**Non-Linear Analysis of Point Processes Seismic Sequences in Guerrero, Mexico: Characterization of Earthquakes and Fractal Properties**

E. Leticia Flores-Marquez and Sharon M. Valverde-Esparza *Instituto de Geofísica, UNAM, Cd. Universitaria, México D.F. Mexico* 

### **1. Introduction**

234 Earthquake Research and Analysis – Seismology, Seismotectonic and Earthquake Geology

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The purpose of this work is to reveal the efficiency of some statistical non-linear methods so as to characterize a seismic zone linked to subduction in Mexico. The Pacific plate subducting into the North American plate produces an important number of earthquakes (EQs), whose magnitudes exceed *Mn* = 5. This region comprises the following States: Jalisco, to the northwest, Michoacan, Guerrero, and Oaxaca, to the southeast; it extends along roughly 1350 km (Figure 1). Therefore, the characterization of this region - in all scopes - is very important. Here, we focus on the application of non-linear methods in the Guerrero State, because it displays an important number of EQs (their magnitudes rise up to 6) and it has a different slip inclination to the rest of the subduction zone, and some authors (Singh et al., 1983; Pardo and Suarez, 1995) have considered that there are some lags of seismicity. The assumptions of the non-linear methods analyzed in this work are: that EQs are stochastic point processes; that the Fano Factor (FF) reveals the fractality of EQs; and that the NHGPPP adjusts to extreme events. The application of these methods to the Guerrero seismic sequence allows us to explain the phenomenological behaviour in the subduction zone.

Traditionally, studies to characterize earthquakes' processes focus on the tectonics mechanism, basically following deterministic approaches. Recently, some studies have investigated the time scale properties of seismic sequences with non-linear statistical approaches so as to understand the dynamics of the process. The deep comprehension of the correlation time structures governing observational time-series can provide information on the dynamical characterization of seismic processes and the underlying geodynamical mechanisms (Telesca et al., 2001).

Scale-invariant processes provide relevant statistical features for characterizing seismic sequences. Since 1944, Gutenberg and Richter have found that earthquake magnitude size follows a power-law distribution. Other scale-invariant features were determined in Kagan (1992, 1994) and Kagan and Jackson (1991). A theory to explain the presence of scaleinvariance was proposed by Bak et al. (1988); they introduced the idea of self-organized criticality (SOC), beginning from a simple cellular automaton model, namely a sand pile (Turcotte, 1990; Telesca et al., 2001).

Sieh (1978) and Stuart and Mavko (1979) proposed that earthquakes are due to a stick-slip process involving the sliding of the crust of the earth along faults. When slip occurs at some location, the strain energy is released and the stress propagates in the vicinity of that position. As such, the SOC concept is well-suited for rationalizing observations of the occurrences and magnitudes of earthquakes (Bak and Tang, 1989). An important part of the relaxation mechanism of the crust of the earth is submitted to inhomogeneous increasing stresses accumulating at continental-plate borders (Sornette and Sornette, 1989). The use of scaling laws concerning earthquakes has been especially used to develop models of seismogenesis, and the efforts of the characterization of EQs in Guerrero state on the part of some authors have been devoted to the shape of waves' propagation in order to reduce the uncertainty in magnitude determination and location (Singh et al., 1983; Pardo and Suarez, 1995).

In addition, many authors using several statistical techniques for specific volcanoes have carried out some studies of volcanic eruption time-series. Most of them have been developed within the scope of statistical distributions. Some of the earliest (Wickman, 1965, 1976; Reyment, 1969; Klein, 1982) employed stochastic principles to analyze eruption patterns. Further studies included the transition probabilities of Markov chains (Carta et al., 1981; Aspinall et al., 2006; Bebbington, 2007), Bayesian analysis of volcanic activity (Ho, 1990; Solow, 2001; Newhall and Hoblitt, 2002; Ho et al., 2006; Marzocchi et al., 2008), homogeneous and non-homogeneous Poisson processes applied to volcanic series (De la Cruz-Reyna, 1991; Ho, 1991), a Weibull renewal model (Bebbington and Lai, 1996a, b), geostatistical hazard-estimation methods (Jaquet et al., 2000; Jaquet and Carniel, 2006), a mixture of Weibull distributions (Turner et al., 2008) and, finally, non-homogeneous statistics to link geological and historical eruption time-series (Mendoza-Rosas and De la Cruz-Reyna, 2008). An exhaustive list of the available literature on this subject is made in Mendoza-Rosas and De la Cruz-Reyna (2009).

Along the same research lines, several distributions have been used to model seismic activity. Among these, the Poisson distribution - which implies the independence of each event from the time elapsed since the previous event - is the most extensively used, since in many cases and for large events a simple discrete Poisson distribution provides a good fit (Boschi et al., 1995).

Like some random phenomena, such as noise and traffic in communication systems (Ryu and Meadows, 1994), biological ion-channel openings (Teich, 1989), trapping times in amorphous semiconductors (Lowen and Teich, 1993a,b), seismic events occur at random locations in time. A stochastic point process is a mathematical description which represents these events as random points on the time axis (Cox and Isham, 1980). Such a process may be called fractal if some relevant statistics display scaling, characterized by power-law behaviour - with related scaling coefficients - that indicates that the represented phenomenon as containing clusters of points over a relatively large set of time scales (Lowen and Teich, 1995). Kagan (1994) and Telesca et al. (1999, 2000a,b, 2010) maintain that an earthquake's occurrence might be characterized by clustering properties with both short and long timescales with temporal correlation among the seismic events.

In this paper, we discuss the estimating of the fractality of a point process modelling a seismic sequence, corresponding to the Guerrero coast (the most seismically active area of the southern coast of Mexico), analyzing the performance of the Fano factor. Afterwards, we look at the extreme-value theory applied to NHGPPP so as to quantitatively evaluate the probabilities of extreme EQ occurrences. This work is organized as follows: first, we present the theoretical concepts of stochastic point processes, fractal analysis by Fano factors and NHGPPP; then we present the EQ data series of the Guerrero region; and finally, we show the results of the analysis of this data when treated as stochastic point processes.

Fig. 1. Four seismicity regions dividing southern Mexico along the Mexican subduction zone, based on the seismicity and shape of the subduction (modified from Singh et al., 1983).
