DETERMINISTIC CHAOS VERSUS STOCHASTICITY IN ANALYSIS AND MODELING OF RAINFALL STRUCTURE

Extended abstract In recent years, new methods for time series analysis, derived from chaos theory, yielded some fascinating results. Among them, the fact that deterministic processes may sometimes be statistically indistinguishable from random noise. Currently, discussion is going on about how to characterize a process and how to recognize whether it is stochastic or deterministic. A popular method for revealing the underlying dynamics, if any, from a time series, is the phase-space reconstruction via time delay embedding. According to this method, the state of a system, as seen from an observable, is approximated by a number of previous values of this observable. A critical issue is how many of these values are necessary, in order to capture efficiently the evolution of the system. In other words, how many variables we need to describe the phasespace in which the phenomenon evolves. The least necessary number of these variables is related to the dimension of the attractor of the system, if such an attractor is identified. The capacity and correlation dimension quantify the nature of the attractor. A variety of studies have shown that in deterministic processes there exists a finite dimension [dk1]while in stochastic processes there is no such a finite value. Therefore the estimation of this dimension provides a way of detecting determinism in a time series. In recent studies, it was found that the correlation dimension of rainfall time series is finite, whereas simple stochastic models (such as ARMA) that are often used to simulate the rainfall process exhibit very different behavior. This indicates the presence of deterministic chaos in the rainfall process. Despite this, the stochastic modeling of the rainfall process still dominates in operational hydrology as it gives sufficient solutions to most problems. Our objective in this study is similar to that of several previous studies: to detect deterministic chaos in historic rainfall series and compare the results with those obtained from synthetic series generated by stochastic models. The difference is that we use a more detailed stochastic model to simulate the rainfall process instead of the simple ARMA models. Specifically, we use the scaling model of storm hyetograph†. This is a simple, parameter parsimonious model for the temporal evolution of rainfall, which limits the need for a stationarity assumption only within the same storm event. The basic hypothesis behind it, is that the process of instantaneous rainfall intensities within a storm is a self-similar (simple scaling) process that scales with the duration of the storm. The model explains reasonably well the dependence of the total storm depth and of the internal structure of a storm on the storm duration. To form a complete rainfall generator, this model was combined with a stochastic process describing storm arrivals, durations and total depths. It is noteworthy that self-similarity that characterizes the stochastic scaling model is often recognized behind chaotic patterns. However, at the generation stage the model behaves like a purely linear stochastic model, which for each storm adapts its parameters according to the storm duration. The historic data set used for the study consists of incremental rainfall depths, measured every quarter of an hour, for an observation period of six years (1984-89), at a station in Ortona, Florida. The synthetic data set was generated using the scaling model, after proper fit. The comparison of the two sets is done by means of time delay embedding and capacity and correlation dimension estimates of the two time series.