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.

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.

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 this chapter, copula modeling is applied to flood analysis with the use of real-world flood data. The chapter is structured in the following sections: (i) an introduction; (ii) at-site flood frequency analysis; (iii) spatial dependence for flood variables; and (iv) concluding remarks.

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.

In this chapter, we apply copulas to network evaluation and design. The network is considered to be comprised of rain gauges that are located in the southwest (seven gauges) and east central (three gauges) parts of Louisiana. To select proper rain gauges for network design, the kernel density is applied to model the marginal rainfall variables as that studied for rainfall analysis in Chapter 10. For the simplicity of illustrating the copula-based network design, meta-elliptical copulas (i.e., meta-Gaussian and meta-Student t) are applied to model the spatial dependence among rain gauges. The network design case study shows the appropriateness of the copula-based network design.

In this chapter, the copula modeling is applied to analyze compound extremes. The number of warm days (NWDs) and monthly precipitation are applied for the case study. The time-varying generalized extreme value (GEV) distribution with a linear trend in the location parameter is applied to model the NWDs after the change. The time-varying copula is applied to model the compound risk of hot and dry, as well as wet and cold days.

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 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 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.

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.

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.