## Abstract

Time series in the Earth Sciences are often characterized as self-affine long-range persistent, where the power spectral density, S, exhibits a power law dependence on frequency, *f*, S(f)~ *f*^{-β}, with β the persistence strength. For modelling purposes, it is important to determine the strength of self-affine long-range persistence β as precisely as possible and to quantify the uncertainty of this estimate. After an extensive review and discussion of asymptotic and the more specific case of self-affine long-range persistence, we compare four common analysis techniques for quantifying self-affine long-range persistence: (a) rescaled range (R/S) analysis, (b) semivariogram analysis, (c) detrended fluctuation analysis, and (d) power spectral analysis. To evaluate these methods, we construct ensembles of synthetic self-affine noises and motions with different (i) time series lengths N = 64,128,256,...,131,072, (ii) modelled persistence strengths β_{model} = -1.0, -0.8,-0.6,...,4.0, and (iii) one–point probability distributions (Gaussian, log-normal: coefficient of variation cv = 0.2‒2.0, Levy: tail parameter a = 1.0‒1.9) and evaluate the four techniques by statistically comparing their performance. Over 17,000 sets of parameters are produced, each characterising a given process; for each process type, 100 realisations are created. The four techniques give the following results in terms of systematic error (bias = average performance test results for β over 100 realisations minus modelled β) and random error (standard deviation of measured β over 100 realisations): (i) Hurst rescaled range (R/S) analysis is not recommended to use due to large systematic errors. (ii) Semivariogram analysis shows no systematic errors but large random errors for self-affine noises with . (iii) Detrended fluctuation analysis is well-suited for time series with thin-tailed probability distributions and for persistence strengths of . (iv) Spectral techniques perform the best of all four techniques: for self-affine noises with positive persistence ( ) and symmetric one–point distributions they have no systematic errors and compared to the other three techniques, small random errors; for anti-persistent self-affine noises ( ) and asymmetric one-point probability distributions, spectral techniques have small systematic and random errors. For quantifying the strength of long-range persistence of a time series, benchmark-based improvements to the estimator predicated on the performance for self-affine noises with the same time series length and one–point probability distribution are proposed. This scheme adjusts for the systematic errors of the considered technique and results in realistic 95% confidence intervals for the estimated strength of persistence. We finish this paper by discussing long-range persistence of three geophysical time series—palaeotemperature, river discharge, and Auroral electrojet index—with the three representing three different types of probability distribution—Gaussian, log-normal, and Levy, respectively.

Original language | English |
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Pages (from-to) | 541–651 |

Number of pages | 111 |

Journal | SURVEYS IN GEOPHYSICS |

Volume | 34 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Sept 2013 |

## Keywords

- fractional noises and motions
- self-affine time series
- long-range persistence
- Hurst rescaled range (R/S) analysi
- semivariogram analysis
- detrended fluctuation analysi
- power-spectral analysi
- random and systematic errors
- root mean squared erro
- confideence intervals
- benchamrk-based imprvoements
- geographysical time series