Is there long-range memory in solar activity on time scales shorter than the sunspot period?

Abstract

The sunspot number (SSN), the total solar irradiance (TSI), a TSI reconstruction, and the solar flare index (SFI), are analyzed for long-range persistence (LRP). Standard Hurst analysis yields $H \approx 0.9$, which suggests strong LRP. However, solar activity time series are non-stationary due to the almost periodic 11 year smooth component, and the analysis does not give the correct $H$ for the stochastic component. Better estimates are obtained by detrended fluctuations analysis (DFA), but estimates are biased and errors are large due to the short time records. These time series can be modeled as a stochastic process of the form $x(t)=y(t)+σ\sqrt{y(t)}\, w_H(t)$, where $y(t)$ is the smooth component, and $w_H(t) $ is a stationary fractional noise with Hurst exponent $H$. From ensembles of numerical solutions to the stochastic model, and application of Bayes' theorem, we can obtain bias and error bars on $H$ and also a test of the hypothesis that a process is uncorrelated ($H=1/2$). The conclusions from the present data sets are that SSN, TSI and TSI reconstruction almost certainly are long-range persistent, but with most probable value $H\approx 0.7$. The SFI process, however, is either very weakly persistent ($H<0.6$) or completely uncorrelated. Some differences between stochastic properties of the TSI and its reconstruction indicate some error in the reconstruction scheme.