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In both our past work and the work in progress we focused on understanding the physics and statistical patterns in earthquake faults and fault systems. Our approach had three key aspects. The first was to look for patterns of seismic activity in earthquake fault systems. The second was to understand the physics of a sequence of models for faults and fault systems that are increasingly more realistic. The third key element was to connect the two previous approaches by investigating specific properties found in models to see if they are indeed properties of real faults. A specific example of how this approach works can be seen in the following: In the papers discussed below, we demonstrated that the cellular automation (CA) versions of the slider block models with long range stress transfer are ergodic and could be described by a Boltzmann-Gibbs distribution in the meanfield limit. The ergodicity follows from the fact that the long range stress transfer makes the model meanfield. The meanfield nature of the CA models, generated by long range stress transfer, also allows a description of the CA models by a Langevin equation. The Langevin equation indicates that evolution of seismicity in the model over relatively short times is linear in time. This appears to be consistent with the success of a forecasting algorithm we have developed that is based on a linear evolution of seismicity patterns. This algorithm has had considerable success in that the regions of the Southern California fault system which have been predicted to have a higher probability of an event greater than magnitude 5 have consistently been the sites where such events occur. These two results have led to the question as to whether the Southern California fault system is ergodic and can be described by a Langevin equation like the model. To answer this question we ran a series of tests for ergodicity very much like the ones run on the models. Our results, which have been accepted for publication in Physical Review Letters (Tiampo et al., in press), demonstrate that the Southern California system is ergodic in the same way that is seen in the models. These results will be discussed in more detail below. However, the point that needs to be emphasized is that it was the combination of model investigation via theory and simulation coupled with assimilation and classification of real data and applying the methods of statistical mechanics to real fault systems that led to both a successful forecasting algorithm and a deeper understanding of the nature of earthquake fault systems. This paper describes in some detail the results obtained in the previous funding period. We present these in three groups. (A) Investigation of statistical physics models and applications. (B) Earthquake fault systems and Greens functions for complex sources and (C) Space time patterns, data analysis and forecasting

Collaborative Research: Analysis and Interpretation of Multi-Scale Phenomena in Crustal Deformation Processes Using Numerical Simulations of Complex Nonlinear Earth Systems