## 2 Data and Method  
The data set used in this study was provided by a temporal network of 7 accelerograph stations installed in Michoacan state, western Mexico (Figure 3). The stations were installed at the following locations: Faro de Brucerias (FMIR), Aguililla (AGM), Apatzingan (APTZ), Taretan (TART), Patzcuaro (PATZ), Morelia (MOIG) and Maravatio (MARM). The seismic array formed a nearly perpendicular line to the subduction trench with a total length of 366 km (Figure 3) with a distance between successive stations between 60 and 80 km. Figure 2: Geological provinces in Michoacan with some focal mechanisms. VF stands for Volcanic Field in Michoacan-Guanajuato VF [12]. The method is based on the superposition of events of a small magnitude, as proposed by [21]. The recording of small and larger nearby events is assumed following the \(\omega^{-2}\) spectral model with constant stress drop [21]. Under these assumptions, the number of necessary sub-events, \(N^{3}\), is obtained from the ratio of the seismic moment of a large earthquake to the seismic moment of the small event. Ref. [20] showed that \(N^{3}\) is equal to the number of subfaults  
\[N^{3}=N_{x}\times N_{w}\times N_{t} \tag{1}\]  
where each sub-fault is divided by \(N_{x}\) and \(N_{w}\) segments in the strike and dip directions, respectively. \(N_{t}\) is the number of time steps in the process. According to [22; 23], the scaling laws imply that  
\[N^{3}=\frac{M_{0}}{m_{0}}=\frac{\bar{U}_{0}}{\bar{u}_{0}} \tag{2}\]  
and  
\[N=\left(\frac{M_{0}}{m_{0}}\right)^{1/3}=\frac{\bar{A}_{0}}{\bar{u}_{0}} \tag{3}\]  
where \(M_{0}\) and \(m_{0}\) are the seismic moments, \(\bar{U}_{0}\) and \(\bar{u}_{0}\) are the flat levels of the displacement spectra, \(\bar{A}_{0}\) and \(\bar{u}_{0}\) are the flat levels of the acceleration spectra of the mainshock and the small event, respectively. The synthetic motion of the large earthquake \(A(t)\) is given in terms of the observed motion of the small earthquake \(a(t)\) by the convolution  
\[A(t)=\sum_{i=1}^{N_{x}}\sum_{j=1}^{N_{w}}\left(\frac{r}{r_{ij}}\right)F(t-t_{ ij})*a(t) \tag{4}\]  
Figure 3: Location of the accelerograph station array (triangles). The locations of two epicentral locations used in this study are also included. where  
\[F(t)=\delta(t-t_{ij})+\frac{1}{n^{\prime}}\sum_{k=1}^{(N-1)n^{\prime}}\delta\bigg{[} t-t_{ij}\frac{(k-1)\tau}{(N-1)n^{\prime}}\bigg{]} \tag{5}\]  
where \(r\) is the hypocentral distance of the small event, \(r_{ij}\) is the distance between the station and the point \((i,j)\) at the fault element of the large earthquake, \(t_{ij}\) is the sum of delay times from the starting point of the rupture element \((i,j)\) to the observation point, \(\tau\) is the rise time of the small event. In order to avoid spurious periodicities [21], an integer \(n^{\prime}\) is introduced, which is adequate to move the fictitious periodicity \(\tau/(N-1)\) into a high-frequency outside the frequency range of engineering interest. \(F(t)\) is a filter introduced to adjust the difference between the time and dislocation of the large and small earthquakes. The empirical Green's function is modified to use a small earthquake motion with a different stress drop from the large event. This difference is corrected by introducing a constant, \(C\)  
\[C=\frac{\Delta\sigma_{le}}{\Delta\sigma_{se}} \tag{6}\]  
where the subscripts \(le\) and \(se\) stand for small earthquakes and large earthquakes, respectively.