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.

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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.

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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)

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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). https://doi.org/10.1007/s10955-012-0629-0

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Keywords

  • Markov process
  • Earthquakes
  • Renewal model