DISCUSSION of Evidence of chaos in the rainfall runoff process * Which chaos in the rainfall runoff process?

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1 Hydrological Sciences Journal des Sciences Hydrologiques, 47(1) February DISCUSSION of Evidence of chaos in the rainfall runoff process * Which chaos in the rainfall runoff process? DANIEL SCHERTZER LMM, Université Pierre et Marie Curie, 4 Place Jussieu, F Paris Cedex 05, France and Météo-France, 1 Quai Branly, F-75007, Paris, France schertze@ccr.jussieu.fr IOULIA TCHIGUIRINSKAIA UMR Sisyphe, LGA, Université Pierre et Marie Curie, 4 Place Jussieu, F Paris Cedex 05, France tchigin@biogeodis.jussieu.fr SHAUN LOVEJOY Physics Department, McGill University, 3600 University Street, Montreal, Quebec H3A 2T8, Canada lovejoy@physics.mcgill.ca PIERRE HUBERT UMR Sisyphe, Ecole des Mines de Paris, 35 Rue St Honoré, F Fontainebleau, France hubert@cig.ensmp.fr HOCINE BENDJOUDI UMR Sisyphe, LGA, Université Pierre et Marie Curie, 4 Place Jussieu, F Paris Cedex 05, France bdj@biogeodis.jussieu.fr MICHELE LARCHEVÊQUE LMM, Université Pierre et Marie Curie, 4 Place Jussieu, F Paris Cedex 05, France larchevq@lmm.jussieu.fr Abstract In the 1980s, there were numerous claims, based on estimates of the correlation dimension, that the variability of various geophysical processes, in particular rainfall, is generated by a low-dimensional deterministic chaos. Due to a recent attempt (Sivakumar et al., 2001) to revive the same approach and with claims of an analogous result for the rainfall runoff process, we think it is necessary to clarify why this approach can be easily misleading. At the same time, we ask which chaos is involved in the rainfall runoff process and what are the prospects for its modelling? Key words rainfall runoff models; (multi-) fractals; chaotic dynamics; stochastic processes; nonlinear analysis; time series analysis; correlation dimension * by Bellie Sivakumar, Ronny Berndtsson, Jonas Olsson & Kenji Jinno, Hydrological Sciences Journal 46(1), Open for discussion until 1 August 2002

2 140 Daniel Schertzer et al. Quel chaos dans le processus pluie débit? Résumé Au cours des années 1980, il a été souvent annoncé, à partir d estimations de la dimension de corrélation, que la variabilité de différents processus géophysiques, en particulier la pluie, était générée par un chaos déterministe de faible dimension. Du fait d une récente tentative (Sivakumar et al., 2001) de ressusciter cette approche et l annonce d un résultat similaire sur le processus pluie débit, nous pensons qu il est nécessaire de clarifier pourquoi cette approche peut être facilement trompeuse. En même temps, nous indiquons quel chaos est en jeu dans le processus pluie débit et quelles sont les perspectives pour sa modélisation. Mots clefs modèles pluie débit; (multi-) fractals; dynamique chaotique; processus stochastiques; analyse nonlinéaire; analyse de séries temporelles; dimension de corrélation INTRODUCTION Without doubt the understanding of the dynamics of the rainfall runoff process constitutes one of the most important and challenging problems in hydrology. Its importance has been emphasized in the prospective of the IAHS decade of Prediction of Ungauged Bassins. Unfortunately, in our opinion, the paper by Sivakumar et al. (2001) does not seem to contribute to a clarification of these issues and may lead to erroneous conclusions. This paper, as well as several similar ones written or co-authored by the same lead author (Sivakumar et al., 1999a,b; Sivakumar, 2000), tries to show with the help of the correlation dimension that hydrological processes have a low-dimensionality (more precisely of the order 6 7), and therefore should fall into the category of (lowdimensional) deterministic chaos. One can recall that it has been known for the last 15 years that such an approach may be misleading. Since Sivakumar et al. (2001) repudiated this knowledge by calling it belief, we will emphasize the well-known fact that an empirical low value of dimension can easily be spurious. It can be an artefact of the finite size of the data set, rather than a reliable estimate of the dimensionality of the process, or it can result from the stochastic nature of the process. With the help of a synthetic series generated by a stochastic process related to the rainfall process, we give a concrete example. We conclude with a discussion on the type of chaos that is involved in the rainfall runoff process. WHAT IS CHAOS? Origin of the chaos concept Chaos is one of the oldest concepts and paradigms. Its origin could be traced back at least to Greek mythology. In this framework, Chaos was the disordered world that preceded Cosmos that is our present ordered world. Therefore, in its wide sense, chaos merely refers to some kind of disorder. And indeed, it has been used for rather distinct types of disorder. For instance, Wiener (1938) called pure chaos the Brownian motion, which he mathematically formalized. More recently, Kahane (1995) referred to Lévy chaos as the extension of this pure chaos to motion, defined with the help of the Lévy variables, which broadly generalize the Gaussian variables.

3 Which chaos in the rainfall runoff process? 141 Deterministic chaos During the last 30 years, chaos took a much more a restrictive meaning, since it became understood as a shorthand for deterministic chaos. The latter denotes the disorder generated by deterministic dynamical systems. Here the adjective deterministic means that the equations do not contain any noise source, and that, in general (but not always), the existence and uniqueness of solutions are mathematically assured. The main defining feature of such a chaotic system is the sensitive dependence of solutions on the initial conditions (and/or boundary conditions for partial differential systems). The prototype example is the celebrated Lorenz model (Lorenz, 1963), which was introduced as a mathematical caricature of atmospheric convection and has the lowest possible dimensionality, i.e. three, for chaotic differential systems. This chaos was initially viewed as a mathematical curiosity, e.g. the Lorenz model attracted little attention until the development of nonlinear time series analysis, about 20 years ago, with the pioneering work by Packard et al. (1980) and the seminal paper by Takens (1980). The core of this approach corresponds to the possible reconstruction of the phase space from (discrete) time series X n = h[x(n t)] for a given scalar observable h of a (vector-valued) trajectory x(t). This can be achieved with the help of the m-dimensional embedding space, E m which is spanned by delay vectors Y n = (X n, X n-τ, X n-2τ,..., X n-(m-1)τ ), for any given (integer) time delay τ. More precisely, a very interesting extension of the classical Witney embedding theorem (for integer dimensional manifolds) was obtained. Indeed, Takens (1980) and Sauer et al. (1991) demonstrated that if the trajectory x(t) is confined on an invariant set A, with box dimension D, then there is a unique and invertible smooth map from A to the delay embedding space E m, for m > 2D. As a consequence, the dynamics can be represented in the m-dimensional delay embedding space E m, i.e. with m independent variables. On the theoretical level, this theorem is extremely appealing, since one needs only a given scalar observable h at discrete times (n t) in order to get the dynamics of a vector-valued process x(t) for continuous time (t). However, the underlying hypothesis is rather demanding (see comment below). Furthermore, for practical reasons (discussed below), its applications were primarily restricted to very low-dimensional systems, say a dimension not much higher than 5, whereas the mathematical theory does not face such a limitation. Indeed, the main mathematical difficulties are related to the transition from finite dimensional systems to infinite systems, e.g. partial differential systems, and the corresponding disorder is often called spatially extended chaos, which is a rather new field. This practical restriction to a very narrow range of (low) dimensions had the consequence that many practitioners believed not only that low dimensionality is a requisite of deterministic chaos, but also that its empirical evidence is an indication of the possible existence of chaos. This corresponds to a misinterpretation of the embedding theorem that is discussed below in more detail. Furthermore, it turns out that nonlinear time series analysis techniques are inherently incapable of distinguishing between low-dimensional deterministic systems and high-dimensional stochastic systems (see stochastic chaos below).

4 142 Daniel Schertzer et al. Stochastic chaos Fortunately, in spite of the big impetus of the deterministic chaos theories, there have also been important developments in stochastic modelling. Some of the reasons need to be reviewed, since Sivakumar et al. (2001) seem to rule out a possible relevance of stochastic modelling to hydrological processes. Firstly, contrary to deterministic chaos, stochastic models can easily simulate spatially extended systems. In particular, this is notable for turbulent fields. Indeed, it has become widely recognized that models with a small number of degrees of freedom are inadequate to model turbulence, except in the low Reynolds number regime below the transition to turbulence. For instance, Ruelle (1989) pointed out the gap dividing simple chaotic systems and fully developed turbulence. Indeed, a qualitative new understanding of the fundamental problem of the intermittency of turbulence, i.e. the fact that only a relatively small fraction of the enormous number of degrees of freedom are effectively active, was obtained with the help of a rather new type of stochastic process. Inspired by the cascade paradigm, the latter are multiplicative rather than additive, as for the classical random walks (and associated diffusive processes). The term multiplicative chaos was indeed proposed by Kahane (1995), although the name multifractal became much more widely used. In any event, stochastic multifractal processes have been increasingly considered in geophysics (Schertzer & Lovejoy, 1991; Schertzer et al., 1997), in particular in hydrology (Lovejoy & Schertzer, 1995). Secondly, the deterministic features of a low-dimensional process can be easily concealed by contamination by a weak noise. The origin of this contamination is not only measurement noise (which could be handled in most cases by specific noise reduction procedures), but also interactions with other scales of the same process or with other processes. In other words, deterministic chaos has the very serious drawback of not being robust! There has therefore been a renewed interest in stochastic differential equations, i.e. dynamic equations that include a noise source term. For instance, it has been known for decades that a local Gaussian perturbation leads to the classical Fokker-Planck equation (e.g. Van Kampen, 1981). But it was only recently demonstrated that strongly non-gaussian perturbations lead to a broad fractional generalization of the Fokker- Planck equation (e.g. Schertzer et al., 2001), i.e. a kinematic equation for the probability, which involves fractional derivatives. Furthermore, there are expectations that this type of equation can generate multifractal fields. Furthermore, multifractal processes help to combine stochastics with dynamics, since they are based on both physics and statistics. This is particularly the case for hydrology, since, as emphasized by Hubert et al. (1993), it reconciles two opposing views on extreme precipitation, the extreme maximum precipitation (PMP) and probability approaches (based on frequency analyses). Indeed, multiplicative cascades account for turbulent processes resulting from nonlinear interactions between different scales and fields and respect a basic symmetry of the nonlinear generating equations, i.e. scale invariance. Furthermore, the details of a multifractal process are theoretically defined by the statistical distribution of the singularities of the generating equations. For applications, there are various reliable techniques to extract this distribution from empirical time and/or space series.

5 Which chaos in the rainfall runoff process? 143 Finally, there is no obvious reason that processes should be run by deterministic equations rather than by stochastic equations, since the former are merely particular cases of the latter. Therefore, one can question the deterministic reductionism that is rather ubiquitous in the natural sciences, i.e. a tendency to look at deterministic systems as if they were the only ones providing causality. It is rather important to appreciate that this tendency corresponds to philosophical bias, rather than to an objective rationality and that, behind the notion of deterministic chaos, there could be a possible resurgence of some former restrictive notions of determinism (see Lovejoy & Schertzer, 1998). THE INTEREST, LIMITATIONS AND PITFALLS OF THE CORRELATION DIMENSION A straightforward method Grassberger & Procaccia (1983) introduced a rather efficient algorithm for dimension estimation, which became extremely popular and therefore available in several numerical packages (e.g. Press et al., 1986). One fundamental reason of its popularity is that, in contrast to a box counting algorithm, the embedding delay space is only implicit rather than explicit, since only the distance between pairs of delay vectors is required. As a consequence, much larger embedding dimensions m can be numerically explored than for a box counting dimension algorithm. Indeed, the correlation dimension D 2 is defined as the scaling exponent of the average number of delay vectors in a sphere of radius r centred on one of them, i.e.: < N(m,r) > r D 2 (m) (1) and this average number is precisely defined with the help of the correlation sum: 2 < N(m,r) > C(m, r) = (N T)(N T 1) i < H(l s i s j ) (2) j T where N is the number of data points, H is the Heaviside function ( x > 0 : H ( x) = 1; x 0 : H ( x) = 0 ) and T = 0 is the Theiler window parameter (Theiler, 1986), which is optionally introduced to suppress trivial pairs having too close time indices. For large embedding dimension m, the estimates D 2 (m) should converge towards the theoretical value D 2. In other words, the curves C(m,r) vs r in a log-log plot should lie on the same straight line (having a slope D 2 ), at least over a given range of r. This is fairly straightforward and it has been extensively used for many different data sets in physics and geophysics. In this respect, Fig. 3(a) and (b) of the paper by Sivakumar et al. (2001) is very suggestive. This is much the same for our Fig. 1, which is discussed below. Theoretical limitations In fact the correlation dimension D 2 and the box dimension D 0 are two special cases of the infinite hierarchy of so-called Renyi dimensions (Grassberger, 1983) that

6 144 Daniel Schertzer et al. characterize the multifractal behaviour of a strange attractor. They are in general distinct and only related by the following inequality: D 0 D 2 (3) Therefore, D 2 yields only a lower bound of the box dimension, whereas the latter is the dimension required for the embedding theorem. Therefore, even for deterministic systems, D 2 may largely underestimate the dimensionality of the dynamics. Contrary to a frequent misinterpretation of this theorem, it is important to note that the theorem hypothesizes that the dynamics are deterministic. Therefore it does not draw any conclusion from the mere determination of a low dimensionality. There are obvious reasons for this. Indeed, there are well known stochastic processes having a low dimensionality. The most celebrated one is Brownian motion (Osborne & Provenzale, 1989; Theiler, 1991), since this additive process is known to have a box-counting dimension D 2 (m) = 2 for any m! Figure 1 corresponds to a correlation-dimension analysis of a synthetic series (displayed in Fig. 2) simulated with the help of a stochastic cascade process. This nonlinear type of process was chosen, because it has been often invoked for rainfall data analysis and simulation, as discussed above. However, to keep this process as simple as possible, so that any curious reader could reproduce it, a discrete (in scale) lognormal cascade was chosen, whereas numerous studies show that the rainfall cascade is continuous in scale and rather log-lévy. Nevertheless, independently of the details of such a cascade process, it has a very large dimensional phase space (infinite dimensional if the cascade process proceeds down to an infinitesimal inner scale) and an infinite dimensional probability space. In spite of the large dimension of the space, the correlation dimension yields a low finite estimate D 2. In our precise example, there are 12 cascade steps, the number of data points is N = 4096 and the mean fractality of the cascade is C 1 = 0.03, and D was obtained numerically Fig. 1 Log C(r,m) vs logr for the synthetic time series displayed in Fig. 2, for embedding dimensions m = 1, 2,..., 20 (top to bottom). The corresponding exponent D 2 (m) (estimated in the scaling range) converges towards a low dimension D 2 2.7, whereas the stochastic process has an infinite dimensional probability space and a very large dimensional phase space.

7 Which chaos in the rainfall runoff process? Fig. 2 Time series (4096 data points) generated by 12 steps of a lognormal cascade, which is defined by its mean fractality C 1 = Pitfalls The ease of the correlation-dimension method is also a source of pitfalls. Indeed, the fact that the embedding space is only implicit may lead one to forget an apparently straightforward, but essential, question: What is really being estimated when the embedding dimension becomes larger and larger? The absence of problems with numerics should not hide the fact that there could be another obvious problem. Indeed, any empirical analysis is performed on a finitely sized sample, and the confinement of empirical points measured by D 2 (m) to a small fraction of the embedding space could be due to the limited number of points rather than the dynamics! In other words, instead of measuring an effect of the dynamics, one is merely evaluating an artefact of the finite sample size! One simple way (e.g. Grassberger, 1986) of estimating the latter is to come back to the box counting dimension. Indeed, according to this notion, the number of points spread homogeneously over a fractal set of dimension D and on a scale ratio λ scales like: N D ( λ) λ (4) Assuming that a decade in scales is a minimum to demonstrate a scaling behaviour, one obtains the celebrated rule of thumb that the minimal number of points to estimate a dimension D is: N D 10 D This rule explained many of the unusual results obtained with the help of the correlation dimension. In their pioneering and influential paper, Nicolis & Nicolis (1984) used 500 values from the isotope record of a deep-sea core to conclude that D According to this estimate, one might be able to create a model predicting climatic changes of the last million years with only 7 8 independent variables. Grassberger (1986) discussed these results. In fact, the 500 values were obtained by interpolation of only 184 actual measurements. Applying the rule of thumb (equation (5)), one obtains that the maximal dimension, which could be safely estimated, is of the order D (5)

8 146 Daniel Schertzer et al. Grassberger (1986) concluded that it is difficult to distinguish these data from those of a random signal series. Nerenberg & Essex (1990) and Essex (1991) refined somewhat the rule of thumb and obtained a slightly more optimistic estimate of the number points required to obtain a reliable dimension: N D D (6) which corresponds to the introduction of an explicit prefactor in equation (5), and to considering a smaller range of scale (λ 2.5). Let us note that in the case of the (stochastic) multiplicative cascade displayed in Fig. 2, the numerical estimate of the corresponding correlation dimension (Fig. 1) D is rather reliable, since , whereas we have points. Chaos and the rainfall runoff process To come back to the question of chaos and of the rainfall runoff process: no-one will question the erratic nature of this process, and therefore its chaotic nature in the wide sense. However, many objections will be raised if chaos should be understood in the narrow sense of deterministic chaos, furthermore with a low dimensionality, say 5 6. Indeed, it flies in the face of common sense that rainfall could be predicted with twelve independent variables. At the very least, one would rather think that some set of partial differential equations is required! The rule of thumb (equation (5)) yields an upper bound for a maximal reliable estimate of the order D 3.2, whereas Sivakumar et al. (2001) claim to have reliable estimates of D 5 6. In other words, they have only 1% of the necessary data set. Nevertheless, the authors are aware of the optimistic evaluation (equation (6)) by Nerenberg & Essex (1990) of the necessary number of data points in order to obtain their estimates in a reliable manner. Corresponding to this estimate, their data contain only about 10% of what would be necessary. They merely repudiate any evaluation of this type, by calling it a belief! They claim that for a particular size, the number of reconstructed vectors may not differ much whether an embedding dimension of, for example, 4 or 10 is used, and, therefore the dimension estimate may not be influenced much. This statement reflects a misunderstanding of the arguments discussed in the previous section. Indeed, the estimate of the necessary number of points (equation (4)) is related to the expected dimension of the attractor as well as to the scale ratio of the scaling range, and not directly to the embedding dimension! Furthermore, they claim that the only important question is to obtain a large scaling region, which indeed corresponds to the second factor, but not the only one, of these arguments. Unfortunately, they forget to evaluate how narrow is their scaling region. For instance, their Fig. 3(a) displays a scaling range that seems to be of the order of λ 3. Let us add that we do not understand why they drop a factor of 2 in the estimate of the necessary number of independent variables, i.e. considering it as D instead of 2D. In any case, Sivakumar et al. (2001) were unable to substantiate their claim of an indication of low-dimensional chaotic behaviour in the rainfall runoff process.

9 Which chaos in the rainfall runoff process? 147 CONCLUSIONS Some time ago, low-dimensional deterministic chaos had been very helpful in order to better understand the limitations of classical methods in analysing and modelling complex systems, in particular in hydrology. This was achieved with the help of an apparently simple caricature of a complex system (e.g. the Lorenz model corresponds to the truncation of the convection of the first three Fourier modes of convection) leading nevertheless to nontrivial behaviours. However, in the name of a mathematical theorem in fact a fundamental misinterpretation of this theorem there had been an awkward tendency to attempt to reduce complex systems to their low-dimensional caricatures. This tendency was reinforced by the apparent success of a fairly straightforward algorithm to estimate rather low dimensionality for various complex systems. However, many of these estimates may easily turn out to be spurious, either because of sample size limitation or of the stochastic nature of the process. The paper by Sivakumar et al. (2001) rather corresponds to a late confirmation of this dead end. In this discussion it has been pointed out that the chaos of spatially extended systems, which include hydrological systems, may require approaches dealing with a very large degree of freedom and that some asymptotic behaviours correspond to infinite numbers. It has also been pointed out that progress in that direction might result from an original blending of stochastics and dynamics. REFERENCES Essex, C. (1991) Correlation dimension and data sample size. In: Non-Linear Variability in Geophysics, Scaling and Fractals (ed. by D. Schertzer & S. Lovejoy). Kluwer, Dordrecht, The Netherlands. Grassberger, P. (1983) Generalized dimensions of strange attractors. Phys. Rev. Lett. A 97, 227. Grassberger, P. (1986) Are there really climate attractors? Nature 322, Grassberger P. & Procaccia, I. (1983) Characterization of strange attractors Phys. Rev. Lett. 50(5), Hubert, P., Tessier, Y., Ladoy, Ph., Lovejoy, S., Schertzer, D., Carbonnel, J. P., Violette, S., Desurosne, I. & Schmitt, F. (1993) Multifractals and extreme rainfall events. Geophys. Res. Lett. 20(10), Kahane, J. P. (1995) Definition of stable laws, infinitely divisible laws, and Lévy processes. In: Lévy Flights and Related Phenomena in Physics (ed. by M. Shlesinger, G. Zaslavsky & U. Frisch), Springer-Verlag, Berlin, Germany. Lovejoy, S. & Schertzer, D. (1995) Multifractals and rain. In: New Uncertainty Concepts in Hydrology and Water Resources (ed. by Z. W. Kundzewicz), Cambridge University Press, Cambridge, UK. Lovejoy, S. & Schertzer, D. (1998) Stochastic chaos, scale invariance, multifractals and our turbulent atmosphere. In: ECO-TEC: Architecture of the In-between (ed. by A. Marras), Storefront Book Series, copublished with Princeton Architectural Press, Princeton, USA. Lorenz, E. N. (1963) Deterministic nonperiodic flow. J. Atmos. Sci. 20, Nerenberg, M. A. H. & Essex, C. (1990) Correlation dimension and systematic geometric effects. Phys. Rev. Lett. A42(12), Nicolis, C. & Nicolis, G. (1984) Is there a climate attractor? Nature 311, Osborne, A. R. & Provenzale, A. (1989) Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35, Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw, R. S. (1980) Geometry from a time series. Phys. Rev. Lett. 45(9), Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. (1986) Numerical Recipes, The Art of Scientific Computing. Cambridge University Press, Cambridge, UK. Ruelle, D. (1989) Chaotic Evolution and Strange Attractors. Cambridge University Press, Cambridge, UK. Sauer, T., Yorke, J. & Casdagli, M. (1991) Embedology. J. Statist. Phys. 65, 579. Schertzer, D. & Lovejoy, S. (eds) (1991) Non-Linear Variability in Geophysics, Scaling and Fractals. Kluwer, Dordrecht, The Netherlands. Schertzer, D., Lovejoy, S., Schmitt, F., Chigirinskaya, Y. & Marsan, D. (1997) Multifractal cascade dynamics and turbulent intermittency. Fractals 5(3), Schertzer, D., Larchevêque, M., Duan, J., Yanovsky, V. V. & Lovejoy, S. (2001) Fractional Fokker Planck equation for nonlinear stochastic differential equations driven by non-gaussian Lévy stable noises. J. Math. Phys. 41, 12.

10 148 Daniel Schertzer et al. Sivakumar, B. (2000) Chaos theory in hydrology: important issues and interpretations. J. Hydrol. 227(1,E4), Sivakumar, B., Liong, S. Y., Liaw, C. Y. & Phoon, K. K. (1999a) Singapore rainfall behaviour: chaotic? J. Hydrol. Engng ASCE 4(1), Sivakumar, B., Phoon, K. K., Liong, S. Y. & Liaw, C. Y. (1999b) A systematic approach to noise reduction in chaotic hydrological time series. J. Hydrol. 219(3/4), Sivakumar, B., Berndtsson, R., Olsson, J. & Jinno, K. (2001) Evidence of chaos in the rainfall runoff process. Hydrol. Sci. J. 46(1), Takens, F. (1980) Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898 (ed. by D. A. Rand & L. S. Young), Springer-Verlag, Berlin, Germany. Theiler, J. (1986) Spurious dimensions from correlation algorithms applied to a limited time-series data. Phys. Rev. Lett. A 34, Theiler, J. (1991) Some comments on the correlation dimension of a 1/f-alpha noise. Phys.Rev. Lett.A 155, 480. Van Kampen, N. G. (1981) Stochastic Processes in Physics and Chemistry. North-Holland Physics Publishing, Amsterdam, The Netherlands. Wiener, N. (1938) The homogeneous chaos. Am. J. Math, 60,

11 Hydrological Sciences Journal des Sciences Hydrologiques, 47(1) February Reply to Which chaos in the rainfall runoff process? BELLIE SIVAKUMAR Department of Land, Air, and Water Resources, University of California, Davis, USA RONNY BERNDTSSON Department of Water Resources Engineering, Lund University, S Lund, Sweden JONAS OLSSON Swedish Hydrological and Meteorological Institute, S Norrköping, Sweden KENJI JINNO Institute of Environmental Systems, Kyushu University, Fukuoka , Japan INTRODUCTION The authors thank Schertzer et al. (2002) for their comment on their work (Sivakumar et al., 2001) on the investigation of the possible presence of low-dimensional chaotic behaviour (the term chaos was used therein to denote the low-dimensional chaos) in the rainfall runoff process in the Göta River basin in Sweden. The authors respond to this discussion with respect to: (a) the purpose of the study by Sivakumar et al. (2001), and the results achieved and conclusions drawn therein; and (b) the argument by Schertzer et al. (2002) regarding the possibility of finite correlation dimensions resulting from stochastic processes when the data size is small. The issue of minimum data size requirement for a reliable estimation of correlation dimension is discussed and the deficiencies in the existing guidelines that relate the data size to the embedding dimension are pointed out. While the authors agree with Schertzer et al. (2002) that low correlation dimensions might also result from stochastic processes, they opine that this is true only when the data size is extremely small. The authors argue that the low correlation dimension obtained by Schertzer et al. (2002) for the stochastic series is not due to the small data size. STUDY BY SIVAKUMAR et al. (2001) The purpose of the study by Sivakumar et al. (2001) was to investigate the possible presence of low-dimensional chaotic behaviour in the rainfall runoff process. The driving force for such a study was the encouraging results (both simulation and prediction) achieved for a variety of hydrological processes and reported in many publications (e.g. Rodriguez-Iturbe et al., 1989; Abarbanel & Lall, 1996; Porporato & Ridolfi, 1997; Krasovskaia et al., 1999; Sivakumar et al., 1999, in press; Jayawardena

12 150 Bellie Sivakumar et al. & Gurung, 2000). For the above purpose, Sivakumar et al. (2001) employed the correlation dimension method to the rainfall runoff process observed in the Göta River basin in Sweden. The rainfall runoff process was analysed separately (using rainfall and runoff series) and jointly (using the runoff coefficient series). It is important to note that Sivakumar et al. (2001) only suggested (neither verified nor confirmed) the possible presence of low-dimensional chaotic behaviour in the rainfall runoff process, based on some preliminary results, i.e. finite and low correlation dimensions obtained. Hence, the possibility of the presence of low-dimensional chaotic behaviour in the rainfall runoff process cannot be excluded. There are several reasons for the failure by Sivakumar et al. (2001) to provide strong conclusions regarding the presence/absence of low-dimensional chaotic behaviour in the rainfall runoff process. The first and foremost reason is the possibility of finite and low correlation dimensions resulting from linear stochastic processes when the data size is small (e.g. Osborne & Provenzale, 1989). Being aware of this and other problems in the correlation dimension method (cf. Sivakumar, 2000, 2001; Sivakumar et al., 1999, in press), Sivakumar et al. (2001) recommended the application of other chaos identification methods to verify the results obtained in their study and to confirm the presence/absence of low-dimensional chaotic behaviour in the rainfall runoff process. The second most important reason is concerned with the data used by Sivakumar et al. (2001) for investigating the presence/absence of low-dimensional chaotic behaviour in the rainfall runoff process. Having at hand only rainfall and runoff time series for the Göta River basin, Sivakumar et al. (2001) based their study on two important assumptions: (a) the individual behaviour of the rainfall (input) and the runoff (output) processes could provide important information about the behaviour of the joint rainfall runoff process (input output relationship); and (b) the runoff coefficient (given by the ratio of runoff to rainfall) could represent the rainfall runoff process as a whole. However, the validity of these assumptions is indeed problematic. Recognizing the concerns about the validity of the assumptions made in their study and the potential uncertainties of the outcomes, Sivakumar et al. (2001) also recommended further studies, such as: (a) a multi-variable analysis involving, for instance, rainfall, runoff, evaporation and infiltration; and (b) the recovery of one variable, which may be difficult to measure but is of direct physical interest, from a relatively easily measurable variable. CORRELATION DIMENSION AND THE ISSUE OF DATA SIZE Schertzer et al. (2002) state that the Grassberger-Procaccia (1983) correlation-dimension algorithm might provide finite and low correlation dimensions even for linear stochastic processes when the data size is not large enough. Schertzer et al. (2002) support their claim by employing the Grassberger-Procaccia algorithm to a synthetic time series (of 4096 data points) generated by a stochastic process, which is claimed to be related to the rainfall process (Schertzer et al., 2002 Fig. 2), and reporting a low correlation dimension of 2.7 ± 0.3 for the series (Schertzer et al., 2002 Fig. 1). Guided by this observation, Schertzer et al. (2002) argue that the rainfall runoff process studied by Sivakumar et al. (2001) could be stochastic rather than low-

13 Which chaos in the rainfall runoff process? Reply 151 dimensional chaotic, and that the low correlation dimensions that resulted may be a result of a small data set (with only 1572 values) used. Since the discussion by Schertzer et al. (2002) revolves essentially around the minimum data size required for the correlation dimension estimation, it is import to discuss this issue in detail here. A fundamental assumption in the development of the correlation dimension method (e.g. Grassberger & Procaccia, 1983) is that the time series is of infinite length. This is not realistic and the data size problem is the basis for the suspicions and criticisms on studies reporting the presence of low-dimensional chaos in real time series. More than 15 years ago, Grassberger (1986) questioned the low correlation dimension result reported by Nicolis & Nicolis (1984) for the isotope record of a deep-sea core, that it was due to the size of data used, i.e. 500 points, rather than the presence of low-dimensional chaos. A number of further studies have raised suspicions on reporting low-dimensional chaos in hydrological series, based on the issue of data size and others (e.g. Ghilardi & Rosso, 1990; Koutsoyiannis & Pachakis, 1996; Schertzer et al., 2002). As an assumption of infinite time series would lead nowhere in the efforts to study practical situations, numerous attempts have been made to provide some useful guidelines on the minimum data size required for a reliable estimation of the correlation dimension. Smith (1988) concluded that the minimum data size N min was equal to 42 m, where m is the smallest integer above the dimension of the attractor. Nerenberg & Essex (1990) demonstrated that Smith s procedure was flawed and that the data requirements might not be so extreme. They suggested that the minimum data size is N min ~ m. Further studies on this issue have reported contrasting results. For instance, Ramsey & Yuan (1990) reported possible underestimation of correlation dimension when a small data size was used, whereas Lorenz (1991) argued that a suitably selected variable could yield a fairly accurate dimension estimate even if the number of points were not large. In spite of the above attempts, a clear-cut guideline or a general consensus on the minimum data size is still eluding. Even though, the theoretical guidelines relating the minimum data size to the embedding dimension (e.g. Smith, 1988; Nerenberg & Essex, 1990; Essex, 1991; Schertzer et al., 2002) could be useful under certain circumstances, blindly following such guidelines for practical situations may be misleading. For instance, if one goes by the guideline given by Schertzer et al. (2002 equation (5)), i.e. N min 10 m, then one requires only 100 points (i.e ) to obtain an accurate estimation of the correlation dimension of the chaotic Henon map (Henon, 1976), whose correlation dimension is 1.22 and can be embedded in a dimension of 2. Similarly, if one goes by the guideline recommended by Essex (1991), i.e. N min [2(m + 1)] m (also Schertzer et al., 2002 equation (6)), then one requires only 36 points to obtain an accurate dimension estimation of the Henon map. That is, accurate modelling and prediction of the Henon map is assumed possible with only 100 points (or even 36 points). This statement, however, is questionable. The authors experience with the study of the Henon map indicates that such a small data size is insufficient to obtain a well-defined attractor in the two-dimensional phase space. Consequently, an accurate estimation of correlation dimension is not possible, because, with such a small data size, the number of reconstructed vectors would be very small, which, in turn, would result in an insufficient number of points in the logc(r) vs logr plot for a reliable selection of the scaling region. On the other hand, if one assumes a hypothetical case

14 152 Bellie Sivakumar et al. involving correlation exponent calculations in a 10-dimensional phase space for the Henon series, then one requires points and points respectively, according to the guidelines recommended by Schertzer et al. (2002) and Essex (1991). [It is not required to carry out the correlation exponent calculations in more than two dimensions, since the attractor can be completely embedded in two dimensions.] However, contrary to these guidelines, with only 5000 points, or even less, a reasonably large scaling region that allows reliable estimation of the correlation exponent is possible even when the embedding dimension is 10 (Sivakumar, 2000 Fig. 1). In fact, the difference between the number of reconstructed vectors in two and 10 dimensions is usually very small. The above observations clearly indicate that the guidelines available for the determination of the minimum data size in terms of the embedding dimension could be misleading; more specifically, such guidelines underestimate or overestimate the data size requirement, for small and large embedding dimensions, respectively. Unfortunately, however, the critiques of reports of low correlation dimensions in hydrological time series, such as the discussion by Schertzer et al. (2002), fail to recognize the above fact. The study by Sivakumar et al. (in press) shows that the correlation-dimension estimate of a runoff series (with only 576 monthly values) is consistent with the optimal embedding dimension obtained using the phase-space reconstruction prediction method and also with the optimal number of inputs obtained using artificial neural networks. It is relevant to note that near-accurate predictions for this runoff series are achieved both with phase-space and neural network approaches, indicating the appropriateness of these approaches for predictions and also for verifying the correlationdimension estimates. Common sense would also support these results, since monthly runoff data observed over a period of 48 years are likely to represent reasonably the changes that the system undergoes with time. In view of the above observations, the authors believe that monthly data over a period as long as 131 years are certainly sufficient to reasonably represent the dynamic changes of hydrological processes in the Göta River basin. Therefore, it is believed that the low correlation dimensions obtained by Sivakumar et al. (2001) for the rainfall runoff process in the Göta River basin are reliable and at least near-accurate. It may be appropriate to note at this point, that use of much shorter data is very common in (stochastic) hydrological data analysis and still the results are considered reliable. DISCUSSION OF RESULTS REPORTED BY SCHERTZER et al. (2002) Underestimation of correlation exponent? If the time series is the result of a stochastic process, then, according to the phasespace reconstruction concept (e.g. Takens, 1981), in all embedding dimensions m, where m = 1, the object is space-filling. Thus, for any value of m, the correlation exponent ν (which is the slope of the logc(r) vs logr plot) is given by ν = m, as shown in Fig. 1. Such a perfect relationship of ν = m is possible only in pure theoretical cases, since it requires an infinite length of time series. In practice, however, small deviations from this relationship may be observed because of the finite length of the time series.

15 Which chaos in the rainfall runoff process? Reply 153 Correlation exponent Hypothetical Stochastic Rainfall (Sivakumar et al., 2001) Runoff (Sivakumar et al., 2001) Runoff Coefficient (Sivakumar et al., 2001) Synthetic Rainfall (Schertzer et al., 2002) Embedding dimension Fig. 1 Relationship between correlation exponent and embedding dimension for: (a) hypothetical stochastic series; (b) monthly rainfall, runoff, and runoff coefficient series from the Göta River basin (Sivakumar et al., 2001); and (c) synthetic stochastic series (Schertzer et al., 2002). A finite series might result in an insufficient number of points in the phase-space reconstruction, which, in turn, might result in only a few points in the logc(r) vs logr plot, thus limiting ones ability to observe a large and clear scaling region. However, this is the case only when the number of points and the embedding dimension are very small and very large, respectively. On the other hand, assuming that a time series is of dimension d = 2.70 (as reported by Schertzer et al., 2002), then, according to the phase-space reconstruction concept, in all embedding dimensions m < 2.70, the object is space-filling. Thus, for m < 3, the correlation exponent is given by ν = m, whereas for m 3, it is given by ν = Thus, the first deviation of the correlation exponent from the diagonal (i.e. ν = 2.70 starting at m = 3 and remaining constant for higher values of m) against the embedding dimension should provide estimation of the correlation dimension. However, this is not the case for the series analysed by Schertzer et al. (2002 Fig. 1). As can be seen, for values of m < 3, ν m (but ν < m), and ν = 2.70 not for all values of m 3, but only when the value of m is large (about 10). In fact, the values of ν are consistently and significantly underestimated for all values of m. Having observed a saturation of ν around a low value of 2.70 ± 0.30, Schertzer et al. (2002) conclude that the (stochastic) series yields a low correlation dimension and that it is due only to the small data size (of 4096 points) used. Why underestimation? Is there an error in the implementation of the procedure? The error in the analysis by Schertzer et al. (2002) may be explained as follows. If one assumes, as argued by Schertzer et al. (2002), that the (small) data size used in their analysis indeed leads to underestimation of the correlation dimension, and if one also assumes that to obtain accurate estimation of correlation dimension one requires a

16 154 Bellie Sivakumar et al. minimum data size N min 10 m (Schertzer et al., 2002) or N min [2(m + 1)] m (Essex, 1991), then with 4096 points analysed by Schertzer et al. (2002) one should be able to accurately estimate the correlation exponents up to an embedding dimension of at least 3, since the data sizes required are equal to or less than only 10, 100, and 1000 for 1, 2, and 3 dimensions, respectively. In other words, as discussed above, one should observe correlation exponent values of 1.0, 2.0, and 3.0 for embedding dimensions of 1, 2, and 3, respectively. However, the correlation exponents obtained from the logc(r) vs logr plots presented by Schertzer et al. (2002 Fig. 1) are 0.3, 0.6, and 0.8 for m = 1, 2, and 3, respectively. These results clearly indicate that: (a) the correlation exponents obtained by Schertzer et al. (2002) are significant underestimations; and (b) the underestimations are not due to the small data size used. To the authors knowledge, the only possible nature of the time series that could lead to such a significant underestimation of the correlation exponent is the presence of a large number of a single value (e.g. zero) in the series (e.g. Tsonis et al., 1994), when the reconstructed hyper-surface in phase space tends to a point and the resulting correlation exponent is a significant underestimation of the dimension of the phase space. However, the synthetic (stochastic) rainfall series analysed by Schertzer et al. (2002 Fig. 2) does not seem to be affected by this problem, as the presence of single values (e.g. zero) is not encountered. Comparison with the results of Sivakumar et al. (2001) Figure 1 presents the correlation exponent vs embedding dimension relationship for the monthly rainfall, runoff, and runoff coefficient series in the Göta River basin studied by Sivakumar et al. (2001). As can be seen, the correlation exponents are in good agreement with the expected values up to an embedding dimension of 3 or 4. For instance, the ν values obtained for the rainfall series are 0.98, 1.92, and 2.81, against the expected values of 1.0, 2.0, and 3.0 for m = 1, 2, and 3, respectively. Therefore, the ν vs m relationships for the three series are consistent with the concept of phase-space reconstruction, indicating that the implementation of the dimension algorithm adopted and the results achieved are indeed correct. A comparison of the correlation exponent results reported by Sivakumar et al. (2001) and those by Schertzer et al. (2002) in terms of data size only raises further questions about the argument made by the latter. Sivakumar et al. (2001) achieved near-accurate ν with only 1572 points, whereas Schertzer et al. (2002) significantly underestimate ν with 4096 points. That is, Schertzer et al. (2002) used a data size about 2.6 times higher than the one used by Sivakumar et al. (2001), while underestimating ν by a factor of more than 3.2. This may indicate that there is an error in the results reported by Schertzer et al. (2002). It is relevant to note, at this point, that the ν values obtained by Sivakumar et al. (2001) are not as accurate as they are supposed to be, in particular at higher embedding dimensions. However, this is due not to the small data size used, but to the space filled by the time series in the embedding phase space. A possible explanation is as follows. As discussed above, if the time series is the result of a stochastic process, then in all embedding dimensions m, the object is space-filling. For instance, the hypothetical stochastic series, for which the correlation exponents are shown in Fig. 1, would fill-up

17 Which chaos in the rainfall runoff process? Reply 155 (a) 250 Rainfall, Xi+1, (mm) (b) Rainfall, X i, (mm) Runoff, Xi+1, (mm) Runoff, X i, (mm) Fig. 2 Phase-space diagram in two dimensions for: (a) monthly rainfall series from Göta River basin; and (b) monthly runoff series from Göta River basin (from Sivakumar et al., 2001). the entire space when embedded in a two-dimensional phase space. However, this may not be the case when one deals with real systems. For example, the monthly rainfall and runoff series in the Göta River basin, when embedded in a two-dimensional phase space, fill up the space only partially (cf. Fig. 2(a) and (b)). The larger the space filled, the higher the ν value, with the highest ν = m when the entire space is filled. This can also be supported, to some extent, from the ν values obtained for the above rainfall and runoff series by Sivakumar et al. (2001). For m = 2 and 3, respectively, the ν values obtained are 1.92 and 2.81, and 1.82 and 2.65, for the rainfall and runoff series, respectively. It is clear from the phase-space plots shown in Fig. 2(a) and (b) that the rainfall series fills a larger space than does the runoff series. The time series studied by Schertzer et al. (2002) seems to fill up a larger space than these rainfall and runoff series and, therefore, is expected to yield higher values of ν. Need for caution in interpreting the results The authors admit that the correlation dimension method possesses certain limitations, but these need not hinder one from viewing hydrological problems from a lowdimensional perspective for a better understanding. In regard to this, the authors would

18 156 Bellie Sivakumar et al. like to bring to the attention of Schertzer et al. (2002) that, in general (unless the data size is extremely small): (a) not all processes yielding finite and low correlation dimensions are stochastic; (b) all low-dimensional chaotic systems yield low correlation dimensions; and (c) many stochastic processes yield infinite correlation dimensions. Therefore, if the dynamic properties of a real hydrological time series are not known a priori, caution is needed while interpreting the dimension results, keeping in mind the limitations of the correlation dimension method. Many studies that have reported the presence of low-dimensional chaos in hydrological processes using the correlation dimension results have also supported their results with the results obtained using other methods (e.g. Sangoyomi et al., 1996; Porporato & Ridolfi, 1997; Sivakumar et al., in press). Unfortunately, however, the studies that criticize the reported results of low-dimensional chaos in hydrological processes often fail to take note of such verifications and confirmations. The authors admit that wrong selection of the parameters in the correlation dimension algorithm and inappropriate implementation of the procedure may wrongly indicate evidence or proof of low-dimensional chaos in a time series (when actually there is none). However, it is important to note that the same errors may also lead one to an inaccurate interpretation of the correlation dimension results and, hence, to the conclusion that there is no chaos (when actually there is). The discussion by Schertzer et al. (2002) is still another case in which it is attempted to prove that a stochastic series (which is claimed to be related to rainfall) results in a low correlation dimension essentially due to the small data size. In view of the above observations, there is a significant difference between the views of Schertzer et al. (2002) and that of the authors in regard to the study of dynamical behaviour of hydrological processes. Schertzer et al. (2002) view hydrological processes as basically influenced by an infinite number of degrees of freedom, whereas the authors view is that hydrological processes may be influenced by any number of degrees of freedom, ranging from very low to infinite, depending upon the underlying system. To prove their point, Schertzer et al. (2002) attempt to provide a counter-example to an established correlation dimension method by implementing the dimension algorithm on a finite synthetic stochastic time series, reporting a low correlation dimension, and finally attributing the low dimension to the small data size. Unfortunately, as discussed above, the results reported by them have eventually failed to support their own argument. CLOSING REMARKS The authors attempt (Sivakumar et al., 2001), which aimed to investigate the possibility of the existence of low-dimensional chaotic behaviour in the rainfall runoff process in the Göta River basin, was successful, as positive evidence of the presence of such behaviour has been observed. Although the authors agreed with the argument by Schertzer et al. (2002) that finite and low correlation dimensions might result from stochastic processes, it was argued that this could be true only when the data size is extremely small. The issue of minimum data size for correlation dimension estimation using both artificial chaotic series (Henon map) and real hydrological (runoff) series was discussed and it was reported that the existing guidelines relating the minimum