A Vine-Copula Model for Time Series of Significant Wave Heights and Mean Zero-Crossing Periods in the North Sea
Publication: ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 3, Issue 4
Abstract
Stochastic descriptions and simulations of oceanographic variables are essential for coastal and marine engineering applications. In the past decade, copula-based approaches have become increasingly popular for estimating the multivariate distribution of some variables at the peak of a storm along with its duration. The modeling of the storm shape, which contributes to its impact, is often simplified. This article proposes a vine-copula approach to characterize hourly significant wave heights and corresponding mean zero-crossing periods as a random process in time. The model is applied to a data set in the North Sea, and time series with the duration of an oceanographic winter are simulated. The synthetic wave scenarios emulate storms as well as daily conditions. The results are useful, for example, as input for coastal risk analyses and for planning offshore operations. Nonetheless, selecting a vine structure, finding appropriate copula families, and estimating parameters is not straightforward. The validity of the model, as well as the conclusions that can be drawn from it, are sensitive to these choices. A valuable byproduct of the proposed vine-copula approach is the bivariate distribution of significant wave heights and corresponding mean zero-crossing periods at the given location. Its dependence structure is approximated by the flexible skew-t copula family and preserves the limiting wave steepness condition.
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Acknowledgments
The authors thank the two anonymous reviewers whose comments and suggestions helped to improve the manuscript. This work was supported by the European Community’s 7th Framework Programme through a grant to RISC-KIT (“Resilience-increasing Strategies for Coasts—Toolkit”), contract 603458.
References
Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009). “Pair-copula constructions of multiple dependence.” Insurance Math. Econ., 44(2), 182–198.
Acar, E. F., Genest, C., and NešLehová, J. (2012). “Beyond simplified pair-copula constructions.” J. Multivariate Anal., 110, 74–90.
Azzalini, A., and Capitanio, A. (2003). “Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution.” J. Royal Stat. Soc. Series B, 65(2), 367–389.
Bedford, T., and Cooke, R. M. (2002). “Vines: A new graphical model for dependent random variables.” Ann. Stat., 30(4), 1031–1068.
Boccotti, P. (2000). Wave mechanics for ocean engineering, Elsevier Science, Amsterdam, Netherlands.
Brechmann, E. C., and Czado, C. (2015). “Copar-multivariate time series modeling using the copula autoregressive model.” Appl. Stochastic Models Bus. Ind., 31(4), 495–514.
Callaghan, D., Nielsen, P., Short, A., and Ranasinghe, R. (2008). “Statistical simulation of wave climate and extreme beach erosion.” Coastal Eng., 55(5), 375–390.
Cooke, R. M. (1997). “Markov and entropy properties of tree-and vine-dependent variables.” Proc., ASA Section of Bayesian Statistical Science, Vol. 27, American Statistical Association, Alexandria, VA.
Corbella, S., and Stretch, D. D. (2012). “Predicting coastal erosion trends using non-stationary statistics and process-based models.” Coastal Eng., 70, 40–49.
Corbella, S., and Stretch, D. D. (2013). “Simulating a multivariate sea storm using archimedean copulas.” Coastal Eng., 76, 68–78.
De Michele, C., Salvadori, G., Passoni, G., and Vezzoli, R. (2007). “A multivariate model of sea storms using copulas.” Coastal Eng., 54(10), 734–751.
Dissmann, J., Brechmann, E. C., Czado, C., and Kurowicka, D. (2013). “Selecting and estimating regular vine copulae and application to financial returns.” Comput. Stat. Data Anal., 59, 52–69.
Faltinsen, O. (1993). Sea loads on ships and offshore structures, Vol. 1, Cambridge University Press, Cambridge, U.K.
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.
Gouldby, B., Méndez, F., Guanche, Y., Rueda, A., and Mìnguez, R. (2014). “A methodology for deriving extreme nearshore sea conditions for structural design and flood risk analysis.” Coastal Eng., 88, 15–26.
Haff, I. H., Aas, K., and Frigessi, A. (2010). “On the simplified pair-copula construction-simply useful or too simplistic?.” J. Multivariate Anal., 101(5), 1296–1310.
Hawkes, P. J., Gouldby, B. P., Tawn, J. A., and Owen, M. W. (2002). “The joint probability of waves and water levels in coastal engineering design.” J. Hydraul. Res., 40(3), 241–251.
Holthuijsen, L. H. (2010). Waves in oceanic and coastal waters, Cambridge University Press, Cambridge, U.K.
Jäger, W., and Morales Nápoles, O. (2015). “Sampling joint time series of significant wave heights and periods in the north sea.” Safety and reliability of complex engineered systems, L. Podofillini, B. Sudret, B. Stojadinović, E. Zio, and W. Kröger, eds., CRC Press, Boca Raton, FL.
Joe, H. (1996). “Families of m-variate distributions with given margins and m bivariate dependence parameters.” Lect. Notes Monogr. Ser., 28, 120–141.
Joe, H. (2014). Dependence modeling with copulas, CRC Press, Boca Raton, FL.
Joe, H., and Krupskii, P. (2014). CopulaModel: Dependence modeling with copulas, CRC Press, Boca Raton, FL.
Karunarathna, H., Pender, D., Ranasinghe, R., Short, A. D., and Reeve, D. E. (2014). “The effects of storm clustering on beach profile variability.” Marine Geol., 348, 103–112.
Kurowicka, D., and Cooke, R. M. (2006). Uncertainty analysis with high dimensional dependence modeling, Wiley, New York.
Leontaris, G., Morales-Nápoles, O., and Wolfert, A. R. (2016). “Probabilistic scheduling of offshore operations using copula based environmental time series-an application for cable installation management for offshore wind farms.” Ocean Eng., 125, 328–341.
Li, F., van Gelder, P., Vrijling, J., Callaghan, D., Jongejan, R., and Ranasinghe, R. (2014a). “Probabilistic estimation of coastal dune erosion and recession by statistical simulation of storm events.” Appl. Ocean Res., 47, 53–62.
Li, F., Van Gelder, P., Ranasinghe, R., Callaghan, D., and Jongejan, R. (2014b). “Probabilistic modelling of extreme storms along the dutch coast.” Coastal Eng., 86, 1–13.
Lin-Ye, J., Garcia-Leon, M., Gracia, V., and Sanchez-Arcilla, A. (2016). “A multivariate statistical model of extreme events: An application to the catalan coast.” Coastal Eng., 117, 138–156.
Masina, M., Lamberti, A., and Archetti, R. (2015). “Coastal flooding: A copula based approach for estimating the joint probability of water levels and waves.” Coastal Eng., 97, 37–52.
Montes-Iturrizaga, R., and Heredia-Zavoni, E. (2016). “Multivariate environmental contours using c-vine copulas.” Ocean Eng., 118, 68–82.
Morales-Nápoles, O. (2010). “Counting vines.” Dependence modeling: Vine copula handbook, World Scientific, Singapore, 189–218.
Morales-Nápoles, O., Cooke, R. M., and Kurowicka, D. (2010). “About the number of vines and regular vines on n nodes.” Delft Univ. of Technology, Delft, Netherlands.
Nai Ruscone, M., and Osmetti, S. A. (2017). “Modelling the dependence in multivariate longitudinal data by pair copula decomposition.” Soft methods data science, Springer, Berlin, 373–380.
Nelsen, R. B. (2013). An introduction to copulas, Vol. 139, Springer Science & Business Media, Berlin.
Poelhekke, L., Jäger, W. S., van Dongeren, A., Plomaritis, T. A., McCall, R., and Ferreira, Ó. (2016). “Predicting coastal hazards for sandy coasts with a bayesian network.” Coastal Eng., 118, 21–34.
Repko, A., Van Gelder, P., Voortman, H., and Vrijling, J. (2004). “Bivariate description of offshore wave conditions with physics-based extreme value statistics.” Appl. Ocean Res., 26(3), 162–170.
Rueda, A., Camus, P., Tomàs, A., Vitousek, S., and Méndez, F. (2016). “A multivariate extreme wave and storm surge climate emulator based on weather patterns.” Ocean Modell., 104, 242–251.
Salvadori, G., Tomasicchio, G., and D’Alessandro, F. (2014). “Practical guidelines for multivariate analysis and design in coastal and off-shore engineering.” Coastal Eng., 88, 1–14.
Schepsmeier, U., Stoeber, J., Brechmann, E. C., Graeler, B., Nagler, T., and Erhardt, T. (2015). VineCopula: Statistical inference of vine copulas. CRC Press, Boca Raton, FL.
Serafin, K. A., and Ruggiero, P. (2014). “Simulating extreme total water levels using a time- dependent, extreme value approach.” J. Geophys. Res. Oceans, 119(9), 6305–6329.
Serinaldi, F., and Grimaldi, S. (2007). “Fully nested 3-copula: Procedure and application on hydrological data.” J. Hydrol. Eng., 420–430.
Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges, Publications de l’Institut Statistique de Université de Paris 8, Paris.
Smith, M., Min, A., Almeida, C., and Czado, C. (2010). “Modeling longitudinal data using a pair-copula decomposition of serial dependence.” J. Am. Stat. Assoc., 105(492), 1467–1479.
Smith, M. S. (2015). “Copula modelling of dependence in multivariate time series.” Int. J. Forecast., 31(3), 815–833.
Tawn, J. A. (1988). “Bivariate extreme value theory: Models and estimation.” Biometrika, 75(3), 397–415.
Vanem, E. (2016). “Joint statistical models for significant wave height and wave period in a changing climate.” Mar. Struct., 49, 180–205.
Van Gent, M., van Thiel de Vries, J., Coeveld, E., De Vroeg, J., and Van de Graaff, J. (2008). “Large-scale dune erosion tests to study the influence of wave periods.” Coastal Eng., 55(12), 1041–1051.
Wahl, T., Mudersbach, C., and Jensen, J. (2011). “Assessing the hydrodynamic boundary conditions for risk analyses in coastal areas: A stochastic storm surge model.” Nat. Hazards Earth Syst. Sci., 11(11), 2925–2939.
Wahl, T., Mudersbach, C., and Jensen, J. (2012). “Assessing the hydrodynamic boundary conditions for risk analyses in coastal areas: A multivariate statistical approach based on copula functions.” Nat. Hazards Earth Syst. Sci., 12(2), 495–510.
Wahl, T., Plant, N. G., and Long, J. W. (2016). “Probabilistic assessment of erosion and flooding risk in the northern gulf of mexico.” J. Geophys. Res. Oceans, 121(5), 3029–3043.
Yoshiba, T. (2015). Maximum likelihood estimation of skew t-copulas, with its applications to stock returns, The Institute of Statistical Mathematics Research Memorandum, Tokyo.
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Published In
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 3 • Issue 4 • December 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Sep 20, 2016
Accepted: Mar 10, 2017
Published online: Jun 14, 2017
Discussion open until: Nov 14, 2017
Published in print: Dec 1, 2017
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