Multivariate Modeling of Annual Instantaneous Maximum Flows Using Copulas
This article has a reply.
VIEW THE REPLYPublication: Journal of Hydrologic Engineering
Volume 23, Issue 3
Abstract
Although copula functions have been successively applied to flood frequency analysis, their application has usually been restricted to modeling bivariate dependence. This is because higher-dimensional expressions are only available for a few copula families, which may not have enough flexibility in modeling complex dependence structures. However, vine copulas, which have been recently introduced as an innovative method, can overcome such limitations. In this study, vine copulas are used for multivariate modeling of annual instantaneous maximum flows of three main tributaries located in the Euphrates River Basin, which is one of the most important sources of water for Turkey. The performance of vine copulas was compared with commonly used Archimedean (Clayton, Frank, and Gumbel-Hougaard) and elliptical (Gaussian and Student’s ) copulas. Statistical tests and tail-dependence assessments found the vine copulas to be most suitable for describing the dependence structure between variables. The developed vine copulas were used to obtain joint and conditional return periods of maximum flows, which can be useful for hydrologic design and management of water resources structures in the basin.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This study was partly supported by the Scientific and Technological Research Council of Turkey (Project No. 115Y673). The authors sincerely thank the General Directorate of State Hydraulic Works, Turkey, for the providing the annual instantaneous maximum flows used in the study. The authors would also to express their gratitude to the editor and reviewers for providing constructive and insightful comments whose inclusion in the revision has led to an improved manuscript.
References
Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009). “Pair-copula constructions of multiple dependence.” Insur. Math. Econ., 44(2), 182–198.
Abdi, A., Hassanzadeh, Y., Talatahari, S., Fakheri-Fard, A., and Mirabbasi, R. (2017). “Parameter estimation of copula functions using an optimization-based method.” Theor. Appl. Climatol., 129(1–2), 21–32.
Adamson, P. T., Metcalfe, A. V., and Parmentier, B. (1999). “Bivariate extreme value distributions: An application of the Gibbs sampler to the analysis of floods.” Water Resour. Res., 35(9), 2825–2832.
Aydogan, D., Kankal, M., and Onsoy, H. (2016). “Regional flood frequency analysis for Coruh Basin of Turkey with l-moments approach.” J. Flood Risk Manage., 9(1), 69–86.
Bedford, T., and Cooke, R. M. (2001). “Probability density decomposition for conditionally dependent random variables modeled by vines.” Ann. Math. Artif. Intel., 32(1–4), 245–268.
Bedford, T., and Cooke, R. M. (2002). “Vines—A new graphical model for dependent random variables.” Ann. Stat., 30(4), 1031–1068.
Bezak, N., Brilly, M., and Sraj, M. (2016). “Flood frequency analyses, statistical trends and seasonality analyses of discharge data: A case study of the Litija station on the Sava River.” J. Flood Risk Manage., 9(2), 154–168.
Brechmann, E. C., and Schepsmeier, U. (2013). “Modeling dependence with c- and d-vine copulas: The R package Cdvine.” J. Stat. Software, 52(3), 1–27.
Can, I., and Tosunoglu, F. (2013). “Estimating T-year flood confidence intervals of rivers in Coruh Basin, Turkey.” J. Flood Risk Manage., 6(3), 186–196.
Chen, L., Singh, V. P., Guo, S. L., Mishra, A. K., and Guo, J. (2013). “Drought analysis using copulas.” J. Hydrol. Eng., 797–808.
Chowdhary, H., Escobar, L. A., and Singh, V. P. (2011). “Identification of suitable copulas for bivariate frequency analysis of flood peak and flood volume data.” Hydrol. Res., 42(2–3), 193–216.
Cigizoglu, H. K., Bayazit, M., and Onoz, B. (2005). “Trends in the maximum, mean, and low flows of Turkish rivers.” J. Hydrometeorol., 6(3), 280–290.
Daneshkhah, A., Remesan, R., Chatrabgoun, O., and Holman, I. P. (2016). “Probabilistic modeling of flood characterizations with parametric and minimum information pair-copula model.” J. Hydrol., 540(9), 469–487.
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.
Durante, F., and Salvadori, G. (2010). “On the construction of multivariate extreme value models via copulas.” Environmetrics, 21(2), 143–161.
Favre, A. C., El Adlouni, S., Perreault, L., Thiemonge, N., and Bobee, B. (2004). “Multivariate hydrological frequency analysis using copulas.” Water Resour. Res., 40(1), W01101.
Ferro, V., and Porto, P. (2006). “Flood frequency analysis for Sicily, Italy.” J. Hydrol. Eng., 110–122.
Fisher, N. I., and Switzer, P. (1985). “Chi-plots for assessing dependence.” Biometrika, 72(2), 253–265.
Fisher, N. I., and Switzer, P. (2001). “Graphical assessment of dependence: Is a picture worth 100 tests?” Am. Stat., 55(3), 233–239.
Frahm, G., Junker, M., and Schmidt, R. (2005). “Estimating the tail-dependence coefficient: Properties and pitfalls.” Insur. Math. Econ., 37(1), 80–100.
Ganguli, P., and Reddy, M. J. (2013). “Probabilistic assessment of flood risks using trivariate copulas.” Theor. Appl. Climatol., 111(1–2), 341–360.
Genest, C., and Boies, J. C. (2003). “Detecting dependence with Kendall plots.” Am. Stat., 57(4), 275–284.
Genest, C., and Favre, A. C. (2007). “Everything you always wanted to know about copula modeling but were afraid to ask.” J. Hydrol. Eng., 347–368.
Genest, C., and Remillard, B. (2008). “Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 44(6), 1096–1127.
Genest, C., Remillard, B., and Beaudoin, D. (2009). “Goodness-of-fit tests for copulas: A review and a power study.” Insur. Math. Econ., 44(2), 199–213.
Graler, B., et al. (2013). “Multivariate return periods in hydrology: A critical and practical review focusing on synthetic design hydrograph estimation.” Hydrol. Earth Syst. Sci., 17(4), 1281–1296.
Griffis, V. W., and Stedinger, J. R. (2007). “Log-Pearson Type 3 distribution and its application in flood frequency analysis. I: Distribution characteristics.” J. Hydrol. Eng., 482–491.
Grimaldi, S., and Serinaldi, F. (2006). “Asymmetric copula in multivariate flood frequency analysis.” Adv. Water Resour., 29(8), 1155–1167.
Haddad, K., and Rahman, A. (2011). “Selection of the best fit flood frequency distribution and parameter estimation procedure: A case study for Tasmania in Australia.” Stoch. Environ. Res. Risk A., 25(3), 415–428.
Haktanir, T., and Horlacher, H. B. (1993). “Evaluation of various distributions for flood frequency-analysis.” Hydrol. Sci. J., 38(1), 15–32.
Joe, H. (1997). Multivariate models and dependence concepts, Chapman & Hall, London.
Kahya, E., and Kalayci, S. (2004). “Trend analysis of streamflow in Turkey.” J. Hydrol., 289(1–4), 128–144.
Karmakar, S., and Simonovic, S. P. (2009). “Bivariate flood frequency analysis. 2: A copula-based approach with mixed marginal distributions.” J. Flood Risk Manage., 2(1), 32–44.
Khedun, C. P., Chowdhary, H., Mishra, A. K., Giardino, J. R., and Singh, V. P. (2013). “Water deficit duration and severity analysis based on runoff derived from Noah land surface model.” J. Hydrol. Eng., 817–833.
Khedun, C. P., Mishra, A. K., Singh, V. P., and Giardino, J. R. (2014). “A copula-based precipitation forecasting model: Investigating the interdecadal modulation of Enso’s impacts on monthly precipitation.” Water Resour. Res., 50(1), 580–600.
Klein, B., Schumann, A. H., and Pahlow, M. (2011). “Copulas-New risk assessment methodology for dam safety.” Flood risk assessment and management, Springer, Dordrecht, Netherlands, 149–185.
Kojadinovic, I., Yan, J., and Holmes, M. (2011). “Fast large-sample goodness-of-fit tests for copulas.” Stat. Sinica., 21(2), 841–871.
Krstanovic, P. F., and Singh, V. P. (1987). “A multivariate stochastic flood analysis using entropy.” Hydrologic frequency modeling, V. P. Singh, ed., Springer, Dordrecht, Netherlands.
Laio, F., Di Baldassarre, G., and Montanari, A. (2009). “Model selection techniques for the frequency analysis of hydrological extremes.” Water Resour. Res., 45(7), W07416.
MATLAB [Computer software]. MathWorks, Natick, MA.
Mirabbasi, R., Fakheri-Fard, A., and Dinpashoh, Y. (2012). “Bivariate drought frequency analysis using the copula method.” Theor. Appl. Climatol., 108(1–2), 191–206.
Nadarajah, S., and Shiau, J. T. (2005). “Analysis of extreme flood events for the Pachang River, Taiwan.” Water Resour. Manage., 19(4), 363–374.
Nelsen, R. B. (2006). An introduction to copulas, Springer, New York.
Nikoloulopoulos, A. K., Joe, H., and Li, H. J. (2012). “Vine copulas with asymmetric tail dependence and applications to financial return data.” Comput. Stat. Data Anal., 56(11), 3659–3673.
Onoz, B., and Bayazit, M. (2003). “The power of statistical tests for trend detection.” Turkish J. Eng. Environ. Sci., 27(4), 247–251.
R [Computer software]. R Foundation for Statistical Computing, Vienna, Austria.
Rajsekhar, D., Singh, V. P., and Mishra, A. K. (2015). “Hydrologic drought atlas for Texas.” J. Hydrol. Eng., 05014023.
Requena, A. I., Mediero, L., and Garrote, L. (2013). “A bivariate return period based on copulas for hydrologic dam design: Accounting for reservoir routing in risk estimation.” Hydrol. Earth Syst. Sci., 17(8), 3023–3038.
Saf, B. (2009). “Regional flood frequency analysis using L-moments for the west Mediterranean region of Turkey.” Water Resour. Manage., 23(3), 531–551.
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.
Salvadori, G., Michele, D. C., Kottegoda, N. T., and Rosso, R. (2007). Extremes in nature-an approach using copulas, Springer, Dordrecht, Netherlands.
Seckin, N., Yurtal, R., Haktanir, T., and Topaloglu, F. (2010). “Regional flood frequency analysis of Ceyhan River Basin in Turkey using L-moments method.” Fresen. Environ. Bull., 19(11A), 2616–2624.
Sen, O., and Kahya, E. (2017). “Determination of flood risk: A case study in the rainiest city of Turkey.” Environ. Modell. Software, 93(3), 296–309.
Serinaldi, F., and Grimaldi, S. (2007). “Fully nested 3-copula: Procedure and application on hydrological data.” J. Hydrol. Eng., 420–430.
Shafaei, M., Fard, A. F., Dinpashoh, Y., Mirabbasi, R., and Michele, D. C. (2017). “Modeling flood event characteristics using d-vine structures.” Theor. Appl. Climatol., 130(3–4), 713–724.
Shiau, J. T. (2003). “Return period of bivariate distributed extreme hydrological events.” Stoch. Environ. Res. Risk A, 17(1–2), 42–57.
Singh, K., and Singh, V. P. (1991). “Derivation of bivariate probability density-functions with exponential marginals.” Stoch. Hydrol. Hydraul., 5(1), 55–68.
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges, Institut de Statistiques de l’Université de Paris, Paris.
Sraj, M., Bezak, N., and Brilly, M. (2015). “Bivariate flood frequency analysis using the copula function: A case study of the Litija station on the Sava River.” Hydrol. Process., 29(2), 225–238.
Tobias, A., Saez, M., and Galan, I. (2007). “Application of the chi-plot to assess for dependence in environmental epidemiology studies.” Environ. Ecol. Stat., 14(2), 181–189.
Tosunoglu, F., and Kisi, O. (2016). “Joint modelling of annual maximum drought severity and corresponding duration.” J. Hydrol., 543(11), 406–422.
Vernieuwe, H., Vandenberghe, S., De Baets, B., and Verhoest, N. E. C. (2015). “A continuous rainfall model based on vine copulas.” Hydrol. Earth Syst. Sci., 19(6), 2685–2699.
Wilks, S. D. (2006). Statistical methods in the atmospheric sciences, Elsevier, Amsterdam, Netherlands.
Yenigun, K., Bilgehan, M., Gerger, R., and Mutlu, M. (2010). “A comparative study on prediction of sediment yield in the Euphrates Basin.” Int. J. Phys. Sci., 5(5), 518–534.
Yilmaz, A. G., and Muttil, N. (2014). “Runoff estimation by machine learning methods and application to the Euphrates Basin in Turkey.” J. Hydrol. Eng., 1015–1025.
Yue, S. (1999). “Applying the bivariate normal distribution to flood frequency analysis.” Water Int., 24(3), 248–254.
Yue, S. (2000). “The bivariate lognormal distribution to model a multivariate flood episode.” Hydrol. Process., 14(14), 2575–2588.
Yue, S. (2001). “A bivariate gamma distribution for use in multivariate flood frequency analysis.” Hydrol. Process., 15(6), 1033–1045.
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). “Trivariate flood frequency analysis using the Gumbel–Hougaard copula.” J. Hydrol. Eng., 431–439.
Zhang, L., and Singh, V. P. (2014). “Trivariate flood frequency analysis using discharge time series with possible different lengths: Cuyahoga River case study.” J. Hydrol. Eng., 05014012.
Information & Authors
Information
Published In
Copyright
©2018 American Society of Civil Engineers.
History
Received: Apr 17, 2017
Accepted: Oct 26, 2017
Published online: Jan 12, 2018
Published in print: Mar 1, 2018
Discussion open until: Jun 12, 2018
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.