Quantifying the non-linear dependence of energetic electron fluxes in the Earth's radiation belts with radial diffusion drivers

In this study, we use mutual information to characterise statistical dependencies of seed and relativistic electron fluxes in the Earth's radiation belts on ultra-low-frequency (ULF) wave power measured on the ground and at geostationary orbit. The benefit of mutual information, in comparison to measures such as the Pearson correlation, lies in the capacity to distinguish non-linear dependencies from linear ones. After reviewing the property of mutual information and its relationship with the Pearson correlation for Gaussian bivariates, we present a methodology to quantify and distinguish linear and non-linear statistical dependencies that can be generalised to a wide range of solar wind drivers and magnetospheric responses. We present an application of the methodology by revisiting the case events studied by Rostoker et al. (1998). Our results corroborate the conclusions of Rostoker et al. (1998) that ULF wave power and relativistic electron fluxes are statistically dependent upon one another. We also estimate that the Pearson correlation is missing between 20 % and 30 % of the statistical dependency between ULF wave power and relativistic electron fluxes. Thus, the Pearson correlation underestimates the impact of ULF waves on energetic electron fluxes. However, we find that observed enhancements in relativistic electron fluxes correlate modestly, both linearly and non-linearly, with the ULF power spectrum when compared with values found in previous studies (Simms et al., 2014) and with correlational values found between seed electrons and ULF wave power for the same case events. Our results are indicative of the importance of incorporating data analysis tools that can quantify linear and non-linear interdependencies of various solar wind drivers.