Advertisement

Modeling of Extremal Earthquakes

  • Margarida BritoEmail author
  • Laura Cavalcante
  • Ana Cristina Moreira Freitas
Part of the CIM Series in Mathematical Sciences book series (CIMSMS, volume 2)

Abstract

Natural hazards, such as big earthquakes, affect the lives of thousands of people at all levels. Extreme-value analysis is an area of statistical analysis particularly concerned with the systematic study of extremes, providing an useful insight to fields where extreme values are probable to occur. The characterization of the extreme seismic activity is a fundamental basis for risk investigation and safety evaluation. Here we study large earthquakes in the scope of the Extreme Value Theory. We focus on the tails of the seismic moment distributions and we propose to estimate relevant parameters, like the tail index and high order quantiles using the geometric-type estimators. In this work we combine two approaches, namely an exploratory oriented analysis and an inferential study. The validity of the assumptions required are verified, and both geometric-type and Hill estimators are applied for the tail index and quantile estimation. A comparison between the estimators is performed, and their application to the considered problem is illustrated and discussed in the corresponding context.

Keywords

Seismic Moment Generalize Extreme Value Tail Index High Quantile Philippines Island 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

ACMF is partially supported by FCT grant SFRH/BPD/66174/2009 and LC is supported by FCT grant SFRH/BD/60642/2009. All three authors are supported by FCT project PTDC/MAT/120346/2010. Research funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst—C/MAT/UI0144/2013. The authors also thank the referees for their comments.

References

  1. 1.
    Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, Chichester (2004)CrossRefGoogle Scholar
  2. 2.
    Brito, M., Freitas, A.C.M.: Limiting behaviour of a geometric estimator for tail indices. Insur. Math. Econ. 33, 221–226 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brito, M., Cavalcante, L., Freitas, A.C.M.: Bias corrected geometric-type estimators. Preprint CMUP 2014-6 (2014)Google Scholar
  4. 4.
    Caeiro, F., Gomes, M.I., Pestana, D.: Direct reduction of bias of the classical Hill estimator. Revstat 3, 113–136 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Corral, A.: Dependence of earthquake recurrence times and independence of magnitudes on seismicity history. Tectonophysics 424, 177–193 (2006)CrossRefGoogle Scholar
  6. 6.
    Csörgő, S., Deheuvels, P., Mason, D.M.: Kernel estimates of the tail index of a distribution. Ann. Stat. 13, 1050–1077 (1985)CrossRefGoogle Scholar
  7. 7.
    Day, R.W.: Geotechnical Earthquake Engineering McGraw-Hill, New york (2002)Google Scholar
  8. 8.
    de Haan, L., Rootzén, H.: On the estimation of high quantiles. J. Stat. Plann. Inference 35, 1–13 (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Deheuvels, P., Haeusler, E., Mason, D.M.: Almost sure convergence of the Hill estimator. Math. Proc. Camb. Philos. Soc. 104, 371–381 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17, 1833–1855 (1989)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dziewonski, A.M., Chou, T.-A., Woodhouse, J.H.: Determination of earthquake source parameters from waveform data for studies of global and regional seismicity. J. Geophys. Res. 86, 2825–2852 (1981)CrossRefGoogle Scholar
  12. 12.
    Ekström, G., Nettles, M., Dziewonski, A.M.: The global CMT project 2004–2010: centroid-moment tensors for 13,017 earthquakes. Phys. Earth Planet. Inter. 200–201, 1–9 (2012)CrossRefGoogle Scholar
  13. 13.
    Fraga Alves, M.I., Gomes, M.I., de Haan, L.: A new class of semi-parametric estimators of the second order parameter. Port. Math. 60, 193–213 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Global CMT Catalogue: Available from http://www.globalcmt.org/ (2013). Last accessed Aug 2013
  15. 15.
    Gomes, M.I., Martins, M.J.: “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5, 5–31 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gomes, M.I, Martins, M.J., Neves, M.: Improving second order reduced bias extreme value index estimation. Revstat 5, 177–207 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Haeusler, E., Teugels, J.L.: On asymptotic normality of Hill’s estimator for the exponent of regular variation. Ann. Stat. 13, 743–756 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hill, B.M.: A simple approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)CrossRefzbMATHGoogle Scholar
  19. 19.
    Howell, B.F. Jr.: An Introduction to Seismological Research. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  20. 20.
    Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–13 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pisarenko, V.F., Sornette, D.: Characterization of the frequency of extreme events by Generalised Pareto Distribution. Pure Appl. Geophys. 160, 2343–2364 (2003)CrossRefGoogle Scholar
  22. 22.
    Pisarenko, V.F., Sornette, D., Rodkin, M.V.: Distribution of maximum earthquake magnitudes in future time intervals, application to the seismicity of Japan (1923–2007). Earth Planets Space 62, 567–578 (2010)CrossRefGoogle Scholar
  23. 23.
    Weissman, I.: Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. 73, 812–815 (1978)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Margarida Brito
    • 1
    Email author
  • Laura Cavalcante
    • 1
  • Ana Cristina Moreira Freitas
    • 2
  1. 1.Faculdade de CiênciasUniversidade do PortoPortoPortugal
  2. 2.Faculdade de EconomiaUniversidade do PortoPortoPortugal

Personalised recommendations