**3. Seismicity evaluation**

Seismicity evaluation is a fundamental part of any seismic hazard assessment where the initial objective is to retrieve the magnitude probability of occurrence. In the case of a time-dependent seismic hazard, once we ensure that the Weibull cumulative distribution fits the subduction seismicity time patterns, we calculate the conditional probability *CP(M)t, Δt* yielding the probability that an earthquake occurs after an elapsed time *Δt* once an earthquake has happened at time *t.* Note that time *t* is the last year in which an earthquake of a specific magnitude M appears in the catalog [27]:

$$(CP(M)\_{t, \Delta t} = \frac{R(t) - R(t + \Delta t)}{R(t)} \tag{6}$$

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

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

where the reliability function is *R t*ðÞ¼ 1 � *F t*ð Þ . **Figure 6** shows the conditional probabilities for each magnitude bin under consideration until 2120. For M 5–5.5 and 5.5–6.0, the conditional probability is practically unity after 2019, while for the rest of the magnitudes, the probabilities increase at a lower rate as the size of the earthquakes increases and time passes. For example, for large shocks with M 7.5–8.12, the probability of occurrence after 2001 (the last destructive earthquake in the region happened that year) yields 0.28, 0.73, and 0.95 for the years 2025, 2050, and 2100, respectively. Such values are suitable for time-dependent seismic hazard assessment [16]. M 6–6.5 and 6.5–7 yield similar conditional probabilities as the elapsed time increases.

We finally test our subduction catalog deriving the classical Gutenberg–Richter (G-R) relationship, yielding:

$$
\log N = A - BM \tag{7}
$$

where *N* is the number of earthquakes per year with a magnitude equal to or above M. *A* and *B* are constants obtained by regression analysis. Note that the values of *N* must be taken after the year of completeness analysis (see Section 1 and **Table 2**) to avoid underestimation of seismicity levels in the magnitude bins. The relationships are obtained for the clustered and the declustered catalog (see **Figure 7a**). The G–R relationships yield log *N* = 6.43–1.01 M, σ = �0.045 and log *N* = 7.73–1.17 M, σ = �0.097 for the declustered and clustered catalogs, respectively. Note that we

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

#### **Figure 6.**

*Weibull conditional probabilities vs. elapsed times (Eq. 6). Note that the probabilities for M 5.5–6 completely overlap the ones from M 5–5.5 since the yield is unity.*

#### **Figure 7.**

*a) Classical Gutenberg–Richter relationships: The observed number of earthquakes per year* N *above a specific magnitude M (Eq. 7). The circles represent observed data; the lines represent* N*'s estimation of the G–R relationships after a regression analysis. The G–R yields log* N *= 6.43–1.01 M, σ = 0.045 and log* N *= 7.73– 1.17 M, σ = 0.097 for the declustered and clustered catalogs, respectively; b) discrete probability function for 20 magnitudes between M 5 and 8.12 for the declustered catalog (Eq. 8).*

retrieved a *B* value of 1.0 with a very low standard deviation σ for the subduction earthquakes when using only the main events in the analysis, which is a characteristic value of tectonic earthquakes worldwide used in several seismic hazard analyses. Note that such G–R relationship must be truncated to the maximum possible magnitude when employing it in a seismic hazard assessment. Since we have yet to compute design spectra, we opt to present our results using the classical relationships based on Eq. (7).

For time-independent seismic hazard analysis, the probability that a magnitude will fall within a magnitude *m1* and *m2* is given by:

$$P(\mathcal{M}) = \int\_{m1}^{m2} f(\mathcal{M})d\mathcal{M} \tag{8}$$

where *f(M)* is the magnitude probability density function based on the constants *A* and *B* in Eq. (7) [28]. We present in **Figure 7b** the magnitude probability for the overall subduction zone suitable for time-independent seismic hazard analysis. The probability of the occurrence of small shocks is higher than the ones of large shocks. The sum of all probabilities is 1.0.
