WATER RESOURCES RESEARCH, VOL. 26, NO. 8, PAGES 1837-1839, AUGUST 1990 Comment on "Chaos in Rainfall" by I. Rodriguez-Iturbe et al. PAOLO GHILARDI 1 AND RENZO Rosso Institute of Hydraulics, Politecnico di Milano, Milan, Italy INTRODUCTION In recent years a deeper insight of a number of natural phenomena, ranging from turbulence to epidemiology, has been achieved by means of chaos theory (see, for example, Holden [1985]). The growing number of these successful applications is encouraging chaos research to approach those geophysical processesthe dynamics of which still remain unexplained in terms of stochasticanalysis. How- ever, it has been pointed out how the pioneer analysis of natural time series was incomplete in some cases or even conceptually wrong (see, for example, Grassberger [1986], Procaccia [1988], and Osborne and Provenzale [1989]). Deterministic chaotic systems are characterized by cha- otic attractors (often called "strange"); an attractor is a subset of the phase space to which almost all sufficiently close trajectories are "attracted" asymptotically after all initial transients die out [Grassberger and Procaccia, 1983a]. When the research on chaotic attractors was still at the pioneering stage,it was very difficultto characterize the shapeof an attractor, even if its underlyingequations were completely known. In fact, a strangeattractor is a fractal object living in a phase spacewith dimensionusually larger than 3; therefore computingits fractal dimensionby means of such traditional procedures as box counting is cumber- some [Greenside et al., 1982]. The alternative method de- veloped by Grassberger and Procaccia [1983a] is both effective and quite simple to apply, so it became perhapsthe most popular procedureto computethe fractal dimension of attractors. This method (hereafter referred to as GP) is based upon the computation of the correlation integral for an increasing embedding dimension E until saturation is achieved;suchconditioncorresponds to E-invariant slopeof the correlation integral plotted in the log-log plane. This method also allows to quantify the degree of chaoticity by computing Kolmogorov entropy [Grassberger and Procac- cia, 1983b]: this is infinite for a stochasticprocessand zero for a purely deterministic process, and it takes a finite positive real value when deterministic chaos occurs. Rodriguez-Iturbe et al. [1989] (hereinafter referred to as RFSG) present some results supporting the presence of chaotic dynamicsin storm rainfall basedon the analysisof a single storm event recorded in Boston. Although they used the GP methodfor the purpose,they do not report estimates of Kolmogorov entropy; they further computed the largest Lyapunov exponent to assess this result. The analysis of rainfall records is a basic requirement in investigating pre- cipitation owing to the lack of knowledge of its underlying 1 Now at Department of Hydraulic and Environmental Engineer- ing, Universith di Pavia, Pavia, Italy. Copyright 1990 by the American Geophysical Union. Paper number 90WR00447. 0043-1397/90/90WR-00447502.00 dynamics; the purpose of the present comment is to point out someproblems arisingin the assessment of deterministic chaos from the analysisof natural time series, with particular emphasis on storm events. CHAOS IN TEMPORAL RAINFALL RECORDS The possibility of chaotic behavior in rainfall was first indicated by Hense [1987] on the basis of a series of 1008 values of monthly rainfall recorded at Nauru island (166øE, 0.5øS). Hense [1987] used the classical method proposed by Packard et al. [1980] to reconstruct the phase space by means of time delays; the application of the GP method to this experimental set of points in the reconstructed phase space yielded a value of fractal correlationdimension D lying between 2.5 and 4.5, and the existence of a strange attractor could be thus hypothesized. However, Hense [1987] noticed that this fractal dimension alone is not able to provide a sufficient condition for chaos, so that deterministic chaos in rainfall could not be clearly assessed. For the same purpose, RFSG analyzed two records, namely, 148 years of weekly rainfall in Genoa, and a single rainstorm event in Boston. Like Hense [1987], RFSG pro- cessedthese time series by reconstructing a phase space by means of time delays; then the fractal dimension of the attractor in this phase spacewas computed by means of the GP method. Because saturation is not achieved up to em- bedding dimensionE = 8, RFSG indicate that rainfall data aggregated at quite large time scales, i.e., weekly, are stochastic in nature. However, RFSG indicated that weekly rainfall in Genoa might be chaotic with attractor dimension larger than 9, i.e., a dimension much larger than the one detected by Hense [1987] for monthly data. The Boston storm was recorded on October 25, 1980; rainfall depthswere measured with sampling frequency of 8 Hz and then aggregated at equally spacedintervals of 15 s. RFSG processed the Boston storm data by computing the correlationintegral for embedding dimensionE rangingfrom 2 to 8; saturation is achieved for E = 4, and an estimate of D = 3.78 is obtained by the GP method. Moreover, RFSG computedthe largest Lyapunov exponent, A•; this is found to be 0.0002 bits/s. Accordingly, RFSG [abstract] came to the conclusion that "the characteristics of the correlation integral and the Lyapunov exponentsof the historical data givepreliminarysupport to the presence of chaoticdynamics with a strange attractor." A deterministic chaotic system is indeed characterized by a positive largest Lyapunov exponent; however, the sign taken by an estimated value A• is not a sufficientproof of chaotic behavior. As a matter of fact, RFSG [p. 1674] clearly stated that "... if the estimation is carried out for a time series stochastically generated it will yield A• > 0. The correlation integral discriminates between stochastic and deterministic chaos .... "Therefore the fractal correlation 1837