Review of Copulas and Their Applications in Water Resources Engineering by Lan Zhang and V. P. Singh

Copulas are statistical entities that enable us to model the dependence structure between two or more random variables. They also allow the distributions of the marginals to be isolated from the modeling of the dependence structure between the variables. The use of Kendall’s $\tau$ and Spearman’s $\rho$, statistics used to measure the association between two variables, facilitates the use of copulas by providing a pathway for hypothesis testing of the dependence structure between variables. Despite their potential being revealed by Sklar’s theorem as early as 1959, hydrologists have been slow to adopt this powerful tool for describing dependence in hydrologic variables. It is only over the last two decades or so that there has been a surge in the use of copulas in hydrology.

The book by Zhang and Singh is a timely and valuable resource for the hydrologic community. The book is broadly divided in two parts. The first part, consisting of nine chapters, focuses on introducing the theoretical underpinnings of copulas. Chapter 1 provides a comprehensive review of applications on copulas in wide-ranging topics such as return periods of hydrologic events, flood and drought variables, and estimation of risk for hydraulic structures. Chapter 2 introduces the reader to preliminary aspects of univariate and bivariate distributions, goodness-of-fit measures, and confidence intervals. Parameter estimation, using method of moments and method of maximum likelihood, is discussed in general terms. Chapter 3 delves into the topic of copulas, various analytical families, and their ranges of applicability. The versatility of copulas for computing joint and conditional probabilities for multiple jointly distributed random variables is revealed here.

Chapter 4 provides a discussion of symmetric Archimedean copulas—perhaps the most popular family of models because they possess several desirable properties such as elegant mathematical representation, simple construction, and a wide range of application. Tables and figures in the chapter illustrate the many mathematical models belonging to the Archimedean family. Chapter 5 leads readers into asymmetric copulas allowing for different pairwise dependencies among three or more variables. The concept of nested copulas is presented here along with details of parameter estimation and simulation and pair-copula decomposition and construction. The role of vine copulas is explored in this context. Chapter 6 takes readers through the subtleties of bivariate and trivariate Plackett copulas.

Chapter 7 is about non-Archimedean copulas, specifically the use of metaelliptical copulas for further flexibility beyond what is afforded by Archimedean models. Chapter 8 discusses copulas that are motivated by entropy considerations where both marginal and dependence structures may be derived from maximum entropy principles. Chapter 9 extends the use of copulas to time series applications, for example, in construction of $k$th-order Markov models and for spatial dependence among different time series.

With this comprehensive coverage into the theory of copulas, the second part consists of eight chapters, each focusing on a particular application in hydrology. Chapter 10 draws on rainfall data from various parts of the United States for depth-duration-frequency relations and other multivariate analyses using Gumbel-Hougaard, asymmetric Archimedean, vine, and metaelliptical copulas. At-site flood frequency analysis and spatial dependence of flood variables are the subject of Chapter 11. Copula mixtures are explored in this chapter for better representation of tail dependence in the data.

In Chapter 12, the authors explore the use of copula-based Markov models for modeling the temporal dependence structure in water quality time series—specifically the deseasonalized series composed of total persulfate nitrogen, dissolved oxygen, temperature, and pH data. The authors show how metaelliptical copulas may be used to model spatial dependence so that quantities at one location may be estimated from observations at a different location.

The entirety of Chapter 13 is devoted to drought studies. Archimedean, metaelliptical, and vine copulas are adopted for bivariate and trivariate analyses of drought severity, duration, interarrival times, and maximum intensity. Conditional return periods of droughts are developed, along with dynamic return periods of particular drought episodes.

The topic of compound extremes, i.e., extremes among multiple hydrologic indicators (e.g., heat waves, temperature, rain, wind) is dealt in Chapter 14, while Chapter 15 focuses on network design using metaelliptical copulas to model spatial dependence of measurements between rain gauges. The use of marginal entropy in network design is explored in this chapter. Suspended sediment yield rating curves and relationships between precipitation, discharge, and sediment yield are subjects of Chapter 16. Finally, Chapter 17 uses copulas to model the dependence between water basin storage options and examines the question of interbasin transfers in this context.

The authors have expended substantial effort to provide detailed case studies along with examples containing real data sets from various parts of the world. This is a particular strength of this book. Readers will find these illustrative examples and data sets very useful for testing their understanding of the subject matter and designing solutions for relevant questions involving multiple hydrologic variables. While there are no exercises offered, there are plenty of applications where users are able to bring their own data sets and use the concepts presented in the book. The book will be useful for beginners and advanced users alike, and is a ready reference for copula enthusiasts in any field. I join the authors in expressing the hope that the book will be useful to graduate students and faculty members.