crues de conception-Rapport final du project C3. INRS-Eau, Rapport de recherché confidential no. R-532, pp. 103, 1998.

T.B.M.J. Ouarda, N.El-Jabi and F. Ashkar, “Flood damage estimation in the residential sector”, Water Recourses and Environmental Hazards: Emphasis on Hydrologic and Cultural insight in the Pacific,” AWRA Technical Publication series, TPS-95-2, pp. 73-82, 1995.

E. J. Gumbel, Statistics of Extremes. Columbia University Press, New York, USA, 1958.




How to Cite

S. Naz, M. J. Iqbal, S. M. Akhter, and I. Hussain, “The Gumbel Mixed Model for Flood Frequency Analysis of Tarbela”, The Nucleus, vol. 53, no. 3, pp. 171–179, Sep. 2016.




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