The term copula is derived from the Latin verb copulare, meaning “to join together.” In the statistics literature, the idea of a copula can be dated back to the nineteenth century in modeling multivariate non-Gaussian distributions. By formulating a theorem, now called Sklar theorem, Sklar (1959) laid the theoretical foundation for the modern copula theory. In general, copulas couple multivariate distribution functions to their one-dimensional marginal distribution functions, which are uniformly distributed in [0, 1]. In other words, copula functions enable us to represent a multivariate distribution with the use of univariate probability distributions (sometimes simply called marginals, or margins), regardless of their forms or types. In this chapter, we will discuss the general concepts of copulas, including their definition, properties, composition and construction, dependence structure, and tail dependence.

Similar to the Archimedean copulas, the non-Archimedean copulas can be classified as one-parameter non-Archimedean bivariate copulas, two-parameter non-Archimedean bivariate copulas, and multivariate (d ≥ 3)$d\ge 3)$ non-Archimedean copulas. In recent years, successful applications of non-Archimedean copulas, such as meta-elliptical copulas and Plackett copulas, have been reported in hydrology and water resources management. In this chapter, we will focus on Plackett copulas and more specifically bivariate and trivariate Plackett copula.

This chapter briefly reviews the development of the copula theory and its applications in the field of water resources engineering (flood, drought, rainfall, groundwater, etc.). It points out the need for applying the copula theory in hydrology and engineering. The chapter is concluded with an outline of the structure of the book.

Meta-elliptical copulas are derived from elliptical distributions. Kotz and Nadarajah (2001) and Nadarajah (2006) made solutions of meta-elliptical copulas available. In this chapter, we will review the definition and probability distributions as well as other properties of meta-elliptical copulas.

Symmetric Archimedean copulas are widely applied for hydrologic analyses for the following reasons: (1) they can be easily constructed with the given generating function; (2) a large variety of copulas belong to this class (Nelsen, 2006); and (3) the Archimedean copulas have nice properties, such as simple and elegant mathematical treatment. This chapter focuses on the symmetric Archimedean copulas.

In this chapter, we focus on the copula applications to at-site bivariate/trivariate drought analysis. In a case study, drought variables are separated from long-term daily streamflow series, i.e., drought severity, drought duration, drought interarrival time, and maximum drought intensity. Drought severity and duration are applied for bivariate drought frequency analysis. Drought severity, duration, and maximum intensity are applied for trivariate drought frequency analysis. The Archimedean, meta-elliptical, and vine copulas are adopted for the bivariate/trivariate analyses. The case study shows that the copula approach may be properly applied for drought analysis.

In previous chapters, we have mainly discussed copula models for bivariate/multivariate random variables. Now we ask two other questions that usually arise in hydrology and water resources engineering. Can we use the stochastic approach to predict streamflow at a downstream location using streamflow at the upstream location? If streamflow is time dependent, then it cannot be considered as a random variable as is done in frequency analysis. Can we model the temporal dependence of an at-site streamflow sequence (e.g., monthly streamflow) more robustly than with the classical time series and Markov modeling approach (e.g., modeling the nonlinearity of time series freely)? This chapter attempts to address these questions and introduces how to model a time series with the use of copula approach.

In this chapter, we will illustrate the application of copulas in rainfall frequency analysis. This chapter is divided into two parts: (1) rainfall depth-duration frequency (DDF) analysis; and (2) multivariate rainfall frequency (i.e., four-dimensional) analysis. The rainfall data from the watersheds in the United States are collected and applied for analyses. The Archimedean, meta-elliptical, and vine copulas are applied to model the dependence among rainfall variables. Application shows that the DDF may be modeled by the Gumbel–Hougaard copula. Both vine and meta-elliptical copulas may be applied to model the spatial dependence of rainfall variables. Compared to the vine copula, modeling is easier to do when applying the meta-elliptical copula.

Much of the literature on copulas, discussed in the previous chapters, is limited to the bivariate cases. The Gaussian and student copulas have been commonly applied to model the dependence in higher dimensions (Genest and Favre, 2007; Genest et al., 2007a). In Chapter 4, we discussed the extension of symmetric bivariate Archimedean copulas as well as their major restrictions to model high-dimensional dependence (i.e., d ≥ 3)$d\ge 3)$. Through the extension of the bivariate Archimedean copula, the multivariate Archimedean copula is symmetric and denoted as exchangeable Archimedean copula (EAC). EAC allows for the specification of only one generating function and only one set of parameters θ. In other words, random variates by pair share the same degree of dependence. Using the trivariate random variable {X1, X2, X3} as an example, {X1, X2}, {X2, X3}, and {X1, X3} should have the same degree of dependence. However, this assumption is rarely valid. This chapter discusses the following two approaches of constructing asymmetric multivariate copulas: nested Archimedean copula construction (NAC) and the vine copulas through pair-copula construction (PCC).

In the previous chapters, we have briefly introduced applications of copulas to analyses of rainfall, streamflow, drought, water quality, and compound extremes, as well as network design. In this chapter, we will introduce suspended sediment transport. Two case studies will be discussed to (i) apply copulas to construct the discharge-sediment rating curve using the Yellow River dataset; and (ii) investigate the dependence among precipitation, discharge, and sediment yield using the event-based dataset retrieved from the flume #3 at Santa Rita experimental watershed.

This chapter discusses how to apply copulas in water quality analysis. For monthly water quality observations, applications will include (i) a copula-based Markov process to study the water quality sequence with temporal dependence; and (ii) a copula-based multivariate water quality time series analysis. This chapter is in line with Chapter 9.

Bivariate or multivariate frequency analysis entails univariate distributions that are determined by empirical fitting to data. The fitting, in turn, requires the determination of distribution parameters and the assessment of the goodness of fit. In practical applications, such as hydrologic design, risk analysis is also needed. The objective of this chapter, therefore, is to briefly discuss these basic elements, which are needed for frequency analysis and will be needed in subsequent chapters.

In this chapter, we will illustrate the application of copulas in rainfall frequency analysis. This chapter is divided into two parts: (1) rainfall depth-duration frequency (DDF) analysis; and (2) multivariate rainfall frequency (i.e., four-dimensional) analysis. The rainfall data from the watersheds in the United States are collected and applied for analyses. The Archimedean, meta-elliptical, and vine copulas are applied to model the dependence among rainfall variables. Application shows that the DDF may be modeled by the Gumbel–Hougaard copula. Both vine and meta-elliptical copulas may be applied to model the spatial dependence of rainfall variables. Compared to the vine copula, modeling is easier to do when applying the meta-elliptical copula.