# Tsallis-Based Nonextensive Analysis of the Southern California Seismicity

## Abstract

**:**

## 1. Introduction

_{th}is log-linearly related with the threshold magnitude M

_{th}by the well known Gutenberg-Richter (GR) law [1]. In terms of energy, this empirical relationship represented one of the features characterizing the self-organizing criticality due to the power-law dependence of the cumulative number of earthquakes with energy [2]. Although this relationship is very important, because it explains from a statistical viewpoint the seismicity occurring in seismic regions, it was not related with general physical principles apart from a recent attempt [3] in which the GR law seems to result from the Maximum Entropy Principle when considering the natural time concept [4], which, among others, allows the determination of an impending mainshock [3] when analyzing in natural time the seismicity that occurs after the recording of a Seismic Electric Signals activity [5,6].

## 2. Nonextensivity in Modeling Earthquakes

^{3}, in agreement with the scaling relationship between seismic moment and rupture length [10]. The maximum entropy principle for the Tsallis’ entropy [8] is given by:

_{σ}(σ) is the probability of finding a fragment of surface σ and q is a real number; k is the Boltzaman constant. It is easy to see that this entropy is the Boltzmann entropy when q→1. Let’s set k = 1 for sake of simplicity.

_{σ}(σ) is obtained maximizing the Tsallis’ entropy under the two constraints:

_{σ}(σ):

_{0}and λ

_{1}are the Lagrange multipliers. Imposing that:

^{3}, the proportionality between the released energy ε and ρ

^{3}becomes:

^{2}and a (the proportionality constant between ε and ρ

^{3}) is proportional to volumetric energy density. Thus, the energy distribution function (EDF) of the earthquakes is obtained in the following manner: using the transformation given by Equation 5, the variable σ is derived as:

_{ε}(ε) = n(ε)/N, where n(ε) is the number of earthquakes of energy ε and N the total number of earthquakes. The normalized cumulative number of earthquakes can be obtained by integrating Equation (10):

_{th}) is the number of earthquakes with energy larger than the threshold ε

_{th}, and thus:

_{th}normalized to the total number of events is given by:

## 3. Application to the Southern California Earthquake Catalog

_{th}for minimum threshold magnitude m

_{0}= 0 and maximum depth h

_{max}= 60 km. The curve fitting the data and representing Equation (16) has nonextensive parameters q = 1.542 and a = 153.127 estimated by the maximum likelihood estimation (MLE) method [18]. The misfit was evaluated by means of the average of the absolute values of the residuals |y-y

_{fit}|, where y indicates the real value and y

_{fit}the predicted value by the fitting. For m

_{0}= 0 the misfit is about 0.1745. The deviation of the fitting curve from the normalized cumulative distribution function at large magnitudes is due to the almost constant value of the distribution within the range of magnitudes from 5.8 to 7.2.

**Figure 1.**Magnitude distribution and fitting with Equation 16 for the whole catalog (depth less than 60 km and minimum magnitude m

_{0}= 0). The black continuous line indicates the nonextensive fitting curve.

_{0}of the nonextensive parameters q and a and the misfit. The dependence of the parameters with the minimum magnitude, revealing that q approximately decreases with m

_{0}, while a increases, is visible. However, the best model is for the minimum magnitude m

_{0}to which corresponds the values of q = 1.506 and a = 438.65 (misfit = 0.146).

**Figure 2.**Variation with the minimum magnitude m

_{0}of q (

**a**), a (

**b**), and misfit (

**c**), for maximum depth h

_{max}= 60 km. The best nonextensive model for the catalogue is for m

_{0}= 2.0 (q = 1.506, a = 438.65).

_{0}and h

_{max}of q, a and the misfit. It is clearly visible that all the curves are almost identical, except for that corresponding to h

_{max}= 10 km.

_{0}= 2.0 was fixed (which corresponded to the best nonextensive model) and one thousand magnitude sequences were simulated by means of the bootstrap method [19] for any value of the maximum depth h

_{max}from 10 km to 60 km. For any of these simulated sequence the nonextensive parameters q

_{S}and a

_{S}were estimated applying the MLE. Then the mean and standard deviation of q

_{S}and a

_{S}were calculated. Figure 4 and Figure 5 show the variation of mean and standard deviation of the nonextensive parameters of q

_{S}and a

_{S}varying the maximum depth h

_{max}. It is clearly shown that the nonextensive parameters estimated for the original series are quasi identical with the average q

_{S}and a

_{S}and the small standard deviation indicates a very good accuracy of the estimates.

**Figure 4.**Results of the application of the bootstrap method with varying the maximum depth h

_{max}(m

_{0}= 2.0). The black squares represent the variation of the q value of the original magnitude sequence; the red circles (bars) are the mean (standard deviation) of the q

_{S}.

**Figure 5.**Results of the application of the bootstrap method with varying the minimum depth h

_{max}(m

_{0}= 2.0). The black squares represent the variation of the a value of the original magnitude sequence; the red circles (bars) are the mean (standard deviation) of the a

_{S}.

**Figure 6.**Yearly variation of the q (

**a**), a (

**b**), number of events (

**c**) and maximal magnitude (

**d**) for the seismicity of Southern California.

## 4. Discussion

_{0}is not an easy task. In the context of the fragment-asperity model, the nonextensivity parameter q quantifies the scale of spatial interactions: for q~1, the spatial correlations are short-ranged and physical state is in quasi-equilibrium. For increasing q, the physical state goes away from equilibrium; and in case of seismicity, this means that the fault planes and fragment filling the gap between them are not in equilibrium, leading to an increased seismic activity to be expected [12]. In [20] the decrease of the nonextensivity parameter q was observed during relatively quiet periods, characterized mainly by the occurrence of small magnitude events; this could reveal that the order within the system of fault is decreased and the amount of accumulated stress is not yet enough to initiate a correlated behavior of the whole system [21]. When a strong earthquake occurs, more correlated behavior of the system constituents is assumed to take place, with the emergence of short and long range correlations, which induce an increase of the nonextensivity parameter q.

- (i)
- a takes the highest values during years in which the events with highest magnitude occurred; a is the volumetric energy density, and its value is large if the energy released is large.
- (ii)
- from 2004 to 2009, the increase of q is reflected by a slight decrease of a; during this period of reduced seismic activity (indicated by a reduced number of events) and without very large earthquakes (indicated by a maximal magnitude ranging between 4.5 and 5.5), the volumetric energy density is reduced, the accumulated stress energy seems to be released mostly through the relative movement of smaller fragments, but the increased degree of the system interactions are mainly governed by the smaller events.

## 5. Conclusions

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**MDPI and ACS Style**

Telesca, L.
Tsallis-Based Nonextensive Analysis of the Southern California Seismicity. *Entropy* **2011**, *13*, 1267-1280.
https://doi.org/10.3390/e13071267

**AMA Style**

Telesca L.
Tsallis-Based Nonextensive Analysis of the Southern California Seismicity. *Entropy*. 2011; 13(7):1267-1280.
https://doi.org/10.3390/e13071267

**Chicago/Turabian Style**

Telesca, Luciano.
2011. "Tsallis-Based Nonextensive Analysis of the Southern California Seismicity" *Entropy* 13, no. 7: 1267-1280.
https://doi.org/10.3390/e13071267