**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 [25–27].

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 [28–30]. The differences were due to looser constraints imposed in the fitting of *μ*.

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 *tf* . Time *t*0 is the beginning of 1951 for New Zealand and 1932 for California, while *tf* 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 *ts* and *tf* , where *ts* is the beginning of 1987 for New Zealand and 1986 for California.

\* Fixed. † Fitted. ! Standard threshold used for CSEP models.



**Figure 4.** Maps of New Zealand seismicity, including the region of surveillance (inner dashed polygon), the search region (outer dotted polygon) and locations of earthquakes with magnitudes (**a**) M > 2.95 with a hypocentral depth ≤45 km from 1951 to 2006 and (**b**) M > 4.95 with a hypocentral depth ≤40 km from 1987 to 2006 in the region of surveillance (158 target earthquakes).

To investigate the space–time trade-off, we varied the EEPAS model parameters in a controlled way. Starting with the parameter sets listed in Table 1, we separately changed the EEPAS\_1F parameters *σA* and *aT* while the other parameters, except the mixing parameter *μ*, remained fixed at their previously fitted values. We changed *σA* in seven steps in either direction away from its optimal value (Table 2) and obtained the corresponding values of the temporal scaling factor *<sup>σ</sup>*2*A*. Subsequently, we changed the *aT* values in a similar manner (Table 3) and obtained the corresponding values of the temporal scaling factor 10*aT*. Over 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 *aT* or *σA*, two free parameters, *μ* and either *σA* or *aT*, were refitted to maximize the likelihood of target earthquakes in the region of surveillance over time period (*ts*, *tf*).

**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 ≤30 km from 1932 to 2004 and (**b**) M > 4.95 and hypocentral depths ≤30 km from 1986 to 2005 in the region of surveillance (155 target earthquakes).


**Table 2.** Controlled values of *σA* in EEPAS\_1F model for New Zealand (NZ) and California.

† Fitted.

9


**Table 3.** Controlled values of *aT* in EEPAS\_1F models for NZ and California.

† Fitted.
