Exploring the physical interpretation of long-term memory in hydrology

Abstract

Long-term memory has been studied for decades and it has long been acknowledged that hydrological and hydro-meteorological time series exhibit this property. Physically, long-term memory is explained by saying that the stronger the long-term memory the more likely a series will ‘remember’ its previous value, in other words, the longer a time series is likely to persist in the proximity of a certain value. To increase the benefit of the study of long-term memory, we investigate the extent to which this explanation is accurate and descriptive of long-term memory by developing a ‘persistence measure’ that quantifies the intuitive description of long-term memory. The ‘persistence measure’ is compared to the scaling exponent (α) which quantifies long-term memory using detrended fluctuation analysis. A total of 17,359 series, including hydrological and hydro-meteorological series, downloaded from online sources and 130 synthetic series are analyzed. The main outcome is that two regimes are found. The first is for series with α ≲ 1.05 where the persistence measure is not sensitive to the scaling exponent. Hence, changes in α does not lead to an increase in the memory strength (i.e. the persistence measure). The second regime is for series with α ≳ 1.05 where a statistically significant positive correlation was found. Hence, for a certain range of values of α the physical explanation of long-term memory holds. Nevertheless, it is proposed that adding additional factors may create a more robust ‘persistence measure’ that can be easily understood and incorporated in hydrologists’ work.