Advertisement

One-Way Markov Process Approach to Repeat Times of Large Earthquakes in Faults

Abstract

One of the uses of Markov Chains is the simulation of the seismic cycle in a fault, i.e. as a renewal model for the repetition of its characteristic earthquakes. This representation is consistent with Reid’s elastic rebound theory. We propose a general one-way Markovian model in which the waiting time distribution, its first moments, coefficient of variation, and functions of error and alarm (related to the predictability of the model) can be obtained analytically. The fact that in any one-way Markov cycle the coefficient of variation of the corresponding distribution of cycle lengths is always lower than one concurs with observations of large earthquakes in seismic faults. The waiting time distribution of one of the limits of this model is the negative binomial distribution; as an application, we use it to fit the Parkfield earthquake series in the San Andreas fault, California.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Abaimov, S., Turcotte, D., Rundle, J.: 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 (2007)

  2. 2.

    Abaimov, S., Turcotte, D., Shcherbakov, R., Rundle, J., Yakovlev, G., Goltz, C., Newman, W.: Earthquakes: recurrence and interoccurrence times. Pure Appl. Geophys. 165, 777–795 (2008)

  3. 3.

    Ammon, C., Kanamori, H., Lay, T.: A great earthquake doublet and seismic stress transfer cycle in the central Kuril islands. Nature 541, 561–565 (2008)

  4. 4.

    Aoi, S., Enescu, B., Suzuki, W., Asano, Y., Obara, K., Kunugi, T., Shiomi, K.: Stress transfer in the Tokai subduction zone from the 2009 Sugura Bay earthquake in Japan. Nat. Geosci. 3, 496–500 (2010)

  5. 5.

    Bakun, W.: Implications for prediction and hazard assessment from the 2004 Parkfield earthquake. Nature 437, 969–974 (2005)

  6. 6.

    Bakun, W., Lindh, A.: The Parkfield, California, earthquake prediction experiment. Science 229, 619–624 (1985)

  7. 7.

    Canavos, G.C.: Applied Probability and Statistical Methods. Little Brown, Boston (1984)

  8. 8.

    Ellsworth, W.L., Matthews, M.V., Nadeau, R.M., Nishenko, S.P., Reasenberg, P.A., Simpson, R.W.: A physically-based earthquake recurrence model for estimation of long-term earthquake probabilities. U.S. Geol. Surv. Open File Rep. 99, 552 (1999)

  9. 9.

    Freed, A.: Earthquake triggering by static, dynamic and postseismic stress transfer. Annu. Rev. Earth Planet. Sci. 33, 335–367 (2005)

  10. 10.

    Gómez, J.B., Pacheco, A.: The minimalist model of characteristic earthquakes as a useful tool for description of the recurrence of large earthquakes. Bull. Seismol. Soc. Am. 94(5), 1960–1967 (2004)

  11. 11.

    González, A., Gómez, J.B., Pacheco, A.F.: The occupation of box as a toy model for the seismic cycle of a fault. Am. J. Phys. 73(10), 946–952 (2005)

  12. 12.

    González, A., Gómez, J.B., Pacheco, A.F.: Updating seismic hazard at Parkfield. J. Seismol. 10, 131–135 (2006)

  13. 13.

    Kagan, Y.: On earthquake predictability measurement: information score and error diagram. Pure Appl. Geophys. 164, 1947–1962 (2007)

  14. 14.

    Kanamori, H., Anderson, D.L.: Theoretical basis of some empirical relations in seismology. Bull. Seismol. Soc. Am. 65, 1073 (1975)

  15. 15.

    Kanamori, H., Brodsky, E.: The physics of earthquakes. Rep. Prog. Phys. 67, 1429–1496 (2004)

  16. 16.

    Keilis-Borok, V., Soloviev, A.: Nonlinear Dynamics of the Lithosphere and Earthquake Prediction. Springer, Berlin (2003)

  17. 17.

    Knopoff, L.: The magnitude distribution of declustered earthquakes in Southern California. Proc. Natl. Acad. Sci. USA 97, 11,880–11,884 (2000)

  18. 18.

    Matthews, M.V., Ellsworth, W.L., Reasenberg, P.A.: A Brownian model for recurrent earthquakes. Bull. Seismol. Soc. Am. 92(6), 2233–2250 (2002)

  19. 19.

    McGuire, J.: Seismic cycles and earthquake predictability on East Pacific Rise transform faults. Bull. Seismol. Soc. Am. 98, 1067–1084 (2008)

  20. 20.

    Michael, A.J.: Viscoelasticity, postseismic slip, fault interactions, and the recurrence of large earthquakes. Bull. Seismol. Soc. Am. 95(5), 1594–1603 (2005)

  21. 21.

    Molchan, G.: Structure of optimal strategies in earthquake prediction. Tectonophysics 193, 267–276 (1991)

  22. 22.

    Molchan, G.M.: Earthquake prediction as a decision-making problem. Pure Appl. Geophys. 149, 233–247 (1997)

  23. 23.

    Newman, W.I., Turcotte, D.L.: A simple model for the earthquake cycle combining self-organized complexity with critical point behavior. Nonlinear Process. Geophys. 9(5/6), 453–461 (2002)

  24. 24.

    Reid, H.F.: The Mechanics of Earthquakes: The California Earthquake of April 18, 1906. Carnegie Institution, Washington (1910).

  25. 25.

    Rikitake, T.: Probability of an earthquake occurrence as estimated from crustal strain. Tectonophysics 23, 299–312 (1974)

  26. 26.

    Scholz, C.H.: The Mechanics of Earthquakes and Faulting, 2nd edn. Cambridge University Press, Cambridge (2002)

  27. 27.

    Schwartz, D.P., Coppersmith, K.J.: Fault behavior and characteristic earthquakes: examples from the Wasatch and San Andreas fault zones. J. Geophys. Res. 89, 5681 (1984)

  28. 28.

    Stein, R.: The role of stress transfer in earthquake occurrence. Nature 402, 605–609 (1999)

  29. 29.

    Sykes, L.R., Menke, W.: Repeat times of large earthquakes: implications for earthquake mechanics and long-term prediction. Bull. Seismol. Soc. Am. 96(5), 1569–1596 (2006)

  30. 30.

    Tejedor, A., Gómez, J., Pacheco, A.: Earthquake size-frequency statistics in a forest-fire model of individual faults. Phys. Rev. E 79(4), 046102 (2009)

  31. 31.

    Toda, S., Stein, R., Richards-Dinger, K., Bozkurt, S.: Forecasting the evolution of seismicity in southern California: animations built on earthquake stress transfer. J. Geophys. Res. 110, 415 (2005). doi:10.1029/2004JB003

  32. 32.

    Utsu, T.: Estimation of parameters for recurrent models of earthquakes. Bull. Earthq. Res. Inst. Univ. Tokyo 59, 53–66 (1984)

  33. 33.

    Vázquez-Prada, M., González, A., Gómez, J.B., Pacheco, A.: A minimalist model of characteristic earthquakes. Nonlinear Process. Geophys. 9(5/6), 513–519 (2002)

  34. 34.

    Vázquez-Prada, M., González, A., Gómez, J.B., Pacheco, A.: Forecasting characteristic earthquakes in a minimalist model. Nonlinear Process. Geophys. 10(6), 565–571 (2003)

  35. 35.

    Wesnousky, S.G.: The Gutenberg-Richter or characteristic earthquake distribution, which is it? Bull. Seismol. Soc. Am. 84, 1940 (1994)

  36. 36.

    Wesnousky, S.G.: Reply to Yan Kagan’s comment on ‘The Gutenberg-Richter or characteristic earthquake distribution, which is it?’. Bull. Seismol. Soc. Am. 86, 286–291 (1996)

  37. 37.

    Yakovlev, G., Turcotte, D., Rundle, J., Rundle, P.: Simulation-based distributions of earthquake recurrence times on the San Andreas fault system. Bull. Seismol. Soc. Am. 96, 1995–2007 (2006)

Download references

Acknowledgements

This work was supported by the Spanish DGICYT (Project FIS2010-19773). AFP would like to thank Jesús Asin, Jesús Bastero, Leandro Moral and Carmen Sanguesa who always help with a smile.

Author information

Correspondence to Alejandro Tejedor.

Appendix: The Negative Binomial Distribution as a Limit: Case N=3

Appendix: The Negative Binomial Distribution as a Limit: Case N=3

In this Appendix we show explicitly that, for N=3, the limit of Eq. (6) when the three parameters are equal is Eq. (8). For simplicity in the notation, let us call a 1=a, a 2=b, a 3=c. Eq. (6) for N=3 reads as follows:

(28)

To carry out the limit, we introduce new variables x and y.

(29)

The limit we seek will be implemented by tending x and y to 1. Substituting the new variables into Eq. (28), the result is:

(30)

Elaborating Eq. (30) slightly, we obtain:

$$ \frac{c^{n-3}}{(x-y)(x-1)(y-1)} \bigl[y\bigl(x^{n-1}-1\bigr)-x\bigl(x^{n-2}-1 \bigr)-y^{n-1}(x-1) \bigr] $$
(31)

Henceforth it is convenient to use the following type of polynomials:

(32)

These polynomials fulfill the so-called cyclotomic property, namely

(33)

So, dividing the second factor in Eq. (31) by (x−1) we obtain

(34)

Now we divide the second factor of Eq. (34) by (xy)

(35)

Returning to Eq. (28), using Eq. (35), and performing the limit x,y→1, we obtain:

(36)

This formula coincides with Eq. (8) when N=3

(37)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tejedor, A., Gomez, J.B. & Pacheco, A.F. One-Way Markov Process Approach to Repeat Times of Large Earthquakes in Faults. J Stat Phys 149, 951–963 (2012) doi:10.1007/s10955-012-0629-0

Download citation

Keywords

  • Markov process
  • Earthquakes
  • Renewal model