# Co-Seismic Magnetic Field Perturbations Detected by Swarm Three-Satellite Constellation
## 4 Discussion and Interpretation
The sum of the residual between the time estimated from the model and the measured one from satellite data was reported. The calculated values for the model with two velocities used the best-obtained velocity estimations. We note that in the second model (b), two cases were correctly described by the model as they are on the ideal trend, while the others were under- or overestimated. As also underlined by the lower value of the sum of the residuals, the model with two velocities estimates better the time of propagation of the disturbance from the epicenter to the satellite, with a lower dispersion of data in the plot and closer to the ideal case (red line). This statistical evidence deposes in favour of our assumption of a gravity wave that mechanically propagates until F2 layer making oscillating the high number of free electrons in this region and starting a scattering that propagates at least to the Swarm satellite altitudes (about 440-500 km above sea level). Considering also some literature [33; 34], we tested the possibility that the second propagation follows the magnetic field lines strictly. We found that the results were worse than the simple straight propagation, thus, even if the perturbation could affect the magnetic field, the disturbance probably did not strictly follow the magnetic field lines. Figure 15: Possible mechanism of propagation of the disturbance from the epicentre to the Swarm satellite. From the lithosphere to the F2 ionospheric layer, an acoustic gravity wave is probably the mechanism, and from F2 to the Swarm position, some electromagnetic scattering could be involved (see the text for further discussion). To finally compare the two approaches describing the data (one-velocity or two-velocity model), we used the Akaike Information Criterion (AIC) developed by [PERSON], 1973 [35]. In case of few parameters a \"corrected\" index AICc can be used. This criterion was used to choose the best model (the one that presents the minimum AIC, or AICc, later described) and select which variables were really fundamental to the system. The AIC is defined as:  
\[AIC=2K-2\ \log(L) \tag{1}\]  
where \(K\) is the number of estimated parameters, i.e., the degree of freedom and \(L\) is the maximum of the likelihood function. For this paper, we used the following equation to estimate the \(\log(L):\)  
\[\log(L)=-\frac{1}{2}n\sigma^{2}=-\frac{1}{2}\pi\Big{(}\frac{RSS}{n}\Big{)}^{2} \tag{2}\]  
where \(n\) is the number of samples and RSS is the Residual Sum of Squares. If the dataset on which the model is evaluated contains a small number of elements, as in our case, a correction factor must be introduced, estimating a second coefficient, called AICc:  
\[AICc=AIC+\frac{2K(K+1)}{n-K-1} \tag{3}\]  
To calculate the values of AIC and AICc for our two models, we followed the approach of [PERSON] and [PERSON], 2014 [36], even if they applied this criterion to a vastly different case study. Table 2 reports the parameters and the AIC and AICc coefficients obtained for the two models. The number of parameters has been increased by one to consider an error coefficient in the model. Despite the height of the F2 layer (hmF2) and the temperature are two parameters used in the second model, they do not add any degree of freedom to the same model, since they are fixed. Figure 14.0 Comparison of the calculated time versus the measured one to propagate from epicenter to the satellite. The red line represents the ideal case, i.e., when the model is identical to the measurements. (a) A model with one velocity. (b) A model with two velocities. Each graph reports the corresponding sum of the residuals of the model with respect to the measured data. The two-velocity model shows the lowest residual. The AICc (and even AIC) suggests that the model with two velocities better describes the magnetic disturbance identified in the 12-track dataset.