Predictability of the large relaxations in a cellular automaton model

Abstract

A simple one-dimensional cellular automaton model with threshold dynamics is introduced. The cumulative distribution of the size of the relaxations is analytically computed and behaves as a power law with an exponent equal to -1. This coincides with the phenomenological Gutenberg-Richter behavior observed in Seismology for the cumulative statistics of earthquakes at the regional or global scale. The key point of the model is the zero-load state of the system after the occurrence of any relaxation, no matter what its size. This leads to an equipartition of probability between all possible load configurations in the system during the successive loading cycles. Each cycle ends with the occurrence of the greatest -or characteristic- relaxation in the system. The duration of the cycles in the model is statistically distributed with a coefficient of variation ranging from 0.5 to 1. The predictability of the characteristic relaxations is evaluated by means of error diagrams. This model illustrates the value of taking into account the refractory periods to obtain a considerable gain in the quality of the predictions.