## 7 Temporal Properties of the Aftershock Sequence
### Scaling of the Aftershocks Focal Zone with Time  
The growth of the aftershock focal zone with time can provide valuable information regarding the triggering mechanisms of earthquake migration. This migration pattern observed in many cases frequently scales as the logarithm of time [97; 98; 99; 100; 101; 102]. Various studies, based on numerical simulations [103; 104], as well as on real cases [97; 105], suggest that this logarithmic migration pattern signifies that aftershocks migration is driven by afterslip. In this case, aftershocks are generated as the outcome of afterslip propagation along the activated fault. To anticipate the growth of the aftershock focal zone with time as the outcome of afterslip, ref. [106] have recently introduced a numerical model. In this model, asperities on a fault are stressed initially by regional creep occurring at a steady deformation rate during the inter-seismic period. As the mainshock occurs, some of the asperities slip co-seismically, transferring large positive Coulomb stresses to the surrounding creeping regions. During the post-seismic phase, the stress-loaded regions can accommodate large amounts of afterslip and when a critical level of afterslip is reached, aftershocks are triggered. Static stress changes in the model are thought to trigger aftershocks only during the early post-seismic phase so that most aftershocks are triggered by afterslip. In this case, the seismicity rate \(R(t)\) can then be proportional to the afterslip rate \(V(t)\)[103; 106; 107]:  
\[V(t)=\frac{V_{+}exp\left(\frac{t}{t_{r}}\right)}{1+\left(\frac{V_{+}}{V_{L}} \right)\left[exp\left(\frac{t}{t_{r}}\right)-1\right]} \tag{12}\]  
where \(V_{+}\) and \(V_{L}\) are the sliding velocity just after the end of co-seismic rupture and the long-term loading velocity after the mainshock and \(t_{r}\) the duration of the post-seismic phase. Considering the previous equation, the seismicity rate \(R(t)\) can then be given by:  
\[R(t)=\frac{R_{+}exp\left(\frac{t}{t_{r}}\right)}{1+\left(\frac{R_{+}}{R_{L}} \right)\left[exp\left(\frac{t}{t_{r}}\right)-1\right]} \tag{13}\]  
where \(R_{+}\) and \(R_{L}\) are the seismicity rates just after the end of co-seismic rupture and the long-term one after the mainshock, respectively. If \(\dot{\tau}\) is the stressing rate and \(\Delta\)CFS the co-seismic Coulomb stress changes induced by the mainshock, then the parameters \(t_{r}\) and \(R_{+}\) are given by \(t_{r}=A\prime/\dot{\tau}\) and \(R_{+}=R_{L}exp(\Delta CFS/A\prime)\), where \(A^{\prime}=(a-b)\sigma\), with \(a\) and \(b\) the rate and state frictional parameters and \(\sigma\) the effective normal stress. For \(t/t_{r}\ll 1\), Equation (13) yields a decay rate for \(R(t)\) proportional to \(1/t\), which is consistent with a modified Omori decay rate with \(p=1\)[87]. With the previous assumptions, the distribution of afterslip velocities can be deduced. Initially, a fault with only depth varying normal stress, stressing rate, and rheological parameter \(A^{\prime}\) is considered. If the initial Coulomb stress field varies with the strike direction \(x\), then aftershocks migrate along \(x\), forming the initial distribution of afterslip velocities. Then, the propagation velocity \(V_{p}\) of the aftershocks focal zone, in the early stage of the post-seismic phase, which typically lasts several weeks or months after the mainshock, is given by [103]:  
\[V_{p}=\frac{A^{\prime}}{t}\times\left(-\frac{\partial\Delta CFS}{\partial x} \right)^{-1} \tag{14}\]  
The expansion of the aftershocks zone \(L_{a}\) between time \(t_{i}\) and \(t\) (\(t>t_{i}\)) is now given by:  
\[\Delta L_{a}(t)=L_{a}(t)-L_{a}(t_{i})=A^{\prime}\times\left(-\frac{\partial \Delta CFS}{\partial x}\right)^{-1}ln\left(\frac{t}{t_{i}}\right) \tag{15}\]  
The latter equation manifests the logarithmic expansion of the aftershocks zone with time and for a smooth co-seismic Coulomb stress field, implies its slow migration.