## 4 Discussion
### Improving the Superposed Epoch Approach by Taking into Account the Rikitake Law  
The results of this paper, especially for the magnetic field anomalies, strongly confirm the empirical Rikitake law [37], which predicts that the logarithm of anticipation time linearly increases with earthquake magnitude. If such a law is true, then the superposed epoch approach would not work due to the mixing of several earthquakes with different magnitudes. It seems to work in the real case, but we think that this is due to the natural distribution of the earthquake magnitudes that follows the Gutenberg-Richter [42] earthquake distribution, privileging the smaller magnitude events. In this paper, we propose a new approach to addressing this issue when the epochs are superposed by \"normalizing\" the time scale using the Rikitike law, with the coefficients (\(a\) and \(b\)) being those proposed by [PERSON] et al. [29] for the _Swarm_ magnetic field and electron density satellite data and referring to an M6.5 earthquake, following the next equation:  
\[\Delta T^{\prime}=\Delta T\cdot\frac{10^{(a+b.6.5)}}{10^{a+b\cdot M}} \tag{8}\]  
where \(\Delta T^{\prime}\) is the normalized anticipation time, \(\Delta T\) is the original time with respect to the earthquake, and \(M\) is the magnitude of the earthquake under analysis. Even if the formula can be, in principle, also applied to post-earthquake anomalies, we think that it is not proper to use it in such a way, as the formula is constructed on possible precursors. The reference magnitude (here, chosen as M6.5) can be, in principle, any magnitude, and the results are not affected by this choice. Its scope is to report the time in a different scale chosen as the number of days that preceded an M6.5 earthquake. M6.5 is an intermediate magnitude with a high number of events in the investigated period to be able to clearly show possible precursors, according to the results illustrated in this paper. To select the window of analysis, we look at the results from [29] and plotted the expected anticipation time as a function of the earthquake magnitude (in Figure 6, as the orange curve). Based on these data, we decided to analyze 150 days before the earthquake  
Figure 5: Receiving Operating Characteristic (ROC) graph for the best _Swarm_ magnetic field and electron density anomalies (i.e., the ones in WSC concentrations with significant statistical parameters \(d\geq 1.5\) and \(n\geq 4.0\)) calculated, varying the alerted time from 10 days to 2500 days. The Area Under the Curve (AUC) is represented as a black dashed pattern for a random case and as a red dashed pattern for the real _Swarm_ case. for the magnetic field data and 85 days for the _Ne_ measurements (when \(M=6.5\), i.e., our reference point). We would underline that, using Equation (4), the analyzed time before the earthquake is not fixed (as in the previous analyses), but it is different for any earthquake as a function of its magnitude, and it is represented as the blue curve in Figure 6, i.e., the result of applying Equation (4)--that is, how many days the normalized time corresponds for other magnitudes. For example, the analyzed time before an M8.0 earthquake is 4037 days for the magnetic field (or 1576 days for _Ne_) according to Equation (4), and the time interval from 4037 days before and until the event will be compressed in the horizontal axis (of Figures 7 and 8) and shown to be equivalent to 150 days to 0 days of an M6.5 earthquake. We note that, for the magnetic field, the larger magnitude earthquakes (M > 8) could show an anticipation time that is greater than that of the 8 years of _Swarm_ data available at the moment (the red dashed horizontal line in Figure 6). We test and apply this approach to the most significant results of the _Swarm_ data, which are the investigations of the magnetic field and _Ne_ data with Method 1 and the _Swarm_ magnetic field anomalies in the band of 2-10 s and the _Ne_ anomalies in the band of 25-50 s, which are those with the best statistical indications from the previous analyses.