Advances in Water Resources 23 (1999) 253±260 Does the river run wild? Assessing chaos in hydrological systems Gregory B. Pasternack 1 Department of Land, Air and Water Resources, University of California, 211 Veihmeyer Hall, Davis, CA 95616-8628, USA Received 7 April 1998; received in revised form 3 November 1998; accepted 3 February 1999 Abstract The standing debate over whether hydrological systems are deterministic or stochastic has been taken to a new level by controversial applications of chaos mathematics. This paper reviews the procedure, constraints, and past usage of a popular chaos time series analysis method, correlation integral analysis, in hydrology and adds a new analysis of daily stream¯ow from a pristine watershed. Signi®cant problems with the use of correlation integral analysis (CIA) were found to include a continued reliance on the original algorithm even though it was corrected subsequently and failure to consider the physics underlying mathematical results. The new analysis of daily stream¯ow reported here found no attractor with D 6 5. Phase randomization of the Fourier Transform of stream¯ow was used to provide a better stochastic surrogate than an Autoregressive Moving Average (ARMA) model or gaussian noise for distinguishing between chaotic and stochastic dynamics. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Chaos; Time series analysis; Stream¯ow analysis; Non-linear dynamics 1. Introduction Chaos mathematics has been increasingly perceived as the de facto tool for studying dynamical systems that deterministic and stochastic models have had limited success with predicting. The ®rst step in applying chaos mathematics is to determine whether a particular hydrologic system is in fact chaotic. This assessment can be accomplished by investigating the limits of predictability and error propagation in current operational forecasting models or by searching for indicators of chaotic dynamics in recorded time series of dynamical system variables. The chaotic nature of instantaneous weather has been ®rmly established by numerical experiments with global circulation models (GCMs). Studies using the most sophisticated GCMs demonstrate that forecasts have a sensitive dependence on their initial conditions [17]. As a result, even if future GCMs perfectly simulate the atmosphere, the predictability of weather variables would approach zero for forecasts beyond two weeks [16,26]. Detailed analyses of simple atmospheric models have been used to study the underlying characteristics of chaotic behavior [15,18,21]. 1 Tel.: +1 530 754 9243; fax: +1 530 752 5262; e-mail: gpast@ucdavis.edu. Because sophisticated dynamical models are not available for many systems of interest, chaos-based time series analysis o€ers an alternative means for identifying chaotic behavior in cases where high quality, long term hydrologic records are available. The primary tool used to look for chaos in time series has been correlation integral analysis (CIA). This fractal scaling method was introduced by Grassberger and Procaccia [10], but was precipitously adopted before important quali®cations (e.g. 9) were publicized. Subsequent attempts at CIA relied on the accuracy of the original algorithm without due consideration of relevant constraints or the physics underlying mathematical results. Consequently, the wave of initial analyses in some ®elds has been followed by a wave of corrections and counterclaims. Unfortunately, use of CIA in hydrology has followed this path, as exempli®ed by the ongoing debate over the nature of rainfall in Boston [8,22,23,27,33]. 2. Correlation integral analysis The correlation dimension (Dc ) is a measure of the dimension (D) of an attractor governing the trajectories of solutions of a dynamical system in phase space. If D is non-integer then the attractor is called a `strange attractor' because it has a complex structure that is selfsimilar at all scales. Strange attractors are examples of 0309-1708/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 0 8 - 1 254 G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 fractals [21]. Because the precise de®nitions of `fractal', `chaotic', and `strange attractor' are still debated, it is not possible to conclude that a strange attractor is necessarily chaotic or that chaos implies fractal geometry [8]. Nevertheless, obtaining a non-integer, ®nite Dc for a time series when Dc for a corresponding control stochastic surrogate (to be discussed) is unbounded demonstrates fractal scaling and suggests chaos. The procedure for computing Dc (see Appendix A) derives from Grassberger and Procaccia [10] but with important additional constraints from the subsequent literature. By this method a time series of a single variable is transformed into a time series of a set of variables by lagging the data m times and assigning the ith lagged series to the ith dimension Fig. 1. The distances between one of the points (the reference) and others in the reconstructed m-dimensional phase space are measured Fig. 2 and compared to the radius, r, of a sphere centered on the reference point. The correlation integral for a given radius, C(r), is the fraction of distances that are less than the radius, after averaging over many di€erent reference points from the set. When ln[C(r)] versus ln[r] is plotted for a given embedding dimension m, the range of ln[r] where the slope of the curve is constant is the scaling region where fractal geometry is indicated. In this region C(r) increases as a power of r, with the scaling exponent being the correlation dimension, Dc . If the time series is chaotic, then for increasing embedding dimension the computed Dc must become independent of m, i.e. `saturate'. The ®nite value of Dc where saturation occurs is the ®nal estimate of D for the attractor, with Dc 6 D. Only a small fraction of reported studies that use CIA to characterize hydrologic systems follow the procedure including all of the necessary precautions. Grassberger and Procaccia [10] state that the embedding time lag (s) may be chosen arbitrarily, but in all applications the amount of data is limited, so s must be long enough for data points to be independent to yield a meaningful attractor reconstruction [30,33]. Where this constraint has been accounted for, suitable lag times have been selected on the basis of the autocorrelation function [5,12,27,33], the mutual information function [7], and the more general redundancy criteria for multidimensional systems [6]. Studies that have sought to assess the Fig. 1. Transformation of (a) measured data into (b) lagged 2-D set. Fig. 2. Illustration of distance measurements made in correlation integral analysis. e€ect of s on Dc (e.g. 14) have used very small ranges of values (e.g. 4±16 days), which are insigni®cant compared to typical decorrelation times in hydrologic systems that may exceed 50±200 days [31]. Furthermore, many authors have neglected the `proximal points' constraint that the distances between points that are closer in time than s should be excluded from the computation of C(r) [5,9,31]. The geometric explanation for this constraint is that points close in time fall on a low-dimensional surface that overcontributes to the correlation integral. Rather than measuring the low dimensional structure along a trajectory, as analysis of proximal points does, appropriate application of CIA seeks to measure the fractal geometry of the distances to other `loops' (Fig. 2). The overcontribution of proximal points arti®cially ¯attens the slope of C(r) and thus depresses the computed Dc . A ®nal important consideration for assessing the reliability of the CIA-computed dimension is the size of the embedded time series, n. In geometric terms, the series must be long enough for the points along one ``edge'' of the attractor to reasonably represent the hypersurface. Wilcox et al. [31] and Tsonis et al. [30] summarize the literature on the number of points needed. Criteria such as 10A or 10 2‡0:4m† data points, where A is the greatest integer lower than Dc and m the embedding dimension, mean that few hydrologic records can be assessed for greater than 5-D attractors since as many as 10,000 points require a 27 yr daily record. Also, di€erent variables of a given system may require di€erent numbers of data points to obtain Dc depending on how each is coupled to the rest of the system and whether each exhibits thresholds in its behavior [12,18,32]. Even if sucient points appear to be available, the number will be substantially diminished by embedding. For example, a data set of 3316 points reduced to 1956 after embedding to m ˆ 10 with a delay corresponding to the time for the autocorrelation function to reach 0.5 [27]. This is a problem because the interval over which G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 scaling exists diminishes as m increases up to a critical embedding dimension, beyond which no scaling region can be accurately de®ned. If the critical embedding dimension for a data set is less than Dc for that data, then Dc cannot be correctly determined [3,33]. The consequence of ignoring these constraints is a signi®cant underestimate of the true dimension of a system. A useful tool for assessing the dimensional limit of a given data set has been used in some studies but not widely discussed. The approach is to conduct a parallel control CIA experiment using a stochastic surrogate derived from the original data. Using CIA on the stochastic surrogate embedded to dimension m, the slope of C(r) for low r will be found to closely approach m given enough data points [22]. In other words, a sucient quantity of random data will ®ll all available dimensions in space. In some cases stochastic systems with an in®nitely large number of degrees of freedom have been found to have ®nite values of Dc , particularly if the data set is too small [8]. However, the slope of C(r) for the control serves as a baseline for comparing real data of the same size to assess whether the underlying system is stochastic or chaotic. If the slope of C(r) tends toward independence from m for the real data faster than that for the stochastic surrogate, then the hydrologic system does not have in®nite degrees of freedom. On the other hand, if slopes for the real data are dependent on m and thus do not become constant before the scaling region vanishes as m increases, then the correlation dimension of the attractor cannot be characterized. Also, if the slope of C(r) for increasing m for the control shows so much deviation from m that the real data cannot be distinguished, then Dc again cannot be computed. These concepts will be illustrated in an example below. Studies of hydrologic records that have included stochastic controls have used either arbitrary random numbers [3,27] or sets generated using Autoregressive Moving Average (ARMA) models (e.g. 14). While the ARMA model does preserve a fraction of the power spectral density of the original data, the most appropriate and objective baseline would be a stochastic time series with the same power spectrum (and thus autocorrelation) as the original series. Such a stochastic surrogate can be generated by calculating the Fourier Transform of the time series, randomly re-assigning phases between 0 and 2p to the transform, and then returning the data to the time domain using the inverse transform algorithm [19]. No CIA studies of hydrologic records have used this approach. 3. Problems encountered with CIA Table 1 lists CIA studies from hydrology (broadly de®ned) and relevant parameters. For comparison, it begins with three well known chaotic systems whose 255 correlation dimensions were computed and compared to known fractal dimensions [10]. Early CIAs examined short oxygen isotope records with additional interpolated values that were highly correlated [4,20]. Grassberger [9] re-analyzed those data and found problems with applying CIA in those instances. Fraedrich [4,5] looked for a winter weather attractor by concatenating November±February data. This novel approach addressed the mathematical needs of CIA. However, in terms of the physical phenomenon, transition periods for the weather trajectory to settle into or out of a winter attractor exist. It is dicult to know when transition periods occurred and for how long, so doing CIA for the Nov-Feb concatenated record may not be indicative of the nature of this type of attractor. Analyses of stream¯ow have been conducted for daily, monthly, and discharge derivative values. Savard [25] analyzed the 27009 data point Merced River discharge derivative (Q t‡1† ÿ Qt ) and found no low dimensional attractor. Wilcox et al. [31] presented a thorough CIA analysis of a standardized (periodicity removed), log-transformed runo€ record and also found no low dimensional attractor. In their analysis, they showed that using a short time lag results in a signi®cant underestimate of Dc . Jayawardena and Lai [14] investigated rivers in Hong Kong, but used very short (2±3 days) time lags. Their reported Dc values of 0.45 violate reality; no fewer than 3 degrees of freedom can generate chaos, and chaotic attractors cannot have D<1, except on a mapping [21]. Physically, Dc ˆ 0.45 means that a river could arbitrarily jump from one location to another and likewise discontinuously vary its velocity and acceleration. Beauvais and Dubois [1] recently investigated monthly discharge, but used only 696 data points with no control experiment. Their 1996 study shows that new research is still applying the original method of Grassberger and Procaccia [10] without the necessary corrections discussed earlier. 4. CIA analysis of Western Run, MD Western Run drains a 155 km2 (59.8 mi2 ) watershed in upper Baltimore County, Maryland in the Gunpowder River basin. Because the river drains into Lock Raven Reservoir, a major water supply for the city of Baltimore, the watershed is kept fairly pristine. As of 1975, more than 80% of the area was reported to be forest, farmland, or pasture [2]. No signi®cant developments have occurred since then. Furthermore, the stream has been well characterized and used to study sediment yield and ¯oodplain evolution [2,13]. The gaging station for Western Run has been measuring the stream's stage and discharge since September 1, 1944, and the record for this analysis extends to September 30, 1993 yielding 17 927 mean daily observations Fig. 3. 256 G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 Table 1 List of correlation integral analyses of interest to hydrologists (data quantity, use of proximal points constraint (ppc), embedding dimension (m) of saturation, claimed correlation dimension (Dc ), and the reference are given for each Data series Data pointsa ppc Henon map Logistic map Lorenz equations 20,000 25,000 15,000 Yes Yes Yes (A) Climate records Oxygen isotopes 1 Oxygen isotopes 1 Oxygen isotopes 2 Tree ring record 500 (184 R) 230 (184 R) 182 7100 No Yes Yes Yes (B) Sea surface conditions Daily temp. 139 Daily pressure 139 Daily temp. 244 Daily pressure 244 13,870 13,855 13,860 13,877 (C) Atmospheric conditions Rel sun duration/day 30 yr Daily record Winter seasons Summer seasons Zonal wave amplitude 10 yr Daily record Winter seasons Summer seasons Surface pressure 15 yr Daily record Winter seasons Summer seasons Winter surface pressure Daily 500 mbar height 0.4s vert. wind vel. 10s vert. wind vel. Daily 500 mbar height Daily surface pressure Daily surface temp. Daily surface temp. Dc Refs. 1.25 0.5 2.05 [10] [10] [10] >10 >15 >12 3.1 >4.5 >4.4 >10 [20] [9] [4] [9] No No No No >19 >19 >19 >19 >8 >8 >8 >8 [33] [33] [33] [33] 10,950 120 (29 ´ C) 120 (30 ´ C) No No No >18 12 10 >8 3.1 4.3 [5] [5] [5] 3650 120 (9 ´ C) 120 (10 ´ C) No No No >15 7 9 >8 3 3.6 [5] [5] [5] 5475 120 (14 ´ C) 120 (15 ´ C) 120 (14 ´ C) 365 1500 3960 12,084 (9 ´ ST) 32,870 36,555 14,245 No No No Yes No No Yes No No No No >15 8 10 >15 9 8 10 8 >19 >19 >19 >9 3.2 3.9 >6.8 5 4 7.3 ~6 >8 >8 >8 [5] [5] [5] [4] [11] [11] [28] [3] [33] [33] [33] m (D) Rainfall records 15s Boston rainfall Time to 0.01 mm rain 1 Time to 0.01 mm rain 2 Time to 0.01 mm rain 3 Daily rainfall Nim Wam Fanling Tai Lam Chung 1990 4000 3991 3316 No No No No 5 10 10 10 3.8 3.3 3.8 3.6 [22] [27] [27] [27] 4015 4015 4015 No No No 29 33 32 0.95 1.76 1.65 [14] [14] [14] (E) Stream¯ow records Shek Pi Tau Tai Tam East Oubangui River South Twin River Merced River Reynolds Mountain Western Run 7300 6205 696 8458 27,009 8800 17,927 No No No Yes Yes Yes Yes 7 10 ? ? ? 20 14 0.455 0.46 3.1 ~8 >10 >4 >5 [14] [14] [1] [24] [25] [31] This study a R ± number of raw data points; C ± multiple 120 day records concatenated and ST ± multiple records space-for-time transformed. The mean discharge recorded at the gage is 1.94 cms (68.4 cfs). At no time has the river run dry, but on occasion the ¯ow has exceeded the range for which the rating curve was calibrated (e.g. Hurricane Agnes in 1972). Using the procedure in the Appendix A, CIA was conducted for the original data set and a phase randomized control. A plot of autocorrelation as a function of lag (in days) shows a small annual periodicity (Fig. 4). To insure linear independence a large time lag G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 Fig. 3. Mean daily discharge record for Western Run, Maryland. The estimated maximum discharge of 7000 cfs occurred during Hurricane Agnes in 1972. Fig. 4. Autocorrelation function for Western Run stream¯ow showing a precipitous initial drop followed by a long time before independence is achieved. A time lag of 146 days corresponding to an autocorrelation of 0.03 was selected to construct an embedded time series. 257 Fig. 5. Calibration of the number of reference points needed to calculate C(r). For this data set as few as 400 points can suce, but a conservative 1500 were used. as r decreases. Also, where scaling regions exist (m ˆ 4, 6, 8), they are larger for the stochastic surrogate than for the real data. However, for both cases the scaling regions vanish beyond m ˆ 8. For m ˆ 4, 6 the real data series shows lower scaling exponents than the stochastic surrogate (Table 2). This weak but tantalizing evidence suggests that if ten times more data were available, a high dimensional attractor might possibly be found. With m ˆ 8 the limited quantity of data (n ˆ 16 905 after embedding) depresses the scaling exponent of the stochastic surrogate to the point that the real data is no longer distinguishable. Thus, the only reliable conclusion is that the mean daily discharge for Western Run cannot be governed by an attractor with D 6 5. 5. Discussion and conclusions (k ˆ 146 days) was selected, yielding an autocorrelation of 0.03 for the daily record. Calibration of the number of reference points (Nref ) showed that as few as 400 points suce to achieve reliable statistics Fig. 5. Past studies that have assessed Nref may have found a higher minimum due to the use of extreme values as references [3]. To be conservative, a value of Nref ˆ 1500 was used in all analyses, with approximately uniform spacing of the selected references through the series. Fig. 6(a) and b plot the correlation integrals of daily discharge and control data for embedding dimensions m ˆ 4, 6, 8, 10, 12 and 14. These graphs do not show any anomalous shoulders like those illustrated by Wilcox et al. [31]. Plots of the local slopes of the correlation integral curves as a function of r highlight the di€erences between the real data and the stochastic surrogate Fig. 6(c) and (d). For all values of m the slopes for the stochastic series more rapidly approach their scaling regions than the corresponding slopes for the real data Despite the improved capabilities of correlation integral analysis over the box-counting method, there has been little success in ®nding attractors for river discharge and other hydrologic variables. Scrutiny of the literature on the fractal dimensions of hydrologic systems reveals inconsistent and unreliable applications of CIA in particular and chaos theory in general. Even though Islam et al. [12] discussed some of the problems with reported CIAs in meteorology, more faulty studies continue to be published. The problem has been compounded by analyses such as that by Tsonis et al. [30] who plotted results from 6 CIA studies to demonstrate their adequacy, even though several of them were highly criticized. For example, Fraedrich [5] demonstrated that his earlier analyses [4] were wrong, but Tsonis et al. [30] used them anyway. When properly applied to undisturbed hydrologic systems such as Western Run, a typical Mid-Atlantic Piedmont river, CIA is not capable of identifying any low dimensional strange attractor. 258 G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 Fig. 6. Correlation integrals for (a) daily discharge and (b) control series; and their local slope curves (c) and (d), respectively. Scaling regions are indicated where local slopes are constant over a signi®cant range of r. Table 2 Assessment of the dependence of Dc on embedding dimension, m (Dc for the real data does not level o€ before it becomes indistinguishable from the control sample, thus indicating the absence of a low dimensional attractor) m Real data (Dc ) Stochastic surrogate (Dc ) 4 6 8 10 12 14 3.83 5.59 7.42 ÿ ÿ ÿ 3.95 5.8 7.46 9.65 10.86 ÿ a a a In the limit. To understand why records of daily river discharge are not likely to yield low dimensional attractors it is useful to consider what happens to the data when it is embedded. Analyzed as if they are random, river discharge data ®t the lognormal distribution, with most points falling in a narrow range of values and a few present as outliers. In the case of Western Run, 0.30% of the data are <3 cms (10 cfs), 84.75% are between 3 and 30.5 cms (10±100 cfs), 14.83% are between 30.5 and 305 cms (100±1000 cfs), and 0.12% are >305 cms (1000 cfs). When this data is embedded into three dimensions, it takes the shape of a spiked sphere, with the number of spikes equal to the number of outliers (Fig. 7). For a low dimensional attractor to exist, each spike must represent a trajectory of the system. Adding more data to the record may help ®ll in an outlier trajectory, but the more data is added, the more poorly populated new spikes form, and with even more extreme trajectories. Furthermore, embedding the data into ever higher dimensions projects the spikes into new regions of sparsely populated phase space. No amount of discharge data will solve these problems, so low dimensional attractors will never be found. However, if the plot of embedded data is viewed as a mapping of a high dimensional system, then the outliers are no longer required to fall on trajectories within the embedding dimension. Even so, outlier regions would have to be visited enough to capture the fractal geometry of the mapping, and this is not happening. New methods for identifying chaos in time series have been forwarded [29], but the original surge of interest primarily stemmed from the hope for low dimensional attractors. Even though high dimensional attractors may exist, at some point, say 100 degrees of freedom, the distinction between chaos and randomness has no value for applying the discoveries of chaos mathematics to understand hydrologic systems. Instead of searching for chaos using these simple time series G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 259 Acknowledgements I gratefully acknowledge Charles Meneveau, Haydee Salmun, and Marc Parlange (Johns Hopkins University) as well as Carlos Puente (University of California, Davis) and anonymous reviewers for discussions and reviews. Appendix A. Correlation integral analysis procedure Begin with a time series with a large number of data points, n. Calculate the autocorrelation rk as a function of lag time, k. Determine s, the decorrelation time needed to achieve linear independence, with rs < X. X should be as close to zero as the data set will allow [5], but no multiples of s may have signi®cant corresponding autocorrelations [31]. Create the data set Y(t) with an embedding dimension m, where Y t† ˆ xt ; x tÿs† ; x tÿ2s† ; . . . ; x‰tÿ mÿ1†sŠ : 1† Calculate the correlation integral C(r) given by C r† ˆ Nref N ÿ
1 X 1X H r ÿ Yi ÿ Y j ; Nref jˆ1 N iˆ1 2† where H is the Heaviside function, with H(u) ˆ 1 for u > 0, and H(u) ˆ 0 for u ˆ 0, r the radius of a sphere centered on Yi , and Nref the calibrated number of reference points in Y(t) that are needed to yield consistent statistics. Do not use data whose values are more than 3 standard deviations from the mean as reference points [3]. For stream¯ow records r values for which C(r) is calculated vary with the data set, but they typically span from 0.1 to 1000. When calculating C(r) only sum H(u) for data points that meet the following proximal points constraint Fig. 7. 2-D reconstruction of attractor for Western Run (a) stream¯ow and (b) logarithm of stream¯ow showing sparsely populated outlier regions. analysis tools, the present analysis suggests that hydrologists would gain more insight into the physics of natural systems by expanding predictability and error propagation experiments using the dynamical models based on ®rst principles. Following this approach, CIA and other methods of chaos-based time series analysis could play an important role in evaluating model output to determine whether it shows the same characteristics as the real data. If model output for a high D hydrologic system was found to be low D (or vice versa), then the model would be missing important dynamics that might lead to overcon®dence in its predictive capabilities. ti ÿ tj > s: 3† (See e.g. [5,9,31]). Repeat the embedding and C(r) computations for many embedding dimensions, e.g. 2 < m < 14. Plot the slope of ln[C(r)] versus ln[r] for each m. The scaling region for a given m is the part of the curve where the slope is constant. If the scaling region vanishes as m increases, then the dimension of the attractor cannot be quanti®ed by this analysis. If a scaling region does exist, then within that region the correlation dimension Dc for Y(t) ®ts the power law C r†  a rDc : 4† Saturation of Dc occurs where it becomes independent of m for increasing embedding dimension. If the saturation Dc for a real data set is not less than that for a corresponding control phase randomized data set, then it is not possible to distinguish between chaotic and stochastic dynamics. 260 G.B. Pasternack / Advances in Water Resources 23 (1999) 253±260 References [1] Beauvais A, Dubois J. Attractor properties of a river discharge dynamical system. Eos Trans AGU 1995;73(46):F234. [2] Costa JE. E€ects of agriculture on erosion and sedimentation in the Piedmont Province, Maryland, Geol Soc Amer Bull 1975;86:1281±6. [3] Essex C, Lookman T, Neremberg MAH. The climate attractor over short timescales. Nature 1987;326:64±6. [4] Fraedrich K. Estimating the dimensions of weather and climate attractors. J Atm Sci 1986;43(5):419±32. [5] Fraedrich K. Estimating weather and climate predictability on attractors. J Atm Sci 1987;44(4):722±8. [6] Frazer AM. Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria. Phys D 1989;34:391±404. [7] Frazer AM, Swinney HL. Independent coordinates for strange attractors from mutual information. Phys Rev A 1986;33(2):1134± 40. [8] Ghilardi P, Rosso R. In: I. Rodriguez-Iturbe et al., editors. Comment on `Chaos in rainfall'. Water Resour Res 1990;26(8):1837±39. [9] Grassberger P. Do climate attractors exist? Nature 1986;323:609± 12. [10] Grassberger P, Procaccia I. Characterization of strange attractors. Phys Rev Lett 1983;50(5):346±9. [11] Henderson HW, Wells R. Obtaining atractor dimensions from meteorological time series. Adv Geophys 1988;30:205±237. [12] Islam S, Bras RL, Rodriguez-Iturbe I. A possible explanation for low dimension estimates for the atmosphere. J Appl Met 1993;32:203±8. [13] Jacobson RB, Coleman DJ. Stratigraphy and recent evolution of Maryland Piedmont ¯ood plains. Amer J Sci 1986;286:617± 37. [14] Jayawardena AW, Lai F. Analysis and prediction of chaos in rainfall and stream¯ow time series. J Hydrol 1994;153:23±52. [15] Lorenz EN. Deterministic nonperiodic ¯ow. J Atm Sci 1963;20(2):130±41. [16] Lorenz EN. Atmospheric predictability experiments with a large numerical model. Tellus 1982;34:505±13. [17] Lorenz EN. Can chaos and intransitivity lead to interannual variability? Tellus 1990;42A:378±89. [18] Lorenz EN. Dimension of weather and climate attractors. Nature, 1991;353:241±4. [19] Newland DE. An introduction to random vibrations and spectral analysis. 2nd ed. New York: Longman Scienti®c and Technical, 1984. [20] Nicolis C, Nicolis G. Is there a climate attractor? Nature 1984;311:529±32. [21] Ott E. Chaos in dynamical systems. New York: Cambridge University Press, 1993. [22] Rodriguez-Iturbe I, De Power BF, Shari® MB, Georgakakos KP. Chaos in rainfall. Water Resour Res 1989;25(7):1667±75. [23] Rodriguez-Iturbe I, De Power BF, Shari® MB, Georgakakos KP. In: I. Rodriguez-Iturbe et al., editors. Reply to Comment on `Chaos in rainfall'. Water Resour Res 1990;26(8):1841±42. [24] Savard CS. Correlation integral analysis of South Twin River stream¯ow, central Nevada: preliminary application of chaos theory. Eos Trans AGU 1990;71(43):1341. [25] Savard CS. Looking for chaos in stream¯ow with dischargederivative data. Eos Trans AGU (Spring Meeting suppl) 1992;73(14):50. [26] Schubert SD, Suarez M. Dynamical predictability in a simple general circulation model: average error growth. J Atm Sci 1989;46(3):353±70. [27] Shari® MB, Georgakakos KP, Rodriguez-Iturbe I. Evidence of deterministic chaos in the pulse of storm rainfall. J Atm Sci 1990;47(7):888±93. [28] Tsonis AA, Elsner JB. The weather attractor over very short time scales. Nature 1988;333:545±7. [29] Tsonis AA, Elsner JB. Nonlinear prediction as a way of distinguishing chaos from random fractal sequences. Nature 1992;358:217±20. [30] Tsonis AA, Elsner JB, Georgakakos KP. Estimating the dimension of weather and climate attractors: important issues about the procedure and interpretation. J Atm Sci 1993;50(15):2549±55. [31] Wilcox BP, Seyfried MS, Matison, TH. Searching for chaotic dynamics in snowmelt runo€. Water Resour Res 1991;27(6):1005± 10. [32] Zeng X, Pielke RA. What does a low dimensional weather attractor mean? Phys Lett A 1993;175:299±304. [33] Zeng X, Pielke RA, Eykholt R. Estimating the fractal dimension and the predictability of the atmosphere. J Atm Sci 1992;49(8):649±59.