# Stochastic Modeling of the Al Hoseima (Morocco) Aftershock Sequences of 1994, 2004 and 2016
## 6 Temporal Stochastic Modeling for the Al Hocima Sequences
[2] proposed a similar model, the RETAS model, based on the assumption that not all events in a series; however, only aftershocks with magnitudes greater than or equal to a threshold value, can directly cause the aftershocks of \"descendants\". This model allows for the inclusion of all potential models between the two limit cases of the MOF and ETAS models, resulting in the conditional intensity expressed as follows:  
\[\text{``}(t|\Im_{t})=\mu+\sum_{ \begin{subarray}{c}i:\ t_{i}<t\\ m_{i}\geq M_{tr}\end{subarray} }\frac{k_{0}e^{\alpha(m_{i}-m_{c})}}{(t-t_{i}+c )^{p}} \tag{9}\]  
It is worth noting that the RETAS model developed by [8] is based on Bath's law, which states that the difference between mainshock magnitude and the strongest aftershock magnitude, ranges between 1.2 to 1.4 units. According to this relationship, ref. [8] argued that the difference between the weaker primary event and the weakest event in the aftershock sequence must be at least 1.2 units, by applying this principle to the subsequences generated by the primary trigger model. Furthermore, the RETAS model has the advantage of examining all potential models between the MOF and ETAS models because the triggering magnitude ranges from the threshold magnitude to the mainshock magnitude. The Akaike information criterion [82], abbreviated as AIC, was used to choose the best-fit model in our case, with the lower AIC value. This is expressed as follows:  
\[\text{AIC}=-2\ \underset{\theta}{max}\ log\ L(\theta;0,T)+2k\text{''} \tag{10}\]  
where \(k\text{''}\) represents the number of parameters of the model, and \(log\ L\) is the logarithm of the likelihood function, given as follows:  
\[log\ L(\theta;0,T)=\sum_{i=1}^{N}log_{10}\text{``}(t_{i}|\Im_{t_{i}})-\int \limits_{0}^{T}\text{``}\theta(s|\Im_{s})ds \tag{11}\]  
In the previous Equation (12), \(N\) is the number of earthquakes with magnitudes greater than or equal to \(m_{c}\), occurring at times \(t_{j}\) (\(j=1,2,\,\ldots,\,N\)), during [0, \(T\)]. Due to the features of the described model, it was used in this study to investigate three aftershock sequences. Table 1 lists the magnitude thresholds used in this study. As a result of the RETAS model analysis, Figure 10 displays the AIC value versus the triggering magnitude value. Analysis of this curve reveals the aftershock clustering type that is most common in the sequence. The AIC parameter was calculated for all the potential models by varying the triggering magnitude from the threshold magnitude to the magnitude of the mainshock. The best-fit model shows the AIC minimum values in Table 3 as well as the triggering magnitude and model parameters. Figure 10: The AIC values of the RETAS model vs. the triggering magnitude for the three sequences. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Sequence** & **Mtr** & **Best Model** & **AIC** & \(\mathbf{K_{0}}\) & \(\mathbf{\alpha}\) & \(\mathbf{c}\) & \(\mathbf{p}\) \\ \hline
1994 & 3.9 & RETAS & \(-12.4\) & 1.727 & 0.0630 & 0.024 & 0.909 \\
2004 & 3.4 & ETAS & \(-1222.5\) & 3.426 & 0.0035 & 0.082 & 1.070 \\
2016 & 2.0 & ETAS & \(-7209.0\) & 1.477 & 0.0380 & 0.039 & 1.183 \\ \hline \end{tabular}
\end{table} 
Table 3: The computed parameters for the RETAS model. When examining the AIC values vs. \(M_{tr}\) curves (Figure 10), we can see that the best-fit model for the minimal AIC value was found to be for a \(M_{tr}\) value of 3.9 for the 1994 sequence. For the 2004 sequence, an \(M_{tr}\) value of 3.4 is the best-fit model obtained and, for the 2016 sequence, a magnitude of 2.0. The best-fit model became an ETAS model when the triggering magnitude \(M_{tr}\) coincided with the completeness threshold magnitude \(m_{c}\) in the last two \(M_{tr}\). Analyzing the AIC vs. \(M_{tr}\) trigger magnitude curve for the 2016 sequence, a monotonic trend exists, resulting in a continuous increase in AIC values, the lowest of which is for \(M_{tr}=m_{c}\).