On potential representations of the distribution law of rare strongest earthquakes

Abstract

Assessment of long-term seismic hazard is critically dependent on the behavior of tail of the distribution function of rare strongest earthquakes. Analyses of empirical data cannot however yield the credible solution of this problem because the instrumental catalogs of earthquake are available only for a rather short time intervals, and the uncertainty in estimations of magnitude of paleoearthquakes is high. From the available data, it was possible only to propose a number of alternative models characterizing the distribution of rare strongest earthquakes. There are the following models: the model based on the Guttenberg – Richter law suggested to be valid until a maximum possible seismic event (Мmах), models of 'bend down' of earthquake recurrence curve, and the characteristic earthquakes model. We discuss these models from the general physical concepts supported by the theory of extreme values (with reference to the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD) and the multiplicative cascade model of seismic regime. In terms of the multiplicative cascade model, seismic regime is treated as a large number of episodes of avalanche-type relaxation of metastable states which take place in a set of metastable sub-systems. The model of magnitude-unlimited continuation of the Guttenberg – Richter law is invalid from the physical point of view because it corresponds to an infinite mean value of seismic energy and infinite capacity of the process generating seismicity. A model of an abrupt cut of this law by a maximum possible event, Мmах is not fully logical either. A model with the 'bend-down' of earthquake recurrence curve can ensure both continuity of the distribution law and finiteness of seismic energy value. Results of studies with the use of the theory of extreme values provide a convincing support to the model of 'bend-down' of earthquakes’ recurrence curve. Moreover they testify also that the 'bend-down' is described by the finite distribution law, i.e. the bend-down occurs more efficiently than it is envisaged in the commonly used model developed by Y. Kagan (which treats the bend-dawn as an exponential decay law). However, despite the finiteness of the distribution law, density of magnitudes decline quite slowly in the area close to the maximum possible Мmах event as (Мmах – M)n, where n varies in the range between 4 and 6 in the majority of cases. As a result Мmах value can be estimated only with a large error. In rare cases, if the space-and-time area under study contains higher number of strongest earthquakes, the empirical distribution law becomes close to the exponential law; in this case n value is quite high, and Мmах values becomes unstable and tend to infinite growth. In our study, the distribution law of strongest earthquakes was investigated by the methods based on the extreme values theory (world data and several regional catalogues were examined), and the results of calculation do not reveal cases of occurrence of characteristic events. However, such a seismic regime was revealed in a number of cases from paleoseismicity data and from some instrumental regional catalogues. Conditions providing for the occurrence of characteristic earthquakes are studied here using the multiplicative cascade model. According to [Rodkin, 2011], this model provides the simulation of all known regularities of seismic regime, such as a decrease in b-value in the vicinity of strong earthquakes, development of aftershock power cascade, and existence of seismic cycle and foreshock activity. This article considers an extension of the cascade model by adding of non-linear members in the kinetic cascade equation in order to describe effects of the 'bend-down' of the earthquake recurrence curve and the characteristic earthquakes occurrence. It is shown that in terms of the multiplicative cascade model, the occurrence of characteristic earthquakes is connected with development of the nonlinear positive feedback between the size of the current rupture zone and the rate of its further growth. The modelling results are compared with data on seismicity of the South-Eastern Asia, which suggest that the regime providing the occurrence of characteristic earthquakes appears to be typical of the seismic regime of subduction zones (while it is not observed outside such zones). It is concluded that the non-linear positive feedback that controls the possibility of occurrence of characteristic earthquakes may be caused with the presence of deep fluids of increased concentration in the subduction zones.