Stochastic Branching Models of Fault Surfaces and Estimated Fractal Dimensions

We discuss simulations of nonplanar fault structures for a variant of the geometric stochastic branching model of Kagan (1982) and perform fractal analyses with 2-D and 3-D box-counting methods on the simulated structures. One goal is to clarify the assumptions associated with the geometric stochastic branching model and the conditions for which it may provide a useful tool in the context of earthquake faults. The primary purpose is to determine whether typical fractal analyses of observed earthquake data are likely to provide an adequate description of the underlying geometrical properties of the structure. The results suggest that stochastic branching structures are more complicated and quite distinct from the mathematical objects that have been used to develop fractal theory. The two families of geometrical structures do not share all of the same generalizations, and observations related to one cannot be used directly to make inferences on the other as has frequently been assumed. The fractal analyses indicate that it is incorrect to infer the fractal dimension of a complex volumetric fault structure from a cross-section such as a fault trace, from projections such as epicenters, or from a sparse number of representative points such as hypocenter distributions.