High‐frequency spectral falloff of earthquakes, fractal dimension of complex rupture, <i>b</i> value, and the scaling of strength on faults

<jats:p>The high‐frequency falloff ω<jats:sup>−γ</jats:sup> of earthquake displacement spectra and the <jats:italic>b</jats:italic> 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 <jats:italic>R</jats:italic> is proportional to <jats:italic>R<jats:sup>−D</jats:sup></jats:italic>, <jats:italic>D</jats:italic> 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 <jats:italic>R</jats:italic><jats:sup>η</jats:sup>, and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with <jats:italic>D</jats:italic> = 2 and stress drop independent of seismic moment (η = 0) produces a main shock with an ω<jats:sup>−2</jats:sup> 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 <jats:italic>D</jats:italic> to the scaling of strength on a fault, such that <jats:italic>D</jats:italic> = 2 − η. Thus <jats:italic>D</jats:italic> = 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 <jats:italic>b</jats:italic> value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. The <jats:italic>b</jats:italic> 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<jats:sup>−2</jats:sup> and <jats:italic>b</jats:italic> values of 1 can be entirely caused by scale‐invariant strength (η = 0) along fault zones.</jats:p>