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Modeling Volcanoes – When will they erupt?

A recent post by the excellent Maths Careers website looked at how we can model volcanic eruptions mathematically. This is an important branch of mathematics – which looks to assign risk to events and these methods are very important to statisticians and insurers. Given that large-scale volcanic eruptions have the potential to end modern civilisation, it’s also useful to know how likely the next large eruption is.

The Guardian has recently run a piece on the dangers that large volcanoes pose to humans.  Iceland’s Eyjafjallajökull volcano which erupted in 2010 caused over 100,000 flights to be grounded and cost the global economy over \$1 billion – and yet this was only a very minor eruption historically speaking.  For example, the Tombora eruption in Indonesia (1815) was so big that the explosion could be heard over 2000km away, and the 200 million tones of sulpher that were emitted spread across the globe, lowering global temperatures by 2 degrees Celsius.  This led to widespread famine as crops failed – and tens of thousands of deaths.

Super volcanoes

Even this destruction is insignificant when compared to the potential damage caused by a super volcano.  These volcanoes, like that underneath Yellowstone Park in America, have the potential to wipe-out millions in the initial explosion and and to send enough sulpher and ash into the air to cause a “volcanic winter” of significantly lower global temperatures.  The graphic above shows that the ash from a Yellowstone eruption could cover the ground of about half the USA. The resultant widespread disruption to global food supplies and travel would be devastating.

So, how can we predict the probability of a volcanic eruption?  The easiest model to use, if we already have an estimated probability of eruption is the Poisson distribution:

$P(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!},$

This formula calculates the probability that X equals a given value of k.  λ is the mean of the distribution.  If X represents the number of volcanic eruptions we have Pr(X ≥1) = 1 – Pr(x = 0).  This gives us a formula for working out the probability of an eruption as 1 -e.  For example, the Yellowstone super volcano erupts around every 600,000 years.  Therefore if λ is the number of eruptions every year, we have λ = 1/600,000  ≈ 0.00000167 and 1 -e also ≈ 0.00000167. This gets more interesting if we then look at the probability over a range of years. We can do this by modifying the formula for probability as 1 -e-tλ where t is the number of years for our range.

So the probability of a Yellowstone eruption in the next 1000 years is 1 -e-0.00167 ≈ 0.00166, and the probability in the next 10,000 years is 1 -e-0.0167 ≈ 0.0164. So we have approximately a 2% chance of this eruption in the next 10,000 years.

A far smaller volcano, like Katla in Iceland has erupted 16 times in the past 1100 years – giving a average eruption every ≈ 70 years. This gives λ = 1/70 ≈ 0.014. So we can expect this to erupt in the next 10 years with probability 1 -e-0.14 ≈ 0.0139. And in the next 30 years with probability 1 -e-0.42 ≈ 0.34.

The models for volcanic eruptions can get a lot more complicated – especially as we often don’t know the accurate data to give us an estimate for the λ.  λ can be estimated using a technique called Maximum Likelihood Estimation – which you can read about here.

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