aftershock: one construct, many systems

An aftershock is one of the smaller earthquakes that follow a larger main shock in the same region, as the crust around the ruptured fault adjusts to its redistributed stress. Their rate is not constant but decays with time roughly as one over elapsed time, a pattern Omori first described in 1894 and that Utsu, Ogata, and Matsu'ura's 1995 The centenary of the Omori formula for a decay law of aftershock activity restated as the modified Omori (Omori-Utsu) law. The referent is the triggered, decaying burst of seismicity following a shock, measured by how the event rate falls off over the days and years afterward.[1]

Unit

The rate of triggered events and its power-law decay with time after a main shock

Recovery means

Background seismicity rate is restored once the aftershock sequence has decayed away

Measured by

Fitted by the modified Omori (Omori-Utsu) law to the declining event rate, with productivity scaling with main-shock magnitude

Econophysicists found the same construct in financial markets: after a large crash, the rate at which volatility (or large price moves) exceeds a threshold does not drop back at once but relaxes slowly, decaying as a power law of the time since the crash. Lillo and Mantegna's 2003 Power-law relaxation in a complex system: Omori law after a financial market crash showed that post-crash index behaviour follows the Omori form borrowed directly from seismology. It is the identical triggered-then-decaying excitation seen after an earthquake, realised in a market rather than the crust, and measured by the same kind of decay exponent fitted to how quickly activity settles.[2]

Unit

The decaying rate of threshold-exceeding price moves after a market crash

Recovery means

Normal volatility returns as the post-crash excitation relaxes along the fitted power law

Measured by

Estimated by fitting an Omori-type power law to the number of above-threshold volatility events versus time after the crash