###### Abstract  
The 'Every Earthquake a Precursor According to Scale' (EEPAS) medium-term earthquake forecasting model is based on the precursory scale increase (\(\Psi\)) phenomenon and associated scaling relations, in which the precursor magnitude \(M_{P}\) is predictive of the mainshock magnitude \(M_{m}\), precursor time \(T_{P}\) and precursory area \(A_{P}\). In early studies of \(\Psi\), a relatively low correlation between \(T_{P}\) and \(A_{P}\) suggested the possibility of a trade-off between time and area as a second-order effect. Here, we investigate the trade-off by means of the EEPAS model. Existing versions of EEPAS in New Zealand and California forecast target earthquakes of magnitudes \(M>4.95\) from input catalogues with \(M>2.95\). We systematically vary one parameter each from the EEPAS distributions for time and location, thereby varying the temporal and spatial scales of these distributions by two orders of magnitude. As one of these parameters is varied, the other is refitted to a 20-year period of each catalogue. The resulting curves of the temporal scaling factor against the spatial scaling factor are consistent with an even trade-off between time and area, given the limited temporal and spatial extent of the input catalogue. Hybrid models are formed by mixing several EEPAS models, with parameter sets chosen from points on the trade-off line. These are tested against the original fitted EEPAS models on a subsequent period of the New Zealand catalogue. The resulting information gains suggest that the space-time trade-off can be exploited to improve forecasting. 1  
[PERSON], [PERSON], A. Space-Time Trade-Off of Precursory Seismicity in New Zealand and California Revealed by a Medium-Term Earthquake Forecasting Model. 11110215110215110215110215110215110215110215102  
precursory magnitude \(M_{P}\), precursor time \(T_{P}\) and precursory area \(A_{P}\) (Figure 1c), within which the precursors, major earthquake and aftershocks all occurred. From the combined identifications of \(\Psi\) from four well-catalogued regions, it was found that \(M_{m}\), \(M_{P}\), \(T_{P}\) and \(A_{P}\) were all positively correlated [12]. In particular, three scaling relations (Figure 2) allowed \(M_{m}\), \(T_{P}\) and \(A_{P}\) to be predicted from \(M_{P}\), defined as the average magnitude of the three largest precursory earthquakes. These three predictive relations became the basis for the 'Every Earthquake a Precursor According to Scale' (EEPAS) medium-term earthquake forecasting model [13]. Although \(M_{m}\), \(M_{P}\), \(T_{P}\) and \(A_{P}\) were all positively correlated, \(A_{P}\) and \(T_{P}\) were less correlated than the other pairs of variables, as shown by the low value of the coefficient of determination \(R^{2}\) in Figure 3a compared with those in Figure 2a-c. In Figure 2, we highlighted the earthquakes for which \(A_{P}\) was high and \(T_{P}\) was low or vice versa relative to the fitted relations, a condition that is not uncommon. The same earthquakes are highlighted in Figure 3. Remarkably, the product of \(T_{P}\) and \(A_{P}\) was highly correlated with \(M_{m}\), as seen in Figure 3b, with \(R^{2}\) being higher than any of those values in Figure 2. These features pointed to a trade-off between \(A_{P}\) and \(T_{P}\). However, the origin of this trade-off was not clear. Could it have a physical origin related to, say, the tectonic setting or seismicity rate [14; 15; 16], or could it be a statistical side-effect? For example, in this case, if \(\log T_{P}\) and \(\log A_{P}\) were independently correlated with \(M_{m}\), then their sum would be correlated even better, such as in Figure 3b. Figure 1: Identification of \(\Psi\) phenomenon for the August 2014 \(M\)6.0 South Napa, California earthquake. (**a**) The precursory area \(A_{P}\) (dashed rectangle) with the epicenters of the precursory seismicity, mainshocks and aftershocks. (**b**) Magnitude versus time of prior and precursory earthquakes. Dashed lines show the precursory increase in magnitude level.