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Math and Volcanic Eruptions

Updated: Jun 14, 2022

By Rosalyn Brady

June 9, 2022


[Photo Credit: “How Does A Volcano Work?”]

Under the earth’s surface, the temperatures are sufficiently high to melt the rock that forms our planet, creating magma - molten rock. This magma is composed of oxygen, silicon, iron, magnesium, aluminum, titanium, manganese, calcium, sodium, and potassium, and is lighter than the rock surrounding it, meaning that buoyancy and the pressures of the gases within the magma force the magma upwards towards Earth’s surface, where it collects in magma chambers and then makes its way through vents and fissures, which we know as volcanoes, in the Earth’s surface as lava.

This is what is known as a volcanic eruption, and these can present themselves in many ways: thin and runny magma from which gases easily escape, which simply flows slowly down the sides of the volcano, providing anybody nearby enough time to evacuate, or thick and sticky magma, which gases cannot escape from, which causes a buildup of pressure that escalates until the gases and the magma escape violently in an explosion. In these eruptions, the magma breaks up into smaller fragments called tephra, which form dense clouds of the rock fragments formed by the cooling magma and then are pulled by gravity down the sides of the volcano to, more often than not, wreak destruction. The smallest tephra, known as ash, can in large enough amounts cover and suffocate plants and animals, and even in the smallest amounts, ash can be carried by the wind several times across the globe before settling on the ground.

Geoscientists - scientists who study the Earth and the rocks that it is composed of - have observed that when underground magma shifts and moves, the ground above it will quiver and flex. Satellites using positioning systems similar to GPS can now observe these movements, and geoscientists are learning how to decipher them, also having developed a new mathematical method to track the flow of the magma. Mary Grace Bato of the Institut des Sciences de la Terre is the team lead for the first geoscience research team to use data assimilation, a technique that combines qualitative data with mathematical representations of the behaviour patterns of the observational subject to estimate the future state of the observational subject, to successfully predict the change in magmatic overpressure - magmatic pressure above what is typical - using GPS and satellite radar data.

In the way that meteorologists use data assimilation to predict the weather using their observations of the ocean and the atmosphere, and climate scientists use it to extrapolate the long-term effects of climate change using carbon emission data, geoscientists may soon use data assimilation to predict volcanic eruptions globally - Mary Grace Bato was only one of the first.

Bato’s experiment simulated a volcano that erupted without any explosivity, and therefore with limited gas pressure buildup. The research team was able to predict the overpressure that drives volcanic eruptions, as well as the speed at which magma flowed into the magma chamber and the magma chamber’s shape, despite the typical depth of magma chambers being many kilometres underground, making them immensely challenging to study with our current technology.

While geoscientists have a long way to go before data assimilation can be used consistently and reliably, Bato’s research team is now taking the next steps in the development of this groundbreaking technology: testing it on Alaska’s Okmok Volcano and Iceland’s Grimsvötn Volcano.

Perhaps someday, we may be able to see a future where data assimilation can be used to predict volcanic activity as easily as it is used to predict the weather, or the outcomes of climate change - always allowing those in the eruption’s path enough time to evacuate and remain safe.


“About Volcanoes | U.S. Geological Survey.” United States Geological Survey,,on%20to%20the%20Earth’s%20surface. Accessed 17 May 2022.,on%20to%20the%20Earth's%20surface.

“How Do Volcanoes Erupt? | U.S. Geological Survey.” United States Geological Survey, 8 May 2019,

Walter, Kenny. “Using Math to Predict Volcanic Eruptions.” Research & Development World, 28 June 2017,

“Predicting Eruptions Using Satellites and Math: Geoscientists Are Learning How to Better Predict Volcanic Eruptions Using GPS and the Same Method That Helps Forecast the Weather.” ScienceDaily, Accessed 17 May 2022.

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