log-periodic behavior of a hierarchical failure model with applications to precursory seismic activation

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Abstract
Seismic activation has been recognized to occur before many major earthquakes including the San Francisco Bay area, prior to the 1906 earthquake. There is a serious concern that the recent series of earthquakes in Southern California is seismic activation prior to a great Southern California earthquake. The seismic activation prior to the Loma Prieta earthquake has been quantified in terms of a power-law increase in the regional Benioff strain release prior to this event and there is an excellent fit to a log-periodic increase in the Benioff strain release. In order to better understand activation a hierarchical seismic failure model has been studied. An array of stress-carrying elements is considered (formally, a cellular automaton or lattice gas, but analogous to the strands of an ideal, frictionless cable). Each element has a time to failure that is dependent on the stress the element carries and has a statistical distribution of values. When an element fails, the stress on the element is transferred to a neighboring element; if two adjacent elements fail, stress is transferred to two neighboring elements; if four elements fail, stress is transferred to four adjacent elemetns, and so forth. When stress is transferred to an element its time to failure is reduced. The intermediate size failure events prior to total failure each have a sequence of precursory failures, and these precursory failures each have an embedded precursory sequence of smaller failures. The total failure of the array appears to be a critical point. There is a sequence of partial failures leading up to the total failure that resembles a log-periodic sequence.