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Let us now turn to the more curiosity-driven aspects of earthquake prediction research. A new trend in recent years has been the involvement of physicists in modeling earthquake phenomena since Bak and Tang  proposed that the Gutenberg-Richter frequency magnitude relation may be attributed to a process described as ``self-organized criticality.'' It was not a surprise for me because it reminded me of the Goishi model of Otsuka  and the branching model of Vere-Jones . The Goishi (stone of the game of go game; a Japanese game played with stones on a board marked into 361 squares) model describes a probabilistic growth of an earthquake fault in the manner similar to the percolation theory. Saito et al.  and Maruyama  demonstrated analytically that the frequency-size distribution becomes a power law similar to the Gutenberg-Richter relation at a critical transition probability. I was reluctant [ Aki, 1981] to accept these models because the elastodynamics of rupture propagation along a fault with velocity comparable to shear waves is essential to the earthquake source, but is missing in these models.
Recent works on seismicity simulation by a group of physicists, however, include the elastodynamic effect based on the block-spring model of Burridge and Knopoff . Most of these works [e.g., Carlson and Langer, 1989, Langer, 1992] are published in Physical Review, but a clear concise description of the model is given in a recent paper by Pepke et al. , who used it to develop effective prediction algorithms. The model includes the velocity weakening friction law and generates complex seismicity from the self-organization of repeated ruptures.
Recently Rice  has raised a serious concern regarding whether the self-organizing explanation can give rise to complexity on a spatially uniform fault. His analysis is based on a three-dimensional vertical strike-slip fault with a rate- and state-dependent friction law [ Dieterich, 1981; Ruina, 1983]. The law includes a characteristic slip distance L for evolution of surface state and slip weakening. He solves the governing equations of elasticity and frictional slip by discretizing with the cell size h. He found an extremely interesting result that the simulated slip shows spatio-temporal complexity when h is greater than a critical size h* called ``nucleation size,'' which scales with L. When h is less than h*, the complexity disappears in favor of simple periodically repeated large earthquakes like the characteristic earthquake discussed earlier.
His concern about the self-organizing explanation of complexity using the Burridge-Knopoff model is that L and h* are zero in the velocity weakening friction law adopted in these models, thus having no well-defined continuum limit as h diminishes. He also points out the significance of failure of these models (restricted to the nearest neighbor interaction) to scale the stress concentration at the edge of a slipping zone with the size of that zone, as required by elasticity theory. This effect makes a dynamic fracture unstoppable once it grows larger on a spatially uniform fault, and leads to the idea that complexity of seismicity comes from fault heterogeneities of multiple scales.
The formulation of Rice  does not include the inertia term, and includes elastodynamic effects only approximately through radiation damping or allowance for dynamic overshoot. Thus, the possibility exists that the proper elastodynamics solutions of uniform fault models with a well defined continuum limit would lead to complex seismicity, as discussed by Madariaga and Cochard  and Cochard and Madariaga . Shaw  also responds to the criticism of Rice  by introducing a viscous damping term into the equation for the Burridge-Knopoff model to make the continuum limit well defined, and demonstrated that complex seismicity can be generated for a uniform fault even when the discretization is made within the continuum limit. Thus, the question of complex seismicity or periodic earthquake for a uniform fault is still open, awaiting for a more complete treatment of dimensionality, friction law and dynamics of the problem.
Whether the complex seismicity is due to fault heterogeneity or non-linear dynamics, the discovery of Rice  that the relative size of discretizing mesh and the nucleation zone strongly affects the spacio-temporal behavior of earthquake occurrence will remain as one of the most important theoretical results upon which we can build a sound physical theory of earthquake prediction.