# Co-Seismic Magnetic Field Perturbations Detected by Swarm Three-Satellite Constellation
## 4 Discussion and Interpretation  
The 12 selected tracks associated to 10 seismic events have been analyzed to search for a possible linear regression between the time interval of the arrival of a supposed co-seismic gravity wave in  
Figure 13: Residual of the Y component of Swarm Charlie magnetic field after the occurrence of the Mw 5.6 2019-04-07 earthquake. The vertical black line in the lower plot represents the point where the satellite crosses the earthquake latitude. The vertical green line is the candidate point where the earthquake-induced phenomena are supposed to start. As in the previous track, this point is the start of the only anomaly presented in the track. The data of Charlie are very similar to the ones of Alpha for this case study. During the time of the occurrence of the earthquake, the satellite was below \(-70^{\circ}\) geographical latitude, and thus, the data are cut. In the Supplementary Materials, more details and the other available Swarm magnetic data are represented (Figure S10). the ionosphere and the distance between the epicentre and the Swarm satellite, as represented in Figure 14. The intercept has been fixed to the origin of the axis as the time to cover a null distance needs to be zero. The obtained slope is (\(0.00163\pm 0.00014\)) min/km that corresponds to a propagation velocity of (\(10.04\pm 0.85\)) km/s. The R\({}^{2}\) = 0.92 denotes a good agreement with this linear regression, confirming that the two quantities plotted in the graph seem to be linearly correlated. Although this correlation is pretty good, the propagation speed for this phenomenon seems to be higher than we might expect for this kind of waves. We propose that actually this could be caused by two different involved phenomena: one from the Earth's surface that reaches the maximum electron density altitude (hmF2 or altitude of ionospheric layer F2), and another physical phenomenon that propagates from this altitude to the satellite position. A sketch of the proposed propagation mechanism is shown in Figure 15. A simple model was implemented with a vertical propagation with velocity v\({}_{1}\) until hmF2 and a second velocity v\({}_{2}\) from the projection of the epicentre at hmF2 until the satellite position. The altitude of the F2 layer was determined by the IRI-2016 model and listed in Table 1. We supposed that the velocity v\({}_{1}\) and v\({}_{2}\) were equal for all the events, and thus, we determined which combination of v\({}_{1}\) and v\({}_{2}\) can reduce the difference with the measured times. \"v\({}_{1}\)\" and \"v\({}_{2}\)\" were selected in the range between 0.10 km/s and 50.00 km/s with a step of 0.01 km/s. We also take into account that the different air temperature \(T\) affects the mechanical velocity of propagation by a factor of \(\sqrt{T}\)[32]. Thus, the surface (2 m height) temperature has been retrieved from the climatological archive ECMWF ERA-Interim for the 10 earthquakes, and the velocity v\({}_{1}\) was multiplied by \(\sqrt{T/298K}\). The division by 298K assures to obtain the result referred to the standard temperature of 25 \({}^{\circ}\)C. The best combination that fits our data (i.e., the one with the minimum residuals) was v\({}_{1}\) = 2.20 km/s (at temperature of 25 \({}^{\circ}\)C) and v\({}_{2}\) = 16.07 km/s. From the dispersion of the measurements versus the model, we estimated an uncertainty on the velocities of about \(\Delta\)v\({}_{1}\) = 0.3 km/s and \(\Delta\)v\({}_{2}\) = 3 km/s. We note that the velocity v\({}_{1}\) is in agreement with the known velocities in the literature for acoustic gravity waves (e.g., [9]). Figure 14: Relation between the time interval of the arrival of the co-seismic gravity wave in the ionosphere in function of the distance between the epicentre and the Swarm satellite. A comparison between the calculated and the measured times needed to propagate from epicentre to the satellite is represented in Figure 16. The red line represents the ideal case, i.e., when the model was identical to the measurements.