Forecasting Aftershock Activity: 3. Bath's Dynamic Law

In seismology according to Bath's well-known law, the magnitude of the strongest aftershock is on average by unity lower than the magnitude of the main shock. At the same time, most of the strongest aftershocks typically occur within a few hours after the main shock. From the practical standpoint, this activity is quite naturally perceived as a direct continuation of the main earthquake. The subsequent strong aftershocks occur against the rarer background shocks, are less expected, and therefore constitute a separate hazard. The average difference in magnitudes between the main shock and the strongest aftershock that occurs a certain time after the main shock gradually increases. In this work, we consider the problem of estimating the magnitudes of the strongest future aftershock at the successive instants of time after the main shock without taking into account the information about the aftershocks that have already occurred before a given time. For these estimates, we construct the theoretical distributions whose shape proves to be independent of time, whereas the time dependence of the shift in the magnitude proves to be known a priori. The predetermination of these dependences at the moment of the strong earthquake gives us grounds to characterize the constructed theoretical model as Bath's dynamic law.