  In many important systems exhibiting crackling noise --- intermittent
avalanche-like relaxation response with power-law and, thus, self-similar
distributed event sizes --- the "laws" for the rate of activity after large
events are not consistent with the overall self-similar behavior expected on
theoretical grounds. This is in particular true for the case of seismicity and
a satisfying solution to this paradox has remained outstanding. Here, we
propose a generalized description of the aftershock rates which is both
self-similar and consistent with all other known self-similar features.
Comparing our theoretical predictions with high resolution earthquake data from
Southern California we find excellent agreement, providing in particular clear
evidence for a unified description of aftershocks and foreshocks. This may
offer an improved way of time-dependent seismic hazard assessment and
earthquake forecasting.
