WATER RESOURCES RESEARCH, VOL. 35, NO. 3, PAGES 895-897, MARCH 1999 Comment on "Nonlinear analysis of river flow time sequences" by Amilcare Porporato and Luca Ridolfi B½11i½ Sivakurnar, Kok-Kwang Phoon, Shi½-Yui Liong, and Chih-Young Liaw Department of Civil Engineering, National University of Singapore, Singapore 1. Introduction One of the assumptions in the development of methods for identification andprediction of chaotic systems is that the time series is noise-free. The presence of noiselimits the perfor- mance of manytechniques of identification and prediction of chaotic systems [e.g.,Schreiber and Kantz, 1996]. Sincereal dataare inherently contaminated bynoise, conclusions regard- ing the existence of chaos and hence the predictability in var- ious physical and naturalsystems are often subject to debate. However, whenthe application of these methods to real data suggests the presence of a strongdeterministic component, then naturallythe next step is to look at the possibility of separating the deterministic signal from the noise toward im- proving theidentification and prediction results. In theirstudy,. Porporato and Ridolfi [1997] provided clues to the existence of a strong deterministic component in the river flow data of Dora Baltea. Subsequently, theyattempted to reduce the noise present in the datausing a nonlinear noise reduction method [Schreiber and Grassberger, 1991]and reported improvements in the estimation of correlation dimension andprediction. The results are encouraging to hydrologists, whooftenhave to deal with field data contaminated by noise. The purposeof the present discussion is to pointout some of the potential prob- lems in the application of such noise reduction methods to the riverflowdata (or anyreal data) andto highlight the existence of one possible approach to overcome these problems. 2. Discussion of Results of River Flow Analysis Two importantfactorsinfluence the effectiveness of the noise reduction method: (1) the selection of the optimal values of theparameters involved in the method and(2) the selection of the number of iterations required to achieve optimalnoise reduction. The authors adopted a trial and error procedure to select the optimal values of the parameters. The optimal stop- pingcriterion for the iteration procedure wasselected asthe point at which the meanof the absolute valueof the correc- tions between successive iterationsbecame insignificant. Pot- potatoand Ridolfi [1997]reported the results achieved after 200 iterations to show the improvements in the estimation of correlation dimension and prediction, sincethey noted that above 200 iterations an unjustifiable calculation time wasnec- essary in order to produce significant corrections. It should be notedthat the optimal values of the parameters depend largely on the leveland the nature of noise. Further- more, in the absence of prior knowledge on the noise levelor the noise-free signal, as in the case of river flow series, the selection of a stopping criterion based on the difference in the mean absolute corrections betweensuccessive iterationsmight Copyright 1999 by the American Geophysical Union. Paper number1998WR900033. 0043-1397/99/1998WR900033509.00 possibly lead to inaccurate results. Depending on the param- eter values chosen, the difference in absolute corrections be- tween successive iterationsmight be (1) still significant even after all the noise is removed, resulting in overcorrection of the dataand (2) insignificant even before all the noise is removed, resulting in underremoval of the noise. The results reported by the authors provide indications to the possible occurrence of one such problem, i.e., over-correction of the data. Porporato and Ridolfi [1997, Figure 12] indicatethat the noise reduction (or mean absolute correction) increases with the growing number of iterations without reaching anyconver- gence, though the increase is rapid duringthe first iterations and is gradual with further increase in the numberof itera- tions. The figure shows that the meanabsolute correction after 200iterations is -4 m3/s, which is approximately equal to 6% of the mean absolute deviation of the river-flow series, and thereforethe noiselevel reduced may be considered as 6%. It shouldbe noted that this 6% noiserefers only to the additive measurementnoise, and therefore the total noise level in the data should be higher than this value, since the system is inherently contaminated by some amount of dynamical noise also, which wasnot dealtwith by Porporato and Ridolfi [1997, p. 1360]. Recent studies [e.g.,Schreiber and Kantz, 1996] havedem- onstratedthat noise is one of the most prominent limiting factors for the predictability of chaotic systems. A noise level of >6% should appreciably affect the prediction results, since the prediction error will increase rapidly with the noise levelbe- cause (1) the prediction errorcannot be smaller thanthe noise levelsince the noise part of the futuremeasurement cannot be predicted and(2) the measured values themselves are contam- inatedby noise, inducing an error proportional to and of the order of the noiselevel [Schreiber and Kantz, 1996].However, the correlation coefficient of the 24-hourforecasts reportedfor the noisy(interpolated) time series is as high as 0.95 [ see, Porporato and Ridolfi, 1997, 1364 and 1365],which is unex- pected considering the presence of >6% noise, as discussed above. Furthermore, the correlation coefficient of the noise- reduced (interpolated) datais reported as0.98 [see Porporato and Ridolfi, 1997p. 1365]indicating only a marginal improve- ment in prediction, which is again unexpected for a noise reduction of >6%. The high correlation coefficient achieved for the noisy time series and the very marginal improvement after noise reduction indicate that the noise present in the data is probably muchsmaller than that removed. Moreover, the improvement in predictionachieved through interpolation seems to be significantly higher than that achieved through noisereduction, casting doubts on the necessity for noisere- duction in the river flow data. In view of these issues, it is essential to estimate the noise level in the data, which could alsoprovidean indication towardthe selection of an appro- priate noisereduction method. Our interpretation, on the basis of the reported results, is 895