Chaos in Hydrology

The dynamics of hydrologic systems are governed by the interactions between climate inputs and the landscape. Due to the spatial and temporal variability in climate inputs and the heterogeneity in the landscape, hydrologic systems exhibit a wide range of characteristics. While some characteristics may be specific to certain systems and situations, most hydrologic systems often exhibit a combination of these characteristics. This chapter discusses many of the salient characteristics of hydrologic systems, including complexity, correlation, trend, periodicity, cyclicity, seasonality, intermittency, stationarity, nonstationarity, linearity, nonlinearity, determinism, randomness, scale and scale-invariance, self-organization and self-organized criticality, threshold, emergence, feedback, and sensitivity to initial conditions. The presentation focuses on the occurrence, form, and role of each of these characteristics in hydrologic system dynamics and the methods for their identification. At the end, a particularly interesting property of hydrologic systems, wherein simple nonlinear deterministic systems with sensitive dependence on initial conditions can give rise to complex and ‘random-looking’ dynamic behavior, and popularly known as ‘chaos,’ is also highlighted.

Hydrology was mainly dominated by deterministic approaches until the mid-twentieth century. However, the deterministic approaches suffered from our lack of knowledge on the exact nature of hydrologic system dynamics and, hence, the exact governing equations required for models. This led to the development and application of stochastic methods in hydrology, which are based on the concepts of probability and statistics. Since the 1950s–1960s, hydrology has witnessed the development of a large number of stochastic time series methods and their applications. The existing stochastic methods can be broadly grouped into two categories: parametric and nonparametric. In the parametric methods, the structure of the models is defined a priori and the number and nature of the parameters are generally fixed in advance. On the other hand, the nonparametric methods make no prior assumptions on the model structure, and it is essentially determined from the data themselves. This chapter presents an overview of stochastic time series methods in hydrology. First, a brief account of the history of development of stochastic methods is presented. Next, the concept of time series and relevant statistical characteristics and estimators are described. Finally, several popular parametric and nonparametric methods and their hydrologic applications are discussed.

Advances in computational power, scientific concepts, and data measurements have led to the development of numerous nonlinear methods to study complex systems normally encountered in various scientific fields. These nonlinear methods often have very different conceptual bases and levels of sophistication and have been found suitable for studying many different types of systems and associated problems. Their relevance to hydrologic systems and ability to model and predict the salient characteristics of hydrologic systems have led to their extensive applications in hydrology over the past three decades or so. This chapter presents an overview of some of the very popular nonlinear methods that have found widespread applications in hydrology. The methods include: nonlinear stochastic methods, data-based mechanistic models, artificial neural networks, support vector machines, wavelets, evolutionary computing, fuzzy logic, entropy-based techniques, and chaos theory. For each method, the presentation includes a description of the conceptual basis and examples of applications in hydrology.

Almost all natural, physical, and socio-economic systems are inherently nonlinear. Nonlinear systems display a very broad range of characteristics. The property of “chaos” refers to the combined existence of nonlinear interdependence, determinism and order, and sensitive dependence in systems. Chaotic systems typically have a ‘random-looking’ structure. However, their determinism allows accurate predictions in the short term, although long-term predictions are not possible. Since ‘random-looking’ structures are a common encounter in numerous systems, the concepts of chaos theory have gained considerable attention in various scientific fields. This chapter discusses the fundamentals of chaos theory. First, a brief account of the definition and history of the development of chaos theory is presented. Next, several basic properties and concepts of chaotic systems are described, including attractors, bifurcations, interaction and interdependence, state phase and phase space, and fractals. Finally, four examples of chaotic dynamic systems are presented to illustrate how simple nonlinear deterministic equations can generate highly complex and random-looking structures.

Considerable interest in studying the chaotic behavior of natural, physical, and socio-economic systems have led to the development of many different methods for identification and prediction of chaos. An important commonality among almost all of these methods is the concept of phase space reconstruction. Other than this, the methods largely have different bases and approaches and often aim to identify different measures of chaos. All these methods have been successfully applied in many different scientific fields. This chapter describes some of the most popular methods for chaos identification and prediction, especially those that have found applications in hydrology. These methods include: phase space reconstruction, correlation dimension method, false nearest neighbor method, Lyapunov exponent method, Kolmogorov entropy method, surrogate data method, Poincaré maps, close returns plot, and nonlinear local approximation prediction method. To put the utility of these methods in a proper perspective in the identification of chaos, the superiority of two of these methods (phase space reconstruction and correlation dimension) over two commonly used linear tools for system identification (autocorrelation function and power spectrum) is also demonstrated. Further, as the correlation dimension method has been the most widely used method for chaos identification, it is discussed in far more detail.

The existing methods for identification and prediction of chaos are generally based on the assumptions that the time series is infinite (or very long) and noise-free. There are also no clear-cut guidelines on the selection of parameters involved in the methods, especially in phase space reconstruction. Since data observed from real systems, such as hydrologic systems, are finite and often short and are always contaminated with noise (e.g. measurement error), there are concerns on the applications of chaos concepts and methods to real systems. Adding to this are complications that potentially arise due to issues that are specific to certain real systems, such as a large number of zeros in rainfall, runoff, and other hydrologic data. Therefore, it is important to study the issues related to methods and data that can potentially influence the outcomes of chaos studies. This chapter addresses four important issues in the applications of chaos methods to real time series, especially those that have particular relevance and gained considerable interest in hydrology: selection of delay time in phase space reconstruction, minimum data size required for correlation dimension estimation, influence of data noise, and influence of the presence of a large number of zeros in the data. Some specific examples are also presented in addressing the issues of data size and data noise.

Over the past three decades, the concepts of chaos theory have been extensively applied in hydrology. Applications of chaos theory in hydrology started with the basic identification of chaos in rainfall data and subsequently explored a wide range of problems in different types of hydrologic data. The problems studied include identification and prediction of chaos, scaling and disaggregation, missing data estimation, and catchment classification, among others. The data studied include rainfall, river flow, rainfall-runoff, lake volume and level, sediment transport, and groundwater, among others. Many studies have also addressed some of the important issues in the applications of chaos methods in hydrology, including delay time, data size, data noise, and presence of zeros in data. This chapter presents an overview of chaos studies in hydrology. The presentation is organized to reflect three stages of development: early stage (1980s–1990s), change of course (2000–2006), and studies on global-scale challenges (since 2007).

Initial applications of the ideas of chaos theory in hydrology were on rainfall data. Early studies essentially addressed the identification and prediction of chaotic behavior of rainfall data. Encouraging outcomes from these studies subsequently led to investigations on the chaotic nature of scaling relationships in rainfall and disaggregation of data, including development of a new chaotic approach for rainfall disaggregation. More recently, some studies have examined the spatial variability and classification of rainfall. In addition to these, a number of studies have also addressed the important methodological and data issues in the applications of chaos methods to rainfall data. This chapter presents a review of chaos studies on rainfall data. The presentation is organized into three parts to address three important problems associated with rainfall: identification and prediction of chaos, scaling and disaggregation, and spatial variability and classification. An example is presented for each of these to demonstrate the utility and effectiveness of chaos concepts and methods to study these different problems.

Chaos theory has found widespread applications in studies on river flow data. Indeed, river flow is the most studied data in the context of chaos studies in hydrology. Early applications mainly focused on identification and prediction of chaotic behavior in river flow dynamics. Later years witnessed studies on a wide range of problems associated with river flow, including scaling and disaggregation, missing data estimation, reconstruction of system equations, multivariable analysis, and spatial variability and classification. Many studies have also addressed the important issues in the applications of chaos theory to river flow data, including data size, data noise, and selection of parameters involved in chaos identification and prediction methods. This chapter presents a review of applications of chaos theory to river flow data. The studies are roughly grouped into three categories to represent the following aspects: identification and prediction, scaling and disaggregation, and spatial variability and classification. These applications are also illustrated through examples.

Following the early chaos studies mainly on rainfall and river flow, the concepts of chaos theory started to find applications in studies on other hydrologic data as well. Although such applications have been noticeably less when compared to those on rainfall and river flow, they have studied various types of hydrologic data. The data studied include rainfall-runoff, lake volume and level, sediment transport, groundwater, and soil moisture, among others. Further, while most of these studies have mainly focused on identification and prediction of chaos and, to some extent, investigation of scaling relationships, several other problems associated with the data have also been addressed. This chapter presents a review of the above studies, with particular focus on rainfall-runoff, lake volume and level, sediment transport, and groundwater. Examples are also provided to illustrate the applications to rainfall-runoff (i.e. runoff coefficient), sediment transport (i.e. flow discharge, suspended sediment concentration, and suspended sediment load), and groundwater (solute transport and arsenic contamination).

Despite the tremendous growth in the applications of chaos theory in hydrology, there have been lingering criticisms. These criticisms have been based on the fundamental assumptions involved in the development of methods for identification and prediction of chaos (e.g. infinite and noise-free time series, lack of clear-cut guidelines on the selection of parameters involved) and/or the limitations of hydrologic data (e.g. short and noisy data, presence of zeros). A number of issues have been raised in this regard, but some have attracted far more attention than the others. This chapter presents a review of studies that have addressed such issues in chaos studies in hydrology. The review mainly focuses on four major issues: selection of an optimum delay time for phase space reconstruction, minimum data size for correlation dimension estimation, effects of data noise, and influence of the presence of zeros in data. Examples are also provided to illustrate how these issues have been addressed to gain more confidence in the applications of the methods and in the interpretation of the outcomes.

A review of chaos studies in hydrology over the past three decades indicates that we have explored a broad spectrum of hydrologic processes, problems, and data issues. The review also reveals that we now possess an ample level of understanding of the concepts and methods and are far more confident in applying the methods and interpreting the outcomes. This chapter discusses the current status of chaos theory in hydrology. In particular, five different aspects are considered for discussion: our ability to reliably identify the presence of chaos in hydrologic data; our ability to obtain better predictions of hydrologic data using chaos methods, especially when compared against other approaches; our success in extending the applications of chaos theory to several problems beyond simple identification and prediction; our knowledge of the limitations and concerns associated with chaos methods; and the discussions and debates that have and continue to improve our understanding of chaos theory.

The tremendous progress that has been achieved, through three decades of research, in the applications of chaos theory in hydrology inevitably leads to questions regarding the future of chaos theory in hydrology. Of particular interest is to identify potential areas for further applications and advancement of the theory and possible ways to achieve fruitful outcomes. This chapter addresses these questions. In light of some of the research questions at the forefront of hydrology at the current time and will be in the future, and also looking at some studies that have already addressed these questions from the perspective of chaos theory (albeit rudimentary), several different areas are identified to further advance chaos theory in hydrology. These are: parameter estimation in hydrologic models, simplification in hydrologic model development, integration of different concepts in hydrology, development of catchment classification framework, extensions of chaos studies using multiple hydrologic variables, reconstruction of hydrologic system equations, and downscaling of global climate models. Finally, the need and the potential to establish reliable links between chaos theory, hydrologic data, and hydrologic system physics are also discussed.

Research on chaos theory in hydrology over the past three decades offers new opportunities as well as challenges. These opportunities and challenges, in turn, provide interesting ways to further explore the relevance and role of chaos theory in hydrology. An obvious question to ask is: if, and how, chaos theory fits within our two dominant, but extreme, views of hydrology of the twentieth century: deterministic and stochastic? This chapter attempts to answer this question, from both philosophical and pragmatic perspectives. It is pointed out that the underpinning concepts of nonlinear interdependence, hidden determinism and order, and sensitivity to initial conditions of chaos theory provide the necessary means to represent both the deterministic and the stochastic characteristics of hydrologic systems. This also leads to the argument that chaos theory offers a balanced middle ground to bridge the gap between the two extreme views of determinism and stochasticity and, therefore, serves as a coupled deterministic-stochastic paradigm to study hydrology in a holistic manner.