ON POTENTIAL REPRESENTATIONS OF THE DISTRIBUTION LAW OF RARE STRONGEST EARTHQUAKES

Assessment of long-term seismic hazard is critically dependent on the behavior of tail of the distribution function of rare strongest earthquakes. Analyses of empirical data cannot however yield the credible solution of this problem because the instrumental catalogs of earthquake are available only for a rather short time intervals, and the uncertainty in estimations of magnitude of paleoearthquakes is high. From the available data, it was possible only to propose a number of alternative models characterizing the distribution of rare strongest earthquakes. There are the following models: the model based on the
Guttenberg – Richter law suggested to be valid until a maximum possible seismic event (Мmах), models of 'bend down' of earthquake recurrence curve, and the characteristic earthquakes model. We discuss these models from the general physical concepts supported by the theory of extreme values (with reference to the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD) and the multiplicative cascade model of seismic regime. In terms of the multiplicative cascade model, seismic regime is treated as a large number of episodes of avalanche-type relaxation of metastable states which take place in a set of metastable sub-systems.

The model of magnitude-unlimited continuation of the Guttenberg – Richter law is invalid from the physical point of view because it corresponds to an infinite mean value of seismic energy and infinite capacity of the process generating seismicity. A model of an abrupt cut of this law by a maximum possible event, Мmах is not fully logical either.

A model with the 'bend-down' of earthquake recurrence curve can ensure both continuity of the distribution law and finiteness of seismic energy value. Results of studies with the use of the theory of extreme values provide a convincing support to the model of 'bend-down' of earthquakes’ recurrence curve. Moreover they testify also that the 'bend-down' is described by the finite distribution law, i.e. the bend-down occurs more efficiently than it is envisaged in the commonly used model developed by Y. Kagan (which treats the bend-dawn as an exponential decay law). However, despite the finiteness of the distribution law, density of magnitudes decline quite slowly in the area close to the maximum possible Мmах event as (Мmах – M)n, where n varies in the range between 4 and 6 in the majority of cases. As a result Мmах value can be estimated only with a large error. In rare cases, if the space-and-time area under study contains higher number of strongest earthquakes, the empirical distribution law becomes close to the exponential law; in this case n value is quite high, and Мmах values becomes unstable and tend to infinite growth.

In our study, the distribution law of strongest earthquakes was investigated by the methods based on the extreme values theory (world data and several regional catalogues were examined), and the results of calculation do not reveal cases of  occurrence of characteristic events. However, such a seismic regime was revealed in a number of cases from paleoseismicity data and from some instrumental regional catalogues. Conditions providing for the occurrence of characteristic earthquakes are studied here using the multiplicative cascade model. According to [Rodkin, 2011], this model provides the simulation of all known regularities of seismic regime, such as a decrease in b-value in the vicinity of strong earthquakes, development of aftershock power cascade, and existence of seismic cycle and foreshock activity. This article considers an extension of the cascade model by adding of non-linear members in the kinetic cascade equation in order to describe effects of the 'bend-down' of the earthquake recurrence curve and the characteristic earthquakes occurrence. It is shown that in terms of the multiplicative cascade model, the occurrence of characteristic earthquakes is connected with development of the nonlinear positive feedback between the size of the current rupture zone and the rate of its further growth.

The modelling results are compared with data on seismicity of the South-Eastern Asia, which suggest that the regime providing the occurrence of characteristic earthquakes appears to be typical of the seismic regime of subduction zones (while it is not observed outside such zones). It is concluded that the non-linear positive feedback that controls the possibility of occurrence of characteristic earthquakes may be caused with the presence of deep fluids of increased concentration in the subduction zones.

1. Bak P., Tang C., Wiesenfeld K., 1988. Self-organised criticality. Physical Review A: Atomic, Molecular, and Optical Physics 38, 364–374. http://dx.doi.org/10.1103/PhysRevA.38.364.

2. Kagan Y.Y., 1993. Statistics of characteristic earthquakes. Bulletin of the Seismological Society of America 83 (1), 7–24.

3. Kagan Y.Y., 1994. Observational evidence for earthquakes as a nonlinear dynamic process. Physica D: Nonlinear Phenomena 77 (1), 160–192. http://dx.doi.org/10.1016/0167-2789(94)90132-5.

4. Kagan Y.Y., 1999. Universality of the seismic moment-frequency relation. In: Seismicity Patterns, their Statistical Significance and Physical Meaning. Pageoph Topical Volume, p. 537–573. http://dx.doi.org/10.1007/978-3-0348-8677-2_16.

5. Laherrere J., Sornette D., 1998. Streched exponential distributions in nature and economy: «fat-tails» with characteristic scales. The European Physical Journal B-Condensed Matter and Complex Systems 2 (4), 525–539. http://dx.doi.org/10. 1007/s100510050276.

6. Myachkin V.I., Kostrov B.V., Sobolev G.А., Shamina О.G., 1975. Fundamentals of physics of foci and precursors of earthquake. In: Physics of Earthquake Foci. Nauka, Moscow, p. 6–29 (in Russian) [Мячкин В.И., Костров Б.В., Соболев Г.А., Щамина О.Г. Основы физики очага и предвестники землетрясений // Физика очага землетрясения. М.: Наука, 1975. С. 6–29].

7. Ogata Y., 1988. Statistical models for earthquake occurrence and residual analysis for point processes. Journal of the Ameri-can Statistical Association 83 (401), 9–27. http://dx.doi.org/10.1080/01621459.1988.10478560.

8. Ogata Y., 1998. Space-time point-process models for earthquake occurrence. Annals of the Institute of Statistical Mathematics 50 (2), 379–402. http://dx.doi.org/10.1023/A:1003403601725.

9. Pacheco J.F., Scholz C., Sykes L., 1992. Changes in frequency-size relationship from small to large earthquakes. Nature 355 (6355), 71–73. http://dx.doi.org/10.1038/355071a0.

10. Pisarenko V.F., Rodkin M.V., 2007. Distributions with Large Tails: Application to Catastrophe Analysis. Computational Seismology, Issue 38. GEOS, Moscow, 240 p. (in Russian) [Писаренко В.Ф., Родкин М.В. Распределения с тяжелыми хвостами: приложения к анализу катастроф // Вычислительная сейсмология. Вып. 38. М.: ГЕОС, 2007, 240 с.].

11. Pisarenko V., Rodkin M., 2010. Heavy-Tailed Distributions in Disaster Analysis. Advances in Natural and Technological Hazards Research, Vol. 30. Springer, 190 p. http://dx.doi.org/10.1007/978-90-481-9171-0.

12. Pisarenko V., Rodkin M., 2014. Statistical Analysis of Natural Disasters and Related Losses. SpringerBriefs in Earth Sciences. Springer, 89 p.

13. Pisarenko V.F., Rodkin M.V., Rukavishnikova T.A., 2014. Estimation of the probability of strongest seismic disasters based on the extreme value theory. Izvestiya. Physics of the Solid Earth 50 (3), 311–324 http://dx.doi.org/10.1134/S106935 1314030070.

14. Pisarenko V.F., Sornette D., 2003. Characterization of the frequency of extreme earthquake events by the generalized pareto distribution. Pure and Applied Geophysics 160 (12), 2343–2364. http://dx.doi.org/10.1007/s00024-003-2397-x.

15. Rodkin M.V., 2011. Alternative to SOC concept-model of seismic regime as a set of episodes of random avalanche-like releases occurring on a set of metastable subsystems. Izvestiya. Physics of the Solid Earth 47 (11), 966–973 http://dx.doi. org/10.1134/S1069351311100107.

16. Rodkin M.V., Gvishiani A.D., Labuntsova L.M., 2008. Models of generation of power laws of distribution in the processes of seismicity and in formation of oil fields and ore deposits. Russian Journal of Earth Sciences 10 (5), ES5004. http://dx.doi.org/10.2205/2007ES000282.

17. Rodkin M.V., Shatakhtsyan A.R., 2013. Statistical analysis of catalogs of large and superlarge ore deposits: empirical regularities and their interpretation. Geoinformatika (4), 25–32 (in Russian) [Родкин М.В., Шатахцян А.Р. Статистический анализ данных по крупным и суперкрупным месторождениям: эмпирические закономерности и интерпретация // Геоинформатика. 2013. № 4. С. 25–32].

18. Rogozhin E.A., Novikov S.S., Rodina S.N., 2010. Paleo-earthquakes and long-term seismic mode of the Koryak upland region. Geofizicheskie Issledovaniya 11 (4), 35–43 (in Russian) [Рогожин Е.А., Новиков С.С., Родина С.Н. Палеоземлетря-сения и долговременный сейсмический режим Корякского нагорья // Геофизические исследования. 2010. Т. 11. № 4. С. 35–43].

19. Rogozhin E.A., Rodina S.N., 2012. Paleoseismic studies and the long-term seismic regime in the North of Sakhalin Island. Seismic Instruments 48 (4), 333–341. http://dx.doi.org/10.3103/S0747923912040032.

20. Rogozhin E.A., Shen J., Rodina S.N., 2013. Comparison of seismotectonic peculiarities of Altai Mountains and Mongolian Altai. Seismic Instruments 49 (4), 285–296. http://dx.doi.org/10.3103/S0747923913040063.

21. Sadovsky M.A. (Ed.), 1989. Discrete Properties of Geophysical Medium. Nauka, Moscow, 176 p. (in Russian) [Дискретные свойства геофизической среды / Отв. ред. М.А. Садовский. М.: Наука, 1989. 176 с.].

22. Sobolev G.A., 2010. The earthquake predictability concept based on seismicity dynamics under triggering impact. In: Extreme natural phenomena and catastrophes. V. 1. Assessment and Ways to Mitigation of Negative Consequences of Extreme Natural Phenomena. Institute of the Earth's Physics RAS, Moscow, p. 15–43 (in Russian) [Соболев Г.А. Концепция предсказуемости землетрясений на основе динамики сейсмичности при триггерном воздействии // Экстре-мальные природные явления и катастрофы. Т. 1. Оценка и пути снижения негативных последствий экстремальных природных явлений. М.: ИФЗ РАН, 2010. С. 15–43].

23. Sobolev G.А., Ponomarev V.А., 2003. Physics of Earthquakes and Precursors. Nauka, Moscow, 270 p. (in Russian) [Соболев Г.А., Пономарев А.В. Физика землетрясений и предвестники. М.: Наука, 2003. 270 с.].

24. Sornette D., 2006. Critical Phenomena in Natural Sciences. Chaos, fractals, selforganization and disorder: concepts and tools. Springer, Berlin, 450 p. Turcotte D.L., 1999. Seismicity and self-organized criticality. Physics of the Earth and Planetary Interiors 111 (3–4), 275–

25. http://dx.doi.org/10.1016/S0031-9201(98)00167-8.

26. Ulomov V.I., Bogdanov M.I., 2013. A new set of the seismic zoning maps of the Russian Federation (GSZ-2012). Inzhenernye Izyskaniya (8), 30–39 (in Russian) [Уломов В.И., Богданов М.И. Новый комплект карт общего сейсмического районирования территории Российской Федерации (ОСР-2012) // Инженерные изыскания. 2013. № 8. С. 30–39].

27. Wesnousky S.G., Scholz C.H., Shimazaki K., Matsuda T., 1983. Earthquake frequency distribution and the mechanics of faulting. Journal Geophysical Research 88 (B11), 9331–9340. http://dx.doi.org/10.1029/JB088iB11p09331.