Describing Seismic Pattern Dynamics by Means of Ising Cellular Automata
This chapter is dedicated to the description and testing of a new method of obtaining Probabilistic Activation Maps of seismic activity. This method is based upon two major concepts: Cellular Automata (CA) and Information Theory. The proposed method can b
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Department of Earth Sciences Biological and Geological Sciences, University of Western Ontario, London, Canada, [email protected]; [email protected] Department of Applied Physics, University of Almer´ıa, Spain [email protected]
Abstract. This chapter is dedicated to the description and testing of a new method of obtaining Probabilistic Activation Maps of seismic activity. This method is based upon two major concepts: Cellular Automata (CA) and Information Theory. The proposed method can be used in other fields, as long as the spatially extended system is described in terms of a Cellular Automata with two available states, +1 and −1, as in the Ising case described here. The crucial point is to obtain the rules of an Ising Cellular Automata that maps one pattern into its future state by means of an entropic principle. We have already applied this technique to the seismicity in two regions: Greece and the Iberian Peninsula. In this chapter, we study other regions to test if the observed behavior holds in general. For this purpose, we will discuss the results for California, Turkey and Western Canada. The Cellular Automaton rules obtained from the correponding catalogs are found to be well described by an Ising scheme. When these rules are applied to the most recent pattern, we obtain a Probabilistic Activation Map, where the probability of surpassing a certain energy (equivalent to a certain magnitude) in the next interval of time is represented, which is a useful information for seismic hazard assessment. Keywords: Cellular automata, Probabilistic seismic hazard, Information theory
1 Introduction When looking for a framework that allows for studying nonlinearity and stochasticity at the same time, Information Theory is one of the most natural candidates. Information Theory confronts the problem of constructing models from experimental time series. These models are used to make predictions, but the underlying dynamics is unknown (or known but with a high dimensionality and thus incomputable). Information Theory was described (for the first time) by Shannon [1, 2] and Shannon and Weaver [3]. This formalism was
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later used by Shaw [4] to study the time series produced by a drop of water that falls from a faucet not properly turned off. He established an alternative way to deal with complex problems in the phase space. The behavior and evolution of a system in which a series of states is known that occurred at times T1 , T2 , ..., Tn , can be characterized by a return map; that is, by representing the state at Ti in the x-axis, and the one at Ti+1 in the y-axis, and the process goes on until the adequate dimension is obtained. Shaw used the resulting scatter plots and the concept of information based on Shannon’s entropy in order to study the knowledge of the future states from the present and the past states. This knowledge can be (among others) characterized by a quantity called mutual information, which will be described later. In a nonlinear, spatially extended system, information can be produced i
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