Analysis of time series in hydrological processes using chaos theory (Case study: Monthly rainfall of Urmia Lake)

Scientific method for correct recognition and understanding of the hydrologic phenomena is investigation of their simple models. Generally, a model is a simple representation of a complex system. In mathematical models, behavior of the system is described by a series of mathematical equations, along with logical relationships between variables and parameters. Despite the various proposed mathematical models for modeling of the hydrologic phenomena, there is not a unique approach in this respect. This might be due to spatial and temporal variability of hydrologic phenomena and also lack of mathematical tools for extraction of proper structure for these phenomena. These variations are the result of dependability of the phenomena on different components. This problem has caused the past researches on hydrologic modeling to view the situation as random and probabilistic. The performance of most natural phenomena, including hydrologic problems, in short time scales, to be viewed as completely random and without any trend. But, with a change in time scale, and using sophisticated models, a type of interval and order will be observed. Nowadays, researchers believe that hydrologic phenomena, which have dynamic and nonlinear nature, could be better analyzed by nonlinear and deterministic chaotic models. Hydrologic components in lakes have non-linear and dynamic nature. But, since the changes that these components create in the lakes don’t happen suddenly, it is possible to study and predict some of these elements in the hydrologic cycle. Nowadays, with the evolution of computer models, it is believed that analysis, modeling and control of complex natural phenomena, including hydrologic processes, could be better performed with chaotic models than probabilistic models. Studying the hydrologic components in analysis of the water resources systems, such as lakes, is very important in their quantitative and qualitative operation and management. Due to the importance of precipitation in variations of water level in Urmia Lake, located in north-western Iran, the chaos theory could be a powerful approach to analyze and model the complex behavior of such phenomena. Investigation of chaotic or random behavior of rainfall time series in the lake, for the choice of the best suited rainfall simulation approach, is an important and controversial issue that has been dealt with in this research. First, using the correlation dimension method (CDM), the monthly rainfall time series of Urmia Lake was analyzed over a period of 40 years (1967-2007). After calculating the delay time using average mutual information (AMI) and also calculation of the embedding dimension using false nearest neighbor (FNN) algorithm, the phase space was reconstructed and then the correlation dimension was determined. Then, by using Lyapunof exponent and Fourier power spectrum, the existence of chaos was investigated. Results revealed that presence of chaos in the rainfall time series of Urmia Lake is evident with the non-integer CDM of 2.56, positive value of Lyapunof exponent (maximum of about 2.5) and broad band Fourier power spectrum. Consequently, the system behavior is regular; in other words, the system is not considered random. In such a system, chaos theory has the ability to extract short-term time series from long-term records. In addition, the existence of low-dimensional chaos implies the possibility of accurate short-term predictions of precipitation.