Impact craters are rare

There are at least two important things to think about when considering an interpretation:

How well does this match the model? (In this case, how much does it look like an impact structure?) How likely are we to see an instance of this model in this dataset? (What’s the base rate of impact structures here?)

Interpreters often forget about the second part. (There’s another part too: How reliable are my interpretations? Let’s leave that for another day, but you can read Bond et al. 2007 as homework if you like.)

The problem is that impact structures, or astroblemes, are pretty rare on Earth. The atmosphere takes care of most would-be meteorites, and then there’s the oceans, weather, tectonics and so on. The result is that the earth’s record of surface events is quite irregular compared to, say, the moon’s. But they certainly exist, and occasionally pop up in seismic data.

In my 2011 post Reliable predictions of unlikely geology, I described how skeptical we have to be when predicting rare things (‘wotsits’). Bayes’ theorem tells us that we must modify our assigned probability (let’s say I’m 80% sure it’s a wotsit) with the prior probability (let’s pretend a 1% a priori chance of there being a wotsit in my dataset). Here’s the maths:

\( \ \ \ P = \frac{0.8 \times 0.01}{0.8 \times 0.01\ +\ 0.2 \times 0.99} = 0.0388 \)

In other words, the conditional probability of the feature being a rare wotsit, given my 80%-sure interpretation, is 0.0388 or just under 4%.

As cool as it would be to find a rare wotsit, I probably need a back-up hypothesis. Now, what’s that base rate for astroblemes? (Spoiler: it’s much less than 1%.)

Just how rare are astroblemes?

First things first. If you’re interpreting circular structures in seismic, you need to read Simon Stewart’s paper on the subject (Stewart 1999), and his follow-up impact crater paper (Stewart 2003), which expands on the topic. Notwithstanding Stewart’s disputed interpretation of the Silverpit not-a-crater structure in the North Sea, these two papers are two of my favourites.

According to Stewart, the probability P of encountering r craters of diameter d or more in an area A over a time period t years is given by:

\( \ \ \ P(r) = \mathrm{e}^{-\lambda A}\frac{(\lambda A)^r}{r!} \)

where

\( \ \ \ \lambda = t n \)

and

\( \ \ \ \log n = - (11.67 \pm 0.21) - (2.01 \pm 0.13) \log d \)