Using Chaos Theory approach for River flow Time Series Analysis: (Case Study: Kashkan River)

Random-like behaviors in various natural phenomena led researchers to apply more accurate forecasting methods. While Statistical models are more traditional to use for such complex behaviors, Chaos theory has been paved a new way to hydrologists and water engineers. In Chaos theory viewpoint, random-like behavior can be related to a simple determinism which is hidden in the background of system dynamics and can be shown in an optimized phase space. If conditions of a chaotic system which Chaos Theory has been stated dominate the system behavior, dynamics in the phase space follows a fracatal pattern which is aforementioned hidden determinism and called the attractor. The case study is the Kashkan River which is located in southwestern of Iran, Lorestan province where semi arid climate is predominant. While number of dry days reaches to 185 in year, average precipitation is 375.3 mm during a water year. The daily runoff time series of the Kashkan River have been analyzed using Chaos theory, following its observed random-like behavior. To perform chaotic analysis, a phase space should be reconstructed by determining optimize time delay and embedding dimension.Various methods has been suggested to calculate the time delay including the Average Mutual Information method which have been gained more popularity among the others. In this paper, a new method has been presented to estimate optimize time delay in the base of the AMI method. Therefore, instead of using first local minimum of Mutual Information Function, its overall minimum has been considered to estimate optimize time delay. This method has been applied to daily runoff time series of the Kashkan River and its efficiency has been studied in fractal dimension estimation methods. While False Nearest Neighbors and Correlation Dimension methods have been employed to evaluate fractal dimension of system attractor, Sensitivity to initial conditions have been studied using Lyapunov Exponent and Kolmogorov Entropy methods as the other majar feature of chaotic behavior.
At first part of this study, AMI method and Mutual Information Function have been examined theoretically by evaluating its performance in a chaotic map called Rossler Map. It has been shown that first minimum of MIF can be an optimize option to form best illustration of system chaotic attractor.While the first local minimum is more effective to mathematical functions such as RM, some other issues should be considere when the case study is a natural system where MIF can have many minimums in different local sections. As mentioned earlier, overall minimum of MIF have been used to determine time delay as a new method phase space reconstruction which is obtained τ =107 days. False Nearest Neighbors and Correlation Dimension method have been used to evaluate fractal dimension, which showed the existence of a fractal attractor in phase space, and also the superiority of new method for phase space reconstruction. Positive Lyapunov Exponent is a sign of sinstivity to initial conditions and therefore, chaotic behavior in system, which is more emphasized by calculating Kolmogorov Entropy in 3 different radius. Regardless of the radius is selected, KE reaches to a finite number which is an evidence of chaotic behavior. While a certain number can not be calculated for Kolmogorov Entropy, it can be observed that by selecting a
1 - MSc Student of River Engineering, Jundi-Shapur University of Technology, Dezful, Iran.
2 - Assistant Professor of Civil Engineering, Jundi-Shapur University of Technology, Dezful, Iran.
*- Corresponding author: moshfegh@jsu.ac.ir, P. O. Box: 64615-334
Received: 2009/08/08 Accepted: 2010/11/11
smaller radius, KE reaches to a fininite positive number. Hence, system entropy is consistent and follows a chaotic pattern.
Results indicated Chaos existence in this time series and suitability of Chaos theory-based models for governing system of flow in the Kashkan River. Sensitivity to initial conditions and fractal attracor as two major characteristics of chaotic system has been observed. In addition, a new AMI-based method to determine optimize time delay has been showed better indication of system chaotic behavior in phase space. Compared with traditional approach, the attractor which has been formed in reconstructed phase space behaves more constantly by increasing embedding dimension in both FNN and CD methods. In comparison to similar studies with times series with different lengths, it can be concluded that length of a time series can not effectively adjust overall conclusion in chaotic analysis of the Kashkan river flow governing system. Generally, evidences of chatic behavior in the Kashkan riverflow time series has been confirmed. Hence, employing chaotic models can be very helpful to forecast.