Is the streamflow process chaotic?

In this paper, the existence of low-dimensional chaos is investigated for the streamflow series of the upper Yellow River and the lower Rhine River with the correlation dimension method. No finite correlation dimension, which is crucial for identifying a chaotic system, is found for the existence of lowdimensional chaos in the streamflow series under study. Because streamflow processes are inevitably influenced by measurement and dynamical noses, to investigate the impacts of dynamical noises on the identification of streamflow dynamics, we analyzed three known chaotic system corrupted with different levels of dynamic noise. It is found that when noise level is low, the chaotic attractor can still be well preserved and we can give basically correct estimate of correlation dimension. That indicates that even if we found the evidence of the existence of chaos in a time series, it does not necessarily mean determinism. A stochastic system could present chaotic behavior. Therefore, even if the streamflow process is chaotic, it does not necessarily mean determinism. On the other hand, in the presence of high-level dynamical noise, it is hard or even impossible to identify a chaotic system. Because the streamflow process usually suffers from strong natural and anthropogenic disturbances which are composed of both stochastic and deterministic components, consequently, it is not likely to identify the chaotic dynamics even if the streamflow process is indeed low-dimension chaotic process under ideal circumstances (i.e., without any or only with small enough stochastic disturbances). observed also for stochastic processes, such as ARMA processes (e.g. Jayawardena & Lai, 1994). Random noises with power law spectra may provide convergence of the Kolmogorov entropy, which implies that the observation of a finite or null value of the K2 entropy in the analysis of data is not enough to infer that the system is dominated by a deterministic process such as low-dimensional chaos. (e.g. Provenzale et al., 1991). A stochastic process, be it linear or nonlinear, can also produce accurate shortterm prediction but not long-term predictability, which is a typical characteristic of a chaotic process. Surrogate data method, which has a typical null hypothesis of Gaussian linear stochastic process, is not specifically designed for detecting chaos. In addition, phase-randomized surrogates may produce spurious identifications of non-random structure (e.g. Rapp et al., 1994). Consequently, none of the above methods can provide a conclusive resolution of whether a given data set is chaotic. What makes it worse for testing for chaos is the inevitable presence of noise in the observed data sets (e.g., Schertzer, 1997), such as hydrological time series. While we cannot prove the existence of chaos with the above methods, we can conclude the absence of chaos if there is no finite correlation dimension for the data set of interest, because it is well recognized that chaotic processes should have finite correlation dimension, provided that the data set is of sufficient size. Therefore, it is crucial to estimate correlation dimension for detecting the presence or absence of chaos. For this reason, correlation exponent method is used by almost all the researchers for detecting chaos in hydrological processes (e.g., Jayawardena & Lai, 1994; Porporato & Ridolfi, 1997; Pasternack, 1999; Bordignon & Lisi, 2000; Elshorbagy et al., 2002, Khan et al., 2005). In this study, we will investigate whether or not chaos exists in streamflow processes with correlation exponent method, and whether it is possible to identify a chaotic system in the presence of strong dynamical noises. The paper is organized as follows. After a brief describtion of the data used in this study in Section 2, correlation exponent method will be applied to test for the presence of chaos in the streamflow series of two rivers in Section 3. In Section 4, the influence of dynamical noises on streamflow dynamics will be discussed. Finally, the paper ends with a conclusion in Section 5.