Abstract
Copula functions are often used for multivariate frequency analyses, but discharge and suspended sediment concentrations have not yet been modelled together with the use of 3-dimensional copula functions. One hydrological station from Slovenia and five stations from USA with watershed areas from 920 km2 to 24,996 km2 were used for trivariate frequency analyses of peak discharges, hydrograph volumes and suspended sediment concentrations. Different parametric marginal distributions were applied and parameters were estimated with the method of L-moments. Maximum pseudo-likelihood method was used for copula parameters estimation. With the use of statistical and graphical tests we selected the most appropriate copula model. Symmetric and asymmetric versions of Archimedean copulas were applied according to the dependence characteristics of the individual stations. We selected Gumbel-Hougaard copula as the most appropriate model for all discussed stations. Primary joint return periods OR and secondary Kendall’s return periods were calculated and comparison between selected copula functions was made. We can conclude that copula functions are useful mathematical tool, which can also be used for modelling variables that are presented in this paper.
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Balistrocchi M, Bacchi B (2011) Modelling the statistical dependence of rainfall event variables through copula functions. Hydrol Earth Syst Sci 15:1959–1977. doi:10.5194/hess-15-1959-2011
Bardossy A (2011) Interpolation of groundwater quality parameters with some values below the detection limit. Hydrol Earth Syst Sci 15:2763–2775. doi:10.5194/hess-15-2763-2011
Bardossy A, Li J (2008) Geostatistical interpolation using copulas. Water Resour Res 44. doi:10.1029/2007wr006115
Benkhaled A, Higgins H, Chebana F, Necir A (2013) Frequency analysis of annual maximum suspended sediment concentrations in Abiod wadi, Biskra (Algeria). Hydrol Process. doi:10.1002/hyp.9880
Bezak N, Brilly M, Sraj M (2013a) Comparison between the peaks over threshold method and the annual maximum method for flood frequency analyses. Hydrol Sci J. doi:10.1080/02626667.2013.831174
Bezak N, Sraj M, Mikos M (2013b) Overview of suspended sediments measurements in Slovenia and an example of data analysis. Gradbeni Vestnik 62:274–280 (In Slovene)
Bonacci O, Oskorus D (2010) The changes in the lower Drava River water level, discharge and suspended sediment regime. Environ Earth Sci 59:1661–1670. doi:10.1007/s12665-009-0148-8
Box GEP, Pierce DA (1970) Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. J Am Stat Assoc 65:1509–1526. doi:10.1080/01621459.1970.10481180
Burn DH (1997) Catchment similarity for regional flood frequency analysis using seasonality measures. J Hydrol 202:212–230. doi:10.1016/s0022-1694(97)00068-1
Chen L, Singh VP, Guo SL, Hao ZC, Li TY (2012) Flood coincidence risk analysis using multivariate copula functions. J Hydrol Eng 17:742–755. doi:10.1061/(asce)he.1943-5584.0000504
De Michele C, Salvadori G, Canossi M, Petaccia A, Rosso R (2005) Bivariate statistical approach to check adequacy of dam spillway. J Hydrol Eng 10:50–57. doi:10.1061/(asce)1084-0699(2005)10:1(50)
Favre AC, El Adlouni S, Perreault L, Thiemonge N, Bobee B (2004) Multivariate hydrological frequency analysis using copulas. Water Resour Res 40. doi:10.1029/2003wr002456
Fisher NI, Switzer P (1985) Chi-plots for assessing dependence. Biometrika 72:253–265. doi:10.1093/biomet/72.2.253
Fisher NI, Switzer P (2001) Graphical assessment of dependence: is a picture worth 100 tests? Am Stat 55:233–239. doi:10.1198/000313001317098248
Ganguli P, Reddy MJ (2012) Risk assessment of droughts in Gujarat using bivariate copulas. Water Resour Manag 26:3301–3327. doi:10.1007/s11269-012-0073-6
Genest C, Boies JC (2003) Detecting dependence with Kendall plots. Am Stat 57:275–284. doi:10.1198/0003130032431
Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12:347–368. doi:10.1061/(asce)1084-0699(2007)12:4(347)
Genest C, Remillard B (2008) Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann Instit Henri Poincare Probabilites Stat 44:1096–1127. doi:10.1214/07-aihp148
Genest C, Ghoudi K, Rivest LP (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82:543–552. doi:10.1093/biomet/82.3.543
Genest C, Remillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insur Math Econ 44:199–213. doi:10.1016/j.insmatheco.2007.10.005
Grimaldi S, Serinaldi F (2006) Asymmetric copula in multivariate flood frequency analysis. Adv Water Resour 29:1155–1167. doi:10.1016/j.advwatres.2005.09.005
Holtschlag DJ (2001) Optimal estimation of suspended-sediment concentrations in streams. Hydrol Process 15:1133–1155. doi:10.1002/hyp.207
Hosking JRM, Wallis JR (1997) Regional frequency analysis: an approach based on L-moments. Cambridge University Press, Cambridge
Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London
Kendall MG (1975) Multivariate analysis. Griffin, London
Koffler D, Laaha G (2012) LFSTAT- an R-package for low-flow analysis. EGU General Assembly, Vienna 22–27.4
Kojadinovic I, Yan J, Holmes M (2011) Fast large-sample goodness-of-fit tests for copulas. Stat Sin 21:841–871. doi:10.1007/s11222-009-9142-y
Ma MW, Song SB, Ren LL, Jiang SH, Song JL (2013) Multivariate drought characteristics using trivariate Gaussian and Student t copulas. Hydrol Process 27:1175–1190. doi:10.1002/hyp.8432
Nelsen RB (1999) An introduction to copulas. Springer, New York
Parajka J, Viglione A, Rogger M, Salinas JL, Sivapalan M, Bloschl G (2013) Comparative assessment of predictions in ungauged basins—part 1: runoff-hydrograph studies. Hydrol Earth Syst Sci 17:1783–1795. doi:10.5194/hess-17-1783-2013
Poulin A, Huard D, Favre AC, Pugin S (2007) Importance of tail dependence in bivariate frequency analysis. J Hydrol Eng 12:394–403. doi:10.1061/(asce)1084-0699(2007)12:4(394)
Reddy MJ, Ganguli P (2012) Bivariate flood frequency analysis of Upper Godavari River flows using Archimedean copulas. Water Resour Manag 26:3995–4018. doi:10.1007/s11269-012-0124-z
Rodríguez-Blanco ML, Taboada-Castro MM, Palleiro L, Taboada-Castro MT (2010) Temporal changes in suspended sediment transport in an Atlantic catchment, NW Spain. Geomorphology 123:181–188. doi:10.1016/j.geomorph.2010.07.015
Salvadori G, De Michele C (2004) Frequency analysis via copulas: Theoretical aspects and applications to hydrological events. Water Resour Res 40. doi:10.1029/2004wr003133
Salvadori G, De Michele C, Kottegoda NT, Rosso R (2007) Extremes in nature an approach using copulas. Springer, Dordrecht
Salvadori G, De Michele C, Durante F (2011) On the return period and design in a multivariate framework. Hydrol Earth Syst Sci 15:3293–3305. doi:10.5194/hess-15-3293-2011
Serinaldi F, Grimaldi S (2007) Fully nested 3-copula: procedure and application on hydrological data. J Hydrol Eng 12:420–430. doi:10.1061/(asce)1084-0699(2007)12:4(420)
Sraj M, Bezak N, Brilly M (2014) Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River. Hydrol Process. doi:10.1002/hyp.10145
Tena A, Batalla RJ, Vericat D, Lopez-Tarazon JA (2011) Suspended sediment dynamics in a large regulated river over a 10-year period (the lower Ebro, NE Iberian Peninsula). Geomorphology 125:73–84. doi:10.1016/j.geomorph.2010.07.029
Tramblay Y, St-Hilaire A, Ouarda T (2008) Frequency analysis of maximum annual suspended sediment concentrations in North America. Hydrol Sci J 53:236–252. doi:10.1623/hysj.53.1.236
Tramblay Y, Ouarda T, St-Hilaire A, Poulin J (2010) Regional estimation of extreme suspended sediment concentrations using watershed characteristics. J Hydrol 380:305–317. doi:10.1016/j.jhydrol.2009.11.006
Vandenberghe S, Verhoest NEC, Onof C, 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. doi:10.1029/2009wr008388
Wang C, Chang NB, Yeh GT (2009) Copula-based flood frequency (COFF) analysis at the confluences of river systems. Hydrol Process 23:1471–1486. doi:10.1002/hyp.7273
Wong G, Lambert MF, Leonard M, Metcalfe AV (2010) Drought analysis using trivariate copulas conditional on climatic states. J Hydrol Eng 15:129–141. doi:10.1061/(asce)he.1943-5584.0000169
Yusof F, Hui-Mean F, Suhaila J, Yusof Z (2013) Characterisation of drought properties with bivariate copula analysis. Water Resour Manag 27:4183–4207. doi:10.1007/s11269-013-0402-4
Zhang L, Singh VP (2006) Bivariate flood frequency analysis using the copula method. J Hydrol Eng 11:150–164. doi:10.1061/(asce)1084-0699(2006)11:2(150)
Zhang L, Singh VP (2007a) Bivariate rainfall frequency distributions using Archimedean copulas. J Hydrol 332:93–109. doi:10.1016/j.jhydrol.2006.06.033
Zhang L, Singh VP (2007b) Trivariate flood frequency analysis using the Gumbel-Hougaard copula. J Hydrol Eng 12:431–439. doi:10.1061/(asce)1084-0699(2007)12:4(431)
Acknowledgement
We wish to thank the Environmental Agency of the Republic of Slovenia (ARSO) for data provision. We would also like to express our thanks to the United States Geological Survey (USGS) for making the hydrological data available to the public on their web site. The results of the study are part of the Faculty of Civil and Geodetic engineering (UL FGG) work on the Slovenian national research project J2-4096 and on the international research project SedAlp, which is financed by the European Union through the Alpine Space program. The critical and useful comments of three anonymous reviewers and associate editor helped to improve this manuscript, for which the authors are very grateful.
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Bezak, N., Mikoš, M. & Šraj, M. Trivariate Frequency Analyses of Peak Discharge, Hydrograph Volume and Suspended Sediment Concentration Data Using Copulas. Water Resour Manage 28, 2195–2212 (2014). https://doi.org/10.1007/s11269-014-0606-2
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DOI: https://doi.org/10.1007/s11269-014-0606-2