Ahmet CELIKOGLU and Ugur TIRNAKLI

Transcription

1 Acta Geophysica vol. 6, no. 3, Jun. 212, pp DOI:.2478/s y Earthquakes, Model Systems and Connections to q-statistics Ahmet CELIKOGLU and Ugur TIRNAKLI Department of Physics, Faculty of Science, Ege University, Izmir, Turkey s: (corresponding author) A b s t r a c t In this work, we make an attempt to review some of the recent studies on earthquakes using either real catalogs or synthetic data coming from some model systems. A common feature of all these works is the use of q -statistics as a tool. Key words: earthquake models, q -statistics, statistical mechanics of earthquakes. 1. INTRODUCTION Earthquakes exhibit spatio-temporal features with complex dynamics which have been subject to active study for more than a century (Kagan and Jackson 1991, Turcotte 1997, Sornette 1999). Despite the long time period, our knowledge about earthquakes roughly consists of three empirical laws. One of them is the Gutenberg Richter (GR) law, stating that the distribution of cumulative number of earthquakes with energy greater than E behaves as a power-law, namely, N( E) E b, (1) where constant b varies from.8 to 1.2. This law can be better formulated by means of earthquake magnitudes m, which is defined to be m = log E, as 212 Institute of Geophysics, Polish Academy of Sciences

2 536 A. CELIKOGLU and U. TIRNAKLI N( m) = a bm (2) where constant a is given by the logarithm of the number of earthquakes with magnitude greater than zero (Gutenberg and Richter 1944). The second empirical scaling relation detected for earthquakes describes the temporal behavior of aftershock activity. This law, known as the Omori law, states that the rate of the occurrence of aftershocks after a mainshock decreases as a power law, namely dn dt = K(C + t) p, (3) where C and K are positive constants and p is typically between.7 and 1.5 (Omori 1894). The last scaling law for earthquakes is called Bath s law (Bath 1965), which states that if one defines the difference in magnitude between a mainshock with magnitude m mainshock and its largest detected aftershock with magnitude m aftershock as m = m mainshock m aftershock, (4) this difference appears to be a constant independent of the mainshock magnitude. Over the last decades, some properties of geophysical systems have been studied in the context of q -statistics (Tsallis 1988, 29). This formalism seems to be a good candidate to be used for earthquake analysis since longe-range spatio-temporal correlations and complex dynamics are natural ingredients of earthquakes. In this paper we summarize some of the recent works on this line and emphasize the use of model systems for earthquakes and recent results obtained from such systems. At this point, we should mention that we prefer to leave some very interesting recent developments on the use of q - statistics in the analysis of plate tectonics (Vallianatos 29, Vallianatos and Sammonds 2) and fault-population (Vallianatos and Sammonds 211) as untouched since there will be separate works in this issue analyzing these points. 2. GUTENBERG RICHTER LAW AND SP MODEL One of the interesting examples for the use of q-statistics is Sotolongo-Costa and Posadas (SP) model (Sotolongo-Costa and Posadas 24). This model tries to establish a relation between Gutenberg Richter law and general principles of physics. Earthquakes originate by relative motion of fault planes. Two planes interact with each other via fragments which are filling the gap between them and these fragments resist the motion. When the accumulated stress exceeds

3 EARTHQUAKES, MODEL SYSTEMS AND CONNECTIONS TO q-statistics 537 breaking force of the fragments, earthquake is triggered and accumulated energy is released. They assumed that the released energy is proportional to the size of the fragment. In this regards the authors found the probability of finding a fragment with surface size σ by using nonadditive Tsallis entropy which has the following form S q = k 1 p q (σ)dσ, (5) q 1 where q is the entropic index. There are two conditions required for maximization of Tsallis entropy. The first one is the normalization condition of p(σ) given by and the other condition p(σ)dσ = 1 (6) σp q (σ)dσ = σ q (7) is related to q mean value of σ. Authors have obtained the energy distribution function of earthquakes as p(ε)dε = C 1 εdε [1 + C 2 ε 2 ] 1/(q 1) (8) by using the size distribution of fragments and an assumption that σ scales as r 2. In this equation, p (ε) is the probability of energy, C 1 and C 2 are constants. This simple approach about fault dynamics gives rise to very interesting results. They have managed to modify the well-known GR law and establish a connection with principles of statistical physics. The modified version of GR law as a function of magnitude m has been given as ( ) 2 q [ log [N(> m)] = log N + log 1 + a(q 1)(2 q) (1 q)/(q 2) 2m], (9) 1 q where a is a constant of proportionality between ε and r. The modified GR law was tested by using Iberian Peninsula, Andalucia and California catalogs. Interestingly, the value of q parameter was obtained to be approximately 1.66 for all these three catalogs (Sotolongo-Costa and Posadas 24). After the appearance of SP model, two different revisions have been proposed in the literature (Silva et al. 26, Darooneh and Mehri 2). These revisions focus on the definition of the expectation value and relation between energy and the linear dimension r of the fragment. Silva et al. approach the

4 538 A. CELIKOGLU and U. TIRNAKLI problem by redefining the dependency of ε on r as ε r 3 and q -expectation value as σ q = <σ> q = where P q is the escort distribution (Abe 23) given by P q = σp q (σ)dσ, () p q (σ) p q (σ)dσ. (11) The cumulative number of earthquakes (Sotolongo-Costa and Posadas 24) N >ε N = ε p(ε)dε (12) used to test the validity of energy distribution function which is now given by p(ε)dε = Cε 1/3 dε [1 + C ε 2/3 ] 1/(q 1) (13) where N ε> is the number of earthquakes with energy greater than ε and N is the total number of earthquakes, C and C are constants which include proportionality constant a and parameter q. Silva et al. (26) derived new version of the modified GR law as ( ) [ ( ) ( )] 2 q 1 q 2m log(n >m ) = log N + log 1 (14) 1 q 2 q by calculating the integral given in Eq. (12). This new version of the modified GR law actually amounts to make the substitution ε ε 1/3 in the SP model and was tested by using seismic data from Samambaia, New Madrid and Anatolian faults and it was shown that it is viable to wider detectable range of magnitude. Although these two versions of the modified GR law, very similar to each other, provide almost same q value and very good fit to the real data from different catalogs, they extremely differ from each other, especially as concerns energy density. Recently, Darooneh and Mehri discussed the same problem again in (Darooneh and Mehri 2). They argued that the idea of previously modified laws does not apply all over the whole magnitude regime and fractal shape of the fragments. They redefine the relation between surface size of fragments and released energy as ε exp (σ 1/a ) inspired by the fractal nature of the fragments. The same constraints used in Silva et al. (26) have been also used in the maximization of Tsallis entropy, and the cumulative distribution of earthquakes as a function of magnitude has been derived as a 2/3

5 EARTHQUAKES, MODEL SYSTEMS AND CONNECTIONS TO q-statistics 539 P > (m) = N(m) N = m Π(m)dm = exp q ( βm a ), (15) where Π(m) is the escort probability of m, β is a parameter, and exp q the q-exponential function which is given by is 1 exp q (x) [1 + (1 q)x] (1 q) [exp 1 (x) = exp(x)]. (16) This modified version of GR law, which has q-stretch exponential form, has also been compared to the other two versions described above by using real data from Iran and California catalogs and it has been argued that this version fits the real data better than others. At this point, it is worth mentioning that in order to better understand which version is better, one can resort to earthquake models for generating arbitrarily long synthetic time series from where a better comparison among proposed versions can be made. 3. AGING PHENOMENON As we have already mentioned, temporal patterns of aftershocks follow an empirical law known as Omori law. This law states that the rate of the aftershocks after a mainshock decreases as power law generically given by Eq. (3). In 24 Abe and Suzuki analyzed the temporal correlations in earthquake phenomena using California catalog by taking into account the events both inside and outside the Omori regimes (Abe and Suziki 24). They observed that correlations among earthquakes exhibit aging inside the Omori regime and this behavior disappears outside the Omori regime. Moreover they determined the scaling function of the aging phenomenon. In order to understand whether the earthquake models where Omori law holds, exhibit similar behavior, properties of correlation among events are studied by using Coherent Noise (CN) (Newman and Sneppen 1996, Sneppen and Newman 1997) and Olami Feder Christensen (OFC) (Olami et al. 1992, Christensen and Olami 1992) models in Tirnakli and Abe (24) and in Tirnakli (24), respectively. On contrary to OFC, the CN model does not have any geometric configuration space but it can still describe some important properties of real earthquakes. In CN model the system consists of N agents. The dynamics of the CN model needs an external stress which acts coherently to all agents. Each agent has a threshold value which is replaced by the new one when the applied external stress is greater than this threshold. External stress and thresholds are chosen randomly from exponential and uniform distributions, respectively. The number of replaced agents determine the size of an event. In order to provide continuity of avalanche generation, a small fraction of N agents

6 54 A. CELIKOGLU and U. TIRNAKLI 1 C(n+n w, n w ) CN Model n / n w 1.5 n w =25 n w =5 n w = n w =2 n w = n Fig. 1. The behavior of correlation function as a function of natural time n. Ensemble average is taken from 12 realizations. ( Data ) collapse is also given in the inset. The solid line corresponds to exp q.7 n/n 1.5 with q w is also chosen randomly and new thresholds are assigned again at each time step. Tirnakli and Abe used the same idea in Abe and Suzuki (24) to analyze the properties of event correlations. The correlation function was defined as C(n + n w, n w ) = t n+n w t nw t n+nw t nw (σ 2 n+n w σ 2 n w ) 1/2, (17) by using basic physical random variable t n of nth aftershock with an arbitrary avalanche size, where n is the natural time of an aftershock, n w is the natural waiting time. Since the system is nonergodic, taking the averages in Eq. (17) as the time average or the ensemble average would yield different results. In Tirnakli and Abe (24), Tirnakli (24), ensemble average which is obtained by using a large number of realization has been performed. The variances are given by σm 2 = t 2 m tm 2. As seen in Fig. 1, since the correlation function depends on both n and n w, it is plotted for various n w ; it is observed that aging appears inside the Omori regime, whereas it disappears outside the Omori regime. Moreover, one can propose a scaling relation of the form ( ) C(n + n w, n w ) = C n n α, (18) w

7 EARTHQUAKES, MODEL SYSTEMS AND CONNECTIONS TO q-statistics 541 where α is numerically obtained as α 1.5 and C is a scaling function. Scaled correlation function is shown in the inset where the solid line corresponds to exp q (.7n/n 1.5 w ) with q In Tirnakli (24) one of us has reported the same behavior by using another well-known earthquake model known as OFC model (Olami et al. 1992, Christensen and Olami 1992). Briefly, the dynamics of the model is as follows: The model consists of discrete systems of blocks, each of them carries a stress (force) F i. When the force F i acting on any of the blocks, is greater than a threshold F th, this block become unstable and earthquake is triggered. Each unstable block relaxes the zero and transfers a portion of own stress F i to its neighbors, so the avalanche starts and continues until the force of all blocks becomes less than or equal to the threshold force. The measure of the size of the avalanche is given by the total number of the blocks which are relaxed during the avalanche. This mechanism is given mathematically as { Fi, F i F th (19) F nn F nn + αf i where α is the conservation parameter and nn denotes neighbors of ith block. As can be seen in Fig. 2, the obtained results are very similar to that of CN model and it is evident that the scaling function given by Eq. (18) works for the OFC model as well. In this context, it should be noted that the same C n (n+n w, n w ) 1 1 OFC Model n w =25 n w =5 n w = n w =2 n w = n / n w n Fig. 2. The behavior of correlation function as a function of natural time n. Ensemble average is taken from 2 realizations. ( Data ) collapse is also given in the inset. The solid line corresponds to exp q.67 n/n 1.5 w with q 2.9.

8 A. CELIKOGLU and U. TIRNAKLI 542 model systems must also be studied using time averages as done for real earthquakes in Abe and Suzuki (24) to see whether they exhibit similar aging properties. 4. RETURN DISTRIBUTION op y Another important quantity to help us to understand the physics of earthquakes is the return distribution. A return is defined as the difference between the sizes of successive avalanches, namely, if the size of an avalanche at time t is denoted by s(t) then the difference between two successive avalanches is given by s = s(t + 1) s(t). Recently, we have studied the return distributions of CN Model for various model parameter values of σ which is related to the external stress distribution (Newman 1996, Newman and Sneppen 1996, Sneppen and Newman 1997) and obtained that these distributions are in the shape of q -Gaussian which is defined to be 1/(1 q) P (x) = P () 1 β(1 q)x2, rc (2) A ut ho where β is a parameter. A representative case with σ =.5 can be seen in Fig. 3 (Celikoglu et al. 2). The q -Gaussian distributions are standard distributions that naturally emerge in q -statistics. An exact relation connecting -1-2 P(x) / P() σ=.5 q-gaussian; q=2.9, β= x P() Fig. 3. Return distributions for σ =.5 case. Two zooms of the central part are given in the insets for better visualization.

9 EARTHQUAKES, MODEL SYSTEMS AND CONNECTIONS TO q-statistics 543 the well-known exponent τ of avalanche size distribution to the q value of the q -Gaussian has been proposed as q = τ + 2. (21) τ This relation is important since it rescues q from being a fitting parameter and now one can determine a priori the value of q that must be used in the q-gaussian distribution if the τ exponent of the system is known. Since the coherent noise model has the advantages both producing different τ values by varying a model parameter σ and simulating in N limit, in Celikoglu et al. (2) we have checked the validity of the proposed relation for various cases and obtained clear numerical evidences on the validity of this relation. We have also studied the finite size effects by simulating directly the model dynamics with different number of agents (different N ) and obtained that the return distributions deviate from q-gaussian when N values are small. As seen in Fig. 4, when the system is getting closer to thermodynamic limit (increasing N), the return distributions better approach the q-gaussian. An attempt was made to explain this behavior using a crossover formula, discussed in Tsallis (1988), Tsallis et al. (1999), Tsallis and Tirnakli (2), of the form dy d(x 2 ) = a ry r (a q a r )y q [a q a r, q > r, y() = 1]. (22) P(x) / P() a a 1 ~N -2 N N= N=5 N= N=2 eq.(23); q=1.95,a q =55,a 1 =.1 eq.(23); q=1.98,a q =55,a 1 =.42 eq.(23); q=1.99,a q =55,a 1 =. eq.(23); q=2.1,a q =55,a 1 =.24 q-gaussian; q=2.9, β=55 σ= x P() Fig. 4. Return distributions of the various N values. In the inset, N dependence of the parameter a 1 is given. Disappearance of the finite size effects, as N, is found to obey a 1 = 11 8N 2.

10 544 A. CELIKOGLU and U. TIRNAKLI This equation has different solutions depending on the choice of r and q values. Let us restrict ourselves to the solution of q > 1 and r = 1. In this case, the solution of the equation is given by y = [ 1 a q + a ] 1/1 q q e (q 1)a1x 2. (23) a 1 a 1 If a 1 =, the solution has the form of q-gaussian, whereas it is Gaussian when a q = a 1, in the other words, when q 1. The crossover appears between them if a q > a 1 > and q > 1. The exponential decaying part in the avalanche size distribution is dominant when the system size is small. Since this part corresponds to Gaussian shape in the return distributions and disappears as N is increasing and the crossover behavior is directly related to the system size, this mathematical model clearly demonstrates and explains the finite size effects. The return distributions are also studied recently in a modified OFC model and same type of finite size behavior is again seen (Zhang et al. 211). In this model, the authors added heterogeneity of energy transmission by redefining transmission parameter α j as F i (t) + F j (t) α j (t) = a 4F i + jɛnn F j(t), (24) where F i and F j denote stress values at each site and a is a control parameter. In order to obtain long-range effects, a small fraction of long-range links is introduced into the lattice which is controlled by long-range parameter p add. When.88 a < 1, the model exhibits both power law behavior and finite size scaling and it is shown that the modified OFC model exhibits self-organized criticality. The model has two different types of avalanche definition. One of them generates avalanches so that each site can topple more than once during an avalanche. The authors obtained that the return distributions of this model can also be given by the crossover formula in Eq. (23). The return distributions converge to the appropriate q-gaussian as the system size N increases (Zhang et al. 211). 5. CONCLUSIONS From the results of all the above-mentioned works one can conclude that q-statistics could be considered as one of the most important candidates to be used for understanding and explaining the physics of complex systems such as earthquakes. In order to better elaborate this point, works including more detailed analysis will surely be appreciated.

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