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Pure and Applied Geophysics

, Volume 172, Issue 1, pp 23–31 | Cite as

The Negative Binomial Distribution as a Renewal Model for the Recurrence of Large Earthquakes

  • Alejandro Tejedor
  • Javier B. GómezEmail author
  • Amalio F. Pacheco
Article

Abstract

The negative binomial distribution is presented as the waiting time distribution of a cyclic Markov model. This cycle simulates the seismic cycle in a fault. As an example, this model, which can describe recurrences with aperiodicities between 0 and 0.5, is used to fit the Parkfield, California earthquake series in the San Andreas Fault. The performance of the model in the forecasting is expressed in terms of error diagrams and compared with other recurrence models from literature.

Key words

Negative binomial distribution renewal process seismic cycle earthquake forecasting 

References

  1. Abaimov, S.G., Turcotte, D.L. and Rundle, J.B. (2007), Recurrence-time and frequency-slip statistics of slip events on the creeping section of the San Andreas fault in central California, Geophys. J. Int. 170, 1289–1299.Google Scholar
  2. Abaimov, S.G., Turcotte, D.L., Shcherbakov, R, Rundle, J.B. Yakovlev, G., Goltz, C., and Newman, W.I. (2008), Earthquakes: Recurrence and Interoccurrence Times, Pure Appl. Geophys. 165, 777–795.Google Scholar
  3. Bakun, W.H. (1988), History of significant earthquakes in the Parkfield area, Earthq. Volcano. 20, 45–51.Google Scholar
  4. Bakun, W.H., and Lindh, A.G. (1985), The Parkfield, California, earthquake prediction experiment, Science 229, 619–624.Google Scholar
  5. Ellsworth, W.L., Matthews, M.V., Nadeau, R.M., Nishenko, S.P., Reasenberg, P.A., Simpson, R.W. (1999), A physically-based earthquake recurrence model for estimation of long-term earthquake earthquake probabilities. United States Geological Survey Open-File Report 99, 552pp.Google Scholar
  6. Ferráes, S. (2003), The conditional probability of earthquake occurrence and the next large earthquake in Tokyo, Japan, J. Seismol. 7, 145–153.Google Scholar
  7. Ferráes, S. (2005), A probabilistic prediction of the next strong earthquake in the Acapulco-San Marcos segment, Mexico, Geofísica Internacional 44(4), 347–353.Google Scholar
  8. Goltz, C., Turcotte, D.L., Abaimov, S.G., Nadeau, R.M., Uchida, N., and Matsuzawa, T. (2009), Rescaled earthquake recurrence time statistics: application to microrepeaters, Geophys. J. Int. 176, 256–264.Google Scholar
  9. Gómez, J.B. and Pacheco, A.F. (2004), The Minimalist Model of characteristic earthquakes as a useful tool for description of the recurrence of large earthquakes, Bull. Seismol. Soc. Am. 94, 1960–1967.Google Scholar
  10. González, Á., Gómez, J.B. and Pacheco, A.F. (2005), The occupation of a box as a toy model for the seismic cycle of a fault, Am. J. Phys. 73, 946–952.Google Scholar
  11. Keilis-Bork D. V. and Soloviev A. (2003), Nonlinear Dynamics of the Lithosphere and Earthquake Prediction, Springer Verlag, Berlin.Google Scholar
  12. Matthews, M.V., Ellsworth, W.L. and Reasenberg, P.A. (2002), A Brownian model for recurrent earthquakes, Bull. Seismol. Soc. Am. 92, 2233–2250.Google Scholar
  13. Michael, A.J. and Jones, L.M. (1998), Seismicity alert probabilities at Parkfield, California, revisited, Bull. Seismol. Soc. Am. 88, 117–130.Google Scholar
  14. Michael, A.J. (2005), Viscoelasticity, postseismic slip, fault interactions, and the recurrence of large earthquakes, Bull. Seismol. Soc. Am. 95, 1594–1603.Google Scholar
  15. Molchan, G.M. (1997), Earthquake prediction as a decision-making problem, Pure Appl. Geophys. 149(1), 233–247.Google Scholar
  16. Newman W. I. and Turcotte D.L. (1992), A simple model for the earthquake cycle combining self-organized complexity with critical point behavior, Nonlinear Process. Geophys. 9, 453–61.Google Scholar
  17. Reid, H.F. (1910), The mechanics of the earthquake, In: The California Earthquake of April 18, 1906, Report of the State Earthquake Investigation Commission, Carnegie Institution, Washington, DC, Vol. 2, pp. 1–192.Google Scholar
  18. Rikitake, T. (1974), Probability of earthquake occurrence as estimated from crustal strain, Tectonophysics 23(3), 299–312.Google Scholar
  19. Scholz, C.H. (2002), The Mechanics of Earthquakes and Faulting, Cambridge University Press.Google Scholar
  20. Sornette, D. and Knopoff, L. (1997). The paradox of the expected time until the next earthquake. Bull. Seismol. Soc. Am. 87, 789–798.Google Scholar
  21. Sykes, L.R., and Menke, W. (2006), Repeat Times of Large Earthquakes: Implications for Earthquake Mechanics and Long-Term Prediction. Bull. Seismol. Soc. Am. 96(5), 1569–1596.Google Scholar
  22. Tejedor, A., Gómez, J.B., and Pacheco, A.F. (2009), Earthquake size-frequency statistics in a forest-fire model of individual faults, Physical Review E 79, 046102.Google Scholar
  23. Tejedor, A., Gómez, J.B., and Pacheco, A.F. (2012), One-way Markov process approach to repeat times of large earthquakes in faults, J. Stat. Phys. 149(5), 951–963.Google Scholar
  24. Utsu, T. (1984), Estimation of parameters for recurrence models of earthquakes, Bull. Earthq. Res. Inst. Univ. Tokyo 59, 53–66.Google Scholar
  25. Vázquez-Prada, M., González, Á., Gómez, J.B. and Pacheco, A.F. (2002), A minimalist model of characteristic earthquakes. Nonlinear. Process. Geophys. 9, 513–519.Google Scholar
  26. Working Group on California Earthquake Probabilities (2003), Earthquake Probabilities in the San Francisco Bay Region: 2002–2031, United States Geological Survey Open-File Report 03-214, 234 p.Google Scholar
  27. Yakovlev, G., Turcotte, D.L., Rundle, J.B., and Rundle, P.B. (2006), Simulation-Based Distributions of Earthquake Recurrence Times on the San Andreas Fault System, Bull. Seismol. Soc. Am. 96(6), 1995–2007.Google Scholar
  28. Zöller, G., Hainzl, S., and Holschneider, M. (2008), Recurrent Large Earthquakes in a Fault Region: What Can Be Inferred from Small and Intermediate Events?, Bull. Seismol. Soc. Am. 98, 2641–2651.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Alejandro Tejedor
    • 1
  • Javier B. Gómez
    • 2
    Email author
  • Amalio F. Pacheco
    • 3
  1. 1.Saint Anthony Falls Laboratory, Department of Civil EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Earth SciencesUniversity of ZaragozaZaragozaSpain
  3. 3.Department of Theoretical PhysicsUniversity of ZaragozaZaragozaSpain

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