Bath's law, correlations and magnitude distributions

The empirical Bath's law is derived from the magnitude-difference statistical distribution of earthquake pairs. The pair distribution related to earthquake correlations is presented. The single-event distribution of dynamically correlated earthquakes is derived, by means of the geometric-growth model of energy accumulation in the focal region. The dynamical correlations may account, at least partially, for the roll-off effect in the Gutenberg-Richter distributions. The seismic activity which accompanies a main shock, including both the aftershocks and the foreshocks, can be viewed as fluctuations in magnitude. The extension of the magnitude difference to negative values leads to a vanishing mean value of the fluctuations and to the standard deviation as a measure of these fluctuations. It is suggested that the standard deviation of the magnitude difference is the average difference in magnitude between the main shock and its largest aftershock (foreshock), thus providing an insight into the nature and the origin of the Bath's law. It is shown that moderate-magnitude doublets may be viewed as Bath partners. Deterministic time-magnitude correlations of the accompanying seismic activity are also presented.