High-frequency spectral falloff of earthquakes, fractal dimension of complex rupture, b value, and the scaling of strength on faults

The high-frequency falloff ω^−γ of earthquake displacement spectra and the b value of aftershock sequences are attributed to the character of spatially varying strength along fault zones. I assume that the high frequency energy of a main shock is produced by a self-similar distribution of subevents, where the number of subevents with radii greater than R is proportional to R^−D, D being the fractal dimension. In this model, an earthquake is composed of a hierarchical set of smaller earthquakes. The static stress drop is parameterized to be proportional to R^η, and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with D = 2 and stress drop independent of seismic moment (η = 0) produces a main shock with an ω⁻² falloff, if the subevent areas fill the rupture area of the main shock. By equating subevents to “islands” of high stress of a random, self-similar stress field on a fault, I relate D to the scaling of strength on a fault, such that D = 2 − η. Thus D = 2 corresponds to constant stress drop scaling (η = 0) and scale-invariant fault strength. A self-similar model of aftershock rupture zones on a fault is used to determine the relationship between the b value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. The b value for aftershock sequences on a fault is found to equal (3 − 1.5η)/(3 + η). Therefore this model indicates that the typically observed spectral falloffs of ω⁻² and b values of 1 can be entirely caused by scale-invariant strength (η = 0) along fault zones.