According to the Omori-Utsu law, the rate of aftershocks after a mainshock
decays as a power law with an exponent close to 1. This well-established law
was intensively used in the past to study and model the statistical properties
of earthquakes. Moreover, according to the so-called inverse Omori law, the
rate of earthquakes should also increase prior to a mainshock -- this law has
received much less attention due to its large uncertainty. Here, we mainly
study the inverse Omori law based on a highly detailed Southern California
earthquake catalog, which is complete for magnitudes larger than M>0.3. First,
we develop a technique to identify mainshocks, foreshocks, and aftershocks. We
then find, based on a statistical procedure we developed, that the rate of
earthquakes is higher a few days prior to a mainshock. We find that this
increase is much smaller for a catalog with a magnitude threshold of m over 2.5
and for the Epidemic-Type Aftershocks Sequence (ETAS) model catalogs, even when
used with a small magnitude threshold. We also analyze the rate of aftershocks
after mainshocks and find that the Omori-Utsu law does not hold for many
individual mainshocks and that it may be valid only statistically when
considering many mainshocks together. Yet, the analysis of the ETAS model based
on the Omori-Utsu law exhibits similar behavior as that of the real catalogs,
indicating the validity of this law.Comment: 19 pages, 7 figure