Bivariate Frequency Analysis of Annual Maximum Rainfall Event Series in Seoul, Korea
Publication: Journal of Hydrologic Engineering
Volume 19, Issue 6
Abstract
The return period of a rainfall event is estimated by the frequency analysis for a given rainfall duration. Thus, it is possible to derive different return periods with different rainfall durations for a given rainfall event. The longest derived return period is generally cited to represent the rainfall event. However, it is not clear if the longest derived return period is a representative measure of the given rainfall event. In this study, as a solution for this problem, a bivariate frequency analysis was introduced. As a first step, annual maximum rainfall events were selected by applying a bivariate exponential distribution. As an application, a total of 1,534 rainfall events observed in Seoul, Korea, over the last 46 years were analyzed. The annual maximum rainfall event series were then analyzed by applying a bivariate logistic model. The results were also compared with those from a conventional univariate frequency analysis. The findings of this study are summarized as follows: (1) the bivariate exponential distribution satisfactorily represented the duration and total rainfall depth data of all independent rainfall events, and the annually estimated parameters of the bivariate exponential distribution were more reasonable with respect to annual changes in the climatic conditions than those for the entire data period; (2) by using the bivariate logistic model, the return period was able to be assigned to each annual maximum rainfall event; and (3) rainfall quartiles of the univariate frequency analysis were bigger than those from the bivariate frequency analysis for rather short return periods of less than 30 years, but smaller for rather long return periods exceeding 100 years, primarily attributable to the smaller variance of the univariate annual maximum series.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This study was supported by the 2005 SOC Project (05-GIBANGUCHUK-D03-01) through the Design Criteria Research Center for Abnormal Weather-Disaster Prevention (DCRC-AWDP) in KICTTEP of MOCT, KOREA. The comments from three anonymous reviewers are acknowledged as very helpful to improve the manuscript.
References
Adams, B. J., and Howard, C. D. D. (1986). “Pathology of design storms.” Can. Water Resour. J., 11(3), 49–55.
Adams, B. J., and Papa, F. (2000). Urban stormwater management planning with analytical probabilistic models, Wiley, New York, 55–68.
Anderson, T. W., and Darling, D. A. (1952). “Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes.” Ann. Math. Stat., 23(2), 193–212.
Ang, A. H.-S., and Tang, W. H. (2007). Probability concepts in engineering, 2nd Ed., Wiley, New York.
Arnold, B. C., and Strauss, D. (1988). “Bivariate distributions with exponential conditionals.” J. Am. Stat. Assoc., 83(402), 522–527.
Bacchi, B., Becciu, G., and Kottegoga, N. T. (1994). “Bivariate exponential model applied to intensities and durations of extreme rainfall.” J. Hydrol., 155(1–2), 225–236.
Box, G. E. P., and Cox, D. (1964). “An analysis of transformations.” J. R. Stat. Soc., Ser. B, 26(2), 211–252.
Coles, S. (1993). “Regional modeling of extreme storms via max-stable processes.” J. R. Stat. Soc., Ser. B, 55(4), 797–816.
Coles, S., and Tawn, J. A. (1991). “Modeling extreme multivariate events.” J. R. Stat. Soc., Ser. B, 53(2), 377–392.
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., and Salvadori, G. (2003). “A generalized Pareto intensity-duration model of storm rainfall exploiting 2-copulas.” J. Geophys. Res., 108(D2), 1–11.
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., 50–57.
Eagleson, P. S. (1972). “Dynamics of flood frequency.” Water Resour. Res., 8(4), 878–898.
Frechet, M. (1951). “Sur les tableaux de correlation dont les marges sont donnees.” Annales de l’Universite de Lyon, Sec. A, 9, 53–77 (in French).
Freund, J. (1961). “A bivariate extension of the exponential distribution.” J. Am. Stat. Assoc., 56(296), 971–977.
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(1–2), 56–67.
Gonzalez, J., and Valdes, J. B. (2003). “Bivariate drought recurrence analysis using tree rings reconstructions.” J. Hydrol. Eng., 247–258.
Grimaldi, S., and Serinaldi, F. (2006). “Asymmetric copula in multivariate flood frequency analysis.” Adv. Water Resour., 29(8), 1155–1167.
Gumbel, E. J. (1960). “Bivariate exponential distributions.” J. Am. Stat. Assoc., 55(292), 698–707.
Gumbel, E. J. (1965). “Two systems of bivariate extremal distribution.” 35th Session of the Int. Statistical Institute, Beograd, No. 69, International Statistical Institute, Hague-Leidschenveen, Netherlands.
Hayakawa, Y. (1994). “The construction of new bivariate exponential distributions from a Bayesian perspective.” J. Am. Stat. Assoc., 89(427), 1044–1049.
Justel, A., Peña, D., and Zamar, R. (1997). “A multivariate Kolmogorov-Smirnov test of goodness of fit.” Stat. Prob. Lett., 35(3), 251–259.
Kao, S. C., and Govindaraju, R. S. (2007a). “A bivariate frequency analysis of extreme rainfall with implication for design.” J. Geophys. Res., 112(D13), D13119.
Kao, S. C., and Govindaraju, R. S. (2007b). “Probabilistic structure of storm surface runoff considering the dependence average intensity and storm duration of rainfall events.” Water Resour. Res., 43(6), W06410.
Kelly, K. S., and Krzysztofowicz, R. (1997). “A bivariate meta-Gaussian density for use in hydrology.” Stochastic Hydrol. Hydraul., 11(1), 17–31.
Kim, N. W. (2000). “Temporal distribution of design rainfall events.” Proc., Korea Water Resources Association, Korea Water Resources Association, Seoul, Korea.
Kim, T.-W., Valdes, J. B., and Yoo, C. (2006). “Nonparametric approach for bivariate drought characterization using Palmer drought index.” J. Hydrol. Eng., 134–143.
Korea Institute of Construction Technology (KICT). (2002). 2002 Flood in Gangwon-do Region by the Typhoon Rusa, Goyang, Korea.
Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000). Continuous multivariate distributions volume 1: Models and applications, Wiley, New York, 350–362.
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.
Kwon, J. H. (2003). “Rainfall analysis to estimate the amount of nonpoint source pollution.” M.S. thesis, Korea Univ., Seoul.
Lee, C. H., Kim, T. W., Chung, G., Choi, M., and Yoo, C. (2010). “Application of bivariate frequency analysis to the derivation of rainfall-frequency curves.” Stochastic Environ. Res. Risk Assess., 24(3), 389–397.
Lee, D. R., and Jeong, S. M. (1992). “A spatial-temporal characteristics of rainfall in the Han River Basins.” J. Korean Assoc. Hydrol. Sci., 25(4), 75–85.
Madadgar, S., and Moradkhani, H. (2013). “Drought analysis under climate change using copula.” J. Hydrol. Eng., 746–759.
Madadgar, S., Moradkhani, H., and Garen, D. (2014). “Towards improved post-processing of hydrologic forecast ensembles.” Hydrol. Process., 28(1), 104–122.
Ministry of Construction, and Transportation (MOCT). (2000). Annual report on water resources management technology development: Volume 1. Rainfall frequency atlas in Korea, Seoul, 192.
Ministry of Construction, and Transportation (MOCT). (2006). Field survey report: Damages caused by the typhoon and the heavy rainfall in July, 2006, Seoul.
Muller, A., Bacro, J.-N., and Lang, M. (2008). “Bayesian comparison of different rainfall depth-duration-frequency relationships.” Stochastic Environ. Res. Risk Assess., 22(1), 33–46.
National Institute for Disaster Prevention (NIDP). (2002). The field survey report: Damages caused by the Typhoon RUSA in 2002, Seoul.
National Institute for Disaster Prevention (NIDP). (2003a). Field survey report: Damages Caused by the Typhoon Maemi in 2003, Seoul.
National Institute for Disaster Prevention (NIDP). (2003b). A study on the integrated countermeasures for repetitive flooded area, Seoul.
Park, C. (2012). “Frequency analysis of independent rainfall events and runoff analysis.” M.S. thesis, Korea Univ., Seoul.
Poulin, A., Huad, D., Favre, A.-C., and Pugin, S. (2007). “Importance of tail dependence in bivariate frequency analysis.” J. Hydrol. Eng., 394–403.
Rao, A. R., and Hamed, K. H. (2000). Flood frequency analysis, CRC, Boca Raton, FL, 227–228.
Restrepo-Posada, P. J., and Eagleson, P. S. (1982). “Identification of independent rainstorms.” J. Hydrol., 55(1–4), 303–319.
Rosbjerg, D. (1987). “On the annual maximum distribution in dependent partial duration series.” Stochastic Hydrol. Hydraul., 1(1), 3–16.
Salvadori, G., and De Michele, C. (2004). “Frequency analysis via copulas: Theoretical aspects and applications to hydrological events.” Water Resour. Res., 40(12), W12511.
Salvadori, G., and De Michele, C. (2007). “On the use of copulas in hydrology: Theory and practice.” J. Hydrol. Eng., 369–380.
Seoul Metropolitan City. (1999). “Flood disasters in Seoul.” Seoul.
Serinaldi, F., and Grimaldi, S. (2007). “Fully nested 3-copula: Procedure and application on hydrological data.” J. Hydrol. Eng., 420–430.
Shiau, J. T. (2003). “Return period of bivariate distributed extreme hydrological event.” Stochastic Environ. Res. Risk Assess., 17(1–2), 42–57.
Shukla, R. K., Trivedi, M., and Kumar, M. (2010). “On the proficient use of GEV distribution: A case study of subtropical monsoon region in India.” Annals of University of Tibiscus Computer Science Series, 8(1), 81–92.
Singh, K., and Singh, V. P. (1991). “Derivation of bivariate probability density functions with exponential marginals.” Stochastic Hydrol. Hydraul., 5(1), 55–68.
Singh, V. P., and Zhang, L. (2007). “IDF curves using Frank Archimedean copula.” J. Hydrol. Eng., 651–662.
Smith, E. (2005). “Bayesian modeling of extreme rainfall data.” Ph.D. thesis, Univ. of Newcastle, Tyne, Australia.
Tawn, J. A. (1988). “Bivariate extreme value theory: Models and estimation.” Biometrica, 75(3), 397–415.
Tawn, J. A. (1990). “Modelling multivariate extreme value distribution.” Biometrica, 77(2), 245–253.
Vannitsem, S., and Naveau, P. (2007). “Spatial dependences among precipitation maxima over Belgium.” Nonlinear Process. Geophys., 14, 621–630.
Yoo, C. (2011). “How long is the return period of the rainfall event on September 21, 2010?” Mag. Korea Water Resour. Assoc., 44(1), 67–72.
Yue, S. (1999). “Applying bivariate normal distribution to flood frequency analysis.” Water Int., 24(3), 248–254.
Yue, S. (2000a). “Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls.” Hydrol. Sci. J., 45(2), 315–326.
Yue, S. (2000b). “The Gumbel logistic model for representing a multivariate rainfall event.” Adv. Water Resour., 24(2), 179–185.
Yue, S. (2001a). “A bivariate gamma distribution for use in multivariate flood frequency analysis.” Hydrol. Process., 15(6), 1033–1045.
Yue, S. (2001b). “A statistical measure of severity of El Niňo events.” Stochastic Environ. Res. Risk Assess., 15(2), 153–172.
Yue, S., Ouarda, T. B. M. J., Bobée, B., Gegendre, P., and Bruneau, P. (1999). “The Gumbel mixed model.” J. Hydrol., 226(1–2), 88–100.
Yue, S., and Rasmussen, P. (2002). “Bivariate frequency analysis: Discussion of some useful concepts in hydrological application.” Hydrol. Process., 16(14), 2881–2898.
Yue, S., and Wang, C. Y. (2004). “A comparison of two bivariate extreme value distribution.” Stochastic Environ. Res. Risk Assess., 18(2), 61–66.
Zhang, L., and Singh, V. P. (2006). “Bivariate flood frequency analysis using the copula method.” J. Hydrol. Eng., 150–164.
Zhang, L., and Singh, V. P. (2007). “Bivariate rainfall frequency distributions using Archimedean copulas.” J. Hydrol., 332(1–2), 93–109.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 19 • Issue 6 • June 2014
Pages: 1080 - 1088
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Aug 1, 2012
Accepted: Aug 2, 2013
Published online: Aug 5, 2013
Discussion open until: Jan 5, 2014
Published in print: Jun 1, 2014
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.