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

Description of Seismic Regime With the Use of Multiplicative Cascade Model. Probabilistic and Deterministic Prognosis of Earthquakes

[7]  Distribution of a number of earthquakes of different size (seismic moment and/or seismic energy values) obeys the Guttenberg-Richter law of earthquake recurrence. Different authors discussed the nature of recurrence law and besides the prevailing now SOC-concept other approaches to the problem were proposed also [Golitsyn, 2001; Grigorian, 1988; Rodkin, 2001; and others]. One of the models formulated in a general way corresponds to an occurrence of a power distribution in a result of a large number of episodes of development of stochastic avalanche-like processes when the velocity of process increase is statistically proportional to its current value [Rodkin, 2001].

eq001.gif(1)

where k is a random value with positive mean value, and the avalanche-like process (1) at each step may continue with probability p or to be interrupted with probability ( 1-p ).

[8]  It can be easily shown that the set of values Xi obtained as a result of a series of processes (1) will be distributed according to a power law. In fact, solving (1), we obtain values of individual events x, which realize in the result of n steps of the process:

eq002.gif(2)

where x0 is the initial value, n is a step number, Dt is a step length. The probability of process interruption at step number n and accordingly of formation of event of value x is equal to

eq003.gif(3)

[9]  Thus we obtain

eq004.gif(4)

where infinite geometric progression is summed up:

(1-p) pn + (1-p) pn+1 + ldots + (1-p) pa.

From (4) it follows

eq005.gif(5)

eq006.gif

[10]  In transition to continuous process that may be interrupted with equal probability at any arbitrary time moment, probability p of process development continuation at an arbitrary Dt may be written as p0Dt, where p0 is a probability of continuation of process development for the step of single-unit duration. Taking it into account we obtain from (5)

eq007.gif(6)

[11]  From (6) it can be seen that model (1) leads to a power-law distribution of a number of events in relation to their size x.

[12]  As applied to earthquake model, let us imagine seismic process as a set of episodes of avalanche-like relaxation of elastic energy accumulated before (or relaxation of inner energy of rocks, for example of energy of metastable mineral assemblies). Characteristics of such model are the intensity of events flow N and two parameters: mean values of parameter k and probability of cessation of an avalanche-like process in time unit p. Parameters k and p in combination determine the slope of the recurrence plot b of a number of events from their values Xi (magnitudes of "earthquakes'') in double-logarithmic coordinates

eq008.gif(7)

[13]  Using (7), it is easy to select values of parameters k and p, with which the obtained values of slope of recurrence plot b and magnitude m (for example m= lg(Xi)) correspond to values typical of seismic process. Specifically, if we assume initial values of Xi equal to one and mean values p=0.5 and k=1, then we obtain the recurrence plot slope b=1.

[14]  If we preset the average number N of such avalanche-like processes in time unit and a certain regularity of change of parameters k and p with time, the model in question will produce a sequence of values of model magnitudes of events lg(Xi) similar to the earthquake magnitudes in an actual seismic process.

2007ES000282-fig01
Figure 1
[15]  To model change in seismic process we preset weak periodic variations in recurrence plot slope b (with amplitude 0.2 and period equal to 1000 time intervals) and assume seismic flow intensity to be N=500 events in unit time. In Figure 1, a typical realization of a model process in a time section as long as 5000 conventional time intervals is given. The obtained this way change of maximum values of magnitude M max(t) and recurrence plot slope b -value( t ) are similar to typical values of an actual seismic process (apart from preset of a periodic character of variation of recurrence plot slope b with time).

2007ES000282-fig02
Figure 2
[16]  The discussed simple and purely stochastic model produces a well-known "prognostic'' indicator, that is time intervals of strong earthquake occurrence M max are preceded in average by decrease in recurrence plot slope b -values. To show this dependence clear in Figure 2 the plot is shown of magnitude maximum values M max= lg(Xi) related to values of recurrence plot slope b in the time interval preceding the time interval to define M max. As it can be seen in Figure 2 these parameters are correlated: peak events M max are realized (statistically) with relatively lower preceding recurrence plot b -value. The mechanism that causes such a correlation is clear. Indeed, generation of earthquake with large magnitude value corresponds statistically to the values of p and k parameters that at the same time correspond to lower values of recurrence plot slope b -value.

[17]  It should be emphasized that in the model under consideration the effect of decrease in the b -value is not a prognostic indicator of a strong event being prepared (it is not correct to speak about strong event preparation as applied to a sequence of independent events) but a parameter related to the probability of occurrence of a strong event. Statistically such an anomaly is characteristic of a time interval before, during and after the occurrence of the strong event. Interpreting the obtained result as applied to the problem of prognosis of strong earthquakes, we obtain that statistical prognosis of strong event occurrence is possible but this prognosis has a stochastic character. Each individual event is a random phenomenon.

[18]  Similar situation may take place in the case of actual seismic regime. In this case, the prediction suitable for practical use may be possible but not in the sense it is commonly understood. In terms of the used model the time intervals when the probability of strong earthquake occurrence is greater can be indicated whereas the physical "process of strong earthquake preparation'' as such is absent. This situation differs essentially from the case when "the process of strong earthquake preparation'' exists actually. Indeed, if an actual process of earthquake preparation is under way, we can reveal its new features and as the amount of data grows and research progresses the prognosis will become more and more exact. In the terms of the model described above, possibilities of improvement of prognosis are limited from the very beginning owing to random character of earthquake occurrence.

[19]  Note that at present the earthquake prediction takes place in statistical sense but it is borne in mind that the reliability of prediction can be improved very significantly with progress in seismology. The model described above suggests that these hopes may by unjustified and an alternative situation without an essential progress can take place.


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

Powered by TeXWeb (Win32, v.2.0).