A probabilistic approach for the classification of earthquakes as ‘triggered’ or ‘not triggered’

Abstract

The occurrence time of earthquakes can be anticipated or delayed by external phenomena that induce strain energy changes on the faults. ‘Anticipated’ earthquakes are generally called ‘triggered’; however, it can be controversial to label a specific earthquake as such, mostly because of the stochastic nature of earthquake occurrence and of the large uncertainties usually associated to stress modelling. Here we introduce a combined statistical and physical approach to quantify the probability that a given earthquake was triggered by a given stress-inducing phenomenon. As an example, we consider an earthquake that was likely triggered by a natural event: the M = 6.2 13 Jan 1976 Kópasker earthquake on the Grímsey lineament (Tjörnes Fracture Zone, Iceland), which occurred about 3 weeks after a large dike injection in the nearby Krafla fissure swarm. By using Coulomb stress calculations and the rate-and-state earthquake nucleation theory, we calculate the likelihood of the earthquake in a scenario that contains only the tectonic background and excludes the dike and in a scenario that includes the dike but excludes the background. Applying the Bayes’ theorem, we obtain that the probability that the earthquake was indeed triggered by the dike, rather than purely due to the accumulation of tectonic strain, is about 60 to 90 %. This methodology allows us to assign quantitative probabilities to different scenarios and can help in classifying earthquakes as triggered or not triggered by natural or human-induced changes of stress in the crust.

Appendix: Rate-and-state earthquake nucleation model

The rate-and-state earthquake nucleation model relates earthquake production per unit area on a population of faults to the evolution of a state variable, γ:

$$ R = \frac{{\rm d}N}{{\rm d}t}=\frac{r}{\gamma \dot \tau} $$

(18)

where N is the number of events per unit area in the time interval [0, T]. For a constant shear stressing rate, γ is equal to $\gamma_0 = 1/\dot\tau$, which yields R = r (steady state): The seismicity rate remains at its constant background value until an external stress perturbation intervenes. The state variable γ evolves with time as:

$$ {\rm d}\gamma = \frac{1}{A\sigma}\left({\rm d}t - \gamma {\rm d}S \right) $$

(19)

where A is a constitutive parameter, σ is the effective normal stress and dS = ΔCFF is the Coulomb stress change (Eq. 8). Step-like stress perturbations are accounted for by setting dt = 0 in Eq. 19 and integrating:

$$ \gamma_1 = \gamma_{0}\exp \left( {\frac{-{\rm d}S}{A\sigma}}\right) $$

(20)

where γ ₁ and γ ₀ are the state variable values before and after the stress perturbation, respectively. By substituting Eqs. 20 in 18 and eliminating the variable γ, the peak in seismicity rate right after the stress change is obtained explicitly:

$$ R_1 = r\exp \left( {\frac{\Delta {\rm CFF}}{A\sigma}}\right) $$

(21)

After a step-like stress change, the tectonic stressing rate dominates in controlling the seismicity rate, which experiences a transient until it reaches again steady state. Setting ${\rm d}S=\dot\tau\,{\rm d}t$ Eq. 19 and integrating:

$$ \gamma(t) = \frac{1}{\dot\tau}+\left(\gamma_1 - \frac{1}{\dot \tau}\right) \exp \left( -\frac{\Delta t \dot\tau}{A \sigma} \right) $$

(22)

which implies that the time needed for R to return to steady state is inversely proportional to $\dot\tau$. The state variable γ can be eliminated by combining Eqs. 18, 20 and 22:

$$ R(t)=r\frac{1}{1+\left[\exp\left(-\frac{\Delta\mathrm{CFF}}{A\sigma}\right)-1\right]\exp\left(-\frac{\dot\tau t}{A\sigma}\right)}$$

(23)

The expected number of events for a given region in a time window Δt = t ₂ − t ₁ can be calculated by integrating the seismicity rate over the time interval:

$$ N_{E} (\Delta t) = \int_{t_1}^{t_2} R(t) {\rm d}t. $$

(24)