Abstract
The analysis of joint probability distributions of rainfall characteristics such as severity and duration is important in water resources management. Deriving their distributions using standard statistical techniques are often problematical due to its complexity. Standard methods usually assume that the rainfall characteristics are independent or that their marginal distributions belong to the same family of distributions. The use of copulas based methodologies can circumvent these restrictions and are therefore increasingly popular. However, the copulas and marginal distributions that are commonly used belong to specific parametric families and their adoption could lead to spurious inferences if the underlying assumptions are violated. For this reason, we recommend a nonparametric or semiparametric approach to estimate the joint distribution of rainfall characteristics. In this paper, we introduce and compare several copula–based approaches, each involving a combination of parametric or nonparametric marginal distributions conjoined by a parametric or nonparametric copula. An empirical illustration of the different approaches using rainfall data collected from six stations in the state of Victoria, Australia, demonstrated that a nonparametric approach can often give better results than a purely parametric approach.
Similar content being viewed by others
References
Abdul Rauf U, Zeephongsekul P (2011) Modelling rainfall severity and duration in north-eastern victoria using copulas. Proceedings of the19th International Congress on Modelling and Simulation,Perth
Abdul Rauf UF, Zeephongsekul P (2014) Copula based analysis of rainfall severity and duration: a case study. Theor Appl Climatol 115(1-2):153–166
Brown BM, Chen SX (1999) Beta-bernstein smoothing for regression curves with compact support. Scand J Stat 26(1):47–59
Cancelliere A, Mauro G, Bonaccorso B, Rossi G (2007) Drought forecasting using the standardized precipitation index. Water Resour Manag 21(5):801–819. doi:10.1007/s11269-006-9062-y
Chappell A, Renzullo LJ, Raupach TH, Haylock M (2013) Evaluating geostatistical methods of blending satellite and gauge data to estimate near real-time daily rainfall for australia. J Hydrol 493(0):105–114
Charpentier A, Fermanian J, Scaillet O (2006) Copulas: from theory to application in finance, 1st edn, Risk Books, Torquay, UK, chap The Estimation of Copulas: Theory and Practice
Chen SX (1999) Beta kernel estimators for density functions. Comput Stat Data Anal 31(2):131–145
Chen SX (2000) Beta kernel smoothers for regression curves. Stat Sin 10:73–91
Chen SX, Huang TM (2007) Nonparametric estimation of copula functions for dependence modelling. Can J Stat 35(2):265–282
De Michele C, Salvadori G (2003) A generalized pareto intensity-duration model of storm rainfall exploiting 2-copulas. J Geophys Res Atmos 108 (D2). doi:10.1029/2002JD002534
Deheuvels P (1979) La fonction de dependance empirique et ses proprietes: un test non paramtrique d’independance. Acad Roy Bull Bull Cl Sci 65:274–292
Genest C, Favre A (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368
Genest C, Ghoudi K, Rivest LP (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3):543–552
Genest C, Rumillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: A review and a power study. Insur Math Econ 44(2):199–213
Guttman NB (1998) Comparing the palmer drought index and the standardized precipitation index1. JAWRA J Am Water Works Assoc 34(1):113–121
Haghighat jou P, Akhoond-Ali A, Nazemosadat M (2013) Nonparametric kernel estimation of annual precipitation over iran. Theor Appl Climatol 112(1-2):193–200. doi:10.1007/s00704-012-0727-6
Harrell FE, Davis CE (1982) A new distribution-free quantile estimator. Biometrika 69(3):635–640
Joe H (1997) Multivariate Models and Dependence Concepts. Chapman and Hall London
Kao SC, Govindaraju RS (2007) A bivariate frequency analysis of extreme rainfall with implications for design. J Geophys Res Atmos 112 (D13). doi:10.1029/2007JD008522
Kao SC, Govindaraju RS (2008) Trivariate statistical analysis of extreme rainfall events via the plackett family of copulas. Water Res Res:44
Kao SC, Govindaraju RS (2010) A copula-based joint deficit index for droughts. J Hydrol 380(12):121–134
Kim TW (2003) Nonparametric approach for estimating return periods of droughts in arid regions. J Hydrol Eng 8(5):237–246
Kim T W, JB Valds, Nijssen B, Roncayolo D (2006) Quantification of linkages between large-scale climatic patterns and precipitation in the colorado river basin. J Hydrol 321(14):173–186
McBride JL, Nicholls N (1983) Seasonal relationships between australian rainfall and the southern oscillation. Mon Weather Rev 3:1998–2004
McKee T, Doesken N, Kleist J (1993) The relationship of drought frequency and duration to time scales. 8th Conference on Applied Climatology, Anaheim, California
Mehrotra R, Sharma A (2011) Impact of atmospheric moisture in a rainfall downscaling framework for catchment-scale climate change impact assessment. Int J Climatol 31(3):431–450
Mirabbasi R, Fakheri-Fard A, Dinpashoh Y (2012) Bivariate drought frequency analysis using the copula method. Theor Appl Climatol 108:191–206
Nelsen R B (2006) Introduction to copulas, lecture notes statistics, vol 139, 2nd edn. Springer-Verlag
Nicholls N, Wong K K (1990) Dependence of rainfall variability on mean rainfall, latitude, and the southern oscillation. J Clim 3:163170
Pui A, Lal A, Sharma A (2011) How does the interdecadal pacific oscillation affect design floods in australia Water Resour Res 47 (5). doi:10.1029/2010WR009420
Reddy M, Ganguli P (2012) Bivariate flood frequency analysis of upper godavari river flows using archimedean copulas. Water Resour Manag 26(14):3995–4018. doi:10.1007/s11269-012-0124-z
Scaillet O, Fermanian J D (2002) Nonparametric estimation of copulas for time series. FAME Research Paper (57)
Serinaldi F, Bonaccorso B, Cancelliere A, Grimaldi S (2009) Probabilistic characterization of drought properties through copulas. Physics and Chemistry of the Earth. Parts A/B/C 34(10-12):596– 605
Sharma A (2000) Seasonal to interannual rainfall probabilistic forecasts for improved water supply management: Part 3 a nonparametric probabilistic forecast model. J Hydrol 239(14):249–258
Sharma A, Lall U (1999) A nonparametric approach for daily rainfall simulation. Math Comput Simul 48(46):361–371
Shiau J (2006) Fitting drought duration and severity with two-dimensional copulas. Water Resour Manag 20:795–815
Shiau J, Shen H (2001) Recurrence analysis of hydrologic droughts of differing severity. J Water Resour Plan Manag 127(1):30–40
Shiau JT (2003) Return period of bivariate distributed extreme hydrological events. Stoch Env Res Risk A 17:42–57
Shiau JT, Modarres R (2009) Copula-based drought severity-duration-frequency analysis in iran. Meteorol Appl 16(4):481–489
Silverman B (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall
Wand M, Jones M (1995) Kernel Smoothing. Chapman and Hall
Yue S (2000) Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls. Hydrol Sci J 45(2):315–326
Yusof F, Hui-Mean F, Suhaila J, Yusof Z (2013) Characterisation of drought properties with bivariate copula analysis. Water Resour Manag 27(12):4183–4207. doi:10.1007/s11269-013-0402-4
Zhang L, Singh VP (2007) Bivariate rainfall frequency distributions using archimedean copulas. J Hydrol 332(12):93–109
Zhao F, Zhang L, Chiew FH, Vaze J, Cheng L (2013) The effect of spatial rainfall variability on water balance modelling for south-eastern australian catchments. J Hydrol 493(0):16–29. doi:10.1016/j.jhydrol.2013.04.028
Acknowledgments
The authors sincerely acknowledge the Bureau of Meteorology (BOM), Australia, for providing the complete monthly precipitation data that been used in this study. The work is financed by SLAB Scholarship provided by the Ministry of Higher Education of Malaysia and National Defence University of Malaysia.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abdul Rauf, U.F., Zeephongsekul, P. Analysis of Rainfall Severity and Duration in Victoria, Australia using Non-parametric Copulas and Marginal Distributions. Water Resour Manage 28, 4835–4856 (2014). https://doi.org/10.1007/s11269-014-0779-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11269-014-0779-8