Review of Chaos in Hydrology: Bridging Determinism and Stochasticity by Bellie Sivakumar

Defining chaos, outlining its three fundamental properties, and contrasting it with randomness in the preface, Professor Sivakumar sets the stage for what the book is about and what to expect therein. Given the increasing emphasis on dynamical systems these days, the book is highly timely and is the first book in hydrology on this topic. Therefore, it unfolds a new vista for hydrologic modeling.

The subject matter of the book is divided into four parts encompassing 15 chapters. Part I on hydrologic systems and modeling comprises four chapters (1–4). The first chapter is a prelude to hydrology and contains the ingredients essential for connecting to the chaos theory and its application later in the book, including the hydrologic cycle, the hydrologic system concept, hydrologic models, hydrologic data, and time series modeling and its physical basis. The chapter has extensive references and is well prepared.

Chapter 2 deals with the characteristics of hydrologic data and systems, such as correlation, complexity, trend, periodicity, cyclicity, seasonality, intermittency, stationarity and nonstationarity, linearity and nonlinearity, determinism and randomness, scaling and scale-invariance, self-organization, threshold, feedback, sensitivity to initial conditions, and class of nonlinear determinism. The discussion in the chapter is easy to understand and has well cited literature.

Stochastic time series methods are presented in Chapter 3. The chapter is an excellent prelude to the basic elements of time series analysis that are used in hydrology. It covers relevant statistical characteristics, such as mean, variance, coefficient of variation, skewness coefficient, autocorrelation function, and power spectrum; parametric methods, including autoregressive models, moving average models, autoregressive moving average models, periodic models, disaggregation models, Markov chain models, point process models, and other models; and nonparametric models, including bootstrap and block bootstrap, kernel density estimate, $k$-nearest neighbor resampling, and $k$-nearest neighbors with local polynomial regression. The chapter is concluded with a brief summary and extensive cited literature. On the whole, it is a well crafted chapter.

Modern nonlinear time series methods constitute the subject matter of Chapter 4. Providing a brief introduction to the nonlinear nature of hydrologic systems, nonlinear stochastic methods, and data-based mechanistic models, the chapter goes on to discussing artificial neural networks, support vector machines, wavelets, evolutionary computing, fuzzy logic, entropy-based methods, and nonlinear dynamics and chaos. It is concluded with a summary and an extensive list of references.

Part II on nonlinear dynamics and chaos is comprised of three chapters (5–7). Chapter 5 deals with fundamentals of chaos theory. Beginning with a discussion of nonlinear dynamics with a focus on nonlinearity, instability, and uncertainty, it defines chaos, presents a brief history of development of chaos theory, and discusses dynamical systems and stability analysis, attractors, bifurcation, interaction and interdependence, sensitivity to initial conditions, state space and phase space, fractal and fractal dimension, and examples of dynamic systems. A brief summary and literature cited conclude the chapter. It is a very insightful chapter.

Chapter 6 discusses chaos identification and prediction methods, including linear tools and limitations, phase space reconstruction, correlation dimension method, false nearest neighbor algorithm, Lyapunov exponent method, Kolmogorov entropy method, surrogate data method, Poincare maps, close returns plot, and nonlinear prediction. A brief summary and a lot of very relevant references end the chapter. The discussion is clear and informative.

Issues in chaos identification and prediction are treated in Chapter 7. They include delay time with delay time selection and delay window selection; data size with the effect of data size and minimum data size; data noise with the effect of noise, noise level determination, noise reduction, and coupled noise level determination and reduction through an example; zeros in data; and other issues. The chapter is concluded with a brief summary and an extensive list of references. The treatment in the chapter is concise and clear.

Part III, dealing with applications of chaos theory in hydrology, contains five chapters (8–12). Chapter 8 provides an overview presenting a historical perspective on applications of chaos theory. Listing chaos studies in hydrology, it discusses the early stage (1980s–1990s), change in course (2000–2006), and looking at global scale challenges (2007–), and concludes with a summary and an extensive literature cited.

Applications to rainfall are discussed in Chapter 9. Starting with a discussion on identification and prediction, it presents a chaos analysis of rainfall with an example from the Gota River basin, scaling, and disaggregation and downscaling with an example of chaotic scale invariance and an example of chaotic disaggregation from the Leaf River basin; mentions other studies; deals with spatial variability and classification; and briefly considers data size, noise reduction, and zeros. The chapter concludes with a brief summary and literature cited. Examples presented in the chapter are very instructive.

Chapter 10 contains applications to river flow data. The discussion in the chapter includes identification and prediction; chaos analysis in river flow with an example from the Coaracy Nunes/Araguary River basin; scaling and disaggregation with a focus on chaos and scaling and chaotic disaggregation of river flow; spatial variability and classification; and other studies. A brief summary and plenty of references conclude the chapter. It is a concise and well-written chapter.

Applications to other hydrologic data are presented in Chapter 11. These include rainfall-runoff, lake volume and level, sediment transport, groundwater, solute transport, and others. The chapter concludes with a summary and literature cited. It shows the breadth and scope of chaos theory applications. Chapter 12 deals with studies on hydrologic data issues, such as delay time, data size, data noise, and zeros in data. A brief summary and literature cited end the chapter. It is a good chapter capping Part III.

Part IV is a look ahead and comprises three chapters. Chapter 13 on the current status reflects on reliable identification, encouraging predictions, successful extensions, limitations and concerns, and discussions and debates. It concludes with a summary and plenty of references. Chapter 14 is on the future, discussing parameter estimation, model simplification, integration of concepts, catchment classification framework, multivariable analysis, reconstruction of system equations, downscaling of global climate model outputs, and linking theory, data, and physics. A brief summary and plenty of literature cited end the chapter. The chapter is full of insights and is rich in ideas. The concluding Chapter 15 is on final thoughts reflecting philosophy and pragmatism. It is concluded with closing remarks and relevant references.

On the whole, this book is well written and will be an excellent reference and an insightful source of information for students and faculty wishing to work in the area of chaos theory applications. Professor Sivakumar has brought forth his more than two decades of research experience to bear on the excellent quality of the book. He deserves a lot of applause for writing this book.