Multifractal scaling properties of a growing fault population

A numerical rupture model, introduced in Cowie, Vanneste & Sornette (1993), is used to simulate the growth of faults in a tectonic plate driven by a constant plate boundary velocity. We find that the plate initially deforms by uncorrelated nucleation of small faults reflecting the distribution of material properties. With increasing strain, growth and coalescence of existing faults dominate over nucleation, a power-law distribution of fault sizes appears, and the fault pattern is fractal. Furthermore, the combined effect of fault clustering and the correlation between fault displacement and fault size leads to a strongly multifractal deformation pattern. We show theoretically that the multifractal spectrum depends explicitly on the exponent c, which defines the size distribution of the faults, as size and displacement are correlated. For different realizations of the numerical model, we calculate the exponent c, and fractal structure of the deformation through time as strain accumulates. We explore in detail the time evolution of the capacity (D0), information (D1), and correlation (D2) fractal dimensions. We relate these scaling parameters to the physical mechanisms of fault nucleation, growth and linkage during different phases of the deformation and discuss the factors that determine the values of the exponents. A consistently observed systematic decrease in the values of c, D1 and D2 through time indicates that the relative strain contribution of the smallest faults decreases as the total strain increases, a signature of the localization of faulting.