Determining brittle extension and shear strain using fault-length and displacement systematics: Part I: Theory

We derive the exact equations by which the continuum approximation to the extensional and shear strains can be determined from measurements of fault-lengths or fault-displacement in a faulted domain. We develop the theory by which we can infer the extensional and shear strain in a volume of brittlely deformed crust from an incomplete inventory of the faults. To that end, we use empirical power-law relationships between fault-length and fault-displacement, and the power-law cumulative frequency distribution for each of these variables, for sampling domains of one, two, and three dimensions. The theory 1) defines the relationships among the parameters in these power-laws, which allows the self-consistency of results from fault-length and fault-displacement studies in domains of one, two, and three dimensions to be evaluated; 2) defines constraints on the relative sizes of the sampling domain and the largest fault that can be included in an analysis using fault systematics; 3) shows that extensional and shear strains in faulted crust can be inferred knowing only an independent set of the parameters defining the population systematics plus the magnitude of either the displacement or the length for the largest fault in the domain; and 4) defines the constraints on the three-dimensional strain imposed by sampling in one- and two-dimensional domains.