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.
- Markov process
- Renewal model