Bivariate £ood frequency analysis: Part1. Determination of marginals by parametric and nonparametric techniques S. Karmakar 1 and S.P. Simonovic 2 1 Centre for Environmental Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, India 2 Department of Civil and Environmental Engineering, Director Engineering Studies, Institute for Catastrophic Loss Reduction, The University of Western Ontario, London, ON, Canada Correspondence: Slobodan P. Simonovic, Department of Civil and Environmental Engineering, Director Engineering Studies, Institute for Catastrophic Loss Reduction, The University of Western Ontario, London, ON, Canada N6A 5B9 Tel.: 1519 661 4075 Fax: 1519 661 3779 Email: simonovic@uwo.ca DOI:10.1111/j.1753-318X.2008.00022.x Key words Flood; frequency analysis; kernel density; nonparametric; orthonormal series; parametric. Abstract In flood frequency analysis, a flood event is mainly characterized by peak flow, volume and duration. These three variables or characteristics of floods are random in nature and mutually correlated. In this article, an effort is made to find out appropriate marginal distribution of the flood characteristics considering a set of parametric and nonparametric distributions, and further mathematically model the correlated nature among them. A set of parametric distribution functions and nonparametric methods based on kernel density estimation and orthonormal series are used to determine the marginal distribution functions for peak flow, volume and duration. In conventional methods of flood frequency analysis, the marginal distribution functions of peak flow, volume and duration are assumed to follow some specific parametric distribution function. The present work performs a better selection of marginal distribution functions for flood characteristics as both parametric and nonparametric estimation procedures are extensively fol- lowed. The methodology is demonstrated with 70-year stream flow data of Red River at Grand Forks of North Dakota, USA. Introduction Most hydrologic design, planning and management problems require a detailed knowledge of flood event characteristics, i.e. flood peak flow, volume and duration. These are random in nature and mutually correlated. Flood frequency analysis defines the severity of a flood event by summarizing the characteristics of flood, and by finding out their mutual dependence structure. A number of methodologies have been developed to perform univariate (Kite, 1978; Cunnane, 1987; Rao and Hamed, 2000) and multivariate flood frequency analysis (Ashkar and Rousselle, 1982; Krstanovic and Singh, 1987; Sackl and Bergmann, 1987; Singh and Singh, 1991; Yue et al., 1999; Yue, 2001) but with many restrictive assumptions (Zhang and Singh, 2006). In hydrologic planning and design for flood management, it is not enough to know information about flood peak flow only, but it is also necessary to statistically value flood volume and duration. In conventional methods of flood frequency analysis, the marginal distribution functions of peak flow, volume and duration are assumed to follow some specific family of parametric distribution functions, for example gamma, lognormal, exponential, extreme value distribution, etc. In most real case studies, the best-fitted marginal distribution for peak flow, volume and duration need not be from the same family of probability distribution functions. The frequency analysis is primarily based on the estimation of the probability density function (pdf). The parametric approaches for estimating the pdf must assume that the data are drawn from a known parametric family of distributions. However, many studies on frequency analysis indicate that there is no universally accepted distribution for representing the hydrologic variables (Adamowski, 1985, 1996; Silver- man, 1986; Yue et al., 1999; Smakhtin, 2001). It is evident that the parametric method, which depends on prior knowl- edge of the particular distribution function, has its limita- tions (Cunnane, 1985; Adamowski, 1989) and, as pointed out by Dooge (1986), ‘no amount of statistical refinement can overcome the disadvantage of not knowing the fre- quency distribution involved’. To overcome some of the limitations of parametric methods, nonparametric density function estimations have been explored in hydrologic frequency estimation (Lall, 1995). A nonparametric method does not require the assumption of any particular form of density function. Using the weighted moving averages of the records from a small neighbourhood of the point of estimation, nonparametric function estimations have the advantage that they always reproduce the attributes J Flood Risk Management 1 (2008) 190–200 c  2008 The Authors Journal Compilation c  2008 Blackwell Publishing Ltd