# Stochastic Modeling of the Al Hoseima (Morocco) Aftershock Sequences of 1994, 2004 and 2016
## 6 Temporal Stochastic Modeling for the Al Hocima Sequences  
This section focuses on the point process modeling of the studied aftershock sequences. Many approaches for modeling this gradual decrease in aftershock frequency have been presented in the literature. The most commonly used model is the Omori law [3], which was improved by [4] into the modified Omori formula (MOF). According to [4], the decay rate of the aftershock per unit time is given as follows:  
\[n(t)=\frac{K}{(t+c)^{p}} \tag{5}\]  
where \(t\) is the time after the occurrence of the mainshock, and the parameter \(K\) is related to the mainshock event and threshold magnitude. The \(c\) parameter is a debatable number [73; 74], with the early stages of imprecise detection of low events in the sequence having a significant effect [75]. Finally, the \(p\) parameter is a decay constant, and it is quite likely the most important parameter for understanding the behavior of the sequence. The \(p\)-value varies from sequence to sequence and typically ranges from 0.5 to 1.8 [73]. This variation may be related to tectonic activity in the area. However, the elements that influence the \(p\)-value remain unclear [76; 77]. The quantity \(n(t)\) allows us to connect with the point process model by considering it a conditional intensity. \[n(t)\approx\ \tilde{\ }(t), \tag{6}\]  
bearing in mind that [5]:  
\[P\ \{\text{an event occurs in}\ (t,t+dt)|\Im_{t}\}=\ \tilde{\ }(t|\Im_{t})\ dt+o(dt), \tag{7}\]  
where \(\Im_{t}\) denotes the internal history of the occurrence process at time t, \(\tilde{\ }(t|\Im_{t})\) is the conditional intensity function [5; 78], and \(o(dt)\), in the Landau notation, is a function of a lower order than the function \(\lambda\), i.e., \(o(dt)\) being negligible. The MOF model includes only the mainshock occurrence time because it is based on the concept that the entire relaxation process is driven by the stress changes induced by the mainshock alone. The aftershocks are conditionally independent and follow a non-stationary Poisson process. The MOF model fits data for simple aftershock sequences well; however, secondary clustering is common when there are secondary aftershocks triggered by strong earthquakes in a sequence. The authors in [5] argued that aftershock clustering is a self-similar process in which all aftershocks might induce other aftershocks, with a triggering capacity dependent on their magnitudes, because of these complex situations involving one or more secondary events. The model was named ETAS, and its conditional intensity function is given as follows [79]:  
\[\tilde{\ }(t|\Im_{t})=\mu+\sum_{i;\ t_{i}<t}\frac{k_{0}e^{\alpha(m_{i}-m_{c})}}{( t-t_{i}+c)^{p}} \tag{8}\]  
where \(\mu\) is the background seismicity rate. The internal history \(\Im_{t}\) includes the time occurrence \(t_{i}\) (in days after the mainshock) and magnitude events \(m_{i}\) of all the events occurring before time \(t\). The summation includes all events with occurrence times \(t_{i}\) and magnitudes equal to or greater than the lower cut-off \(m_{c}\). The \(c\) and \(p\) model parameters were the same as those used in the MOF model. Furthermore, \(k_{0}\) is a parameter that affects total aftershock productivity and is common to all aftershocks. According to [80], every term of the summation in Equation (8) indicates the contribution of a prior event to the occurrence probability of subsequent events at time t. The exponential term is controlled by two factors: (a) the temporal rate decrease, as presented by the MOF model and (b) the exponential term chosen because the logarithm of the aftershock area and the magnitude of the mainshock present a linear relationship [81]. Parameter \(\alpha\) measures the effect of the magnitude of the production of 'children' events, also called \"descendants\". The MOF and ETAS models, as expressed by Equations (5) and (8), respectively, present two limited cases for modeling the temporal distribution of an aftershock sequence. Ref.