Chaos Theory for Hydrologic Modeling and Forecasting: Progress and Challenges

Chaos Theory for Hydrologic Modeling and Forecasting: Progress and Challenges

Bellie Sivakumar (The University of New South Wales, Australia)
DOI: 10.4018/978-1-61520-907-1.ch010
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Abstract

In hydrology, two modeling approaches have been prevalent: deterministic and stochastic. The ‘permanent’ nature of the Earth, ocean, and the atmosphere and the ‘cyclical’ nature of the associated mechanisms support the deterministic approach. The ‘highly irregular and complex’ nature of hydrologic processes and our ‘limited ability to observe’ the details favor the stochastic approach. In view of these, the question of whether a deterministic approach or a stochastic approach is better is meaningless. Indeed, for most hydrologic systems and processes, both the deterministic approach and the stochastic approach are complementary to each other and, thus, an approach that can couple these two and serve as a middle-ground would often be the most appropriate. ‘Chaos theory’ can offer such a coupled deterministic-stochastic approach, since its underlying concepts of nonlinear interdependence, hidden determinism and order, sensitivity to initial conditions are highly relevant in hydrology. The last two decades have witnessed numerous applications of chaos theory in hydrology. The outcomes of these studies are encouraging, but many challenges also remain. This chapter is intended: (1) to provide a comprehensive review of chaos theory applications in hydrology; and (2) to discuss the challenges that lie ahead and the scope for the future.
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Introduction

Hydrologic processes arise as a result of interactions between climate inputs and landscape characteristics that occur over a wide range of space and time scales. The tremendous changes in climate inputs and heterogeneities in landscape characteristics sometimes give rise to processes that are highly irregular, complex, and random, but at other times result in processes that are regular, simple, and deterministic. When, where, why, and how either of these situations occurs remains largely a mystery, despite our intense research efforts over a century or more.

Traditionally, two broad approaches have been adopted to study hydrologic systems and processes: deterministic and stochastic. Either of these approaches has its own merits, having solid foundations in scientific principles/philosophies, verifiable assumptions for specific situations, and the ability to provide reliable results. For example, the deterministic approach has merits considering the ‘permanent’ nature of the Earth, ocean, and the atmosphere and the ‘cyclical’ nature of the associated mechanisms, whereas the merits of the stochastic approach lie in the facts that hydrologic processes exhibit ‘highly irregular and complex’ structures and that we have only ‘limited ability to observe’ their detailed variations.

In light of these, the general question of whether the deterministic approach or the stochastic approach is better for hydrologic modeling is meaningless. Such a question is really a philosophical one that has no general answer, but it is better viewed as a pragmatic one, which has an answer only in terms of specific situations (Gelhar, 1993). These specific situations must be viewed in terms of the process, scale (time and/or space), and the purpose of interest, which may collectively form the ‘hydrologic system’ (Sivakumar, 2008a). For some situations, both the deterministic approach and the stochastic approach may be equally appropriate; for some other situations, the deterministic approach may be more appropriate; and for still others, the stochastic approach may be more appropriate. It is also reasonable to contend that, for most (if not all) hydrologic systems and processes, both the deterministic approach and the stochastic approach are actually complementary to each other. This may be supported by our observation of both deterministic and stochastic properties at one or more scales in time and/or space of a hydrologic process. For example, it is common to observe a significant deterministic nature in river flow in the form of seasonality and annual cycle, while the interactions of the various mechanisms involved and their various degrees of nonlinearity bring stochasticity. The point is that use of a coupled deterministic–stochastic approach, incorporating both the deterministic and the stochastic components, will likely yield a higher ‘probability of success’ compared to either approach when adopted independently. An immediate question is: how to devise such an approach? The answer may well lie in ‘chaos theory’ (e.g. Lorenz, 1963).

In the dynamic systems science literature, the term ‘chaos’ is used to refer to situations where complex and random-looking behaviors arise from simple nonlinear deterministic systems with sensitive dependence on initial conditions (the converse also applies). The three fundamental properties inherent in this definition, namely (a) nonlinear interdependence; (b) hidden determinism and order; and (c) sensitivity to initial conditions, are highly relevant in hydrologic systems and processes. For example: (a) components and mechanisms involved in the hydrologic cycle act in a nonlinear manner and are also interdependent; (b) the daily cycle in temperature and annual cycle in river flow possess determinism and order; and (c) contaminant transport in surface and sub-surface waters largely depends upon the time (i.e. rainy or dry season) at which the contaminants are released. The first property represents the ‘general’ nature of hydrologic processes, whereas the second and third represent their ‘deterministic’ and ‘stochastic’ natures, respectively. Furthermore, despite their complexity and random-looking behavior, hydrologic processes may be governed only by a few degrees of freedom (e.g. runoff in a well-developed urban catchment depends essentially on rainfall), another basic idea of chaos theory.

Key Terms in this Chapter

Stochasticity: A situation in which the progression from earlier to later states is not completely determined by any law.

Autocorrelation Function: A normalized measure of the linear correlation among successive values in a time series.

Determinism: A situation in which later states evolve from earlier ones according to a fixed law.

Chaos: A situation where a complex and random-looking behavior arises from simple nonlinear deterministic systems with sensitive dependence on initial conditions.

Attractor: A geometric object that characterizes the long-term behavior of a system in the phase space.

Phase Space: A graph or a co-ordinate diagram, whose co-ordinates represent the variables necessary to completely describe the state of a system at any moment.

Lyapunov Exponent: The average exponential rate of divergence or convergence of nearby orbits in the phase space.

Correlation Dimension: A measure of the extent to which the presence of a data point affects the position of the other points lying on the attractor in a multi-dimensional phase space.

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