Surveys in Geophysics

, Volume 34, Issue 5, pp 541-651

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Quantification of Long-Range Persistence in Geophysical Time Series: Conventional and Benchmark-Based Improvement Techniques

  • Annette WittAffiliated withDepartment of Nonlinear Dynamics, Max–Planck Institute for Dynamics and Self-OrganizationDepartment of Geography, Earth and Environmental Dynamics Research Group, King’s College London Email author 
  • , Bruce D. MalamudAffiliated withDepartment of Geography, Earth and Environmental Dynamics Research Group, King’s College London


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 (1) time series lengths N = 64, 128, 256, …, 131,072, (2) modelled persistence strengths β model = −1.0, −0.8, −0.6, …, 4.0, and (3) one-point probability distributions (Gaussian, log-normal: coefficient of variation c v = 0.0 to 2.0, Levy: tail parameter a = 1.0 to 2.0) and evaluate the four techniques by statistically comparing their performance. Over 17,000 sets of parameters are produced, each characterizing a given process; for each process type, 100 realizations are created. The four techniques give the following results in terms of systematic error (bias = average performance test results for β over 100 realizations minus modelled β) and random error (standard deviation of measured β over 100 realizations): (1) Hurst rescaled range (R/S) analysis is not recommended to use due to large systematic errors. (2) Semivariogram analysis shows no systematic errors but large random errors for self-affine noises with 1.2 ≤ β ≤ 2.8. (3) Detrended fluctuation analysis is well suited for time series with thin-tailed probability distributions and for persistence strengths of β ≥ 0.0. (4) Spectral techniques perform the best of all four techniques: for self-affine noises with positive persistence (β ≥ 0.0) 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 (β < 0.0) 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 quantifying long-range persistence (and corresponding uncertainties) 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.


Fractional noises and motions Self-affine time series Long-range persistence Hurst rescaled range (R/S) analysis Semivariogram analysis Detrended fluctuation analysis Power spectral analysis Random and systematic errors Root-mean-squared error Confidence intervals Benchmark-based improvements Geophysical time series