## 7 Temporal Properties of the Aftershock Sequence  
### Aftershock Production Rate and Modelling  
It has long been recognized that the number of aftershocks following a major event resembles a power law decay with the time that expresses the relaxation process after the mainshock. This mathematical relationship takes the form of the so-called modified Omori's scaling law [87, 88]:  
\[n(t)=K(t+c)^{-p} \tag{7}\]  
where \(n(t)\) is the production rate of aftershocks \(n(t)=dN(t)/dt\), \(N(t)\) is the number of aftershocks in time \(t\) after the mainshock, \(K\) and \(c\) are constants and \(p\) the power law exponent that usually takes values in the range 0.9 \(<\)\(p\) < 1.6 [89]. Moreover, the cumulative frequency of aftershocks \(N(t)\) is estimated from \(n(t)\) as:  
\[N(t)=\int\limits_{0}^{t}n(s)ds=\begin{cases}K\Big{[}c^{1-p}-\left(t+c\right)^{ 1-p}\Big{]}/(p-1),\text{for }p\
eq 1\\ Kln(t/c+1),\text{for }p=1\end{cases} \tag{8}\]  
Herein, we use the previous equation to model the evolution of the aftershocks in terms of the cumulative frequency of aftershock activity that followed the Mw6.0 mainshock. We focus on the two major aftershocks clusters, the NE and SW spatial clusters that were discussed previously, and apply the maximum likelihood method to estimate the model parameters of Equation (8) [90]. In Figure 16 the cumulative number of aftershocks (for \(M\geq M_{c}\)) with time is shown for the two spatial clusters along with the modified Omori's law (Equation (8)), which generally provides a fair fit for the parameter values given in Table 7. However, large aftershocks may trigger secondary aftershock sequences embedded in the aftershock sequence of the mainshock. In this case, several Omori regimes may be used to model the aftershocks production rate \(n(t)\)[89, 90, 91]:  
\[n(t)=K_{1}(t+c_{1})^{-p_{1}}+H(t-t_{2})K_{2}(t-t_{2}+c_{2})^{-p_{2}}+H(t-t_{3} )K_{3}(t-t_{3}+c_{3})^{-p_{3}} \tag{9}\]  
Figure 15: The frequency-magnitude distribution of earthquakes (squares) for the (**a**) foreshock sequence, (**b**) aftershock sequence, (**c**) NE aftershocks cluster, (**d**) SW aftershocks cluster. The corresponding fit according to Equation (6) is shown with the solid line, for the parameter values shown in the down left corner and Table 6. The dotted lines represent the 95% confidence intervals.  
where \(H(\cdot)\) denotes a unit step function and \(t_{2}\), \(t_{3}\) indicates the occurrence times of secondary aftershock sequences. In Figure 16, breaks are observed in the cumulative number of aftershocks for both spatial clusters that are associated with strong aftershocks and the generation of secondary aftershock sequences. Hence, we investigate if the composite model of Equation (9) fits better the observed distribution. By setting \(t_{2}\) = 22.8 days and \(t_{3}\) = 67.4 days that designate the occurrence times of strong aftershocks following the mainshock for the NE cluster (Table 7), we find that the composite model provides a better fit to the observed distribution (Figure 16), which is further confirmed by the smaller Akaike Information Criterion (AIC) value in comparison to the single Omori regime (Table 7).  
A similar result is obtained for the SW cluster. In this case, we used \(t_{2}\) = 24.1 days and \(t_{3}\) = 32.8 days which mark the occurrence times of strong aftershocks in the SW cluster (Table 7). The composite model provides a better fit to the observed distribution, in comparison to the single Omori regime, for the parameter values given in Table 7.