RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES5004, doi:10.2205/2007ES000282, 2008
[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].
![]() | (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:
![]() | (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
![]() | (3) |
[9] Thus we obtain
![]() | (4) |
where infinite geometric progression is summed up:
![]() | (5) |
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[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)
![]() | (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
![]() | (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.
![]() |
Figure 1 |
![]() |
Figure 2 |
[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.
Citation: 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.
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