tokens: which Geotail was inside the solar wind, produced by the NASA at the GFSC ( ftp://spdf.gsfc.nasa.gov/pub/data/geotail/merged/sw_min_merged_ascii ), and by assuming a shock-simplified model (from Cairns et al. 1995 ; Farris et al. 1999 ) we considered only the time intervals in which the spacecraft was located inside the solar wind but outside the foreshock. These conditions are verified by examining magnetic field data from both Geotail and IMP-8 and by comparing these measurements with corresponding observations made by both ACE and WIND , which are located inside the solar wind. Table 2 reports the time periods when Geotail was in the solar wind. Table 2 Time Periods When Geotail Was in the Solar Wind In particular, we used the Geotail EPIC-ICS experiment data related to the proton flux measured in 9 different energy channels, from 58 keV to 3 MeV, characterized by 16 azimutal sectors (Williams et al. 1994 ). By averaging the fluxes over all azimutal sectors we computed the omnidirectional fluxes at hourly resolution. Table 1 shows the details of the EPIC-ICS energy ranges. Figure 3 shows the proton flux for the different energy channels. As expected, low-energy channels measure greater proton fluxes than high-energy ones: typically, G1–G4 show one (up to two) order of magnitude proton flux increases with respect to G5–G9. Finally, a reduction in the proton flux during the period 1995–1997, corresponding to the minimum phase of SC No. 22, is observed at high energies. Figure 3. Time history of the proton flux measured in the nine energy channels (G1–G9) of the EPIC-ICS experiments on board Geotail . The x -axis labels refer to the time periods when Geotail was in the solar wind. 3. Background Proton Fluxes: CDF Analysis To investigate the variability of the background proton fluxes we used a statistical approach based on the evaluation of the CDF for each energy channel. For each considered data set from ACE , IMP-8 , and Geotail , the PDFs, from which CDFs can be obtained, have been evaluated using a non-parametric approach based on the kernel density estimator technique (e.g., Silverman 1998 ; Alberti et al. 2014 , 2017b , for more details) with the definition of a non-negative function, called a kernel function, with zero mean and integrating to one, and by choosing an appropriate smoothing parameter, called the bandwidth. From the class of kernel functions (Gaussian, Epanechnikov, Triangular, and so on) a good choice is the Epanechnikov kernel, which is optimal in a minimum variance sense. Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. Thus, when PDFs have been evaluated, CDFs are obtained as the probability that F ′ will take a value less than or equal to a value F (with being F the proton flux). In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to F :In
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