Seismic Cycles and the Evolution of Stress Correlation in Cellular Automaton Models of Finite Fault Networks

A cellular automaton is used to study the relation between the structure of a regional fault network and the temporal and spatial patterns of regional seismicity. Automata in which the cell sizes form discrete fractal hierarchies are compared with those having a uniform cell size. Conservative models in which all the stress is transferred at each step of a cascade are compared with nonconservative ("lossy") models in which a specified fraction of the stress energy is lost from each step. Particular attention is given to the behavior of the system as it is driven toward the critical state by uniform external loading. All automata exhibit a scaling region at times close to the critical state in which the events become larger and energy release increases as a power-law of the time to the critical state. For the hierarchical fractal automata, this power-law behavior is often modulated by fluctuations that are periodic in the logarithm of the time to criticality. These fluctuations are enhanced in the nonconservative models, but are not robust. The degree to which they develop appears to depend on the particular distribution of stresses in the larger cells which varies from cycle to cycle. Once the critical state is reached, seismicity in the uniform conservative automaton remains random in time, space, and magnitude. Large events do not significantly perturb the stress distribution in the system. However, large events in the nonconservative uniform automaton and in the fractal systems produce large stress perturbations that move the system out of the critical state. The result is a seismic cycle in which a large event is followed by a shadow period of quiescence and then a new approach back toward the critical state. This seismic cycle does not depend on the fractal structure, but is a direct consequence of large-scale heterogeneity of these systems in which the size of the largest cell (or the size of the largest nonconservative event) is a significant fraction of the size of the network. In essence, seismic cycles in these models are boundary effects. The largest events tend to cluster in time and the rate of small events remains relatively constant throughout a cycle in agreement with observed seismicity.