The time dependence of the parameter of the Gutenberg-Richter (GR) magnitude distributions is computed for correlated foreshock sequences of earthquakes, by using the geometric-growth model of earthquake focus, the magnitude distribution of correlated earthquakes and the time-magnitude correlations, derived recently. It is shown that this parameter decreases in time in the foreshock sequence, from the background values down to the main shock. If correlations are present, this time dependence and the time-magnitude correlations provide a tool of monitoring the foreshock seismic activity. We discuss a possibility of estimating the moment of occurrence of a main shock by such an analysis of a foreshock sequence. The discussion is applied to two Vrancea main shocks. The appreciable limitations of such an estimation are discussed.

The empirical Bath’s law is derived from the statistical distribution in magnitude difference of pairs of earthquakes. It is shown that earthquake correlations can be expressed by means of the magnitude-difference distribution. We introduce a distinction between dynamical correlations, which imply an “earthquake interaction”, and purely statistical correlations, generated by other, unknown, causes. The distribution of dynamically correlated earthquakes is derived from the statistical fluctuations of the accumulation time, by means of the geometric-growth model of energy accumulation in the focal region. The derivation of the Gutenberg-Richter statistical distributions in energy and magnitude is presented, as resulting from this model. It is shown that the most suitable framework for understanding the origin of the Bath’s law is the extension of the statistical distributions to pairs of earthquakes, where the difference in magnitude is allowed to take negative values. 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. Time correlations of the accompanying seismic activity are also presented.

The empirical Bath’s law is derived from the statistical Gutenberg-Richter distribution in magnitude difference of pairs of earthquakes. The derivation of the statistical Gutenberg-Richter distributions in energy and magnitude is presented, as resulting from a geometric-growth model of energy accumulation in the focal region. It is shown that the most suitable framework of understanding the origin of the Bath’s law is the extension of the statistical distributions to pairs of earthquakes, where the difference in magnitude is allowed to take negative values. If the seismic activity which accompanies a main shock is viewed as a relaxation process, then we need to include both the aftershocks and the foreshocks in this accompanying seismic activity, and to view it as fluctuations in magnitude. The extension of the magnitude difference to negative values leads to a vanishing mean value of the fluctuations and to accepting 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. The geometric-growth model of energy accumulation in the focal region induces a lower bound to the magnitudes of the largest aftershocks (foreshocks), such that the (average) reference value $\Delta M=1.2$ between the magnitudes of the main shock and the largest accompanying seismic event corresponds to the smallest aftershock (foreshock) in the whole set of the largest aftershocks (foreshocks).