Trivariate Flood Frequency Analysis Using Discharge Time Series with Possible Different Lengths: Cuyahoga River Case Study
Publication: Journal of Hydrologic Engineering
Volume 19, Issue 10
Abstract
A common approach to multivariate flood frequency analysis is to apply discharge time series records with the same length (i.e., one upstream and one downstream discharge gauge in the same river for bivariate flood frequency analysis, and multivariate flood frequency analysis at the confluence of river systems). However, in reality the gauged discharge time series records may have different lengths due to different activation times of the gauges. In addition, there exists one common assumption for flood frequency analysis, that is, the discharge time series may be considered as a stationary signal. However, due to land-use and land-cover (LULC) and climate changes, the stationary assumption may need to be justified. To answer the above questions, this paper investigates (1) the full-length discharge record at each discharge gauge; (2) the dependence structure of bivariate and multivariate discharge time series with different lengths using the copula theory; (3) employment of the vine copula for multivariate flood frequency analysis (i.e., ); and (4) the validation of the proposed method and comparison of its performance with asymmetric, symmetric Archimedean, and meta-elliptical copulas using the discharge time series from the Cuyahoga River basin, Ohio, as a case study.
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References
Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009). “Pair-copula constructions of multiple dependence.” Insur. Math. Econ., 44(2), 182–198.
Bárdossy, A. (2006). “Copula-based geostatistical models for groundwater quality parameters.” Water Resour. Res., 42(11), W11416.
Bárdossy, A., and Li, J. (2008). “Geostatistical interpolation using copulas.” Water Resour. Res., 44(7), W07412.
Bartels, R. (1982). “The rank version of von Neumann’s ratio test for randomness.” J. Am. Stat. Assoc., 77(377), 40–46.
Beighley, R. E., and Moglen, E. (2002). “Trend assessment in rainfall-runoff behavior in urbanizing watersheds.” J. Hydrol. Eng., 27–34.
Beighley, R. E., and Moglen, G. E. (2003). “Adjusting measured peak discharges for an urbanizing watershed to reflect a stationary land use signal.” Water Resour. Res., 39(4), 27–34.
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (2008). Time series analysis: Forecasting and control, 4th Ed., Wiley, Hoboken, NJ.
Chen, Y. Q., Zhang, Q., Xiao, M. Z., and Singh, V. P. (2013). “Evaluation of risk of hydrological droughts by the trivariate Plackett copula in the East River basin (China).” Nat. Hazards, 68(2), 529–547.
Cunnane, C. (1987). “Review of statistical models for flood frequency estimation.” Hydrologic frequency modeling, V. P. Singh, ed., Reidel, Dordrecht, 49–95.
Czado, C. (2010). “Pair-copula construction of multivariate copulas.” Copula theory and its applications, lecture notes in statistics 198, P. Jawoski, et al., eds., Springer, New York, NY.
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.
Evin, G., and Favre, A.-C. (2008). “A new rainfall model based on the Neyman-Scott process using cubic copulas.” Water Resour. Res., 44(3), W03433.
Favre, A.-C., Adlouni, S. E., Perreault, L., Thiémonge, N., and Bobée, B. (2004). “Multivariate hydrological frequency analysis using copulas.” Water Resour. Res., 40(1), W01101.
Genest, C., Favre, A.-C., Béliveau, J., and Jacques, C. (2007a). “Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data.” Water Resour. Res., 43(9), W09401.
Genest, C., and Mackay, L. (1986). “The joy of copulas: Bivariate distributions with uniform marginals.” Am. Stat., 40(4), 280–283.
Genest, C., Quessy, J. -F., and Rémillard, B. (2006). “Goodness-of-fit procedures for copula models based on the probability integral transformation.” Scand. J. Stat., 33(2), 337–366,.
Genest, C., Rémillard, B., and Beaudoin, D. (2009). “Goodness-of-fit tests for copulas: A review and a power study.” Insur. Math. Econ., 44(2), 199–213.
Genest, C., and Rivest, L. (1993). “Statistical inference procedures for bivariate Archimedean copulas.” J. Am. Stat. Assoc., 88(423), 1034–1043.
Gyasi-Agyei, Y. (2012). “Use the observed scaled daily storm profiles in a copula based rainfall disaggregation model.” Adv. Water Resour., 45, 26–36.
Hipel, K. W., and McLeod, A. I. (1994). Time series modeling of water resources and environmental systems, Elsevier Sciences, Amsterdam, Netherlands.
Joe, H. (1997). Multivariate models and dependence concepts—Monographs on statistics and applied probability, Chapman and Hall, London.
Kao, S.-H., and Govindaraju, R. S. (2007). “A bivariate frequency analysis of extreme rainfall with implications for design.” J. Geophys. Res., 112(D13), D13119.
Kao, S.-H., and Govindaraju, R. S. (2008). “Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas.” Water Resour. Res., 44(2), W02415.
Kite, G. W. (1978). Frequency and risk analysis in hydrology, Water Resources Publications, Fort Collins, CO.
Kurowicka, D., and Cooke, R. (2006). Uncertainty analysis: With high dimensional dependence modeling, Wiley, England.
Laux, P., Vogl, S., Qiu, W., Knoche, H. R., and Kunstmann, H. (2011). “Copula-based statistical refinement of precipitation in RCM simulations over complex terrain.” Hydrol. Earth Syst. Sci., 15, 2401–2419.
Li, Q., Sun, T. R., and Chen, Y. (2007). “Great Salt Lake surface level forecasting using RIARCH modeling.” Proc., ASME 2007 Int. Design Engineering Technical Conf. & Computers and Information in Engineering Conf., Las Vegas, NV.
Mann, H. B., and Whitney, D. R. (1947). “On a test of whether one of two random variables is stochastically larger than the other.” Ann. Math. Stat., 18(1), 50–60.
McCuen, R. H. (1998). Hydrologic analysis and design, 2nd Ed., Prentice-Hall, Old Tappan, NJ.
Moglen, G. E., and Shivers, D. E. (2006). “Methods for adjusting U.S. geological survey rural regression peak discharges in an urban setting.”, U.S. Dept. of the Interior, U.S. Geological Survey, Reston, VA.
Nelsen, R. B. (2006). An introduction to copula, 2nd Ed., Springer, New York.
Patton, A. J. (2006). “Estimation of multivariate models for time series of possibly different lengths.” J. Appl. Econ., 21, 147–173.
Rao, A. R., and Hamed, K. H. (2000). Flood frequency analysis, CRC, Boca Raton, FL.
Robson, A. J., Jones, T. K., Reed, D. W., and Bayliss, A. C. (1998). “A study of national trend and variation in UK floods.” Int. J. Climatol., 18(2), 165–182.
Rosenblatt, M. (1952). “Remarks on a multivariate transformation.” Ann. Math. Stat., 23(3), 470–472.
Salas, J. D. (1993). “Analysis and modeling of hydrologic time series.” Handbook of hydrology, D. R. Maidment, ed., McGraw-Hall, New York.
Salvadori, G., and De Michele, C. (2004). “Frequency analysis via copulas: Theoretical aspects and applications to hydrological events.” Water Resour. Res., 40(12), W12511.
Samaniego, L., Bárdossy, A., and Kumar, R. (2010). “Streamflow prediction in ungagged catchments using copula-based dissimilarity measures.” Water Resour. Res., 46(2), W02506.
Sankarasubramanian, A., Vogel, R. M., and Limburner, J. F. (2001). “Climate elasticity of streamflow in the United States.” Water Resour. Res., 37(6), 1771–1781.
Serinaldi, F. (2009). “A multisite daily rainfall generator driven by bivariate copula-based mixed distributions.” J. Geophys. Res., 114(D10), D10103.
Serinaldi, F., and Grimaldi, S. (2007). “Fully nested 3-copula: Procedure and application on hydrological data.” J. Hydrol. Eng., 420–430.
Sklar, A. (1959). “Fonctions de repartition à n dimensions et leurs marges.” Pub. Inst. Stat. Univ. Paris, 8, 229–231.
Solari, S., and Losada, M. A. (2011). “Non-stationary wave height climate modeling and simulation.” J. Geophys. Res., 116(C9), C09032.
Song, S., and Singh, V. P. (2010). “Frequency analysis of drought using the Plackett copula and parameter estimation by genetic algorithm.” Stochastic Environ. Res. Risk Assess., 24(5), 783–805.
U.S. Water Resources Council. (1981). “Guidelines for determining flood flow frequency.” Bulletin 17B (revised), Hydrology Committee, Washington, DC.
Vandenberghe, S., Verhoest, N. E. C., Onof, C., and De Baets, B. (2011). “A comparative copula-based bivariate frequency analysis of observed and simulated storm events: A case study on Bartlett-Lewis modeled rainfall.” Water Resour. Res., 47(7), W07529.
Wald, A., and Wolfowitz, J. (1940). “On a test whether two samples are from the same population.” Ann. Math Stat., 11(2), 147–162.
Wang, C., Chang, N.-B., and Yeh, G.-T. (2009). “Copula-based flood frequency (COEF) analysis at the confluences of river systems.” Hydrol. Processes, 23, 1471–1486.
Wang, W. P., Van Gelder, P. H. J. M., Vrijling, J. K., and Ma, J. (2005). “Testing and modeling autoregressive conditional heteroskedasticity of streamflow processes.” Nonlinear Processes Geophys., 12, 55–66,.
Widder, D. V. (1941). The Laplace transform, Princeton University Press, Princeton, NJ.
Zhang, H., Zhang, X. L., Zhu, R. R., Liu, C. M., Sato, Y., and Furushima, Y. (2009). “Response of streamflow to climate and land surface change in the headwaters of the Yellow River basin.” Water Resour. Res., 45(7), W00A19.
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. (2007a). “Bivariate rainfall frequency analysis using Archimedean copulas.” J. Hydrol., 332(1–2), 93–109.
Zhang, L., and Singh, V. P. (2007b). “Gumbel-Hougaard copula for trivariate rainfall frequency analysis.” J. Hydrol. Eng., 409–419.
Zhang, L., and Singh, V. P. (2007c). “Trivariate flood frequency analysis using the Gumbel-Hougaard copula.” J. Hydrol. Eng., 431–439.
Zhang, Q., Li, J. F., and Singh, V. P. (2012a). “Application of Archimedean copulas in the analysis of the precipitation extremes: Effects of precipitation changes.” Theor. Appl. Climatol., 107(1–2), 255–264.
Zhang, Q., Singh, V. P., Li, J. F., Jiang, F. Q., and Bai, Y. G. (2012b). “Spatio-temporal variations of precipitation extremes in Xinjiang, China.” J. Hydrol., 434–435(1–2), 255–264.
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Journal of Hydrologic Engineering
Volume 19 • Issue 10 • October 2014
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© 2014 American Society of Civil Engineers.
History
Received: Mar 23, 2013
Accepted: Mar 20, 2014
Published online: Mar 22, 2014
Published in print: Oct 1, 2014
Discussion open until: Nov 25, 2014
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