Studying the hierarchical structure of the aftershock sequence of the three largest earthquakes of the last decade, we show that the number of offspring events counted in a fixed magnitude band with respect to the magnitude of the parent events follows an exponential distribution. Such an exponential productivity law is coherent with the exponential decays inferred from largest earthquakes worldwide. Epidemic Type Aftershock Sequences (ETAS) are the most popular stochastic models of seismicity and they are all based on a Poisson distribution of the earthquake productivity with a pronounced non-zero mode. We construct here an alternative model incorporating an exponential productivity law. For the three aftershock sequences. we estimate parameters of both models using aftershocks occuring during the first 2 days. We simulate a set of synthetic earthquake catalogues for both models and compare the average cumulated number of events with respect to time. In all cases, the ETAS model overestimates the number of events in the interval from 2 to 365 days. For the same time period, the exponential ETAS model gives a satisfactory cumulative number of events. We conclude that exponential distribution of the earthquake productivity seems to be an important property of the seismic relaxation process.
Strong earthquakes, aftershocks, productivity, ETAS model, EP model, forecast
1. Baiesi, M., M. Paczuski (2004) , Scale-free networks of earthquakes and aftershocks, Phys. Rev. E, 69, p. 066106, https://doi.org/10.1103/PhysRevE.69.066106.
2. Baranov, S. V., V. A. Pavlenko, P. N. Shebalin (2019) , Forecasting Aftershock Activity: 4. Estimating the Maximum Magnitude of Future Aftershocks, Izv., Phys. Solid Earth, 55, p. 548-562, https://doi.org/10.1134/S1069351319040013.
3. Baranov, S. V., P. N. Shebalin (2017) , Forecasting aftershock activity: 2. Estimating the area prone to strong aftershocks, Izv., Phys. Solid Earth, 53, p. 366-384, https://doi.org/10.1134/S1069351317020021.
4. Console, R., M. Murru, A. M. Lombardi (2003) , Refining earthquake clustering models, J. Geophys. Res.: Solid Earth, 108, https://doi.org/10.1029/2002JB002130.
5. Gvishiani, A., M. Dobrovolsky, S. Agayan, et al. (2013a) , Fuzzy-based clustering of epicenters and strong earthquake-prone areas, Environmental Engineering and Management Journal, 12, no. 1, p. 1-10, https://doi.org/10.30638/eemj.2013.001.
6. Gvishiani, A., B. Dzeboev, S. Agayan (2013b) , A new approach to recognition of the earthquake-prone areas in the Caucasus, Izvestiya, Physics of the Solid Earth, 49, no. 6, p. 747-766, https://doi.org/10.1134/S1069351313060049.
7. Gvishiani, A., B. Dzeboev, S. Agayan (2016) , FCAZm intelligent recognition system for locating areas prone to strong earthquakes in the Andean and Caucasian mountain belts, Izvestiya. Physics of the Solid Earth, 52, no. 4, p. 461-491, https://doi.org/10.1134/S1069351316040017.
8. Hainzl, S., D. Marsan (2008) , Dependence of the omori-utsu law parameters on main shock magnitude: Observations and modeling, J. Geophys. Res.: Solid Earth, 113, https://doi.org/10.1029/2007JB005492.
9. Hainzl, S., O. Zakharova, D. Marsan (2013) , Impact of aseismic transients on the estimation of aftershock productivity parameters, Bull. Seismol. Soc. Am., 103, p. 1723-1732, https://doi.org/10.1785/0120120247.
10. Helmstetter, A. (2003) , Is earthquake triggering driven by small earthquakes?, Phys. Rev. Lett., 91, p. 058501, https://doi.org/10.1103/PhysRevLett.91.058501.
11. Helmstetter, A., D. Sornette (2002) , Subcritical and supercritical regimes in epidemic models of earthquake aftershocks, J. Geophys. Res.: Solid Earth, 107, p. 2237, https://doi.org/10.1029/2001JB001580.
12. Holschneider, M., G. Zöller, S. Hainzl (2011) , Estimation of the maximum possible magnitude in the framework of the doubly-truncated gutenberg-richter model, Bull. Seimol. Soc. Am., 112, p. 1649-1659, https://doi.org/10.1785/0120100289.
13. Kagan, Y. Y., L. Knopoff (1981) , Stochastic synthesis of earthquake catalogs, J. Geophys. Res.: Solid Earth, 86, p. 2853-2862, https://doi.org/10.1029/JB086iB04p02853.
14. Marsan, D., A. Helmstetter (2017) , Single-link cluster analysis as a method to evaluate spatial and temporal properties of earthquake catalogues, J. Geophys. Res.: Solid Earth, 122, p. 5544-5560, https://doi.org/10.1002/2016JB013807.
15. Marsan, D., O. Lenglin{é} (2008) , Extending earthquakes{\\textquoteright} reach through cascading, Science, 319, p. 1076-1079, https://doi.org/10.1126/science.1148783.
16. Ogata, Y. (1989) , Statistical model for standard seismicity and detection of anomalies by residual analysis, Tectonophysics, 169, p. 159-174, https://doi.org/10.1016/0040-1951(89)90191-1.
17. Shebalin, P. N., S. V. Baranov (2019) , Forecasting Aftershock Activity: 5. Estimating the Duration of a Hazardous Period, Izv., Phys. Solid Earth, 55, p. 719-732, https://doi.org/10.1134/S1069351319050112.
18. Shebalin, P. N., S. V. Baranov, B. A. Dzeboev (2018) , The Law of the Repeatability of the Number of Aftershocks, Dokl. Earth Sc., 481, p. 963-966, https://doi.org/10.1134/S1028334X18070280.
19. Utsu, T. (1965) , A method for determining the value of b in a formula {logn=a-bM} showing the magnitude-frequency relation for earthquakes (with English summary), Geophys Bull. Hokkaido Univ., 13, p. 99-103.
20. Utsu, T. (1970) , Aftershocks and earthquake statistics (ii): Further investigation of aftershocks and other earthquake sequences based on a new classification of earthquake sequences, J. Faculty Sci., Hokkaido University, Ser. VII (Geophys.), 3, p. 197-266.
21. Wang, Q., F. P. Schoenberg, D. D. Jackson (2010) , Standard errors of parameter estimates in the etas model, Bull. Seismol. Soc. Am., 100, p. 1989-2001, https://doi.org/10.1785/0120100001.
22. Werner, M. J., D. Sornette (2008) , Magnitude uncertainties impact seismic rate estimates, forecasts, and predictability experiments, J. Geophys. Res.: Solid Earth, 113, https://doi.org/10.1029/2007JB005427.
23. Zaliapin, I., Y. Ben-Zion (2013) , Earthquake clusters in southern california i: Identification and stability, J. Geophys. Res.: Solid Earth, 118, p. 2847-2864, https://doi.org/10.1002/jgrb.50179.
24. Zaliapin, I., A. Gabrielov, V. Keilis-Borok, H. Wong (2008) , Clustering analysis of seismicity and aftershock identification, Phys. Rev. Lett., 101, p. 018501, https://doi.org/10.1103/PhysRevLett.101.018501.
25. Zhuang, J., C.-P. Chang, Y. Ogata, et al. (2005) , A study on the background and clustering seismicity in the taiwan region by using point process models, J. Geophys. Res.: Solid Earth, 110, https://doi.org/10.1029/2004JB003157.
26. Zhuang, J., Y. Ogata, D. Vere-Jones (2002) , Stochastic declustering of space-time earthquake occurrences, J. Am. Stat. Ass., 97, p. 369-380, https://doi.org/10.1198/016214502760046925.
27. Zhuang, J., Y. Ogata, D. Vere-Jones (2004) , Analyzing earthquake clustering features by using stochastic reconstruction, J. Geophys. Res.: Solid Earth, 109, https://doi.org/10.1029/2003JB002879.
