$1/f$ Noise in Synthetic and Solar Wind Data: Superposition Principles

TL;DR

The paper investigates the solar-origin hypothesis for interplanetary $1/f$ noise by testing the superposition principle, whereby exponential-type fluctuations with a scale-invariant or log-normal distribution of correlation times can yield a $1/f$ spectrum. Using synthetic time series and ACE/1 au data, it demonstrates that averaging correlations (or concatenating time series) reliably produces a $1/f$ regime, while averaging the raw series does not, and that the $1/f$ band broadens with the span of correlation times. The ACE observations corroborate the synthetic results, showing a log-normal distribution of local turbulence correlation times and a $1/f$ spectrum that extends to very low frequencies ($\sim 10^{-6}$ Hz), implicating long-range, mixed solar-wind signals including solar rotation. These findings support a solar-origin, large-scale mechanism for $1/f$ noise and provide a framework for interpreting long-duration heliospheric data, with implications for future missions and multi-spacecraft analyses.

Abstract

The interplanetary magnetic field exhibits a distinctive $1/f$ spectral density from frequencies of around $\unit[10^{-6}]{Hz}$ to around $\unit[10^{-4}]{Hz}$, ranging from harmonics of the solar rotation to the reciprocal of the turbulence correlation time in the spacecraft frame. Various theories have been proposed to explain its origin, typically invoking either processes in the lower corona or in the solar interior, or local interplanetary dynamics. Here, we investigate the {\it superposition principle} that underlies explanations of the solar/coronal types, which in principle can generate the full observed range of $1/f$ noise. Using synthetic time series with scale-invariant or log-normal distributions of correlation times, we examine the efficacy of several superposition approaches in generating a $1/f$ regime. The persistence of $1/f$ spectrum is further illustrated with decade-long {\it in situ} magnetic field measurements from the ACE spacecraft. Together, these results help explain the ubiquity of $1/f$ noise under the unavoidable superposition inherent in long-duration heliospheric data.

Figures (5)

• Figure 1: Top panel: power spectra resulting from scale-invariant correlation times with $\tau_1, \tau_2 = 3, 150$, obtained through superposition methods 1 (in red), 2 (in green), 3 (in yellow), and 4 (in blue). Bottom panel: compensated spectra corresponding to those on top panel. Dotted vertical lines mark frequencies $1/2\pi \tau$ associated with the minimum and maximum correlation times, respectively. Dashed gray lines are guiding power-laws with index $-1$ and $-5/3$, respectively.
• Figure 2: Top panel: power spectra resulting from scale-invariant correlation times with $\tau_1, \tau_2 = 20, 150$, obtained through superposition methods 1 (in red), 2 (in green), 3 (in yellow), and 4 (in blue). Bottom panel: compensated spectra corresponding to those on top panel. Dotted vertical lines mark frequencies $1/2\pi \tau$ associated with the minimum and maximum correlation times, respectively. Dashed gray lines are guiding power-laws with index $-1$ and $-5/3$, respectively.
• Figure 3: Top panel: power spectra resulting from log-normal correlation times with $\mu, \sigma^2 = 4, 1$, obtained through superposition methods 1 (in red), 2 (in green), 3 (in yellow), and 4 (in blue). Bottom panel: compensated spectra corresponding to those on top panel. Dotted vertical lines mark $2\pi$-scaled boundaries of scale-invariant correlation time portion with $\theta = 0.5$ (see text). Dashed gray lines are guiding power-laws with index $-1$ and $-5/3$, respectively.
• Figure 4: Distribution of magnetic field correlation time evaluated from 24-hour intervals. Ensemble expectation value (E) and standard deviation (Std) are listed in legend. Solid curve represents best-fit log-normal distribution with parameters also reported in legend.
• Figure 5: Recovery of 1/f with different superposition methods. Method 1: Averaged-then-normalized spectrum from 10-day intervals (red); normalized-then-averaged spectrum (dotted purple); averaged-then-normalized spectrum from randomly selected 10-day intervals (green); averaged-then-normalized spectrum from 1-day intervals (orange). Method 2: Spectrum from averaged 10-day intervals (black). Method 3: Spectrum from unsegmented 12-year data (gray); integrated spectrum and associated uncertainty from 12-year data Wang26. Dashed lines indicate constant and $f^{-2/3}$ power-laws. Vertical line indicates 27-day solar rotation frequency.