**2. Point process**

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

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.,

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

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

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

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

long timescales with temporal correlation among the seismic events.

1983; Pardo and Suarez, 1995).

Mendoza-Rosas and De la Cruz-Reyna (2009).

(Boschi et al., 1995).

A stochastic point process was described by some authors (Telesca et al., 2001; Cox and Isham, 1980; Lowen and Teich, 1995) in terms of a mathematical description which represents the 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 timescales (Lowen and Teich, 1995).

In this work, any earthquake sequence is assumed to be a realization of a point process, with events occurring at some random locations in time, and it is completely defined by the set of event times - or equivalently - by the set of inter-event intervals. Over a continuous time process, events can occur anywhere on the time axis. In a discrete time point process, the occurrence of events occurs at equally spaced increments. The continuous time point process is a simple Poisson process. If the point process is Poissonian, the occurrence times are uncorrelated; for this memoryless process, the inter-event interval probability density function *P(t)* behaves as a decreasing exponential function *P(t) = λe−λt* , for *t* ≥ 0, with *λ* as the mean rate of the process.

If the point process is characterized by fractal behaviour, the inter-event interval probability density function *P(t)* generally decreases as a power-law function of the inter-event time, (1 ) ( ) *P t kt* , with the so-called fractal exponent (Thurner et al., 1997). The exponent measures the strength of the clustering and represents the scaling coefficient of the decreasing power-law spectral density of the process ( ) *S f f* (Lowen and Teich, 1993a,b). The power spectral density furnishes information about how the power of the process is concentrated at various frequency bands (Papoulis, 1990) and it provides information about the nature of the temporal fluctuations of the process.

In recent studies, some authors (Bodri, 1993; Luongo et al., 1996) have focused their attention on the observational evidence of time-clustering properties in earthquake sequences of different seismic areas, demonstrating the existence of a range of time scales with scaling behaviour. The method that they used - the Cantor dust method (Mandelbrot, 1983) - consists of dividing the time interval *T*, over which *N* earthquake occur, into a series of *n* smaller intervals of length *t Tn* with *n* = 2, 3, 4, . . . and computing the number *R* of intervals of length *t* which contain at least one event. If the distribution of events has a fractal structure (Smalley et al., 1987) then *R t1−D* , where *D* is the fractal dimension, which has sub-unitary values: the clustering is higher as D approaches to 0, while a value of 1 corresponds to an uniform distribution (events equally spaced in time). But the parameter *R* does not give information about the temporal fluctuations, because it is not directly correlated to the power spectral density *S(f)* of the process itself.

#### **3. Fractal analysis (Fano factor)**

The self-organized critical systems reach the critical steady state with temporary fluctuations in their events characterized by the energy they release. To detect the presence of clustering of events in a time series, several methods can be used among which is the Fano factor calculation, which estimates the value of the fractal exponent of the study process.

According to such authors as Telesca et al. (2004), assuming a sequence of events is the result of a point process defined by the set of occurrence times. You can use a statistical measure such as the Fano factor *FF*( ) to characterize the process.

For fractal process, that displays clustering properties, *P*(*t*) generally behaves as a powerlaw function of the inter-event time *t* with exponent (1 ) , were is called fractal exponent, which characterizes the clustering of the process.

The representation of a point process is given by dividing the time axis into equally spaced contiguous counting windows of duration , and producing a sequence of counts *Nk* ( ) , with ( ) *Nk* denoting the number of earthquakes in the kth window:

$$N\_k\left(\tau\right) = \int\_{t\_{k-1}}^{t\_k} \sum\_{j=1}^n \delta\left(t - t\_j\right) dt\tag{1}$$

The sequence is a discrete-random process of natural numbers.

The *FF*( ) (Thurner et al., 1997) is a measure of correlation over different timescales. It is defined as the variance of the number of events in a specified counting time divided by the mean number of events in that counting time, that is:

$$FF(\tau) = \frac{\left\langle N\_k^2(\tau) - N\_k(\tau) \right\rangle^2}{\left\langle N\_k(\tau) \right\rangle} \tag{2}$$

where denotes the expectation value.

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

1993a,b). The power spectral density furnishes information about how the power of the process is concentrated at various frequency bands (Papoulis, 1990) and it provides

In recent studies, some authors (Bodri, 1993; Luongo et al., 1996) have focused their attention on the observational evidence of time-clustering properties in earthquake sequences of different seismic areas, demonstrating the existence of a range of time scales with scaling behaviour. The method that they used - the Cantor dust method (Mandelbrot, 1983) - consists of dividing the time interval *T*, over which *N* earthquake occur, into a series of *n* smaller intervals of length *t Tn* with *n* = 2, 3, 4, . . . and computing the number *R* of intervals of length *t* which contain at least one event. If the distribution of events has a

has sub-unitary values: the clustering is higher as D approaches to 0, while a value of 1 corresponds to an uniform distribution (events equally spaced in time). But the parameter *R* does not give information about the temporal fluctuations, because it is not directly

The self-organized critical systems reach the critical steady state with temporary fluctuations in their events characterized by the energy they release. To detect the presence of clustering of events in a time series, several methods can be used among which is the Fano factor

According to such authors as Telesca et al. (2004), assuming a sequence of events is the result of a point process defined by the set of occurrence times. You can use a statistical

For fractal process, that displays clustering properties, *P*(*t*) generally behaves as a power-

The representation of a point process is given by dividing the time axis into equally spaced

 1 1

() () ( ) ( ) *k k k*

*N N*

*N* 

*dt*

 

(Thurner et al., 1997) is a measure of correlation over different timescales. It is

<sup>2</sup> <sup>2</sup>

 

*<sup>t</sup> <sup>n</sup> t t <sup>k</sup> <sup>j</sup> <sup>j</sup> <sup>t</sup>*

*k*

*k*

denoting the number of earthquakes in the kth window:

defined as the variance of the number of events in a specified counting time

*N* 

*FF*

The sequence is a discrete-random process of natural numbers.

the mean number of events in that counting time, that is:

to characterize the process.

 *t1−D* , where *D* is the fractal dimension, which

of the study process.

is called fractal

,

divided by

, were

, and producing a sequence of counts *Nk* ( )

(1)

(2)

(Lowen and Teich,

decreasing power-law spectral density of the process ( ) *S f f*

information about the nature of the temporal fluctuations of the process.

correlated to the power spectral density *S(f)* of the process itself.

calculation, which estimates the value of the fractal exponent

law function of the inter-event time *t* with exponent (1 )

exponent, which characterizes the clustering of the process.

fractal structure (Smalley et al., 1987) then *R* 

**3. Fractal analysis (Fano factor)** 

measure such as the Fano factor *FF*( )

contiguous counting windows of duration

with ( ) *Nk* 

The *FF*( ) 

The *FF* varies as a function of counting time . The exception is the Homogeneous Poisson Point Process (HPP). For an HPP, the variance-to-mean ratio is always unity for any counting time . Any deviation from unity in the value of *FF*( ) therefore indicates that the point process in question is not a homogenous Poisson in nature. An excess greater than the unit reveals that a sequence is less ordered than an HPP, while values below the unit signify sequences that are more ordered.

The *FF*( ) of a fractal point process with 0 1 varies as a function of counting time as:

$$FF(\tau) = 1 + \left(\frac{\tau}{\tau\_0}\right)^{\alpha} \tag{3}$$

The monotonic power-law increase is representative of the presence of fluctuations on many timescales (Lowen and Teich, 1995); 0 is the fractal onset time and it marks the lower limit for significant scaling behavior in the *FF*( ) (Teich et al., 1996). Therefore a straight-line fit to an estimate of *FF*( ) vs. on a doubly logarithmic plot can also be used to estimate the fractal exponent. However, the estimated slope of the FF saturates at unity so that this measure finds its principal applicability for processes with 1 .
