The hurst effect and its implications for extrapolating process rates

Abstract

AbstractIn long‐term records of climatic or hydrological events, persistent periods of high and low values are measured by the ‘rescaled range’, which is defined for a sequence of n consecutive values. As n increases the rescaled range for historic sequences increases as nH for 0·5 < H < = 1. This is the Hurst effect. Sequences of independent random events give H = 0·5, while sequences in which the values are successive positions of a random walk (i.e. the sum of independent events) give H = 1. Intermediate values of H may be obtained from fractal sequences, from sequences with a steadily changing mean, or through a storage model. In all but the fractal case H exceeds 0·5 for a time related to the duration of the changing mean or the response time of the store.An exponential storage model has been explored in some detail. Storage models have the advantage of offering a plausible physical mechanism for persistence. Analysis of simulated sequences shows that the Hurst effect is evident for about 30 x the response time of the store, so that relatively modest store sizes are enough to explain observed Hurst effects over available periods of record. Comparison with empirical values shows that high H values seem to be associated with large physical stores.Whether the Hurst effect is due to storage effects or climatic or other change imposed from outside, the effect on longterm process rates is strong. Short‐term measurements of process rates show compressed variances and simple extrapolation is likely to produce large errors. If it is assumed that long‐term rates are strongly influenced by climatic variables, then measurements may be used to obtain parameters of a climatic model for differences in process rates. Errors in extrapolation will be least where the model has a sound physical basis.A first implication for magnitude and frequency distributions is that the sequence of events may be as important as their distribution over time in determining process impact. A second conclusion is that the geomorphic history of an area is inevitably dominated by a sequence of larger and larger events receding into the past, and that the Hurst exponent over geologic time‐spans measures the extent of that domination.