Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws

By Agnes Helmstetter, Didier Sornette, Jean-Robert Grasso. 2002

191 citations recorded by the source corpus.

[1] 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/(tc − t)p′ of the time to the mainshock occurring at tc. Here, we show that this law results from the direct Omori law for aftershocks describing the power law decay ∼ 1/(t − tc)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. In particular, we predict and verify by numerical simulations on the epidemic-type aftershock sequence (ETAS) model that p′ is always smaller than or equal to p and a function of p, of the b-value of the Gutenberg–Richter law (GR), and of a parameter quantifying the number of direct aftershocks as a function of the magnitude of the mainshock. The often documented apparent decrease of the b-value of the GR law at the approach to the mainshock results straightforwardly from the conditioning of the path of seismic activity culminating at the mainshock. However, we predict that the GR law is not modified simply by a change of b-value but that a more accurate statement is that the GR law gets an additive (or deviatoric) power law contribution with exponent smaller than b and with an amplitude growing as a power law of the time to 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 toward 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. Foreshocks are not just statistical creatures but are genuine forerunners of large shocks as shown by the large prediction gains obtained using several of their qualifiers.

Subfields: Geophysics

View Paper: https://www.semanticscholar.org/paper/fb7509399d7d7d200d57e9015f6225b85af746e9

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In Geophysics

Citations in the corpus

Brittle creep, damage and time to failure in rocks (2005, background): “However, for earthquakes, this precursory acceleration of seismicity is not systematic, but is only significant when averaging over a large number of sequences, and can be explained by the properties of earthquake triggering [Helmstetter et al., 2003].”
Earthquake clusters in southern California I: Identification and stability (2013, background): “[48] The above results are consistent with the Omori-Utsu law for the intensity of both aftershocks and foreshocks [Omori, 1894; Utsu, 1961; Papazachos, 1973; Jones and Molnar, 1979; Utsu et al., 1995; Helmstetter et al., 2003]:”
Foreshocks explained by cascades of triggered seismicity (2002, methodology): “We refer to [Helmstetter et al., 2003] for the derivation of the results stated below, which is too involved to be reported here.”
Seismicity rate immediately before and after main shock rupture from high-frequency waveforms in Japan (2007, methodology): “observed exponent p in this model is smaller than 1 at short times, and increases slowly with time [Helmstetter and Sornette, 2002; Helmstetter et al., 2003], as observed in our”