Explaining the seismic moment of large earthquakes by heavy and extremely heavy tailed models
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Explaining the seismic moment of large earthquakes by heavy and extremely heavy tailed models Miguel Martins Felgueiras
Received: 19 September 2011 / Accepted: 27 July 2012 / Published online: 15 August 2012 © Springer-Verlag 2012
Abstract The search of physical laws that explain the energy released by the great magnitude earthquakes is a relevant question, since as a rule they cause heavy losses. Several statistical distributions have been considered in this process, namely heavy tailed laws, like the Pareto distribution with shape parameter α ≈ 0.6667. Yet, for the usually considered Californian region (where earthquakes with moment magnitude, MW , greater than 7.9 were never registered) the Pareto distribution with index near the above mentioned seems to have a “too heavy” tail for explaining the bigger earthquakes seismic moments. Usually an exponential tapper is applied to the distribution right tail (above the so called corner seismic moment), or another distribution is considered to explain these high seismic moment data (like another Pareto with different shape parameter). The situation is different for other regions where seisms of larger magnitudes do occur, leading to data sets for which heavy or even extremely heavy tailed models are appropriated. The purpose of this paper is to reduce the seismic moment, M0 , of the very large earthquakes to particular heavy and extremely heavy tailed distributions. Using world seismic moment information, we apply Pareto, Log-Pareto and extended slash Pareto distributions to the data, truncated for M0 ≥ 1021 N m and for M0 ≥ 1021.25 N m. For these great seisms we conclude that extended slash Pareto is a promising alternative to the more traditional Pareto and Log-Pareto distributions as a candidate to the real model underlying the data.
This research was partially sponsored by national funds through the Fundação Nacional para a Ciência e Tecnologia, Portugal FCT under the project (PEst-OE/MAT/UI0006/2011). M. M. Felgueiras (B) CEAUL Lisbon and ESTG, CIIC, Polytechnic Institute of Leiria, Leiria, Portugal e-mail: [email protected]
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Int J Geomath (2012) 3:209–222
Keywords Pareto · Log-Pareto · Extended slash Pareto · Large earthquake · Seismic moments fitting Mathematics Subject Classification
60E05 · 62P35
1 Introduction The great earthquakes are a rare phenomenon, but with terrible consequences. For this reason, the modelling of the energy released by them is an interesting problem, often treated in specialized literature. Before the introduction of the Moment Magnitude Scale (MW ) , by Hanks and Kanamori (1979), many others scales were considered, like the Local Magnitude Scale, the Surface Wave Magnitude scale or the Body Wave Magnitude Scale (Kanamori 1983). These previously used scales shared a common problem: they saturate for the bigger earthquakes. For example, seisms with magnitude higher than 8.8 were never registered using those scales, and seisms with magnitude above 8.0 were extremely rare events. On the other hand, the MW and the seismic mom
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