## 6 Discussion
### Inverse Problem  
The inverse problem of the physics of earthquake source is to determine the phenomenological coefficients from the observation data of aftershocks. [PERSON] law in the form (2) makes it possible to formulate and solve the problem of determining the deactivation coefficient \(\sigma\) from the observation data of the frequency of aftershocks \(n\). The auxiliary value \(g\), which we introduced in Section 3, is conveniently written in the form of \(g=(n_{0}-n)/nm_{0}\). It is easy to see that  
\[\sigma=\frac{d}{dt}\langle g\rangle. \tag{21}\]  
Here, the angle brackets denote the regularization of the function \(g(t)\) calculated from observation \(n(t)\). Regularization is reduced in this case to the smoothing procedure. Our experience indicates that the deactivation coefficient \(\sigma\) undergoes complex variations over time [8; 20]. However (and this seems to us extremely important) at the first stages of evolution \(\sigma=const\). The time interval during which \(\sigma=const\), we called the Omori epoch. In the Omori epoch, the classical Omori law is fulfilled (1), according to which the frequency of aftershocks hyperbolically decreases over time. In our experience, the duration of the Omori epoch varies from case to case from a few days to two or three months. We noticed a tendency for the duration of the Omori epoch to increase with the increase in the magnitude of the main shock. An interesting prediction follows from the existence of the Omori epoch, which has been reliably confirmed by experience [21]. Namely, the question of dependence of the deactivation factor on the magnitude of main shock is analyzed theoretically and experimentally. A monotonic decrease in the deactivation factor with an increase in the magnitude of the main shock, \(M_{0}\), has been reliably established. Figure 3 shows the result of measurements of \(\sigma\) at different values of \(M_{0}\). To measure the deactivation factor, we used the USGS/NEIC earthquake catalog and the technique developed during the compilation of the Atlas of Aftershocks [20]. We see that, on average, \(\sigma\) decreases monotonically with the increase in \(M_{0}\). The dependence \(\sigma(M_{0})\) is approximated by the formula  
\[\sigma=A-BM_{0}, \tag{22}\]  
where \(A\)= 0.64, \(B\) = 0.07 with a sufficiently high coefficient of determination \(R^{2}\) = 0.82. Thus, the theoretical inequality \(\mathrm{d}\sigma/\mathrm{d}M_{0}\) is reliably confirmed by direct measurements. We have a wonderful harmony between theory and experiment. It is quite clear that the question of how best to formulate the Omori law, in the form (1) or (2), could only be solved by observation and experience. Equation (2) turned out to be more effective, since it made it possible to introduce a simple and useful concept of deactivation of the source, to pose the inverse problem of the source, and to reveal the existence of the Omori epoch. In addition to this, we have shown that with the help of (2) one can make a meaningful statement about the deactivation coefficient and, moreover, check this statement experimentally. Finally, the question of whether it is not better to use the one-parameter Formula (2) for modeling aftershocks than the two-parameter Hirano-Utsu formula \(n=k/(c+t)^{p}\)[2] deserves discussion. We give preference to Formula (2), since the inverse problem solved on its basis indicates the existence of the Omori epoch [8; 9]. The Hirano-Utsu formula is unacceptable, since it contradicts the existence of the Omori epoch at \(p\
eq 1\), and at \(p=1\) it coincides with the Omori Formula (1). Perhaps it would be appropriate to draw a distant historical analogy here. According to the law of gravitation, the interaction potential \(\varphi\propto 1/r\) leads to the ellipticity of the planetary orbit. The deviation from ellipticity, for example, of the orbit of Mercury, could in no way serve as a reason for choosing the interaction, say, in the form \(\propto 1/r^{p}\).