**5. Data processing and analysis**

In the Mexican Republic - including its territorial sea - five tectonic plates converge: the North American, the Pacific, the Caribbean, the Cocos plate and the plate Rivera (Nava, 1987; Kostoglodov et al., 2001).

The subduction zone includes the entire Pacific coast between Puerto Vallarta in the state of Jalisco to Tapachula in Chiapas state. This extension has produced the largest earthquakes to have occurred this century in Mexico (Kostoglodov et al., 2001). Subduction earthquakes occur mainly in the coastal state of Guerrero. This type of earthquakes is rated as the most dangerous and they deform the ocean floor and generate tsunamis. According to the above description, it is possible to locate Guerrero in a territorial space latent to earthquakes' presence, not only with it being built on ground plate convergence but also by the presence of seismic gaps.

The Guerrero gap is one of the main concerns of researchers, being a rupture zone which for more than ninety years has not recorded an earthquake. Considering the magnitude of past earthquakes, these have ranged between 7.5 and 7.9 degrees and so - according to the elapsed time - an earthquake is expected to peak at 8.4 degrees, which would represent a greater seismic event than occurred in 1985, which is why in this study we have provided the data epicentres located off the coast of Guerrero - in the area between 15.5 -17.5 N and 98.0-102.0 W for a period of 16 years from 1990 to 2005, as provided by the National Seismological Service (NSS, Servicio Sismologico Nacional).

Figure 2 represents the 4700 epicentres located in the study area. The figure shows that the number of small and medium earthquakes increases towards the state of Oaxaca and we can distinguish a swarm near the Guerrero gap. Note also that the EQs have a greater magnitude away from the coast.

Fig. 2. Location of the 4700 epicenters in the study area (15.5-17.5 N and 98.0-102.0 W) for the 16 year period, 1990-2005, such that the size of each circle is proportional to the magnitude of the event.

#### **5.1 Magnitudes of seismic data for years**

First we made a preliminarily analysis of the data per year (Fig. 3 shows two years as an example) in order to determine the extreme events of the series.

elapsed time - an earthquake is expected to peak at 8.4 degrees, which would represent a greater seismic event than occurred in 1985, which is why in this study we have provided the data epicentres located off the coast of Guerrero - in the area between 15.5 -17.5 N and 98.0-102.0 W for a period of 16 years from 1990 to 2005, as provided by the National

Figure 2 represents the 4700 epicentres located in the study area. The figure shows that the number of small and medium earthquakes increases towards the state of Oaxaca and we can distinguish a swarm near the Guerrero gap. Note also that the EQs have a greater

Fig. 2. Location of the 4700 epicenters in the study area (15.5-17.5 N and 98.0-102.0 W) for the 16 year period, 1990-2005, such that the size of each circle is proportional to the magnitude

First we made a preliminarily analysis of the data per year (Fig. 3 shows two years as an

Seismological Service (NSS, Servicio Sismologico Nacional).

magnitude away from the coast.

of the event.

**5.1 Magnitudes of seismic data for years** 

example) in order to determine the extreme events of the series.

Fig. 3. An example of the EQs' series occurring per year in terms of its magnitude. The selected years are: a) 2003 b) 2004. The green open circles focus the EQs with 5.0 6.0 *Mn* and magenta circles shows EQs with 6.0 *Mn* .

The analysis made of the data finds that the overall average magnitude earthquakes can be considered as 3.9 0.4, and the number of EQs with a magnitude greater than the threshold 5 is between 0-7 per year. However, the years 1997 and 2002 have 9 and 14 events respectively, which doubles the number of events of 5 *Mn* occurring in the area. The influence of the type of instrumentation used to record the earthquakes is obvious because we observe that events which have a magnitude of less than two are not registered. Equally, as to the implementation of broadband seismographs in 1992, the number of records of EQ magnitudes of less than 3 increased significantly. We also observed that for the analyzed period there are no EQs with 7.5 *Mn* , so we decided to take thresholds corresponding to magnitudes: 3, 4, 5, 6, and 7 for this analysis. This data can be seen concentrated in Table 1, which also shows the dependence between the existence of large-scale EQs and the total number of events.


Table 1. Extreme events and maximum number of EQs occurring per month and per year

#### **5.2 Temporal clustering analysis of EQs**

The self-organized critical systems reach this condition due to temporary fluctuations in their events, where they release much of their energy. For this reason, it is necessary to obtain the Fano factor for calculating the fractal exponent ( ), so that it can detect the temporal clustering of events characteristic of the type of event detected.

The exponent of Fano is an estimate of the fractal exponent of the power law that characterizes the density spectrum of a process with scaling properties. The value of indicates the degree of clustering in a process according to Thurner et al. (1997). Over long time scales, the curve behaves essentially as ~ *T* and the curve can be fit by a straight line of the slope . When 0 , the point process is a Homogeneous Poisson and the occurrence times are uncorrelated. However, if 0 , the point process is Non-Homogeneous Poisson and has scaling properties. The value of obtained for the studied area - as is shown in figure 4 ( 0.6653 ) - indicates, as expected, the presence of scaling behavior of the occurrence of the earthquakes.

(year) EQ (month)

Year 5 *Mn* No. Max

Table 1. Extreme events and maximum number of EQs occurring per month and per year

The self-organized critical systems reach this condition due to temporary fluctuations in their events, where they release much of their energy. For this reason, it is necessary to

characterizes the density spectrum of a process with scaling properties. The value of

indicates the degree of clustering in a process according to Thurner et al. (1997). Over long

0 , the point process is a Homogeneous Poisson and the

and the curve can be fit by a straight line

0 , the point process is

0.6653 ) - indicates, as expected, the presence of scaling

), so that it can detect the

of the power law that

obtained for the studied

Non-

**5.2 Temporal clustering analysis of EQs** 

time scales, the curve behaves essentially as ~ *T*

behavior of the occurrence of the earthquakes.

occurrence times are uncorrelated. However, if

of the slope

. When

area - as is shown in figure 4 (

obtain the Fano factor for calculating the fractal exponent (

The exponent of Fano is an estimate of the fractal exponent

Homogeneous Poisson and has scaling properties. The value of

temporal clustering of events characteristic of the type of event detected.

Fig. 4. *FF* curve for 0.6653 . The estimation of the fractal exponent - indicative of the degree of clusterization - is carried out estimating the slope of the curve plotted in bilogarithmic scale in the linear range (in red) of the counting times.

#### **5.3 Setting a process of Non-Homogeneous Poisson Pareto**

Since the temporal clustering analysis of the EQ series indicates that it is a Nonhomogeneous Poisson process, then the methodology - discussed in section 3 - for the analysis NHGPPP is appropriated so as to apply to these data series and in order to calculate the probability of the occurrence of extreme EQs. For the property of the stability threshold it is known that the excesses can be fitted to a GPD, and to validate the method we proceed to make an analysis of the excesses.

To compute the mean excess (that is, the sum of positive differences in the magnitude of the fixed threshold and the magnitude of earthquakes that exceed the threshold, per number of excesses), and the mean exceedance (the sum of the magnitude of the earthquakes that exceed the threshold fixed by the number of the exceedance) we used equations 6 and 7, respectively.

First, you get the graph average exceedance over a threshold and check the feasibility of the linear fit of the observed data with <sup>2</sup> *R* so as to be close to the unit, in this case for the mean exceedance <sup>2</sup> *R* 0.9832 , which is indicative of the reliability and applicability of the proposed method. Next, we proceed to obtain the shape and scale parameters of the GPD (i.e. *k* and *a*, respectively), and for this - as shown in Figure 5 - we fit a straight line by a linear regression in the plot of the mean of the excesses against the magnitude of the fixed threshold.

Fig. 5. Fitting of the mean exceedance for the period 1990-2005, with <sup>2</sup> *R* 0.9832 for the mean of exceedance and <sup>2</sup> *R* 0.4629 for the mean of excesses.

Once the linear regression is fitted, we get the shape and scale parameters. This procedure was done for earthquakes occurring by year, and the parameters of the GPD for each year of the studied period are given in Table 2; the parameters were also calculated for the whole period - repeating the procedure already explained - and obtaining *k* and *a* for the whole period. The comparison between the parameters computed annually and those for the global setting allows us to conclude that the global *k* and *a* can be used for all the data in the studied region, as it is shown in figure 6.

The obtained parameters *k* and *a* were used to fit the GPD to the excess. Figure 6 presents this comparison between the distribution obtained by *k* and *a* per year and for whole period for two selected years as an example of good fitting (year 2004) and the worst of them (year 2003).

In most of the years that were analyzed, the parameters of the GPD average reproduce the behaviour of the data; however, the years 2002, 2003 and 2005 show clear differences in the settings, as these years have atypical features of the studied area, so we will proceed to

Fig. 5. Fitting of the mean exceedance for the period 1990-2005, with <sup>2</sup> *R* 0.9832 for the

Once the linear regression is fitted, we get the shape and scale parameters. This procedure was done for earthquakes occurring by year, and the parameters of the GPD for each year of the studied period are given in Table 2; the parameters were also calculated for the whole period - repeating the procedure already explained - and obtaining *k* and *a* for the whole period. The comparison between the parameters computed annually and those for the global setting allows us to conclude that the global *k* and *a* can be used for all the data in the

The obtained parameters *k* and *a* were used to fit the GPD to the excess. Figure 6 presents this comparison between the distribution obtained by *k* and *a* per year and for whole period for two selected years as an example of good fitting (year 2004) and the worst of

In most of the years that were analyzed, the parameters of the GPD average reproduce the behaviour of the data; however, the years 2002, 2003 and 2005 show clear differences in the settings, as these years have atypical features of the studied area, so we will proceed to

mean of exceedance and <sup>2</sup> *R* 0.4629 for the mean of excesses.

studied region, as it is shown in figure 6.

them (year 2003).

calculate the probabilities of the whole area for periods of up to 100 years, with the parameters obtained from the full term.


Table 2. Parameters *k* and *a* obtained of linear fitting per year and for the complete data series.

To calculate the intensity distribution of the NHGPPP and to obtain the probabilities of the EQs' occurrence, we use equation 8. Since the approach of exceedance implicitly assumes that the scale inherent to the phenomena is open, we force the magnitude scale to ends at 9 *Mn* and we then subtract the probabilities of the EQs exceeding that magnitude from the probabilities of the lower magnitudes. As it is, the probabilities of occurrence lower than one event of 5 *Mn* for a period of 100 years is always 1. Using the same equation (8) we compute the values of the intensity function for a threshold of 5 *Mn* for a period of 100 years, and we obtain *P M <sup>n</sup>* 7 1 .

Fig. 6. Graph of the GPD fitted to the excesses using the parameters for each year compared to the excesses GPD adjusted for the period of 16 years, for the years: a) 2003 and b) 2004, providing examples of good and worst adjust.

#### **6. Discussions**

One of the goals in describing geological processes is to be able to predict their future behaviour. However, this has not been possible despite the many efforts being made in science. So, a small step is made by the characterization of such processes as earthquakes and volcanic eruptions, allowing for a step towards prediction.

As has been indicated, the data analysis was conducted over a period of 16 years - from 1990 to 2005 - looking at the earthquakes of the Guerrero state recorded by the NSS seismic web.

From the preliminary analysis of the EQ series, it was observed that the influence of the kind of instrumentation used to record EQ events is evident throughout the years, as with the implementation of broadband seismographs in 1992 where the registration of the number of EQ magnitudes of less than 3 increases considerably - and so the behaviour of the data is affected. The mean EQ magnitudes of 3.9 0.4 was computed for all the data, and the number of EQ with 5 *Mn* is between 0 - 7 per year; however, the years 1997 and 2002 have 9 and 14 events respectively, which doubles the number of events of that magnitude which occurred in the area. In addition, the number of small and medium earthquakes increases towards the state of Oaxaca and we can distinguish a swarm near the Guerrero gap; also, we note that the EQs are of a greater magnitude away from the coast.

Following this, in order to characterize the EQ events as punctual point process series, the clusterization of this data was allowed by the Fano factor, which indicates an value of 0.6653. This points out that the EQs follow a Non-Homogeneous Poisson Process.

Next, for the property of the stability threshold, it is known that the excesses can be fitted to a GPD and so we next proceed to an analysis of the excess, obtaining the parameters *k* and *a* to fit again to a GPD. The adjusted parameters were computed in two ways: one analysis of the EQs covered each year and the other the whole period (16 years). In most of the studied years, the adjustment of the mean of the GPD parameters reproduced the behaviour of all the data, except for the years 2002, 2003 and 2005, which show clear differences in the settings (as these years have atypical features of the area, the calculation was done with the average parameters).

The last step was to calculate the intensity distribution of the NHGPPP and to obtain the probabilities of the EQs' occurrence with the specific magnitudes in which we are interested. Because the approach of exceedance implicitly assumes an open scale of the phenomena, we assume that the magnitude scale ends at 9 *Mn* , and we compute the values of the intensity function for a threshold of 7 *Mn* for a period of 100 years, and so we obtain *P M <sup>n</sup>* 7 1 .
