Mainshocks are Aftershocks of Conditional Foreshocks: How do foreshock statistical properties emerge from aftershock laws

The inverse Omori's law for foreshocks discovered in the 1970s states that the rate of earthquakes prior to a mainshock increases on average as a power law $\propto 1/(t_c-t)^{p'}$ of the time to the mainshock occurring at $t_c$. Here, we show that this law results from the direct Omori's law for aftershocks describing the power law decay $\sim 1/(t-t_c)^{p}$ of seismicity after an earthquake, provided that any earthquake can trigger its suit of aftershocks. The inverse Omori's 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. Foreshocks are not just statistical creatures, they are genuine forerunners of large shocks as shown by the large prediction gains obtained using several of their qualifiers.