On the Spatial Scale Dependence of Long‐Term Persistence in Global Annual Precipitation Data and the Hurst Phenomenon

Abstract

Long-term persistence (LTP) in geophysical time series, or the tendency of above or below average runs of years to be unusually long, was first quantified by Hurst (Hurst, 1951(Hurst, , 1956 using a coefficient H which characterizes LTP in the range 0.5 < H < 1, with H = 0.5 corresponding to the independent (white noise) case, that is, no persistence. Hurst analyzed a wide set of geophysical times series and found an average value of H = 0.73, with a standard deviation of 0.09. In particular, he reported a value of H = 0.9 for annual river Nile flows at Aswan, reflecting strong LTP. The disparity between these results, and the then current theory that predicted H = 0.5 based on the increments of Brownian motion, has come to be known as the Hurst Phenomenon. Over the years, a number of stochastic approaches to modeling LTP have emerged (e.g., fractional Gaussian noise (Mandelbrot & Wallis, 1968, 1969, ARMA models (O'Connell, 1974a(O'Connell, , 1974b, shifting mean models (Boes & Salas, 1978) and fractionally differenced models (Hosking, 1984)). Koutsoyiannis (2011a) has shown that those models exhibiting Hurst behavior asymptotically can be encapsulated within a Hurst -Kolmogorov (HK) stochastic dynamics framework characterized by a simple scaling law, acknowledging the contribution

The Spatial Scale Dependence of The Hurst Coefficient in Global Annual Precipitation Data, and Its Role in Characterising Regional Precipitation Deficits within a Naturally Changing Climate

“…The Hurst coefficient was estimated using aggregated variance plot analysis [16,25], which is based on a property of the change in the statistical characteristics of a Hurst -Kolmogorov (HK) process with respect to the time scale of averaging [25]. The summary here is taken from [16].…”

“…Cumulative departure from the mean (CDM) plots are a useful diagnostic tool for differentiating time series with upward and downward fluctuations that can result in similar H estimates. They are used in analysing the spatial scale dependence of average regional precipitation [16]. The CDM is defined as…”

“…Each window must lie entirely within the region. The Hurst coefficient was estimated for the average precipitation for each tile (in calculating the regional average precipitation values, the grid scale values were weighted by their projected earth surface areas) [16]. A box plot is used to display the sample frequency distribution of H against increasing scale.…”

“…Bunde, et al [13] questioned whether LTP/memory exists in precipitation, while Mudelsee [14] attributed the higher H values observed for large river basin flows to the aggregation process performed by the river network (essentially the summation of tributary runoffs, modelled as HK stochastic processes, from upstream to downstream), dismissing the argument of Potter [15] that an explanation of the Hurst phenomenon must lie in precipitation. Recently, it has been shown that for selected climatic regions, LTP and H for annual precipitation increases with the spatial scale of averaging, thereby providing the evidence that LTP in regionally-averaged precipitation can be strong, and associated with large-scale, long-term modes of fluctuation in the global climate system that appear to be linked with sea surface temperatures [16]. It was suggested that this finding can account for strong LTP in the annual flows of large river basins, and this was demonstrated for the case of the river Nile, thereby explaining the Hurst phenomenon in that case.…”

“…The causal link between LTP in basin average precipitation and river flows is analysed for two large river basins. The crossing properties of regional annual precipitation time series (basically the durations and volumes of precipitation deficits below a prescribed level) for the set of regions exhibiting strong LTP [16] are analysed, and the use of the Hurst coefficient H to characterize precipitation deficits is investigated. We explore whether or not there is evidence that ACC has led to the intensification of regional precipitation deficits in recent decades.…”