Bath's law, correlations and magnitude distributions

Abstract

The empirical Bath's law is derived from the magnitude-difference statistical distribution of earthquake pairs. 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 pair distribution related to earthquake correlations is presented. The singleevent 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. The dynamical correlations may account, at least partially, for the roff-off effect in the Gutenberg-Richter distributions. 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 earthquake pairs, 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