Exploring complexity in the structure of rainfall

Exploring complexity in the structure of rainfall Ignacio Rodriguez-lturbe lnstituto lnternacional de Estudios Aranzados P.O. Box 17606, Parque Central Caracas, Vene'zuela

The enormous complexity embedded in the rainfall structure is studied through the analysis of data sets of different characteristics. It is shown that despite the highly fluctuating character of precipitation, there exists an organizing structure hidden in the irregular patterns. The presence of self-similarity in the time senses of storm is discussed, as well as the fractal dimension which characterizes the irregularity of the process. Rainfall data from a highly sensitive raingauge provides some evidence regarding the existence of deterministic chaos in the fine structure of storm rainfall. The dynamics appears to be characterized by a strange attractor with a correlation dimension of less than 4. Key Words: Deterministic chaos, randomness, self-similarity, and rainfall

F L U C T U A T I O N S IN R A I N F A L L Rainfall, when considered either in the form of storm events or as an aggregated process (e.g. weekly rainfall at a point), shows large dispersions from a mean motion similar to those exhibited by a stochastic process. Such a variability is illustrated in Fig. 1 for part of the rainfall recorded at Denver, Colorado (USA). This type of behaviour may result either from the random probabilistic structure in the amounts of rainfall depth or from a highly non-linear deterministic system highly sensitive to the initial conditions. In the second case, the underlying dynamics are perfectly deterministic although the appearance is similar to that of a stochastic process. The strong non-linear character of the process is the quality that allows the resulting values of this kind of dynamics to change greatly through time and/or space, making it feasible to have values which appear unrelated one to another. The coupling of non-linear features with the sensitive dependence on initial conditions produces very complicated dynamical features in the output of even some of the apparently more simple mathematical models. In fact, the output is undistinguishable from a stochastic process by the standard statistical techniques. One then talks of deterministic chaos. Figure 2 shows 148 years of annual rainfall in Genoa and one again observes a large degree of variability. This data was kindly provided by Prof. Franco Siccardi of the University of Genoa, and Prof. Renzo Rosso of the Politecnico di Milano. One way to quantify the variability or erraticness of the record is through its variance which, for the case of the Genoa record, is plotted in Fig. 4, both for annual and weekly rainfall as a function of the record length. In the case of annual rainfall, the first point is the variance of the first 10 years, the second point is the variance of the first 20 years, and so on. In the case of weekly rainfall, the first point is the variance of the first 50 weeks, the second point is the variance of the first 100 weeks, and so on. Even for such large sample sizes one P a p e r a c c e p t e d N o v e m b e r 1990. Discussion closes F e b r u a r y 1992.

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notices the variance estimate is not stable but rather has a small but noticeably positive trend which in most cases is caused by a small number of large jumps occurring at different moments throughout the historic record. Storm event data is also quite erratic as can be seen in Fig. 3 which shows part of the record for the storm which took place in Boston from the 12:27 hours to the 20:45 hours in October 25th, 1980. Rainfall depths were recorded with a highly sensitive tipping bucket raingauge capable of making measurements every one-eighth of a second. The gauge is located on the roof of a tall building in the M.I.T. campus and sends a signal every time it collects a depth of one-hundredth of a millimetre of rain. For the particular storm of October, 1980, we have a record of 1990 measurements of rain at equally spaced intervals of 5O V l hour folnloll

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t5 seconds. The x-axis in Fig. 3 represents time (hours) and the y-axis represents 100 x millimetres of rainfall collected in 15-second intervals. Figure 4c shows the variance of this record as a function of the number of measurements from the beginning of the storm. The first point is the variance of the first 50 periods of 15" rainfall, the second point is the variance of the first 100 periods of 15" rainfall and so on. The variance increases with the record length due mainly to one sudden j u m p resulting from the clustering of highly erratic rainfall values. Figure 5 shows the autocorrelations of the weekly Genoa rainfall and the storm of October 1980, in Boston. In the case of the weekly rainfall, the lags are in weeks; for the storm in Boston, the lags are in 15" intervals. The autocorrelation is a normalized measure of the linear dependence among successive values of the process. For the weekly rainfall data, one observes a very fast decay of the autocorrelation which very quickly starts to oscillate around zero in a cyclic manner which, even if it has a small amplitude, has also a clearly distinguishable period of 52 weeks reflecting the presence of the annual rainfall cycle. The autocorrelation of the storm event is very different in nature; it decays much more slowly and, what is more important, after a considerable number of lags it tends to stabilize itself at a small value clearly larger than zero. This can be corroborated because at lags of 500 and larger the autocorrelation does not oscillate around zero with positive and negative values which would be the case if its structure was so that rainfall

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The large jumps seen in the rainfall variance as function of the record length imply the possible existence of enormous variations in rainfall rates at different periods in time. Thus, one could entertain the hypothesis that for the process of rainfall changes the variance does not exist, being in fact infinite. This will occur when the probability distribution of rainfall changes is a hyperbolic one, meaning that Pr[AR > Ar] ~ Ar-". In other words, the probability that the change of rainfall intensity AR between two instants separated by a time AT is larger

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than any wtlue Ar, is proportional to Ar". tf the distribution of AR, or at least its upper tail, is a hyperbolic one, the variance will not exist when a > 2. Mandelbrot called this the Syndrome of Infinite Variance. Hyperbolic distributions plot as a straight line in double logarithmic paper. Figure 6 shows the results of the above type of analysis for the weekly rainfall data at Genoa and the 15 seconds storm data at Boston. For the case of the weekly rainfall, the intervals between measurements are one, two, ten and twenty weeks, Ar is in millimetres. For the storm in Boston, the intervals between measurements are 15", 30", 60", 120", Ar is in 100x millimetres of rainfall collected in 15" intervals. The weekly data shows a strong curvature meaning that the distributions are not hyperbolic, also there tends to be a superposition and crossing among the lines. On the other hand, the storm data yields a slope of approximately 1.54 for those AT's used in the analysis. This would imply that the variance does not exist and since a sample variance is always finite the emergence of < 2 may trouble the mind of the modeller. In truth, it only expresses the fact that the sample variance may become arbitrarily large without converging to any specific value when the sample size keeps growing indefinitely.

SELF-SIMILARITY IN THE STRUCTURE OF R A I N F A L L EVENTS Although the temporal rainfall pattern of a storm is highly jagged and irregular, that is a code that brings unity and

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order to such a structure. The key lies in the existing relationship between large scale and small scale irregularities. Small scale irregularities are given by the population of rainfall changes A R ( A T ) = r ( t 2 ) - R ( t l ) when AT = t2 -- t~ represents a short span in time. Large scale irregularities represent rainfall fluctuations between instants with considerable separation in time. Figure 6 shows the distribution of At(AT) for four different values of AT:15", 30% 60" and 120" for the Boston storm of October 1980. It embodies a strikingly surprising feature which is defined as 'self-similarity'. Doubling AT does not change the separation between the lines which have a slope of approximately 1.54. This means that the distribution of AR(2AT) is identical to the distribution of 2UAR(AT) irrespective of the value of AT. If we take measurements of 0.1 in the probability axis for the cases

Exploring complexity in the structure o/r~lin[all. l. Rodriguez-lturhe AT = 15" and 30", the corresponding values of AR are approximately 0.70 and 1.05. thus 1.05 = 0.70 x 2 ~ with H = 0.59. Generally one says that R(t) is scaling, or self-similar, because (AR),;.AT is identical in distribution to ).nAR(AT). There is symmetry across scales in the irregularities &R(A T) of the rainfall process R(t) reflecting an underlying structural symmetry which goes beyond the trivial symmetry of size relations among similar elements and which in apparently patternless processes links fluctuations among large and small scales. It is the above characteristic irregularity of the process what seems to be crucial for its description. This characteristic irregularity we describe in terms of what is called the 'fractal dimension'. The property of selfsimilarity implies that, even in periods where the process seems to behave without major irregularities, if the scale of observation is changed, one will be able to detect fluctuations which look statistically similar to those of more erratic periods except for a factor of scale. Similarly. the closer we look at the more erratic periods, the more lapses of a more tranquil behaviour will be discovered, but the process has wrinkles and peaks at all scales of observations. The process reveals more and more detail as it is increasingly magnified. Thus, self-similarity implies a characteristic irregularity across scales which for the temporal storm pattern of October, 1980, in Boston, corresponds to a dimension larger than that of a one-dimensional line and power than the one of a plane filling figure. The exponent H previously used in the probabilistic analysis of rainfall fluctuations AR(AT), is called in hydrology the Hurst coefficient and it is related to the fractal dimension, D, by the equation D = 2 - H. For the Boston storm one gets an approximate fractal dimension of D = 1.41. A fractal or self-similar process is also peculiar in its autocorrelation and power spectrum. Since the irregularities are linked across different time scales, AT, the autocorrelation does not go to zero for larger and larger lags but rather approaches non-zero type of asymptote. The power s p e c t r u m - - F o u r i e r transform of the autocorrelation function--shows a behaviour GO')~fXbfor high frequencies f

p-dimensional vectors which represent the p-dimensional portrait of the system. One can now draw a circle la sphere for p = 3. or hypersphere for p > 4) of radius r centred about an arbitrary point of the p-dimensional set of vectors and count the number of points Nit) located inside the circle. Normalizing such count one obtains what is called the correlation integral of the process, C~r): 1

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ON RANDOMNESS AND DETERMINISTIC CHAOS If a system is deterministic, a precise and complete specification of the initial conditions will imply a unique trajectory for the future. Deterministic chaos means that very close initial conditions may lead to completely different paths of the system after a certain time in its evolution. Small perturbations may have dramatically large effects. The behaviour of the system does not converge either to a fixed equilibrium point or to limit cycles, but rather, never repeats itself. It is characterized by strange attractors with fractal geometry. Such a behaviour may be studied in the phase-space representation of the process. Starting with one time-dependent variable, R(t), like rainfall as function of time, we can construct a trajectory in a p-dimensional phase space by taking as coordinates R(t), R ( t + r ) , R ( t + 2 z ) . . . . . R(t + (p - 1)r) where r is an appropriate delay time. Thus moving along with time t, we obtain a series of

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where i and j are indices ordering the points along a trajectory containing a total of m points. Suppose R(t) is a white noise process, in this case N(r) and thus also C(r) will be proportional to r 2 in the 2-dimensional representation, proportional to r 3 in the 3-dimensional case and in general C(r) ~ r" where in this

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case v = p. There are no strict rules for the choice of the delay "v. Usually several values are tried so to assure the results are not sensitive to the chosen delay. Schuster 4 suggests the choice of ~ so that the correlation function takes the value 0.5. It is difficult to analyze rainfall for the existence of strange attractors because conventional sensors yield short time series. To study correlation dimensions there should exist several thousand data points in the time series under study for dimensions less than 4 (Ref. 2). Rodriguez-lturbe et al? and Sbarifi et al. 5 carried out the analysis for three storms collected with the highly sensitive gauge at M.I.T. described earlier in this paper. The data sets consisted of the times the raingauge signalled the collection of one-hundredth of a millimetre of rain. The storms description is provided in Sharifi et al., 5 the number of data points per storm was 4000, 3991 and 3316. Figure 7 shows the spectral density functions for the three storms. The frequency is in cycles per count. The behaviour C ( r ) - v s - r is shown in Fig. 8 together with the linear segment on each line used for the determination ofv. For storm 1, ~ = 4, for storm 2, ~ = 16 and for storm 3, -r= 134. The v - u s - p plots are presented in Fig. 9 where the last graph corresponds to the results from a white noise process. The saturation value of v obtained for the storm data was in the range 3.3 to 3.8, suggesting a low-dimensional strange attractor in the 0.01-ram pulses of rainfall for the storms examined. The possibility of a low-dimensional attractor in the time series of storm events presents an opportunity for challenging research towards the determination of the limits of the predictability of the process. Physically based rainfall models whose non-linear dynamics result in the chaotic evolution detected in the data constitute a new frontier in hydrometeorology.

Exploring c o m p l e x i t y in the structure ACKNOWLEDGEMENTS T h i s is a r e v i e w p a p e r p r e s e n t e d as a K e y n o t e L e c t u r e in I n t e r n a t i o n a l C o n g r e s s o n N u m e r i c a l M e t h o d s in W a t e r R e s o u r c e s at Venice, Italy in J u n e 1990. M o s t of this p a p e r is a d a p t e d f r o m the references 3 a n d 5 herein.

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REFERENCES

1

Mandelbrot, B. B., The Fractal Geometry of Nature, New York, Freeman and Co., 1982

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Ramse~. J. B. and Yuan. H. J. The Statistical Properties of Dimension Calculations Using, Small Data Sets, Research Report No. 87-20, C.V. Start Center For Applied Economics, New York University, New York. 1987 Rodriguez-lturbe, I.. Febres de Power, B., Sha~ifi, M. B. and Georgakakos, K. P. Chaos in rainfall. Water Resources Reseurch, 1989 25 (71 1667-1676 Schuster, H. G. Deterministic Chaos, VCH Weinheim, F.D.R., 1988 Sharifi, M. B., Georgakakos, K. P. and Rodriguez-Iturbe, 1. Evidence of deterministic chaos in the pulse of storm rainfall, Journal of .qtmosptzeric Sciences', 1990 45 (7)

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