RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES5004, doi:10.2205/2007ES000282, 2008

Introduction

[2]  A characteristic feature of dynamic system is the fulfillment of a power law of distribution,that corresponds to a high degree of concentration of a total effect in a small number of a few strongest events. Guttenberg-Richter earthquakes recurrence law is a well known example of the power law distribution of a number of earthquakes from their energy and seismic moment values. Power distributions are frequently associated with fractal geometry of corresponding objects. They are abundant in nature and are typical of dynamic systems of different physical nature. The distribution of blocks number (tectonic plates) in relation to their size, the distribution of deposit number in relation to the stock values, self-similar character of the geometry of micro-crack structure in samples and tectonic faults in the lithosphere and others are examples that can be treated as power-law distributions [Bak et al., 1987, 1988; Burshtein, 2006; Czechowski, 2003; Kontorovich et al., 1985; Mandelbrot, 1982; Sadovskiy, 1989; Sornette, 2000; Sornette and Pisarenko, 2003; Turcotte, 1997; and others].

[3]  Realization of self-similar power distributions in geophysical systems is treated ordinary by analogy with the critical type of behavior [Bak et al., 1987, 1988; Ito and Matsuzaki, 1990; Klimontovich, 1995; Sornette, 2000; Tyupkin, 2007; Zaliapin et al., 2002; and others]. It is corroborated by similarity noted in the behavior of such systems with properties occurring in the vicinity of critical points and in the vicinity of second order phase transitions, since power laws describe the variation of physical characteristics in the vicinity of critical points [Ma, 1980; and others]. Such behavior occurring in critical points and during second order phase transitions is explained by the growth of fluctuations (their magnitude and spatial scale) as the critical point (phase transition) is approaching and individual elements of the system behave in a more and more cooperative way.

[4]  The analogy between the process of seismogenesis and second order phase transition is widely used in modern geophysics to understand the nature of seismic process and in elaboration of algorithms of earthquake prognosis [Ito and Matsuzaki, 1990; Tyupkin, 2002, 2007; Zaliapin et al., 2002; and others]. The effect of increasing of seismic regime correlation, which by this analogy may be expected during strong earthquake preparation, was revealed from the results of earthquake catalog studies and in model experiments. This effect manifests itself in the following features: in an increase of in a number of swarms of earthquakes and in appearance of remote earthquakes and earthquake chains. Indicators of such increase in correlation of seismic regime were used in elaboration of new algorithms of prognosis of strong earthquakes [Rundle et el., 2000; Sobolev and Ponomarev, 2003; Tyupkin and Giovambattista, 2005; Zaliapin et al., 2002; and others].

[5]  However in the use of the analogy between seismic regime and second order phase transition, the important differences between these two phenomena should be taken into account. These differences may be reduced to two major moments. Firstly, geophysical systems and specifically seismic regime are described by power self-similar laws everywhere and not only in the vicinity of some special points as in the case of critical phenomena and phase transitions. This difference is minimized in the model of self-organized criticality, the SOC-conception. The SOC model describes and even partly postulates the process of spontaneous evolution of a complex system in the direction of developing of a self-similarity [Bak et al., 1987, 1988]. Secondly, the critical phenomena and second order phase transitions (as distinct from first order phase transitions) occur without energy release (or absorption). But geophysical self-similar processes, specifically earthquakes commonly release large amount of energy stored up in the system before.

[6]  The differences noted above are significant. Therefore, besides the SOC model, other models that could describe the origin of self-similar power distributions in geophysical systems are to be proposed. Some versions are given in [Czechowski, 2001, 2003]. Below, we discuss the approach based on the model of power law distribution appearing in a result of a set of episodes of avalanche-like relaxation of a set of formed before metastable situations (metastable sub-systems) [Rodkin, 2001]. This model formally corresponds to a multiplicative cascade model was used before in [Rodkin, 2002b] to model a regime of natural disasters. In this paper, we apply this model to describe a seismic process and origin of power-law distribution in seismic process and to model the empirical distribution of a number of deposits from the amount of resources [Burshtein, 2006; Kontorovich et al., 1985; Sornette, 2000; Turcotte, 1997]. This model [Rodkin, 2001, 2002b] allows us to advance to a better insight in the variety of conditions in which power law distributions can be formed, in conditions of realizing of deterministic and/or statistical prediction of strong earthquakes, and in processes of formation of empirical distributions of number of deposits in relation to their resources.


RJES

Citation: Rodkin, M. V., A. D. Gvishiani, and L. M. Labuntsova (2008), Models of generation of power laws of distribution in the processes of seismicity and in formation of oil fields and ore deposits, Russ. J. Earth Sci., 10, ES5004, doi:10.2205/2007ES000282.

Copyright 2008 by the Russian Journal of Earth Sciences

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