Båth's law and the self‐similarity of earthquakes

Abstract

[1] We revisit the issue of the so-called Båth's law concerning the difference D 1 between the magnitude of the main shock and the second largest shock in the same sequence. A mathematical formulation of the problem is developed with the only assumption being that all the events belong to the same self-similar set of earthquakes following the GutenbergRichter magnitude distribution. This model shows a substantial dependence of D 1 on the magnitude thresholds chosen for the main shocks and the aftershocks and in this way partly explains the large D 1 values reported in the past. Analysis of the New Zealand and Preliminary Determination of Epicenters (PDE) catalogs of shallow earthquakes demonstrates a rough agreement between the average D 1 values predicted by the theoretical model and those observed. Limiting our attention to the average D 1 values, Båth's law does not seem to strongly contradict the Gutenberg-Richter law. Nevertheless, a detailed analysis of the D 1 distribution shows that the Gutenberg-Richter hypothesis with a constant b-value does not fully explain the experimental observations. The theoretical distribution has a larger proportion of low D 1 values and a smaller proportion of high D 1 values than the experimental observations. Thus, Båth's law and the Gutenberg-Richter law cannot be completely reconciled, although based on this analysis the mismatch is not as great as has sometimes been supposed.

Aftershock Statistics

“…A number of extensive studies of the statistical variability of Dm have been carried out (VERE-JONES, 1969;KISSLINGER and JONES, 1991;TSAPANOS, 1990;FELZER et al, 2002FELZER et al, , 2003CONSOLE et al, 2003;. Despite progress in understanding the nature of Ba˚th's law, its validity still remains an open question.…”

Is Earthquake Triggering Driven by Small Earthquakes?

“…These studies yield α-value close to 1, but the limited range of the mainshock magnitude considered and the large scatter of the number of aftershocks per mainshock do not allow an accurate estimation of α. The case α = b also explains another well documented property of aftershocks, known as Bath's Law [Bath, 1965;Drakatos et al, 2001;Felzer et al, 2002;Console et al, 2002], which states that the difference between the mainshock magnitude and its largest aftershock is on average close to 1, independently of the mainshock magnitude. Again, the limited range of mainshock magnitudes used in these studies and possible biases in the method of data selection [VereJones, 1969;Console et al, 2002] does not allow one to test the dependence of the magnitude difference as a function of the mainshock magnitude.…”

Tectonic implications and seismicity triggering during the 2008 Baluchistan, Pakistan earthquake sequence

“…According to Bath (1965), generally, this size difference shows a constant value (Dm = 1.2) and does not depend upon the size of main shock which implies that the stress transfer responsible for the occurrence of aftershocks is a self-similar process (Utsu, 2002a;Shcherbakov and Turcotte, 2004). A number of studies have been carried out to show the statistical variability of this law (Kisslinger and Jones, 1991;Tsapanos, 1990;Felzer et al, 2002Felzer et al, , 2003Console et al, 2003;Helmstetter and Sornette, 2003;Yadav et al, , 2011. However, its validity remains an open question, regardless of the progress of understanding the characteristics of Bath's law.…”