**6. Conclusions**

A space–time trade-off of precursory seismicity has been investigated by repeated refitting of the EEPAS earthquake forecasting model to the catalogues of New Zealand and California. In a sequence of controlled fits, the temporal scaling parameter was constrained to vary in steps ranging over two orders of magnitude with the spatial scaling parameter before being refitted, and vice versa. The two resulting curves of the temporal scaling factor against the spatial scaling factor differed depending on which parameter was controlled and which was fitted. However, both curves were consistent with an even trade-off between space and time once the temporal and spatial limits of the contributing earthquake data were considered. As the controlled parameter deviated further from its optimal value, the likelihood of the refitted model decreased. In addition, the refitted model had an increasingly large background component and a diminishing time-varying component.

The trade-off implies that the earliest precursors to a major earthquake tend to occur very close to its source and that the later precursors occupy a wide area around the source. A simple example in which hybrid forecasts were created by mixing several EEPAS models with parameters chosen from the trade-off line suggests that it should be possible to exploit the trade-off for improved forecasting. However, more research is needed to develop a formal method for routinely incorporating the space–time trade-off into medium-term earthquake forecasts.

**Author Contributions:** Conceptualization, S.J.R., D.A.R. and A.C.; methodology, D.A.R.; software, S.J.R. and D.A.R.; formal analysis, S.J.R. and D.A.R.; writing—original draft preparation, S.J.R. and D.A.R.; writing—review and editing, S.J.R., D.A.R. and A.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Strategic Science Investment Fund (SSIF) of the Ministry of Business, Innovation and Employment of New Zealand.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The New Zealand Earthquake Catalogue was obtained from GeoNet. Available online: http://www.geonet.org.nz (accessed 30 September 2021). Earthquake data for the California region came from the Advanced National Seismic System (ANSS) Worldwide Earthquake Catalog, which is contributed to by members of the U.S. Council of the National Seismic System and maintained by the Northern California Earthquake Data Center. Available online: www.ncedc.org/ anss/catalog-search.html (accessed 30 September 2021).

**Acknowledgments:** We acknowledge the New Zealand GeoNet project and its sponsors EQC, GNS Science, LINZ, NEMA and MBIE for providing the data used in this study. We would like to thank Stephen Bannister and Rob Buxton for their constructive internal reviews of an earlier version of this manuscript. We also thank two anonymous reviewers. However, the authors are solely responsible for the final manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
