Does the river run wild? Assessing chaos in hydrological systems
Gregory B. Pasternack
1Department 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 con-troversial 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 withD65. 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 thede factotool 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 predict-ability 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 so-phisticated GCMs demonstrate that forecasts have a sensitive dependence on their initial conditions [17]. As a result, even if future GCMs perfectly simulate the at-mosphere, 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].
Because sophisticated dynamical models are not available for many systems of interest, chaos-based time series analysis oers 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. Unfortu-nately, 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. IfDis non-integer then the attractor is called a `strange attractor' because it has a complex structure that is self-similar at all scales. Strange attractors are examples of
1Tel.: +1 530 754 9243; fax: +1 530 752 5262; e-mail:
gpast@ucda-vis.edu.
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 geome-try [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 vari-able is transformed into a time series of a set of varivari-ables by lagging the datamtimes and assigning theith lagged series to theith dimension Fig. 1. The distances between one of the points (the reference) and others in the re-constructed m-dimensional phase space are measured Fig. 2 and compared to the radius, r, of a sphere cen-tered 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 dierent reference points from the set. When ln[C(r)] versus ln[r] is plotted for a given embedding dimensionm, 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
satu-ration occurs is the ®nal estimate ofDfor the attractor, with Dc6D.
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, sosmust 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 multidimen-sional systems [6]. Studies that have sought to assess the
eect ofsonDc(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 au-thors have neglected the `proximal points' constraint that the distances between points that are closer in time thansshould be excluded from the computation ofC(r) [5,9,31]. The geometric explanation for this constraint is that points close in time fall on a low-dimensional sur-face 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 computedDc.
A ®nal important consideration for assessing the re-liability 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 20:4m data points,
whereAis the greatest integer lower thanDc andmthe
embedding dimension, mean that few hydrologic re-cords can be assessed for greater than 5-D attractors since as many as 10,000 points require a 27 yr daily record. Also, dierent variables of a given system may require dierent 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 be-havior [12,18,32].
Even if sucient 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 tom10 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 Fig. 1. Transformation of (a) measured data into (b) lagged 2-D set.
scaling exists diminishes asmincreases up to a critical embedding dimension, beyond which no scaling region can be accurately de®ned. If the critical embedding di-mension for a data set is less thanDcfor that data, then Dc cannot be correctly determined [3,33]. The
conse-quence of ignoring these constraints is a signi®cant un-derestimate 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 sto-chastic surrogate embedded to dimensionm, the slope of
C(r) for lowrwill be found to closely approachmgiven enough data points [22]. In other words, a sucient 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 ofDc, 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 frommfor 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 onmand thus do not become constant before the scaling region vanishes asmincreases, then the correlation dimension of the attractor cannot be characterized. Also, if the slope of C(r) for increasingmfor the control shows so much deviation from m that the real data cannot be distinguished, thenDcagain 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 appro-priate and objective baseline would be a stochastic time series with the same power spectrum (and thus auto-correlation) 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
correlation dimensions were computed and compared to known fractal dimensions [10]. Early CIAs examined short oxygen isotope records with additional interpo-lated values that were highly correinterpo-lated [4,20]. Grass-berger [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 ad-dressed 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 dicult 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 dis-charge derivative (Q t1ÿQt) and found no low
di-mensional 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]
investi-gated rivers in Hong Kong, but used very short (2±3 days) time lags. Their reportedDc values of0.45
vio-late reality; no fewer than 3 degrees of freedom can generate chaos, and chaotic attractors cannot have
D<1, except on a mapping [21]. Physically, Dc0.45
means that a river could arbitrarily jump from one lo-cation to another and likewise discontinuously vary its velocity and acceleration. Beauvais and Dubois [1] re-cently 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
The mean discharge recorded at the gage is 1.94 cms (68.4 cfs). At no time has the river run dry, but on oc-casion 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 ran-domized 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 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 m D
c Refs.
Henon map 20,000 Yes 1.25 [10]
Logistic map 25,000 Yes 0.5 [10]
Lorenz equations 15,000 Yes 2.05 [10]
(A)Climate records
Oxygen isotopes 1 500 (184R) No 3.1 [20]
Oxygen isotopes 1 230 (184R) Yes >10 >4.5 [9]
Oxygen isotopes 2 182 Yes >15 >4.4 [4]
Tree ring record 7100 Yes >12 >10 [9]
(B)Sea surface conditions
Daily temp. 139 13,870 No >19 >8 [33]
Daily pressure 139 13,855 No >19 >8 [33]
Daily temp. 244 13,860 No >19 >8 [33]
Daily pressure 244 13,877 No >19 >8 [33]
(C)Atmospheric conditions
Rel sun duration/day
30 yr Daily record 10,950 No >18 >8 [5]
Winter seasons 120 (29´C) No 12 3.1 [5]
Summer seasons 120 (30´C) No 10 4.3 [5]
Zonal wave amplitude
10 yr Daily record 3650 No >15 >8 [5]
Winter seasons 120 (9´C) No 7 3 [5]
Summer seasons 120 (10´C) No 9 3.6 [5]
Surface pressure
15 yr Daily record 5475 No >15 >9 [5]
Winter seasons 120 (14´C) No 8 3.2 [5]
Summer seasons 120 (15´C) No 10 3.9 [5]
Winter surface pressure 120 (14´C) Yes >15 >6.8 [4]
Daily 500 mbar height 365 No 9 5 [11]
0.4s vert. wind vel. 1500 No 8 4 [11]
10s vert. wind vel. 3960 Yes 10 7.3 [28]
Daily 500 mbar height 12,084 (9´ST) No 8 ~6 [3]
Daily surface pressure 32,870 No >19 >8 [33]
Daily surface temp. 36,555 No >19 >8 [33]
Daily surface temp. 14,245 No >19 >8 [33]
(D)Rainfall records
15s Boston rainfall 1990 No 5 3.8 [22]
Time to 0.01 mm rain 1 4000 No 10 3.3 [27]
Time to 0.01 mm rain 2 3991 No 10 3.8 [27]
Time to 0.01 mm rain 3 3316 No 10 3.6 [27]
Daily rainfall
Nim Wam 4015 No 29 0.95 [14]
Fanling 4015 No 33 1.76 [14]
Tai Lam Chung 4015 No 32 1.65 [14]
(E)Stream¯ow records
Shek Pi Tau 7300 No 7 0.455 [14]
Tai Tam East 6205 No 10 0.46 [14]
Oubangui River 696 No ? 3.1 [1]
South Twin River 8458 Yes ? ~8 [24]
Merced River 27,009 Yes ? >10 [25]
Reynolds Mountain 8800 Yes 20 >4 [31]
Western Run 17,927 Yes 14 >5 This study
(k146 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 suce to achieve re-liable 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 Nref1500 was used in all analyses, with
ap-proximately 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
m4, 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 in-tegral curves as a function ofrhighlight the dierences between the real data and the stochastic surrogate Fig. 6(c) and (d). For all values ofmthe slopes for the stochastic series more rapidly approach their scaling regions than the corresponding slopes for the real data
as r decreases. Also, where scaling regions exist (m4, 6, 8), they are larger for the stochastic surrogate than for the real data. However, for both cases the scaling re-gions vanish beyond m8. For m4, 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. Withm8 the limited quantity of data (n16 905 after embedding) depresses the scaling exponent of the sto-chastic surrogate to the point that the real data is no longer distinguishable. Thus, the only reliable conclu-sion is that the mean daily discharge for Western Run cannot be governed by an attractor withD65.
5. Discussion and conclusions
Despite the improved capabilities of correlation in-tegral analysis over the box-counting method, there has been little success in ®nding attractors for river dis-charge and other hydrologic variables. Scrutiny of the literature on the fractal dimensions of hydrologic sys-tems 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 com-pounded 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 undis-turbed hydrologic systems such as Western Run, a typical Mid-Atlantic Piedmont river, CIA is not capable of identifying any low dimensional strange attractor. Fig. 5. Calibration of the number of reference points needed to cal-culateC(r). For this data set as few as 400 points can suce, but a conservative 1500 were used.
Fig. 3. Mean daily discharge record for Western Run, Maryland. The estimated maximum discharge of 7000 cfs occurred during Hurricane Agnes in 1972.
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 dis-charge 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 re-cord 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. Fur-thermore, embedding the data into ever higher dimen-sions 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 sys-tem, 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 di-mensional attractors. Even though high didi-mensional attractors may exist, at some point, say 100 degrees of freedom, the distinction between chaos and random-ness has no value for applying the discoveries of chaos mathematics to understand hydrologic systems. Instead of searching for chaos using these simple time series 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 ofr.
Table 2
Assessment of the dependence ofDcon 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 dimen-sional attractor)
m Real data (Dc) Stochastic surrogate (Dc)
4 3.83 3.95
6 5.59 5.8
8 7.42 7.46
10 ÿ 9.65a
12 ÿ 10.86a
14 ÿ ÿ
analysis tools, the present analysis suggests that hydrologists would gain more insight into the physics of natural systems by expanding predictability and er-ror propagation experiments using the dynamical models based on ®rst principles. Following this ap-proach, CIA and other methods of chaos-based time series analysis could play an important role in evalu-ating 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 lowD
(or vice versa), then the model would be missing im-portant dynamics that might lead to overcon®dence in its predictive capabilities.
Acknowledgements
I gratefully acknowledge Charles Meneveau, Haydee Salmun, and Marc Parlange (Johns Hopkins University) as well as Carlos Puente (University of California, Da-vis) and anonymous reviewers for discussions and re-views.
Appendix A. Correlation integral analysis procedure
Begin with a time series with a large number of data points,n. Calculate the autocorrelationrk 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 ofsmay have signi®cant corresponding autocorrelations [31].
Create the data setY(t) with an embedding dimension
m, where
Y t xt;x tÿs;x tÿ2s;. . .;xtÿ mÿ1s: 1
Calculate the correlation integralC(r) given by
C r 1
Nref
XNref
j1
1 N
XN
i1
H rÿ ÿ Yi
ÿYj
; 2
where H is the Heaviside function, with H(u)1 for
u > 0, andH(u)0 foru0, r the radius of a sphere centered on Yi, and Nref the calibrated number of
ref-erence points inY(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 calculatingC(r) only sumH(u) for data points that meet the following proximal points constraint
ti
ÿtj
>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 eachm. The scaling region for a given mis the part of the curve where the slope is constant. If the scaling re-gion 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 dimensionDc forY(t) ®ts the power law
C r arDc: 4 Saturation of Dc occurs where it becomes
indepen-dent of m for increasing embedding dimension. If the saturationDcfor 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.
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