# How to compute the earthquake occurrence probability under ETAS model

I have fitted an ETAS model (unmarked) with an independent exponential magnitude distribution to a set of earthquake data. However, I do not know how to calculate the probability of the earthquake occurrence (or the probability of no events) during a certain time interval.

Is there an explicit formula to compute the probability, like the one in the Poisson model?

• This may be speculation (mostly out of ignorance) but couldn't a probability be generated by running an ensemble of model runs? – BarocliniCplusplus Jan 25 '17 at 21:17
• Check out Grand Solar Minimums. They happen every 206 years or so or less? They all have NAMES; Maunder, Dalton, Centennial as far back as you want to go. We are now in the Eddy Minimum. Check out what happens with earthquakes and volcanism during these cycles! Who needs technology if they have written records and oil paintings about each of the GSMs on this continent? The New Madrid Fault is of special importance. You could probably use this GSM cycle in your own formula if no one else has already done so. John L Casesy and UpHeaval has tons of great data. – stormy Oct 18 '18 at 21:14

An epidemic-type aftershock sequence (ETAS) model is a marked point process model.

Typically, a temporal model has the form (Ogata, 1988):

$$\lambda^*(t) = \mu + \sum k10^{\alpha(M_i-M_c)} (t_i+c)^{-p}$$

Where $$\lambda^*(t)$$ is a conditional intensity function. If you are using no marks (e.g. earthquake magnitudes) you may be dealing with a Hawkse process.

Then the likelihood of an event can be written as:

$$f(t|\mathcal{H}_{t_n}) = \lambda^*(t)e^{-\int^t_{t_n}{\lambda^*(s) ds}}$$

This is in fact what is optimized in the maximum likelihood estimation, typically using the log likelihood of all the events:

$$logL = \sum log\lambda^*(t_i) -\int_0^T \lambda^*(s)ds$$

If I understand the question correctly. The probabibily that an event does not occur in between the time of an event $$t_i$$ and some time $$t$$ can be written as $$1-F(t)$$ where $$F(t)$$ is the cumulative distribution of the event likelihood such that:

$$1-F(t) = \exp\left(-\int_{t_n}^t \lambda^*(s)ds\right)$$

I do not know the exact form of the ETAS model you are using, but it should be reasonably straightforward to implement from here on end.

Refer to this excellent note set for more: https://arxiv.org/pdf/1806.00221.pdf