**2. The Weibull and Poisson cumulative distributions**

A straightforward way to investigate if a specific probability distribution fits the time seismicity patterns in a seismogenic zone is by applying the cumulative probability distribution based on the interoccurrence times (ITs), namely, the time between consecutive earthquakes of a specific magnitude bin. Salazar [16] studied the ITs matching the Weibull and Poisson cumulative distributions for the upper-crustal

*Perspective Chapter: Testing the Interoccurrence Times Probability Distributions… DOI: http://dx.doi.org/10.5772/intechopen.110584*

#### **Figure 1.**

*Earthquake epicenters (circles) for the subduction zone comprising events with moment magnitude M 5.0–8.12, 1609–2019, and focal depth from 1 to 300 km. a) Clustered catalog (all events); the rectangle depicts the area of the cross-section AB-CD in Figure 2 b) Declustered catalog (after removing foreshocks and aftershocks). Blue contour lines depict the top Cocos plate depths every 20 km based on the USGS slab 2.0 model [13].*

#### **Figure 2.**

*El Salvador cross-section illustrates hypocenters for all subduction earthquakes with moment magnitude M 5.0– 8.12. Dashed lines depict the top of the Cocos plate based on the USGS slab 2.0 model [13] for the area AB-CD in Figure 1. Earthquakes with a fixed depth of 33 km are not plotted. The red triangle indicates the position of the volcanic chain in Central America, and the letter T indicates the position of the trench between the Cocos and the Caribbean plates.*


#### **Table 1.**

*Estimated year of completeness for the subduction zone. We noticed that for all magnitude bins, the year of completeness is the same when employing the clustered and the declustered catalogs (see Figure 3).*

volcanic chain earthquakes in El Salvador. Other authors have also studied the ITs in other parts of the globe [19–23]. Note that we count the ITs in a magnitude bin (e.g., 5–5.5, 5.5–6.0) since the final objective is to use the probability of occurrence of events in a seismic hazard assessment.

*Perspective Chapter: Testing the Interoccurrence Times Probability Distributions… DOI: http://dx.doi.org/10.5772/intechopen.110584*

#### **Figure 3.**

*Example of the completeness period for subduction earthquakes within a magnitude bin M 5.0–5.5 employing the a) clustered catalog: Main shocks, foreshocks, and aftershocks; b) declustered catalog: Only main shocks. The catalog is complete for this magnitude interval back to 1975 because the cumulative annual number of earthquakes for this magnitude range is approximately linear back to that date [18]. All other magnitude bins' periods of completeness are presented in Table 1.*

The cumulative *F(t)* Weibull distribution yields [24]:

$$F(t) = 1 - e^{-at^{\beta}}t > 0, a, \beta > 0\tag{1}$$

where *t* is the earthquake ITs, and *α* and *β* are constants found by the non-linear search algorithm of Bean et al. [25]. The Poisson probability cumulative distribution *F (t)* yields:

$$F(t) = \mathbf{1} - e^{-\lambda t} \, t > \mathbf{0}, \lambda > \mathbf{0} \tag{2}$$

where *λ* is the number of earthquakes per unit of time [26], and *t* must be calculated based on the completeness analysis explained in the previous section. **Table 2** shows the *α* and *β* Weibull and the Poisson *λ* constants for the clustered and


**Table 2.**

*Weibull and Poisson cumulative probability distribution parameters (Eqs. 1 and 2).* σ *denotes the standard deviation. M: moment magnitude;* λ *is expressed above as the number of earthquakes per year; however, for the Poisson cumulative probability calculations, the number of earthquakes per day must be used in Eq. (2).*

declustered earthquake catalogs and their corresponding mean and standard deviations for several magnitude bins. The mean *μ* for the Weibull distribution is:

$$
\mu = a^{-1/\beta} \Gamma \left( \mathbf{1} + \frac{\mathbf{1}}{\beta} \right) \tag{3}
$$

and the variance *σ<sup>2</sup>* gives:

$$\sigma^2 = a^{-2/\beta} \left\{ \Gamma \left( \mathbf{1} + \frac{2}{\beta} \right) - \left[ \Gamma \left( \mathbf{1} + \frac{\mathbf{1}}{\beta} \right) \right]^2 \right\} \tag{4}$$

The standard deviation *σ* is the root square of the variance. *Γ* is the gamma function; we also listed the fit error in terms of the root mean square (RMS):

$$RMS = \sqrt{\frac{\sum\_{i=1}^{N} (Prob\\_Obs\_i - Probability\\_Pred\_i)^2}{N}} \tag{5}$$

where *N* is the number of data, Prob\_*Obs*<sup>i</sup> is the observed cumulative probability, and the *Prob*\_*Predi* is the predicted cumulative probability at the *i* event.

We infer by comparing the observed and the predicted ITs using the obtained constants that subduction events pose the Weibull cumulative distribution when considering all the events in the subduction catalog, namely, principal, fore, and aftershocks (clustered catalog) for all magnitude bins with low RMS between 0.01 and 0.085 (**Figure 4a, c, e, g, i**, and **k**). Abaimov et al. [19] suggested that the conventional graphs depicting the cumulative probability *F(t)* vs. the ITs (left panel in **Figure 4**) are not appropriate for judging, in the first instance, the fit between observed and predicted probabilities. Instead, they propose linearizing the cumulative probability, applying -ln (1-*F(t)*). The linearized probability Weibull plots confirm the goodness to fit (**Figure 4b, d, f, h, j**, and **l**) in all cases. However, the Weibull distribution better predicts the probabilities for longer ITs than short ones in one case after 100 days (e.g., M 6.5–7). The α value yields practically zero for M 7.5–8.12 (**Table 2**), arguing a Poisson process tendency for the largest subduction shocks. Indeed, the Poisson and Weibull models predict similar probabilities (see the right panel in **Figure 4j** and **l**). Although the Poisson probability distribution is thought to be applied to independent events only, we also tested the ITs to the clustered catalog to investigate if such distribution fits some magnitude bins under consideration, especially for big events.

The Poisson cumulative distribution does not fit the ITs for smaller events with magnitudes between 5.5 and 7 when using the clustered catalog (all events in the analysis), yielding RMS from 0.035 to 0.064 (**Table 2**). The linearized plots on the right panel in **Figure 4d, f**, and **h** confirm such a statement for M 5.5–6, 6–6.5, and 6.5–7.0, respectively.

However, when applying the Gardner and Knopoff [14] method to remove the foreshocks and aftershocks and conform to the declustered catalog, the Poisson distribution better fits the ITs for M 5.5–6 and 6–6.5 (**Figure 5d** and **f**) but still fail to reproduce the ITs for M 6.5–7.0 (**Figure 5h**), although there are lower RMS yields for

*Perspective Chapter: Testing the Interoccurrence Times Probability Distributions… DOI: http://dx.doi.org/10.5772/intechopen.110584*

**Figure 4.**

*Interoccurrence times vs. cumulative probabilities for several magnitude bins. (a) M 5–5.5 (c) M 5.5–6 (e) M 6– 6.5 (g) M 6.5–7 (i) M 7–7.5 (j) M 7.5–8.12. We compare the observed and the estimated cumulative probability distributions employing the Weibull and Poisson models based on the clustered catalog. The arrow depicts Weibull's mean. The right side (b, d, f, h, j, and k) depicts the linearized probability plots.*

these cases (**Table 2**). The linearized probability plots confirm the goodness of the Poisson fit in most of the magnitude bins under analysis, including large shocks above M 7.5 (**Figure 5j** and **l**). We conclude that the Poisson distributions fit the time seismicity patterns once only when the main shocks are considered in the analysis, except for the magnitude bin between 6.5 and 7.0 when ITs are less than 100 days.
