Gumbel–Hougaard Copula for Trivariate Rainfall Frequency Analysis
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Volume 12, Issue 4
Abstract
Joint distributions of rainfall intensity, duration, and depth or those of rainfall intensity and duration, rainfall depth and duration, and rainfall intensity and depth are important in hydrologic design and floodplain management. Considering the dependence among rainfall intensity, depth, and duration, multivariate rainfall frequency distributions have been derived using one of three fundamental assumptions. Either the rainfall intensity, duration, and depth have been assumed independent, or they each have the same type of marginal probability distribution or they have been assumed to have the normal distribution or have been transformed to have the normal distribution. In reality, however, rainfall intensity, duration, and depth are dependent, do not follow, in general, the normal distribution, and do not have the same type of marginal distributions. This study aims at deriving trivariate rainfall frequency distributions using the Gumbel–Hougaard copula which does not assume the rainfall variables to be independent or normal or have the same type of marginal distributions. The trivariate distribution is then employed to determine joint conditional return periods, and is tested using rainfall data from the Amite River Basin in Louisiana.
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References
Akaike, H. (1974). “A new look at the statistical model identification.” IEEE Trans. Autom. Control, AC-19(6), 716–722.
Bacchi, B., Becciu, G., and Kottegoda, N. T. (1994). “Bivariate exponential model applied to intensities and durations of extreme rainfall.” J. Hydrol., 155, 225–236.
Cordova, J. R., and Rodriguez-Iturbe, I. (1985). “On probabilistic structure of storm surface runoff.” Water Resour. Res., 21(5), 755–763.
De Michele, C., Salvadori, G., Canossi, M., Petaccia, A., and Rosso, R. (2005). “Bivariate statistical approach to check adequacy of dam spillway.” J. Hydrol. Eng., 10(1), 50–57.
Favre, A.-C., El Adlouni, S., Perreault, L., Thiemonge, N., and Bobee, B. (2004). “Multivariate hydrological frequency analysis using copulas.” Water Resour. Res., 40, W01101.
Freund, J. E. (1961). “A bivariate extension of the exponential distribution.” J. Am. Stat. Assoc., 56, 971–977.
Genest, C., Ghoudi, K., and Rivest, L. (1995). “A semiparametric estimation procedure of dependence parameters in multivariate families of distributions.” Biometrika, 82(3), 543–552.
Genest, C., and Mackay, L. (1986). “The joy of copulas: Bivariate distributions with uniform marginals.” Am. Stat., 40, 280–283.
Genest, C., and Rivest, L. (1993). “Statistical inference procedures for bivariate Archimedean copulas.” J. Am. Stat. Assoc., 88, 1034–1043.
Goel, N. K., Kurothe, R. S., Mathur, B. S., and Vogel, R. M. (2000). “A derived flood frequency distribution for correlated rainfall intensity and duration.” J. Hydrol., 228, 56–67.
Griamaldi, S., Serinaldi, F., Napolitano, F., and Ubertini, L. (2005). “A 3-copula function application for design hyetograph analysis.” IAHS Publication, 293, 1–9.
Gringorten, I. I. (1963). “A plotting rule of extreme probability paper.” J. Geophys. Res., 68(3), 813–814.
Hashino, M. (1985). “Formulation of the joint return period of two hydrologic variates associated with a Poisson process.” J. Hydrosci. Hydr. Eng., 3(2), 73–84.
Kelly, K. S., and Krzysoztofowicz, R. (1997). “A bivariate meta-Gaussian density for use in hydrology.” Stochastic Environ. Res. Risk Assess., 11(1), 17–31.
Kurothe, R. S., Goel, N. K., and Mathur, B. S. (1997). “Derived flood frequency distribution of negatively correlated rainfall intensity and duration.” Water Resour. Res., 33(9), 2103–2107.
Long, D., and Krzysoztofowicz, R. (1992). “Farlie–Gumbel–Morgenstern bivariate densities: Are they applicable in hydrology?” Stchastic Hydrol. Hydraul., 6(1), 47–54.
Long, D., and Krzysoztofowicz, R. (1995). “A family of bivariate densities constructed from marginals.” J. Am. Stat. Assoc., 90(340), 739–746.
Nelsen, R. B. (1999). An introduction to copulas, Springer, New York.
Salvatori, G., and De Michele, C. (2004). “Frequency analysis via copulas: Theoretical aspects and applications to hydrological events.” Water Resour. Res., 40(12), W12511.
Singh, K., and Singh, V. P. (1991). “Derivation of bivariate probability density functions with exponential marginals.” Stchastic Hydrol. Hydraul., 5(1), 55–68.
Sklar, A. (1959). “Fonctions de repartition à n dimensions et leurs marges.” Publ. Inst. Stat. Univ. Paris, 8, 229–231.
Yevjevich, V. (1972). Probability and statistics in hydrology, Water Resources Publications, Fort Collins, Colo.
Yue, S. (2000). “Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls.” Hydrol. Sci. J., 45(2), 315–326.
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Information
Published In
Journal of Hydrologic Engineering
Volume 12 • Issue 4 • July 2007
Pages: 409 - 419
Copyright
© 2007 ASCE.
History
Received: Aug 29, 2006
Accepted: Sep 26, 2006
Published online: Jul 1, 2007
Published in print: Jul 2007
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