The Earth's radiation belts are non-linearly driven and weakly collisional plasma environments in which deposited energy and momentum leads to the energisation of electrons to relativistic energies . From a fundamental physics perspective, the acceleration of charged particles to supra-thermal energies is ubiquitous to astrophysical plasma environments. As the closest astrophysical accelerator of particles to the Earth, the radiation belts are amenable to detailed in situ measurements of electromagnetic fields' distribution functions. Their studies are therefore relevant to other astrophysical environments with comparable thermodynamical properties in which particles are confined by large-scale inhomogeneous magnetic fields . From an applied perspective, a wide range of satellites' orbits overlap with the Earth's radiation belts, with the undesirable consequence that the energetic particles can damage the onboard electronics and shorten the lifespan of communication systems . Thus, the main focus of Earth's radiation belts' studies is to quantify the processes scaling from electron kinetic scales to planetary scales that enhance and deplete the plasma .

It has been known for several decades that the Earth's radiation belts were driven far from thermodynamical equilibrium as a result of variable solar wind conditions . This departure from thermodynamical equilibrium results in kinetic distribution functions that are unstable and the production of fluctuations that can thermalise the plasma and accelerate particles. A growing number of in situ measurements and observational studies in the last 2 decades have demonstrated that the Earth's radiation belts' response to solar wind driving and fluctuations can also be non-linear and that non-linearity ought to be accounted for in order to improve prediction capabilities . From a theoretical perspective, every self-consistent set of equations describing fluid- and kinetic-scale plasma physics is inherently non-linear. The departure of linearity in a dynamically evolving plasma translates into the appearance, and therefore measurements, of non-Gaussian fluctuations . Even if a non-linear system is initialised with Gaussian fluctuations, non-Gaussian fluctuations would eventually emerge. It is therefore not surprising that non-Gaussian fluctuations are commonly found across a wide range of astrophysical plasma environments . Taking into account the above theoretical constraints and observational results, one quickly recognises that in order to quantify non-linearity in the Earth's radiation belts, one has to use measures that can be sensitive to non-linear dependencies and are capable of distinguishing them from linear ones.

In this study, we present an application of information theory to the search of dependencies between energetic electron fluxes measured in the Earth's radiation belts and ultra-low-frequency (ULF) wave power measured both at geostationary orbit and on the ground. Unlike more commonly used measures like the Pearson correlation, information-theoretic tools, such as mutual information, have the benefit to distinguish non-linear dependencies from linear ones. In order to demonstrate the value in the use of information-theoretic methods, we revisit the highly cited case studies of . In their study, it was suggested that ULF pulsations can provide energy for acceleration of electrons to relativistic energies based on visual inspection of relativistic electron fluxes at geostationary orbit and ground ULF wave power. It should be stressed that conclusions are cautiously stated and that a value for a correlation or any other measure is not provided. Nonetheless, it is not too uncommon to find citing authors describing their results as compelling and evidence of strong correlation between ULF wave power and relativistic electron fluxes. The impact of ULF fluctuations in the enhancement and loss of energetic electron fluxes also forms the basis of radial diffusion formalisms and is, as of today, understood as one of the two dominant transport mechanisms in planetary radiation belts (see , and references therein).

The application of information-theoretic measures to space plasma problems is not new, but it has recently shown its utility for a wide range of methodologies and problems (see , and references therein). Of particular relevance to our study, presented an application of information-theoretic measures to quantify the dependence of relativistic electron fluxes measured on geostationary orbits to a wide range of solar wind drivers. In their study, demonstrate that the solar wind speed is the main driver and that the effect of the solar wind density, sometimes suggested as a dominant driver for relativistic electron fluxes , holds 30 % lesser information content and operates on a different timescale. The main departure between the work presented hereafter and the study of lies in our introduction of a quantity called information-adjusted correlation and the use of a dataset that has a 1 h resolution of geostationary-measured seed and relativistic electron fluxes. The information-adjusted correlation is defined as the correlation value that would be obtained from the mutual information under the assumption that the dependence between the two variables can be represented as a Gaussian bivariate. The choice of a Gaussian bivariate to distinguish linear and non-linear dependences as hinted above stems from the fact that non-linear equations produce non-Gaussian statistics, even in the instance where a system is initialised with Gaussian-distributed random variables . We therefore present a methodology that allows us to provide clear answers to the following two questions:

Are the events studied by evidence of statistical dependence between ULF wave power and electron fluxes?

Our report is presented as follows. Section 2 presents a brief summary of the tools of information theory used for the analysis. We put a particular emphasis on the application of mutual information to the case of Gaussian random variables of arbitrary correlation, which serves as a benchmark for linear dependencies. In Sect. 3 we describe the dataset used and the associated instruments' specificities relevant to our study. In Sect. 4, we present our results for geostationary-measured seed and relativistic electron fluxes measured during the events presented by . In Sect. 5, we interpret and compare our results in light of previous studies and then conclude with suggestions for future studies and improvement of our methodologies for instances where statistical dependencies are difficult to extract.

In this section we present a definition of mutual information in terms of the Shannon entropy and the specific mutual information of Gaussian bivariate random variables. The Gaussian bivariate case with arbitrary Pearson correlation ρ is used as a toy model to benchmark the numerical estimate of mutual information and to distinguish linear from non-linear statistical dependencies. A detailed description and derivation of mutual information for Gaussian bivariates is provided in the Appendix for the interested reader.

The data used in this study correspond to the two events analysed by . The first period extends from 2 to 31 May 1994 (91 d in total), and the second one spans from 1 to 26 November 1993. During the first period a big geomagnetic storm occurred on 17 April with minimum Dst of -201 nT, and the period featured also several moderate and intense storms. During the second period an intense storm peaked on 4 November with minimum Dst -119 nT. Another significant storm during this latter period was a moderate storm on 18 November with minimum Dst -82 nT. Both periods were thus geomagnetically active. Our choice to revisit the work of through mutual information stems from the fact that such methodology has not been used before and that their study, highly cited in the literature as evidence that radial diffusion is a leading mechanism for the energisation of relativistic electrons, can serve as a benchmark for more involved methodologies. Additionally, we have access to a comparable dataset with better resolution (1 h resolution instead of 1 d), so we cannot only revisit the results of with information theory but find a more accurate time lag for the electron's response to ULF wave power. In the Pc5 ULF measurements were from the Gillam measurement station of the Canadian Auroral Network for the OPEN Program Unified Study Project (CANOPUS), and the electron fluxes (>2 MeV) were from Geostationary Operational Environmental Satellite 7 (GOES 7). The GOES data are the daily average flux, and the ULF data are the average over a 6 h period from dawn to noon.

Figures () and () show the 24 h average of relativistic electron flux indices and ULF power indices as a function of time for the two events studied by . In each figure the panel on the left has the geosynchronous ULF index plotted, whereas the panel on the right has the ground ULF index plotted. We remind the reader that our datasets have different time resolutions from those used by with 24 h resolution, whereas we use 1 h resolution and 24 h moving averages. However, the visual comparison of Figs.  and to Figs. 1 and 2 in Rostoker et al. (1998) shows that they are very similar

Reducing our resolution to 24 h for a strict comparison with is not useful because the values of mutual information and correlation are low, and reducing the number of points would bring both measures to the noise level.

We are now ready to answer the two questions stated in the Introduction. (1) Are the events studied by examples of strong ULF wave power and energetic electron dependence? (2) Is the statistical dependence between ULF wave power and electron fluxes non-linear? In order to answer these two questions, we have tabulated the values of the maximum Pearson correlation and maximum mutual information for all events in Table 1. The columns denote, from the left to the right, the event year, the flux index, the ULF index, the maximum Pearson correlation, the maximum mutual information, the information-adjusted correlation, and the lag for the maximum mutual information, respectively. The information-adjusted correlation is defined as the correlation value that would be obtained from the mutual information under the assumption that the dependence between the two variables can be represented as a Gaussian bivariate (cf. Eq. 3). The choice of a Gaussian bivariate to distinguish linear and non-linear dependences stems from the fact that non-linear equations produce non-Gaussian statistics, even in the instance where a system is initialised with Gaussian-distributed random variables . Mathematically, the information-adjusted correlation can be defined by applying the inverse of Eq. (3): 4ρadj=sign(ρ)1-2-2I. The information-adjusted correlation ρadj allows us to determine whether the Pearson correlation has underestimated the dependence between the random variables due to the presence of non-linearity. The instance in which the adjusted correlation is statistically comparable to the Pearson correlation denotes that a linear dependence between the fluxes and ULF power dominates and that non-linear dependencies are either too weak or non-existent. In the opposite case, an adjusted correlation larger than the Pearson correlation indicates that non-linear dependencies between fluxes and ULF power are statistically significant.

Are the events evidence of strong ULF wave power and energetic electron dependence? For the two events studied, the Pearson correlation and the mutual information are both statistically significant and well above the noise level. However, the maximum correlation values for relativistic electrons range between 0.41 and 0.59, and the maximum mutual information values range between 0.36 and 0.49. For comparisons, the analyses by from 219 storms show values of correlation of 0.65 for ground ULF and 0.50 for GOES ULF with relativistic electron fluxes. Thus, the values for correlation of the 1993 event are showing weaker linear and non-linear statistical dependence, whereas the 1994 event has correlation values on par with events found over 11 years of data . The methodology of separates variables in terms of storm phases and defines a predictor variable, for example, ULF wave power, as an average over an appropriate time period for a given storm phase. Since we are studying case events, the statistical methodology of cannot be explicitly reproduced, but we find comparisons with our results useful in that it gives us a point of reference to judge the strength of the correlation values we found. For another comparison, for all times during the years 1995–2006, found a Pearson correlation coefficient of 0.34 between Fe1.2 and Sgr, whereas they found a higher correlation coefficient of 0.54 between Fe1.2 and the 123 h time integral of Sgr. Similarly, it was found that the correlation coefficient between Fe1.2 and Sgeo was 0.21, whereas the correlation coefficient was 0.25 between Fe1.2 and the 138 h time integral of Sgeo. For all times during the years 1995–2004 Borovsky and Denton (2014) explored correlation coefficients between Sgr and Sgeo and a relativistic electron flux F that was calculated differently from Fe1.2. They found correlations between F and Sgr of 0.36 (with a time lag of 56 h) and between F and the 126 h time integral of Sgr of 0.55. Likewise they found correlations between F and Sgeo of 0.28 (with a time lag of 71 h) and between F and the 156 h time integral of Sgr of 0.32. Our results demonstrate that even though the events appear, at least visually, to show strong correlation between ULF waves and relativistic electron fluxes, quantitatively the dependence is comparable to other values found in the literature but nonetheless modest when compared with the correlation between ULF waves and seed electrons.

Comparing between seed and relativistic electrons, the statistical dependence on ULF wave power of the 130 keV flux is significantly larger than for relativistic fluxes and ranges between 0.54 and 0.68 for the maximum Pearson correlation and 0.44 and 0.67 for the maximum mutual information. We also note that the time lag for the maximum values is comparable whether one uses the mutual information or the Pearson correlation. The 130 keV fluxes have a maximum dependence with time lags of less than a day, whereas the relativistic electrons see a maximum for time lags considerably longer between 42 and 67 h. Moreover, the ground ULF wave power gives a larger dependence than geostationary measured ULF wave power for the 1994 event. For the 1993 event the statistical dependence is the same whether one uses ground or geostationary indices. The ground ULF index spans local daylight hours between 05:00 and 15:00, whereas the GOES ULF covers the full 24 h period. This local time difference between ground and geostationary sampling of wave power makes the latter more susceptible to be influenced by substorm activity and the former by viscous processes and pressure pulses on the dayside magnetosphere during moderate geomagnetic activity . However, and as pointed out by , the most notable difference between ground and GOES data is that the ground magnetometers are better positioned to catch ULF wave activity that would result in radial diffusion transport .

To address the second question, we compare the values of the information-adjusted correlation with the Pearson correlation. We note that the adjusted correlation is significantly larger than the Pearson correlation for all instances. In other words, though constrained to two case studies, our results demonstrate the presence of non-linear statistical dependencies between energetic electron fluxes and ULF wave power. By using information theory we make no assumptions about the functional form of the non-linear dependence between the variables, but we can nonetheless state that non-linearities have to be accounted for. Our results are consistent with the study of , in which they built regression models that assumed a quadratic dependence in the ULF wave power with a 1 d lag. Their results indicate that the response of relativistic electron fluxes can be a combination of linear and non-linear dependence and that incorporating a quadratic term might provide better predictions. Based on the values for the information-adjusted correlation, the Pearson correlation might be missing between 20 % and 30 % of the statistical dependencies between ULF wave power and relativistic electron fluxes.

The Earth's inner magnetosphere is a non-linearly driven plasma environment in which electrons can be collectively energised to relativistic energies by ULF fluctuations . The emergence of non-linear processes translates into non-Gaussian fluctuations in the electromagnetic fields and particle distribution functions. Thus, in order to quantify the processes at play to model the Earth's radiation belts accurately, one needs to determine whether non-linear statistical dependencies between drivers, such as the solar wind speed and the ULF wave power, and quantities in which energy and momentum is deposited, such as electron fluxes, have to be accounted for.

In this study, we described the use of mutual information to characterise statistical dependencies of relativistic electron fluxes on ULF wave power. The benefit of mutual information, in comparison to the Pearson correlation, lies in the capacity to distinguish non-linear dependencies from linear ones. In order to test our methodology, we revisited the case study of , in which two events were shown, from a visual perspective, to indicate strong correlation between the rise of relativistic electron fluxes and ULF wave power. Our application of mutual information to the events presented by indicates that relativistic electron fluxes are linearly and non-linearly dependent on ULF wave power. However, the values that we found for both the Pearson correlation and mutual information of relativistic electron fluxes and ULF wave power are modest when compared to previous statistical results and consistently smaller than the correlational measures found between seed electrons and ULF wave power. This result is counter-intuitive since seed electrons with long azimuthal periods cannot experience drift–orbit resonance with ULF wave fluctuations and should therefore not be correlated with radial diffusion drivers more strongly than relativistic electron fluxes. However, our results do not indicate a necessary causal physical relationship between seed electrons and ULF wave power, but they do point out the necessity of not over-interpreting correlational measures, whether linear or non-linear. The modest dependence of energetic electron fluxes with ULF wave power could also originate in a shared dependence on solar wind drivers, such as the solar wind speed for relativistic electrons and solar wind density for seed electrons. Our results are therefore indicative of the need to incorporate data analysis tools that can distinguish between interdependencies of various solar wind drivers. In the framework of information theory, conditional mutual information is specifically built for that purpose and has been successfully used to resolve a long-standing question about the relative role of solar wind speed and density in driving relativistic electron fluxes . In future studies, we will also apply a comparable methodology presented in to seek dependencies of relativistic electron fluxes on solar wind drivers for given storm phases and build non-parametric estimators for the probability density of random variables that do not require binning .