Eliminating finite-size effects and detecting the amount of white noise in short records with long-term memory

Abstract

Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records and additive white noise in the data. Here we study synthetic Gaussian distributed records ${x}_{i}$ of length $N$ that consist of a long-term correlated component $(1\ensuremath{-}a){y}_{i}$ characterized by a correlation exponent $\ensuremath{\gamma}$, $0<\ensuremath{\gamma}<1$, and a white-noise component $a{\ensuremath{\eta}}_{i}$, $0\ensuremath{\le}a\ensuremath{\le}1$. We show that the autocorrelation function ${C}_{N}(s)$ has the general form ${C}_{N}(s)=[{C}_{\ensuremath{\infty}}(s)\ensuremath{-}{E}_{a}]/(1\ensuremath{-}{E}_{a})$, where ${C}_{\ensuremath{\infty}}(0)=1$, ${C}_{\ensuremath{\infty}}(s>0)={B}_{a}{s}^{\ensuremath{-}\ensuremath{\gamma}}$, and ${E}_{a}={2{B}_{a}/[(2\ensuremath{-}\ensuremath{\gamma})(1\ensuremath{-}\ensuremath{\gamma})]}{N}^{\ensuremath{-}\ensuremath{\gamma}}+O({N}^{\ensuremath{-}1})$. The finite-size parameter ${E}_{a}$ also occurs in related quantities, for example, in the variance ${\ensuremath{\Delta}}_{N}^{2}(s)$ of the local mean in time windows of length $s$: ${\ensuremath{\Delta}}_{N}^{2}(s)=[{\ensuremath{\Delta}}_{\ensuremath{\infty}}^{2}(s)\ensuremath{-}{E}_{a}]/(1\ensuremath{-}{E}_{a})$. For purely long-term correlated data ${B}_{0}\ensuremath{\cong}(2\ensuremath{-}\ensuremath{\gamma})(1\ensuremath{-}\ensuremath{\gamma})/2$ yielding ${E}_{0}\ensuremath{\cong}{N}^{\ensuremath{-}\ensuremath{\gamma}}$, and thus ${C}_{N}(s)=[\frac{(2\ensuremath{-}\ensuremath{\gamma})(1\ensuremath{-}\ensuremath{\gamma})}{2}{s}^{\ensuremath{-}\ensuremath{\gamma}}\ensuremath{-}{N}^{\ensuremath{-}\ensuremath{\gamma}}]/[1\ensuremath{-}{N}^{\ensuremath{-}\ensuremath{\gamma}}]$ and ${\ensuremath{\Delta}}_{N}^{2}(s)=[{s}^{\ensuremath{-}\ensuremath{\gamma}}\ensuremath{-}{N}^{\ensuremath{-}\ensuremath{\gamma}}]/[1\ensuremath{-}{N}^{\ensuremath{-}\ensuremath{\gamma}}]$. We show how to estimate ${E}_{a}$ and ${C}_{\ensuremath{\infty}}(s)$ from a given data set and thus how to obtain accurately the exponent $\ensuremath{\gamma}$ and the amount of white noise $a$.