Advances in Meteorology

Volume 2015, Article ID 195940, 12 pages

http://dx.doi.org/10.1155/2015/195940

## Identifying and Evaluating Chaotic Behavior in Hydro-Meteorological Processes

^{1}Columbia Water Center, Columbia University, New York, NY 10027, USA^{2}Department of Civil Engineering, Inha University, Incheon 402-751, Republic of Korea

Received 20 November 2014; Accepted 7 April 2015

Academic Editor: Ismail Gultepe

Copyright © 2015 Soojun Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this study is to identify and evaluate chaotic behavior in hydro-meteorological processes. This study poses the two hypotheses to identify chaotic behavior of the processes. First, assume that the input data is the significant factor to provide chaotic characteristics to output data. Second, assume that the system itself is the significant factor to provide chaotic characteristics to output data. For solving this issue, hydro-meteorological time series such as precipitation, air temperature, discharge, and storage volume were collected in the Great Salt Lake and Bear River Basin, USA. The time series in the period of approximately one year were extracted from the original series using the wavelet transform. The generated time series from summation of sine functions were fitted to each series and used for investigating the hypotheses. Then artificial neural networks had been built for modeling the reservoir system and the correlation dimension was analyzed for the evaluation of chaotic behavior between inputs and outputs. From the results, we found that the chaotic characteristic of the storage volume which is output is likely a byproduct of the chaotic behavior of the reservoir system itself rather than that of the input data.

#### 1. Introduction

Hydrologic phenomena arise as a result of interactions between climate inputs and landscape characteristics that occur over a wide range of space and time scales. Due to the tremendous heterogeneities in climatic inputs and landscape properties, such phenomena may be highly variable and “complex” at all scales [1]. The nonlinear behavior of hydrologic systems had been known for a long time [2, 3]. The rainfall-runoff process is nonlinear, almost regardless of the basin area, land uses, rainfall intensity, and other influencing factors, which are changing in a highly nonlinear fashion and so are the outputs, often in unknown ways [1].

To study the nonlinear characteristics of natural phenomena, many statisticians and scientists have suggested the chaos theory which analyze and forecast the nonlinear phenomena of the natural system. Lorenz [4] suggested the strange attractor in a simple model of convection roll in the atmosphere. Packard et al. [5] suggested the method of delays and Takens [6] proved the method of delays using differential topology. Grassberger and Procaccia [7] and Farmer et al. [8] demonstrated the estimation of chaotic characterization using correlation dimension. Wolf et al. [9] calculated the largest Lyapunov exponent using the Benettin’s method. Fraser and Swinney [10] suggested a method for the estimation of time delay using the mutual information. Gilmore [11] introduced the topological method for chaos characterization, especially useful for small data sets. Farmer and Sidorowich [12] forecasted the chaotic time series using the local linear approximation. Also, Casdagli [13] forecasted the chaotic time series using the radial basis functions and Casdagli and Weigend [14] modeled and forecasted the chaotic time series using DVS (deterministic versus stochastic) algorithm. Kim et al. [15, 16] suggested a new method for the estimation of delay parameters in chaos analysis. Falanga and Petrosino [17] estimated the complexity of the system by the degrees of freedom necessary to describe the asymptotic dynamics in a reconstructed phase space. The mechanism of stochastic resonance, which is a nonlinear phenomenon, has been applied in the field of the physics of atmosphere since it was introduced by Benzi et al. [18, 19] and Nicolis [20].

Many hydrologists have been also analyzed hydrologic phenomena using nonlinear deterministic chaos to interpret the nonlinear characteristic of the hydrologic system. Rodriguez-Iturbe et al. [21] found the chaotic characteristics in rainfall data recorded with the time interval of 15 seconds using the correlation dimension and the Lyapunov exponent. Wilcox et al. [22] tested the chaotic behavior of daily snowmelt runoff data by correlation dimension. Sangoyomi et al. [23, 24] used the Great Salt Lake volume data recorded with the time interval of 15 days for searching for the chaotic characteristics. Jeong and Rao [25] used 13 tree ring series to determine their chaos characteristics. Rodriguez-Iturbe et al. [26] investigated the nonlinear dynamics of soil moisture using a soil moisture balance equation. Kim et al. [27] searched strange attractor in wastewater flow using the C-C method. Ahn and Kim [28] showed that the nonlinear stochastic model is more valid for the SOI time series analysis and modeling than linear stochastic analog by the BDS statistic. Kim et al. [29] assessed nonlinear deterministic characteristics in hydrologic time series like rainfall, stream flow, and reservoir volume series. Sivakumar et al. [30] examined the utility of nonlinear dynamic concepts for analysis of rainfall variability across Western Australia. Kim et al. [31] assessed the applicability chaotic dynamics and filtering techniques in radar rainfall.

Even though Salas et al. [32] investigated how hydrologic process (e.g., precipitation) which is low-dimensional chaotic is changed by its transformations such as aggregation and sampling, mostly, the single hydrologic time series have been analyzed for investigating its chaotic and nonlinear dynamic characteristics. Therefore, the aim of this study is to identify chaotic behavior for the components in hydro-meteorological processes such as air temperature, precipitation, discharge, and lake storage volume series. The components contribute to hydro-meteorological system as inputs and outputs. For this, the main question is, given that what is the significant factor to provide chaotic characteristics to output data? We pose the following two hypotheses. (1) Assume that the input data is the significant factor to provide chaotic characteristics to output data. (2) Assume that the system itself is the significant factor to provide chaotic characteristics to output data.

This paper is organized to solve the issue as follows. In Section 2, we give brief overview of the methodology to estimate the correlation dimension, which can detect chaotic characteristics of data series. In Section 3, we also give brief overview of the wavelet transform to extract the data of the representative period from the original time series and artificial neural networks (ANN) for modeling the hydro-meteorological system. In Section 4, we apply methods for identifying chaotic behavior of the data series and discuss the results. Finally, in Section 5, we summarize the findings and conclusions.

#### 2. Estimation of Correlation Dimension

##### 2.1. Phase Space Reconstruction

Phase space is a useful tool for representing the evolution of a system in time. It is essentially a graph or a coordinate diagram, whose coordinates represent the variables necessary to completely describe the state of the system at any moment (in other words, the variables that enter the mathematical formulation of the system). The trajectories of the phase space diagram describe the evolution of the system from some initial state, which is assumed to be known, and, hence, represent the history of the system [5]. The “region of attraction” of these trajectories in the phase space provides at least important qualitative information on the “extent of complexity” of the system, which can subsequently be verified quantitatively using methods based on, for example, the concept of dimensionality.

For a dynamic system with known partial differential equations (PDEs), the system can be studied by discretizing the PDEs, and the set of variables at all grid points constitutes a phase space. One difficulty in constructing the phase space for such a system is that the (initial) values of many of the variables may not be known. However, a time series of a single variable of the system may be available, which may allow the attractor (a geometric object that characterizes the long-term behavior of a system in the phase space) to be reconstructed. The idea behind such a reconstruction is that a (nonlinear) system is characterized by self-interaction, so that a time series of a single variable can carry the information about the dynamics of the entire multivariable system. Many methods are available for phase space reconstruction from an available time series. Among these, the method of delays (e.g., [6]) is the most widely used one. According to this method, given a single-variable series, , where , a multidimensional phase space can be reconstructed aswhere ; is the dimension of the vector , called embedding dimension; and is an appropriate delay time (an integer multiple of sampling time). A correct phase space reconstruction in a dimension generally allows interpretation of the system dynamics (if the variable chosen to represent the system is appropriate) in the form of an -dimensional map , given by where and are vectors of dimension , describing the state of the system at times (current state) and (future state), respectively.

##### 2.2. Correlation Integral and Correlation Dimension

The dimension of a time series is, in a way, a representation of the number of variables dominantly governing the underlying system dynamics. Correlation dimension is 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 the phase space. The correlation dimension method uses the correlation integral (or function) for determining the dimension of the attractor and, hence, for distinguishing between low-dimensional chaos and high-dimensional system. The concept of the correlation integral is that a time series arising from deterministic dynamics will have a limited number of degrees of freedom equal to the smallest number of first-order differential equations that capture the dominant features of the dynamics. Thus, when one constructs phase spaces of increasing dimension, a point will be reached where the dimension equals the number of degrees of freedom, beyond which increasing the phase space dimension will not have any significant effect on correlation dimension. Many algorithms have been formulated for the estimation of the correlation dimension. Among these, the Grassberger-Procaccia algorithm [7] has been the most popular. The algorithm uses the concept of phase space reconstruction for representing the dynamics of the system from an available single-variable time series, as presented in (1). For an m-dimensional phase space, the correlation integral or function is given bywhere is the Heaviside step function, with for and for , where , is the vector norm (radius of sphere) centered on or . If the time series is characterized by an attractor, then and are related according towhere is a constant and is the correlation exponent or the slope of the versus plot. The slope is generally estimated by a least square fit of a straight line over a certain range of (scaling regime) or through estimation of local slopes between values. The distinction between low-dimensional (perhaps determinism) and high-dimensional (perhaps stochasticity) can be made using the versus plot. If saturates after a certain and the saturation value is low, then the system is generally considered to exhibit low-dimensional and possibly deterministic dynamics. The saturation value of is defined as the correlation dimension of the attractor, and the nearest integer above this value is generally an indication of the number of variables dominantly governing the dynamics. On the other hand, if increases without bound with increase in , the system under investigation is generally considered to exhibit high-dimensional and possibly stochastic behavior.

#### 3. Wavelet Transform and Artificial Neural Networks

##### 3.1. Wavelet Transform

According to Fourier theory, a signal can be expressed as the sum of a possibly infinite series of sine and cosines, referred to as a Fourier expansion [33]. However, a Fourier expansion has only frequency resolution and not time resolution; that is, no amplitude modulation of the signal at a given frequency is considered. Moving-window Fourier transforms have been used to address this issue, but this method is sensitive to the choice of window width. Alternatively, the wavelet transform [34, 35] enables the identification of frequency components as well as their variation in time. The continuous wavelet transform of a discrete sequence is defined by the convolution of with a scaled and translated wavelet function :where indicates the complex conjugate, is the localized time index, is the scale parameter, and is the number of points in the time series. In this study, we use the Morlet wavelet function defined as , where is a frequency and is a nondimensional “time” parameter. By varying the wavelet scales and translating along the localized time index , one can construct a picture that shows both the amplitude of any features versus the scale and how this amplitude varies with time. A vertical slice through a wavelet plot is a measure of the local spectrum. The time-averaged wavelet spectrum over all the local wavelet spectra gives the global wavelet spectrum:

A more detailed presentation for wavelet transform analysis is referred to read Torrence and Compo [35].

##### 3.2. Artificial Neural Networks

ANN is a model of neurotransmission by a neuron, which is a nerve cell in the human brain. ANN is an empirical pattern search technique that enables the consideration of a nonlinear relationship between input variables and output variables. ANN is used in various areas because of its unique applicability [36, 37]. This includes the field of climate science, where its applicability is proven [38, 39].

Many studies suggest the ANN technique, which is a nonlinear model of the data series, and ANN is better than other techniques by way of systematic evaluation of various techniques [40, 41]. Therefore, this study also applies ANN, which is judged to have superior applicability in the simulation of nonlinear characteristics of the hydro-meteorological system.

#### 4. Applications and Results

##### 4.1. Study Area and Data Series Used

The Bear River Basin, located in northeastern Utah, southeastern Idaho and southwestern Wyoming, comprises 7,500 square miles of mountain and valley lands including 2,700 in Idaho, 3,300 in Utah, and 1,500 in Wyoming. The Bear River crosses state boundaries five times and is the largest stream in the western hemisphere that does not empty into the ocean. It ranges in elevation from over 1,278 to 3,868 feet and is unique in that it is entirely enclosed by mountains, thus forming a huge basin with no external drainage outlets (http://www.greatsaltlakeinfo.org/Background/BearRiver). The Bear River is the largest tributary to the Great Salt Lake (see Figure 1).