Power Law Distributions of Seismic Rates

We report an empirical determination of the probability density functions $P_{\text{data}}(r)$ of the number $r$ of earthquakes in finite space-time windows for the California catalog. We find a stable power law tail $P_{\text{data}}(r) \sim 1/r^{1+\mu}$ with exponent $\mu \approx 1.6$ for all space ($5 \times 5$ to $20 \times 20$ km$^2$) and time intervals (0.1 to 1000 days). These observations, as well as the non-universal dependence on space-time windows for all different space-time windows simultaneously, are explained by solving one of the most used reference model in seismology (ETAS), which assumes that each earthquake can trigger other earthquakes. The data imposes that active seismic regions are Cauchy-like fractals, whose exponent $\delta =0.1 \pm 0.1$ is well-constrained by the seismic rate data.