**5. Discussion**

As seen in Figure 8, the controlled fits produced two curves which did not lie on the even trade-off line but instead had higher or lower slopes. This result can be explained by the limitations on the length of the catalogue and the size of the search region. The fitted parameters could only adjust to the precursors that were contained in the catalogue and not to those that were screened out by such limitations. We now consider in detail the trend of the fitted *σA* value away from the even trade-off line for the controlled values of *aT*. The trend of the other curve can be explained similarly.

As *aT* was stepped down to lower values (i.e., the time scale was shortened), fitting the trade-off required earthquakes at increasingly longer distances from the target earthquakes. However, at longer distances, more precursory events were screened out by the spatial limitation on the input catalogue. The precursors of the largest earthquakes in the target magnitude range would be most affected by the spatial limitation because they had larger precursory areas (Figure 2). The spatial limitation at small *aT* values forced the fitted values of *σA* to increasingly fall below the even trade-off line. On the other hand, as *aT* was stepped up to higher values, the precursory time scale became longer and exceeded the available lead time. This temporal limitation most affected the largest earthquakes in the target magnitude range, which had the longest precursor times (Figure 2). Thus, more and more precursory earthquakes on the specified time scale were screened out by the limited time span of the catalogue. The remaining precursors for fitting *σA* would be those at the lower end of the time distribution. Because of the space–time trade-off, these remaining precursors tended to be at longer distances than the screened-out events. This forced the fitted *σA* to increasingly exceed the even trade-off line.

The space–time trade-off in the EEPAS model shows that as the mean of *f*(*t*|*m*) in Equation (4) increased, the area of the fitted *h*(*<sup>x</sup>*, *y*|*m*) in Equation (5) decreased and vice versa. This phenomenon can be interpreted in terms of the predictive scaling relations on which the EEPAS model is based (Figure 2). Figure 2 shows that *TP* and *AP* both increased with the precursory earthquake magnitude *MP*. Similarly, in the EEPAS model, the mean of the time distribution *f*(*t*|*m*) and the area of the location distribution *h*(*<sup>x</sup>*, *y*|*m*) both increased with *m*. Now, the space–time trade-off observed in the EEPAS model can be interpreted in terms of the space–time distribution of precursors to an individual major earthquake; that is, the earliest precursors tend to occur very close to the source, and the later precursors to occupy a wide area around the source. This interpretation only applies to precursors occurring more than 50 days before the mainshock because of the time lag applied for EEPAS model fitting here.

The existence of this trade-off raises the question of how it can be exploited to improve the performance of the EEPAS model. The EEPAS model treats the time and location as independent variables, but the trade-off implies that they are correlated. We will illustrate how to improve forecasting by forming hybrid models. The hybrid models are mixtures of three EEPAS models with the values of *aT* and *σA* chosen from points on the even trade-off line with a slope of −1. We constructed two models, Hybrid\_1F and Hybrid\_1R, starting from two different EEPAS models: EEPAS\_1F and EEPAS\_1R, respectively. EEPAS\_1R was similar to EEPAS\_1F in nearly all aspects, apart from having fewer optimized parameters. Its fixed and optimized parameters are given in Table 4. An important difference between the two models was that EEPAS\_1F (Table 1) had a larger value of *σT* than EEPAS\_1R

(Table 4). The parameter *σT* was optimized in the fitting of EEPAS\_1F but not in EEPAS\_1R. In prospective testing over 10 years in the New Zealand CSEP testing center, EEPAS\_1F significantly outperformed EEPAS\_1R [29].


**Table 4.** EEPAS\_1R model parameters for New Zealand.

\* Fixed. † Fitted.

To construct the hybrid models, we replaced the time-varying component of each model's rate density with the average rate density of the three models, with the values of *aT* and *σA* chosen from the trade-off line. The three models were the original one and two others formed by an arbitrary increase and decrease in *aT* of Δ = 0.5. For an increase in Δ in *aT*, the corresponding value of *σA* on the trade-off line was found by multiplying the original *σA* by 10−0.5Δ. The other parameters, including *μ* and *σT*, remained unchanged at their values in Tables 1 and 4. Using information gain statistics, we compared the performance of the EEPAS\_1F, EEPAS\_0F, Hybrid\_1F and Hybrid\_1R models. For this, we used a test period from 2007 to 2017, during which there were 259 target earthquakes with magnitudes M > 4.95. Hybrid\_1R outperformed all the other models, and EEPAS\_1R was the weakest model (Figure 9). Figure 9a shows the information gain of EEPAS\_1F, and Figure 9b shows that of Hybrid\_1R over the other models. Both hybrid models and EEPAS\_1F outperformed EEPAS\_1R with 95% confidence according to the *T*-test [24].

**Figure 9.** Information gain per earthquake and 95% confidence interval of the (**a**) EEPAS\_1F model and (**b**) Hybrid\_1R model compared with other models during the test period of 2007–2017 in the New Zealand testing region (259 target earthquakes with M > 4.95).

This simple example of hybrid formation, even without fitting additional parameters, suggests that it might be possible to use the space–time trade-off to improve forecasting. However, much more work needs to be done to construct a formal method for optimal inclusion of the trade-off in the fitting of the EEPAS model. The temporal and spatial limitations of the catalogue are clearly among the issues to be considered. The spatial limitations

can be resolved if a global catalogue is used, but then a higher threshold magnitude of completeness would apply. That in turn imposes further limitations. Additionally, there is evidence that the precursor time distribution is dependent on the strain rate in the vicinity of a target earthquake [17]. This dependence would have to be included in a global model. Temporal limitations can also be partly resolved by introducing a fixed lead time for all target earthquakes and then compensating for the lead time using the method described in [20].
