## 3 Method  
The EEPAS model is usually fitted with a time lag to prevent any influence on the parameters from short-term clustering. Here, a time lag of 50 days was applied. This means that no precursory earthquake contributed to the time-varying rate density until 50 days after its occurrence.  
Two different weighting strategies are commonly adopted in applications of the EEPAS model: equal weighting and down-weighting of aftershocks. For down-weighting of aftershocks, the weight assigned to each earthquake depends on the ratio of the rate density of the background model to the rate density of an epidemic-type aftershock model at the time, magnitude and location of its occurrence. For details, see [13]. The down-weighted aftershocks strategy is preferable for investigating the space-time trade-off because it better respects the hierarchical nature of seismicity, as seen in aftershock occurrence as well as precursory seismicity [25, 26, 27].  
We considered two versions of the down-weighted aftershocks EEPAS model, which we labeled EEPAS_1F. The models were called NZ EEPAS_1F and California EEPAS_1F. The model parameters are listed in Table 1. The values were slightly different from the models previously tested in the New Zealand and California testing centers of the Collaboratory for the Study of Earthquake Predictability (CSEP) since 2008 and 2006, respectively [28, 29, 30]. The differences were due to looser constraints imposed in the fitting of \(\mu\).  
The surveillance and search regions for New Zealand and California are shown in Figures 4 and 5, respectively. Figures 4a and 5a show the locations of earthquakes with magnitudes M > 2.95 contributing to their fitting between times \(t_{0}\) and \(t_{f}\). Time \(t_{0}\) is the beginning of 1951 for New Zealand and 1932 for California, while \(t_{f}\) is the end of 2006 for New Zealand and 2005 for California. Figures 4b and 5b show the locations of the target earthquakes with M > 4.95 between times \(t_{s}\) and \(t_{f}\), where \(t_{s}\) is the beginning of 1987 for New Zealand and 1986 for California.  
seven steps, each of the controlled scaling factors varied by an order of magnitude on either side of the optimal fit. For each controlled value of \(a_{T}\) or \(\sigma_{A}\), two free parameters, \(\mu\) and either \(\sigma_{A}\) or \(a_{T}\), were refitted to maximize the likelihood of target earthquakes in the region of surveillance over time period (\(t_{s}\), \(t_{f}\)).  
\begin{table}
\begin{tabular}{c c} \hline
**NZ EEPAS-1F** & **California EEPAS-1F** \\ \hline
0.34 & 0.26 \\
0.41 & 0.31 \\
0.49 & 0.37 \\
0.58 & 0.44 \\
0.69 & 0.53 \\
0.82 & 0.97 \\
0.97 & 0.74 \\
1.16 \({}^{\dagger}\) & 0.88 \({}^{\ddagger}\) \\
1.38 & 1.05 \\
1.64 & 1.25 \\
1.95 & 1.49 \\
2.32 & 1.77 \\
2.75 & 2.10 \\
3.27 & 2.50 \\
3.89 & 2.98 \\ \hline \end{tabular} \({}^{\dagger}\)Fitted.  
\end{table}
Table 2: Controlled values of \(\sigma_{A}\) in EEPAS_1F model for New Zealand (NZ) and California.  
Figure 5: Maps of California’s seismicity, including the region of surveillance (inner dashed polygon), search region (outer dotted polygon), and locations of earthquakes with magnitudes (**a**) M > 2.95 and hypocentral depths \(\leq\)30 km from 1932 to 2004 and (**b**) M > 4.95 and hypocentral depths \(\leq\)30 km from 1986 to 2005 in the region of surveillance (155 target earthquakes).