# An Analysis of the 2008 \(M_{\rm s}\) 8.0 Wenchuan Earthquake's Aftershock Activity
## 2 Data and Data Completeness
### Completeness Analysis  
Determining the initial completeness magnitude of the aftershock catalog \(M_{c}\) is crucial for studying aftershock activity as it affects the reliability of the study's results. [PERSON] and [PERSON] [4] proposed the following method to estimate the minimum completeness magnitude. Assume that the distribution of earthquake magnitudes conforms to the G-R law. Then, the smallest magnitude \(M_{i}\) is selected; the observed data with \(M\geq M_{i}\) are fitted  
Figure 1: A map of the Wenchuan aftershock sequence. The circles mark the aftershocks, and the symbol size is proportional to the magnitude. The red star marks the mainshock hypocenter location. F1, Longmenshan back fault; F2, Longmenshan central fault; F3, Longmenshan front fault.  
according to the G-R relationship so that the theoretical values for each magnitude interval can be determined using the fitted parameters. Finally, the absolute difference between the theoretical and observed values is obtained as follows:  
\[R=(a,b,M_{i})=100-\left[\frac{\sum\limits_{M_{i}}^{M_{\max}}|B_{i}-S_{i}|}{\sum \limits_{i}B_{i}}100\right] \tag{1}\]  
where \(B_{i}\) and \(S_{i}\) are the observed and theoretical numbers of earthquakes in each magnitude interval, respectively. The value of \(M_{i}\) gradually increases, and \(M_{i}\) is considered the minimum complete magnitude \(M_{c}\) until \(R\geq 95\).  
Using the above method, we explored the pattern of spatiotemporal variation in the minimum completeness magnitude. After an earthquake, seismic monitoring in the epicenter area inevitably strengthens, so the minimum completeness magnitude that can be monitored decreases over time. We investigated the temporal variation in the minimum complete magnitude \(M_{c}\) using a moving time window approach [9]. First, we estimated \(M_{c}\) using 200 aftershocks following the mainshock; then, we moved the window forward in time by 100 earthquakes to successively calculate \(M_{c}\) to obtain the pattern of the temporal variation in \(M_{c}\) (Figure 2). In Figure 2, \(M_{c}\) roughly exhibits an exponential decaying trend over time. In the first few hours after the mainshock, \(M_{c}\) fluctuated around a magnitude of 3.5 and then sharply dropped to approximately 2.5. In the following week, the \(M_{c}\) rebounded to a magnitude of approximately 2.8 due to the strong aftershocks during this period. After approximately ten days, \(M_{c}\) stabilized below a magnitude of 2.5. In addition, the value spatially differed due to the uneven distribution of seismic stations in the study area, so we analyzed the spatial distribution of \(M_{c}\) (Figure 3) by dividing the study area into 0.01\({}^{\circ}\)\(\times\) 0.01\({}^{\circ}\) grids and counting the number of earthquakes within 5 km around each grid point to calculate the \(M_{c}\) value. Figure 3 demonstrates that the \(M_{c}\) value spatially varied between magnitudes of 1.1 and 2.6, indicating that the aftershock records of magnitudes above 2.6 in the study area are basically complete. Therefore, by combining the temporal and spatial distribution patterns of \(M_{c}\), we think that the aftershock records of magnitude above 2.7 are basically complete and can serve as a basis for a subsequent study and analysis.  
Figure 2: The magnitude of completeness, \(M_{c}\), as a function of time for the Wenchuan aftershock sequence.