Discrete Scale Invariance in Geomagnetic Data

The geomagnetic field has recently been shown to exhibit scale-invariant features, most notably in terms of power laws, e.g., in power spectra and ranked intervals between dipole reversals. However, these scaling laws are merely a first approximation, leaving substantial residuals unexplained. With the aid of a novel technique (in principle applicable to any time series of sufficient length), additional information may be extracted that quantifies the characteristic scales of the studied process, that is, the preferred size(s) of the most important fluctuations and their associated timings. This so-called discrete scale invariance (DSI) betrays its presence by a logperiodic modulation of the underlying power law, and can be detected through a bootstrapping approach. Given an observed gamut of fluctuations over a studied period, a set of linearly incremental thresholds of absolute change is first defined, and the associated intervals between successive exceedances recorded, to be ranked by size on a log-log scale (thus yielding as many new datasets as there are thresholds). To all sets satisfying certain basic quality criteria, a power law fit is performed (negative gradient), and its residuals (unequally spaced on a log-scale) subjected to Lomb spectral analysis, which exploits this uneven spread. If a significant, coherent modulation is found, matrix inversion will yield the remaining wave parameters, and the associated topmost scaling stratum can then be identified as the highest point where the modulation intersects the power law from below. This level is stored with the set's (log-scaled) threshold, contributing to a meta-dataset that associates these preferred timings with their fluctuation threshold. After additional quality control of this set, a new power law fit is performed (positive gradient), and its residuals subjected to Lomb analysis and inversion. The resulting wave parameters yield the preferred scaling levels of the original observable and the associated timings. An additional feature of this approach is its potential predictive power above and below the observed range. Three independent geomagnetic datasets were examined to look for signatures of discrete scale invariance: the sequence of dipole reversal intervals from the Mesozoic to the present day (161-0 Ma, Gradstein & Ogg 1996); fluctuations in relative intensity over the last 2 Ma (SINT2000 dipole excursions, Valet, Meynadier & Guyodo 2005); and secular variation as captured by the historical field map gufm1 (1590-1990 A.D., Jackson, Jonkers & Walker 2000). Not only does each set exhibit significant DSI, but the recorded characteristic scales of both reversals and secular variation (individually) are quite well predicted by each of the other two datasets.