Editorial

Much attention has long been focused on hydrologic frequency analysis. The bulk of the literature has dealt with univariate frequency analysis of either one of the flood variables or one of the rainfall variables. The most common flood variables are flood peak, volume, and duration; the most common rainfall variables are rainfall intensity, depth, and duration. In hydrologic design, floodplain management, flood damage assessment, urban planning, drought mitigation, water supply, irrigation scheduling, and so on, joint frequency distributions are needed. For example, in urban drainage design, joint frequency distributions of runoff peak and volume are needed. In damage assessment, joint frequency distributions of flood volume and duration are important. In irrigation scheduling, joint frequency distributions of soil moisture status and the availability of water in the reservoir are needed. For the design of water supply schemes, joint frequency distributions of the rate of water supply and the duration for which water is to be supplied are needed for designing water storage facilities. Thus, the need for multivariate frequency distributions seems ubiquitous. As a result, a number of attempts have been made to perform multivariate flood frequency analyses that take into consideration the dependence among variables (e.g., flood peak, volume, and duration) but with restrictive assumptions. In the majority of studies, multivariate frequency distributions have usually been derived using one or more of three fundamental assumptions: (1) variables (e.g., rainfall intensity, depth, and duration) each have the same type of marginal probability distribution; (2) the variables are assumed to have a joint normal distribution or have been transformed and assumed to have a joint normal distribution; and (3) they have been assumed independent—a trivial case. In reality, however, variables (e.g., rainfall or flood) are dependent, do not follow, in general, the normal distribution, and do not have the same type of marginal distributions.

In the last few years a new method, called the copula method, has been developed. This method permits derivation of bivariate or multivaraiate probability distributions without using the above-mentioned assumptions. A major advantage of this method is that marginal distributions of individual variables can be of any form and the variables can be correlated. The copula method is a potentially powerful method. This special issue is devoted to this method.

This issue is comprised of 9 articles. The first article is by Schweizer, one of the pioneers of the copula method. In his article, Schweizer provides a brief historical perspective of this method. The second article, by Genest and Favre, provides a comprehensive discussion of copula-based modeling and essential steps involved therein. Genest is another pioneer in the copula methodology. The next article by De Michele and Salvatori provides a discussion of the theory and practice of copulas in hydrology. The fourth article by Dupuis discusses benefits, cautions, and issues related to the application of copulas in hydrology. Dupuis is another leader in this area. These four articles lay the foundation for copula-based modeling in general and in hydrology in particular.

The next article by Poulin, Huard, Favre, and Pugin dwells on the importance of tail dependence in bivariate frequency analysis. The sixth article by Gebremichael and Krajewski discusses an application of copulas to modeling temporal sampling errors in satellite–derived rainfall estimates. The next article is by Zhang and Singh, who discuss an application of the Gumbel-Hougaard Copula for trivariate rainfall frequency analysis. The eighth article by Serinaldi and Grimaldi deals with a fully nested 3-copula and discusses its application in hydrology. The last article by Zhang and Singh deals with trivariate flood frequency analysis using the Gumbel-Hougaard Copula. These five articles illustrate applications of copulas.

This issue presents both theoretical concepts of the copula methodology as well as its application to rainfall and flow frequency analysis. It is hoped that this issue will stimulate discussion of this potentially powerful and useful method for a range of applications in hydrology. For example, the copula-based regression approach, where the regression function is used as the conditional expectation employing the information provided by the joint distribution of variables, may find wider application in river flow and precipitation analyses. Similarly, the application of modeling two- (or multi-) dimensional distribution for the same kind of variable at two different points, e.g., inflow of two tributaries to a reservoir, precipitation at two stations within a catchment, and so on, is another example. Examples where copulas can be usefully applied abound in water resources.