Noise reduction in chaotic hydrologic time series: facts and doubts A. Elshorbagy a, * , S.P. Simonovic b , U.S. Panu c a Kentucky Water Research Institute, University of Kentucky, Lexington, KY 40506-0107, USA b Department of Civil and Environmental Engineering, and Institute for Catastrophic Loss Reduction, University of Western Ontario, London, ON, N6A 5B9 Canada c Department of Civil Engineering, Lakehead University, Thunder Bay, ON, P7B 5E1 Canada Received 12 September 2000; revised 27 August 2001; accepted 14 September 2001 Abstract The issues of noise reduction and the reliability of its application to hydrologic time series are discussed. First, the concepts of noise, its effect, and noise reduction are brie¯y presented. Second, a few published articles in hydrology are critically reviewed with regard to the application of noise reduction to hydrologic data. Third, a case study of the English River, Ontario, Canada, is used to support the conclusions. It is found that the commonly used algorithm for noise reduction in hydrologic data might also remove a signi®cant part of the original signal and introduce an arti®cial chaoticity to the data. It is recommended that current noise reduction algorithms should be applied with caution and used only for better estimation of chaotic invariants. The raw data should always be the basis for any further hydrologic analysis. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Chaos theory; Noise reduction; Arti®cial neural networks; Nonlinear time series analysis 1. Introduction In mid eighties and early nineties, when meteorol- ogists and hydrologists (e.g. Fraedrich, 1987; Rodri- guez-Iturbe et al., 1989) imported the principles of chaos theory from physics to apply them to hydrologic time series, caution in both analyses and reporting the results was exercised. In general, one can ®nd state- ments indicating that describing a geophysical time series as a fully chaotic system, might not be feasible. For example, Kantz and Schreiber (1997) mentioned that any time series might have components of both systems, chaotic and stochastic. The analyst has to decide whether the process to be modeled is linear stochastic or deterministic chaos. Such lack of certainty and blurring of the boundaries between the two systems have been expressed in the ®nal remarks of Rodriguez-Iturbe et al. (1989). They raised the question of whether or not the complex multidimen- sional patterns of the rainfall storm can be usefully described by a single attractor in a single state space. Over a period of a decade, a few case studies applying principles of chaos to hydrologic time series have been reported (e.g. among others, Jayawardena and Lai, 1994; Lall et al., 1996; Porporato and Ridol®, 1997; Wang and Gan, 1998). Until 1999 questions such as whether rainfall is chaotic or not was raised by Sivakumar et al. (1999b). The overriding objective of their work was to ensure the existence of some chaos in the time series. Driven by the application-oriented approach of the hydrologists, Jayawardena and Lai (1994) advanced one step further by applying a local nonlinear approx- imation method for short-term prediction of daily rainfall and stream¯ow data. Similarly, a local nonlinear prediction was applied by Sivakumar et al. Journal of Hydrology 256 (2002) 147±165 0022-1694/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S0022-1694(01)00534-0 www.elsevier.com/locate/jhydrol * Corresponding author.