Fractal models of roughness & earthquake statistics

Rock surfaces, irrespective of their origin and composition, contain irregularities or deviations from a prescribed geometrical form at all scales. The protrusions on the surfaces are referred to as asperities, and the dimples are referred to as valleys. When two nominally flat surfaces are placed in contact, surface roughness causes contact to occur at discrete contact spots or junctions. Friction results from the collective and interactive behavior of the two surfaces in contact, each representing roughness and microstructural details of the surface. Experimental evidence has shown that both natural and manufactured surfaces are self-affine, with a broad bandwidth of spatial frequencies and a consistent fractal dimension (Power & Tullis, 1995), leading to local asperities on a variety of scales (Main, 1988).

A fundamental statistical relationship in seismology is the Gutenberg–Richter frequency–magnitude relation. It describes the frequency of earthquakes as a function of their magnitude. It is usually expressed as: log₁₀ N = a - b M, where N is the number of earthquakes with magnitude ≥ M, a is a constant that represents the total seismicity rate in a given region and time period, b is the b-value, typically around 1 for natural earthquakes, which characterizes the relative likelihood of small versus large earthquakes, and M is the earthquake magnitude.  While there have been qualitative attempts to link the b-value with fractal models of roughness, a unified mathematical framework connecting the two has yet to be developed.

Web/Literature References:

Alghannam, M., Nordbotten, J. M., & Juanes, R. (2025). Stick–slip from heterogeneous Coulomb friction. Physical Review E, 111(5), 055505.

Bak, P. (2013). How nature works: The science of self-organized criticality. Springer Science & Business Media.

Burridge, R., & Knopoff, L. (1967). Model and theoretical seismicity. Bulletin of the Seismological Society of America, 57(3), 341–371.

Carlson, J. M., & Langer, J. S. (1989). Properties of earthquakes generated by fault dynamics. Physical Review Letters, 62(22), 2632–2635.

Main, I. (1988). Prediction of failure times in the Earth for a time-varying stress. Geophysical Journal International, 92(3), 455–464.

Power, W., & Tullis, T. (1995). Review of the fractal character of natural fault surfaces with implications for friction and the evolution of fault zones. In Fractals in the earth sciences (pp. 89–105). Springer.

A widely used model in the statistical study of earthquakes is the Burridge–Knopoff model (1967). It is a mechanical model in earthquake physics used to simulate and understand the dynamics of fault slip and earthquake generation. Carlson & Langer (1989) performed the first study of the statistical properties of the B–K model, paying attention to the magnitude distribution of earthquake events and its dependence on friction parameters. They used velocity-weakening friction and were able to produce a frequency–magnitude plot with a b-value around 1. In this project, the student is expected to: (1) review the derivation of the Burridge–Knopoff model (1967), (2) reproduce the results for the frequency–magnitude relation in Carlson & Langer (1989), and (3) explore ways to include fractal models of roughness.