The inverse Omori law for foreshocks discovered in the 1970s states that the
rate of earthquakes prior to a mainshock increases on average as a power law ~
1/(t_c-t)^p' of the time to the mainshock occurring at t_c. Here, we show that
this law results from the direct Omori law for aftershocks describing the power
law decay ~ 1/(t-t_c)^p of seismicity after an earthquake, provided that any
earthquake can trigger its suit of aftershocks. In this picture, the seismic
activity at any time is the sum of the spontaneous tectonic loading and of the
activity triggered by all preceding events weighted by their corresponding
Omori law. The inverse Omori law then emerges as the expected (in a statistical
sense) trajectory of seismicity, conditioned on the fact that it leads to the
burst of seismic activity accompanying the mainshock. The often documented
apparent decrease of the b-value of the GR law at the approach to the main
shock results straightforwardly from the conditioning of the path of seismic
activity culminating at the mainshock. In the space domain, we predict that the
phenomenon of aftershock diffusion must have its mirror process reflected into
an inward migration of foreshocks towards the mainshock. In this model,
foreshock sequences are special aftershock sequences which are modified by the
condition to end up in a burst of seismicity associated with the mainshock.Comment: Latex document of 35 pages, 10 figure
