A qualitative comparison between some synthetic and empirical scaling properties in seismicity

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1 SUPLEMENTO Revista Mexicana de Física S 58 (1) JUNIO 2012 A qualitative comparison between some synthetic and empirical scaling properties in seismicity A. Muñoz-Diosdado Departamento de Ciencias Básicas, Unidad Profesional Interdisciplinaria de Biotecnología, Instituto Politécnico Nacional, Zacatenco, México, amunoz@avantel.net A.H. Rudolf-Navarro and F. Angulo-Brown Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Zacatenco, México. Recibido el 23 de Marzo de 2010; aceptado el 27 de Abril de 2011 Earth s crust can be conceived as a hierarchical set of objects of many shapes and sizes with fractal behavior. In fact, many geological phenomena are scale free. Self similar behavior is reflected in several empirical power-laws in geology and geophysics that have been found during more than a century. The scale invariance requires power law dependence between the number of objects of a specific size with the size. So an earthquake-fault model has to be able to produce these scaling relations. Within the context of self-organized critical systems, Olami, Feder and Christensen (OFC) proposed a nonconservative spring block earthquake model that reproduces some seismic properties such as the Gutenberg-Richter (G-R) law for the size distribution of earthquakes. In the present article we analyze some statistical properties of a spring-block model and we compare them with the corresponding statistical properties of actual seismicity of the Mexican South Pacific Coast. We have showed that by means of the OFC model is possible to find G-R laws and staircase plots that are at least qualitatively analogous to the corresponding G-R laws and staircase plots of actual seismicity. It is remarkable that synthetic OFC-seismicity is characterized by a kind of attractor consisting of a straight line whose slope attracts the long-term cumulative seismicity. Unfortunately, the actual Mexican seismicity catalogues are yet too short to completely contrast this synthetic seismicity property. Keywords: Spring-block model; self-organized criticality; seismicity patterns; precursory seismic quiescence and synthetic seismicity. La corteza terrestre puede concebirse como un conjunto jerarquizado de objetos de muchas formas y tamaños con comportamiento fractal. De hecho, muchos fenómenos geológicos son libres de escala. El comportamiento autosimilar se ve reflejado en varias leyes de potencias empíricas que se ha encontrado durante más de una centuria. La invariancia de escala requiere una dependencia de leyes de potencia entre el número de objetos de determinado tamaño con el tamaño. Por lo tanto un modelo de una falla sísmica tiene que ser capaz de producir tales relaciones de escalamiento. Dentro del contexto de sistema críticamente autoorganizados, Olami, Feder y Christensen (OFC) propusieron un modelo de resorte bloque para sismos que reproduce algunas propiedades sísmicas como la ley de Gutenberg-Richter (G-R) para la distribución de tamaño de los sismos. En el presente artículo se analizan algunas propiedades estadísticas del modelo de resorte bloque y se les compara con las propiedades estadísticas correspondientes de la sismicidad real de la Costa Sur del Pacífico mexicano. Se ha mostrado por medio del modelo OFC que es posible encontrar leyes de G-R y gráficas en escalera que son al menos cualitativamente análogas a las leyes de G-R y a las gráficas en escalera de la sismicidad real. Es sobresaliente que la sismicidad sintética del modelo OFC esté caracterizada por una clase atractor consistente en una línea recta cuya pendiente atrae a la sismicidad acumulada de largo plazo. Desafortunadamente, los catálogos mexicanos actuales de sismicidad son todavía muy cortos para contrastar completamente esta propiedad de la sismicidad sintética. Descriptores: Modelo resorte bloque; criticalidad auto organizada; patrones de sismicidad; quietud sísmica precursora y sismicidad sintética. PACS: Px; j; b 1. Introduction Scaling and fractal properties of many experimental data sets stemming from real-world complex systems have attracted the attention of many researchers during the last decades [1-8]. In particular within Earth s sciences many power-laws describing structural and dynamical properties have been reported [2-5]. Such is the case of the Earth crust, whose dynamical behavior has been analyzed by means of data expressed as electric self-potential time series [3,9-15]. Since the end of the XIX century (Omori s law) until the fifties of the XX century, several empirical power-laws were found within the context of seismology [16-20]. In fact, many geological phenomena are scale free. For example, frequency distribution of the fragment size of rocks, seismic faults, earthquakes and volcanic eruptions [2]. The scale invariance is equivalent to a fractal distribution, which requires power law dependence between the number of objects of a specific size with the size. The theory of plate tectonics states that the lithosphere is broken into about a dozen of major rigid plates and several minor ones. These plates slowly grind against each other, building up stress and cresting faults. A first check on the robustness of an earthquakefault model is that it has to be able to produce these scaling relations. However, the ability to produce a scaling relation does not mean that the model is useful, because it

2 A QUALITATIVE COMPARISON BETWEEN SOME SYNTHETIC AND EMPIRICAL SCALING PROPERTIES IN SEISMICITY 97 also must be able to reproduce other known phenomena and led to predictions that the seismologists can observe on real faults. A relevant example is the Gutenberg-Richter (GR) law obtained from the statistics of frequency and magnitude of earthquakes [21]. The concept of self-organized criticality (SOC) proposed by Bak et al. [16] has been used by many authors [17-20] together with the spring-block model of Burridge and Knopoff [20] to model seismic faults by means of cellular automata with homogeneous distributions, and in all cases they reproduced at least qualitatively the G-R law. As asserted by Bak [16], any theoretical or numerical earthquakes model has to reproduce the empirical G-R law [21]. In the present article we analyze some statistical properties of a SOC spring-block model and we compare them with the corresponding statistical properties of actual seismicity of the Mexican South Pacific Coast, which is a very seismically active region linked to the Middle American trench, which is the border between the Cocos and American tectonic plates. The article is organized as follows: In Sec. 2, we make a resume of some of the main empirical laws found in seismology by means of seismic catalogues; in Sec. 3 we make a brief presentation of the so-called spring-block model for a seismic fault; In Sec. 4 we show a comparison between G-R laws and staircase plots for a SOC spring-block modeled and real seismicity, respectively; and finally in Sec. 5, we discuss on these comparisons and we present our conclusions. 2. Seismological empirical relations In this section we make a brief presentation of some empirical power-laws stemming from the statistical seismology. During the last years, efforts have been made to obtain some of them by using numerical earthquake models [16,18-19,22]. The first empirical law we describe was proposed by Beno Gutenberg and Charles Richter in 1944, it was a relationship between the local magnitude M (Richter magnitude) of earthquakes that occurred in California and the frequency of earthquakes of greater or equal magnitude than M, known today as the Gutenberg-Richter law [21,23-26] log N(M) = a bm (1) where N is the number of earthquakes per unit of time of magnitude greater or equal than M. The Omori law refers to aftershocks. Aftershocks are a universal consequence of earthquakes of large or intermediate magnitude, its occurrence rate decreases according to the empirical Omori law [27-30]: dn dt = A(M) (t t r) α(m) (2) where α varies between 0.9 and 1.8 (originally Omori had obtained α = 1), t r is the occurrence time of the earthquake that causes aftershocks series and the factor A(M) monotonically decreases with magnitude M. In general, larger aftershocks occur at the beginning of the series. In 1954 Utsu and Seki found a linear relationship between the logarithm of the area of aftershocks or rupture area S of an earthquake (in square kilometers) and its magnitude M [30], the proposed relation is log S = 1.02M (3) In 1956 Gutenberg y Richter [24-26] obtained the relationship between the magnitude M S of an earthquake and the total energy released by seismic waves E S. log 10 E S = 1.5M S (4) And finally, we describe the Tsubokawa relation. Early attempts to correlate the duration time of a precursor anomaly of a great seism with its magnitude were made by Tsubokawa in 1969 [31], who proposed the linear relationship between the logarithm of time T (in days) of the duration of the anomalous deformation of the terrain and the earthquake magnitude: log T = 0.79M S (5) In the following sections we mainly focus on the G-R law under the perspective of a SOC-model and its comparison with real seismicity occurring in the Mexican Pacific Coast. 3. The spring-block model The scale-invariant distribution of earthquake sizes is usually reproduced by processes based on avalanches of stress redistribution, following the idea that there is self-organized criticality (SOC) [1-2,7-8]. The precursor of this concept in geophysics has been the slider-block model by Burridge and Knopoff [32]. It is evident from many models that the mechanism of avalanches of relaxations robustly leads to sizefrequency power-laws [33]. The best models to simulate the dynamics of seismic faults are all cellular automata seismicity models. All models consist of a rectangular grid of cells, each of which is assigned a constant, scalar failure threshold and a scalar variable representing shear stress. Various failure threshold distributions are employed in the models: Constant failure thresholds or random distribution of thresholds. Initially, the stress of each cell is assigned a random value selected from a uniform random distribution ranging from zero to the failure threshold of the cell. The increase in stress due to tectonic loading is modeled by incrementing the stress of each cell by a small stress increment. The size of the increment is computed as the increment required to increase the stress of at least one cell to the failure threshold of that cell. When the stress of a cell equals or exceeds its failure threshold, the cell fails (modeling the stick-slip transition) and redistributes stress to neighboring cells. The stress of the failed cell is reduced to a lower value and some stress is dissipated from the model, simulating the loss of energy due

3 98 A. MUÑOZ-DIOSDADO, A.H. RUDOLF-NAVARRO, AND F. ANGULO-BROWN F i±1,j F i±1,j + δf i±1,j F i,j±1 F i,j±1 + δf i,j±1 F i,j 0, (7) where the force increase in the nearest neighbors is given by, δf i±1,j = δf i,j±1 = K 1 2K 1 + 2K 2 + K L F i,j = α 1 F i,j, K 2 2K 1 + 2K 2 + K L F i,j = α 2 F i,j, (8) FIGURE 1. The geometry of the spring-block model. The force on the blocks increases uniformly as a response to the relative movement of the two plates. to seismic wave propagation and heat. Numerous stress redistribution mechanisms have been proposed in the literature [34]. Stress redistribution may trigger failure of neighboring cells, initiating a failure cascade or synthetic earthquake. Failure cascades proceed until no more failures occur, at which time tectonic loading is repeated. In 1992, Olami, Feder and Christensen (OFC) [19,35] introduced a continuous cellular automaton model where the conservation level can be controlled. They obtained a very robust SOC behavior and found results that permitted them to find the G-R law exponents [21]. In the OFC model the fault is represented by a twodimensional array of blocks interconnected by springs with elastic constants K 1, K 2 (Fig. 1). Each block is connected to its four nearest neighbors and also to a driving rigid plate by another set of springs with stiffness K L, and by the friction force to a fixed rigid plate. The blocks are left to move by the relative movement of the two rigid plates. When the force over one of the blocks is larger than some threshold value Fth (the maximal static friction), the block slips. Olami et al. assumed that the block that is moved will slip to the zero force position. The block slipping will redefine the forces in its nearest neighbors. This can result in more slipping and a chain reaction can evolve. If we define an L L arrangement of blocks by (i, j), where i and j are integers whose values are between 1 and L and if the displacement of each block from its relaxed position on the lattice is x i,j, then the total force exerted by the springs on a given block (i, j) is expressed by [34] F i,j = K 1 [2x i,j x i 1,j x i+1,j ] + K 2 [2x i,j x i,j 1 x i,j+1 ] + K L x i,j (6) The redistribution of forces, after a local slip at the position (i, j)due to the force on one of the blocks is larger than the maximal static friction, is given by, αa 1 and αa 2, are the elastic ratios. Observe that the force redistribution is not conservative. Olami et al. first restricted the simulation to the isotropic case, K 1 = K 2 (αa 1 = αa 2 = αa) with a rigid frontier condition, implying that F = 0 in it. They made the mapping of the spring-block model into a continuous, nonconservative cellular automaton modeling earthquakes which is described by the following algorithm. (1) Initialize all the sites to a random value between 0 and F th. (2) Locate the block with the largest force, F max. Add F th F max to all sites (global perturbation). (3) For all F i,j F th redistribute the force in the neighbors of F i,j according to the rule F n,n F n,n + αf i,j F i,j 0 (9) where F n,n are the forces for the four nearest neighbors. An earthquake is in process. (4) Repeat step 3 until the earthquake has totally evolved. One synthetic earthquake is showed in Fig. 2; in this figure we show all the blocks than were relaxed in the process, the position of the first block is the epicenter. The size of this synthetic earthquake is the number n of relaxed blocks. We can define the magnitude in terms of n, as the log x (n), where x is a suitable base of the logarithm. (5) Once the earthquake has thoroughly ended, return to step 2. We can repeat this procedure many times, and for each seism we can count the relaxed blocks, for instance in figure 3 we show events. For these conditions we obtain a robust SOC behavior for the probability distribution of the earthquakes size.

4 A QUALITATIVE COMPARISON BETWEEN SOME SYNTHETIC AND EMPIRICAL SCALING PROPERTIES IN SEISMICITY 99 FIGURE 2. The rupture surface in a synthetic earthquake. The X point is the epicenter. more realistic models, but the complexity of the model should not be very high and the calculations should not be irksome. A seismic fault model must be able to produce power laws of the type of the G-R law [21]. Analyzing successful models, we can arrive to a better theoretical understanding about the earthquakes. Because of this, we have extensively studied the OFC model and we have found that it qualitatively reproduces many properties that are observed in real seismicity. (See for example the references [36-39]). However, in this article we are going to focus in two aspects: First, it is possible to calculate the exponents of the G-R law and second, the staircase graphics of the cumulative seismicity that we obtained when we plot real data of the cumulative seismicity versus time. Although the G-R law relationship (1) is universal, the values of a and b depend on the region. The constant a specifies a regional level of seismicity. The values of bare approximately between 0.75 and Very large earthquakes (m > 7) have different values of b (1.2 < b < 1.54) that the smaller (0.75 < b < 1.2). This difference can be the result of the large earthquakes exclusion to the ductile region of the lithosphere [40]. For instance, in Fig. 4, we display the G- R law for seisms occurred along the Mexican Pacific coast since 1970 to 2000 in magnitude interval 4.3 M 8.0. We repeat the algorithm many times and we obtain that the magnitude of the synthetic earthquakes follows the G-R law (see Fig. 5). But the best of the OFC model is that we can obtain b values that are close to the experimental values, with α-values around 0.2. We can show in Fig. 4 the probability distribution of the earthquakes size that it is the power law FIGURE 3. Time series of synthetic earthquakes (16384 events), α=0.2, L=80. In this figure magnitude is the number of blocks that relaxed. 4. Results 4.1. The catalogues of synthetic seismicity and the G-R law In real seismicity a data summary of the earthquakes including at least minimally the occurrence date, location, size and a brief description of the damage caused, is called a catalogue. It is important to have more complete and exact catalogues, because the analysis of them has many applications. For example, the hypothesis of the seismic gap implies that major earthquakes are waited throughout sections of tectonic plate frontiers in which large earthquakes have been produced and they have not suffered a rupture during the last few years. Synthetic catalogues can be generated for different values of elastic parameters and for different sizes of the system. The analysis of synthetic catalogues will give better results for FIGURE 4. Logarithm of the frequency against the time for seisms occurred along the Mexican Pacific coast since 1970 to 2000 in magnitude interval 4.3 M 8.0. The first part can be adjusted to a straight line whose slope is the b-value of the G-R relation. For earthquakes of larger magnitude we have a change in the slope that in opinion of Pacheco et al. [40] it is probably the result of the large earthquakes exclusion to the ductile region of the lithosphere.

5 100 A. MUÑOZ-DIOSDADO, A.H. RUDOLF-NAVARRO, AND F. ANGULO-BROWN FIGURE 5. Distribution of synthetic earthquake magnitude in a system with open boundary conditions, events. This is the Gutenberg-Richter relation for synthetic earthquakes. Note the finite-size effect for synthetic earthquakes of great magnitude. FIGURE 6. Distribution of synthetic earthquake duration times in a 100x100 system with open boundary conditions, events. law analogous to the G-R law; we can observe as in real seismicity that for the greater magnitudes we have a finite size effect (see Figs. 4 and 5). We also have obtained that the duration of the synthetic earthquakes is also a power law, as indicated in Fig. 6, there is no experimental confirmation of this fact, but it seems logical to us that the duration of real earthquakes should also follows a power law The cumulative seismicity in the OFC model It is usual in the analysis of real cumulative seismicity to take the seismic counting above a threshold magnitude. Typically this kind of plots have a stair-shaped form, in a plot of this kind a quiescence could be recognized easily, a quiescence is presented when a stair step remains almost horizontal during a lot of time. We have noted that in the long-term situation the envelope of this kind of plots seems to tend to a straight FIGURE 7. a) The staircase graphs structure for the cumulative seismicity, α=0.2, L=80. Na is total cumulative seismicity. b) The staircase graphs structure for the cumulative seismicity, α=0.2, L=80. This graph was obtained when we subtract from total cumulative seismicity the events less than 1/16 of the maximum event. c) The staircase graphs structure for the cumulative seismicity, α=0.2, L=80. This graph was obtained when we subtract from total cumulative seismicity all events whose magnitude is less than 1/8 than the maximum. line, so when quiescence is produced, the plot tends to return to the historical slope of the seismological region. If the quiescence is large, the return could be done by two possible ways: first, with a large earthquake and its aftershocks or second, with a swarm of many small earthquakes.

6 A QUALITATIVE COMPARISON BETWEEN SOME SYNTHETIC AND EMPIRICAL SCALING PROPERTIES IN SEISMICITY 101 FIGURE 8. The staircase graphs structure for the total cumulative seismicity, α=0.2, L = 100, events. The fitting with the straight line is very good in the long-term situation. We calculated the cumulative seismicity in the OFC model, and then we plotted it as a function of time. The results show that staircase graphics are a characteristic of the model, for all values of elastic parameters α between 0.10 and 0.25 and for system sizes from to we always obtained such graphics. The graphics of cumulative seismicity of the OFC model for a small number of events have the aspect shown in Fig. 7; similar staircases to those of real seismicity are obtained taking only events above certain magnitude. In Fig. 7, graph a) corresponds to the total cumulative seismicity, graph b) was obtained from graph a) subtracting the events that were less than 1/16 of the maximum event and graph c) corresponds to the subtraction of the total cumulative seismicity of all the events whose magnitude are 1/8 smaller than the maximum. As can be seen in Fig. 7c), some quiescence periods can be identified. We found in all cases the stair-shaped plots, the quiescence can be identified only by making different cuts in the cumulative seismicity just as indicated. In Fig. 7a) the role of the straight line envelope as an attractor of the seismicity is insinuated. In Fig. 8, we show the total cumulative activity for events (α = 0.20). The fitting with the straight line is very good in this longterm situation. Thus, it seems that this is a general property of the synthetic cumulative seismicity obtained from the OFC model, and we can suggest that it is a property of real cumulative seismicity once a source area is well identified. We have tested this property for several values of α in both isotropic and anisotropic cases. In Figs. 9a), 9b), 9c) we show staircase plots for actual seismicity stemming from the Mexican Pacific coast for intervals of coordinates indicated in each figure. The seismic data we taken from the Rudolf- Navarro catalogue [41]. It is remarkable the qualitative similarity between these figures and the Figs. 7a), 7b) and 7c) respectively. Once we were sure that it is possible to assign in the longterm situation a straight line to each stair-shaped plot for the FIGURE 9. a) Cumulative number of seisms with 4 M 8.2 between the 16.1 and 17.8 lat. N and the 99.3 and lon. W, 0 h 60 km, from Jan. 1, 1990 to Jan. 20, b) Cumulative number of seisms with 4.3 M 8.2 between the 17.8 and 19.8 lat. N and the 103 and lon. W, 0 h 60 km, from Jan. 1, 1969 to Jan. 20, c) Cumulative number of seisms with 4.3 M 8.2 between the 16.5 and 19.5 lat. N and the 101 and lon. W, 0 h 60 km, from Jan. 1, 1969 to Jan. 20, synthetic cumulative seismicity, we find the slope of this straight line. When we plot the slope of the stair-shaped curves against the linear system size L, we find in all the studied cases, a behavior as that shown in Fig. 10. That is, apparently the slopes have an absolute maximum. Thus, the cumulative synthetic seismicity cannot be arbitrarily large. If this was not the case, one would find that a relatively small quiescence would generate high seismicity (main shocks plus

7 102 A. MUÑOZ-DIOSDADO, A.H. RUDOLF-NAVARRO, AND F. ANGULO-BROWN aftershocks) for returning to the characteristic slope of each system size. Therefore, we propose that once we found this characteristic historical slope the seismicity associated to a fault remains bounded. When the time interval considered for calculating the slope of the stair-shaped curves of the cumulative synthetic seismicity is small, this slope is sensitive to the size of the time interval, but when the analysis is made over a long time interval (as in Fig. 8) the slope is constant and characterizes the synthetic seismicity of the system. Evidently, this idea cannot be contrasted against real seismicity due to real seismicity catalogues are very short, and their completeness is yet very recent [42]. However, we have divided the seismicity of the Mexican Pacific coast in four seismological regions: Jalisco-Colima, Michoacán, Guerrero and Oaxaca. We show as examples 3 cases: Fig. 9a) belongs to Guerrero, Fig. 9b) belongs to Jalisco-Colima and Fig. 9c) corresponds to Michoacán. Rudolf-Navarro [41] has shown that its catalogue is complete for magnitudes greater or equal to 4.3, and because we have only hundreds of events it is not possible to show the long-term behavior. However, observing carefully figures 9a), 9b) and 9c) it seems to be that that it is possible to have a straight line envelope as an attractor (and superior bound) of the cumulative seismicity. This idea was suggested by McNally for the Mexican seismological regions before 1981 [43]. Having a model with properties that belong to real seismicity could allow us to study synthetic catalogues without the problems that arise in real seismicity; synthetic catalogues have advantages with respect to real catalogues because we can simulate any number of events. They have not problems of underestimation or overestimation, and they are complete. We think that it is possible to obtain valuable insights about real seismicity by studying synthetic catalogues. 5. Concluding remarks FIGURE 10. The slope of the stair-shaped curves against the linear system size L, apparently the slopes have an absolute maximum. Thus, the cumulative synthetic seismicity cannot be arbitrarily large. The earth s crust can be conceived as a hierarchical set of objects of many shapes and sizes suitable for a fractal description [44]. Scale invariance is a well known property of many geological structures and phenomena. Self similar behavior is reflected in several empirical power-laws in geology and geophysics [2]. Seismicity for example, has fractal structure with respect to time, space and magnitude [45]. Bak and Tang [46] suggested that the structure and dynamics of the crust can be interpreted as arising from a self-organized critical process. The critical state is characterized by spatial and temporal power laws. Sornette and Sornette [47] suggested that SOC is relevant for understanding earthquakes as a relaxation mechanism which organizes the crust both at spatial and temporal levels. During several decades, seismology lacked of a solid basis to explain empirical power laws, such as, for example, the laws of Gutenberg-Richter, Omori, and Utsu. It was until the arising of the SOC concept in 1987 [16] that empirical power-laws were obtained from a lot of models, being the OFC model one of the most important SOC models. In the present article we have showed that by means of the OFC- SOC model is possible to find G-R laws and staircase plots at least qualitatively analogous to the corresponding G-R laws and staircase plots of actual seismicity, for the case of seismicity catalogs stemming from the seismicity of the Mexican Pacific coast. It is remarkable that synthetic OFC-seismicity is characterized by a kind of attractor consisting of a straight line of constant slope which attracts the long-term cumulative seismicity. This straight line seemingly bounds the long-term cumulative seismicity becoming relevant to the upper limit of main-event magnitudes. Unfortunately, the actual seismicity catalogues are yet too short to contrast this synthetic seismicity property. On the other hand, the obtaining of both the G-R law and staircase plots by means of OFC-SOC models lead to very reasonable results regarding real seismicity. Acknowledgements The authors wish to acknowledge EDI, COFAA and EDD- IPN for partial support. 1. D. Sornette, Critical Phenomena in natural Sciences (Springer, Heidelberg, Germany, 2000). 2. D.L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge University Press, Cambridge, 1992). 3. A.A. Tsonis and J.B. Elsner (Eds), Nonlinear Dynamics in Geosciences (Springer, New York, 2007).

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