Stress spatial distributions, the Gutenberg–Richter and Omori–Utsu laws

We investigate several earthquake models in one and two dimensions of space and analyze in these models the stress spatial distribution. We show that the statistical properties of stress distribution are responsible for the distribution of earthquake magnitudes, as described by the Gutenberg–Richter (GR) law. A series of predictions is made based on the analogies between the stress profile and one-dimensional random curves or two-dimensional random surfaces. These predictions include the b-value, which determines the ratio of small to large seismic events and, in two-dimensional models, we predict the existence of aftershocks and their temporal distribution, known as the Omori–Utsu law. Both the GR and Omori–Utsu law are properties which have been extensively validated by earthquake observations in nature.