Correlations and Bath’s law

Bath’s empirical law is derived from the magnitude-difference statistical distribution of earthquake pairs. The pair (two-event, bivariate) 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. It is shown that the dynamical correlations may account, at least partially, for the roll-off effect in the Gutenberg-Richter distributions. The extension of the magnitude difference to negative values (by observing the order of the members of the pair) leads to a vanishing mean value of the magnitude difference and to the standard deviation as a measure of its variations. 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 Bath’s law. Earthquake purely statistical correlations and deterministic time-magnitude correlations of the accompanying seismic activity are also presented