INTERACTIVE COMPANION · FIELD NOTE · VOLCANOLOGY

The Deformation Bench

A live model accompanying “A Volcano That Has Been Dead for 250,000 Years Is Still Breathing”

Uturuncu, in southwest Bolivia, last erupted 250,000 years ago. The ground above it has been rising at about a centimetre a year since satellites started watching in 1992, and that rising dome sits inside a wide ring of slow subsidence. The whole pattern is called the sombrero, and its shape is the reason this page exists.

Two benches sit below. The first lets you place pressure sources under the volcano and try to reproduce a fixed sombrero profile, with the misfit updating as you drag. The second does the division that turns an inflation rate into a forecast. Both are the club's own simplified model rather than the paper's analysis, and every number they print can be checked by hand against the formulas beside them.

Model 1. Try to build a moat

A Mogi source is a small pressurised sphere buried in elastic rock. Inflate it by a volume ΔV at depth d and the surface above moves by

u_z(r) = (1 − ν) · ΔV · dπ · (r² + d²)^3/2 with ν = 0.25

Look at what is in that expression. The depth d is positive because the source is underground. The bracket (r² + d²) is a sum of squares, so it is positive, and so is any power of it. π is positive. (1 − ν) is positive. So the sign of u_z is the sign of ΔV and nothing else, at every radius out to infinity.

Which means an inflating source lifts the whole surface and a deflating one lowers all of it. Neither can lift the middle and drop the rim.

The bench starts where the article's grid search ended: the single best source, at 12.25 km, inflating at 6.62 × 10⁶ m³ a year. That is the best a lone source can do against this profile, and the readout says how well it does. Then switch the second source on and go looking for the moat.

Source 1 depth 12.25 km Slide it anywhere from 2 to 60 km and watch the misfit. It never drops below about 1.29 mm/yr, because depth changes the width of the bump and nothing else. A deep source makes a broad bump, a shallow one a narrow bump, and neither one makes a dip. Source 1 volume rate 6.62 × 10⁶ m³/yr This scales the whole curve up and down without changing its shape. Try dragging it to zero: the misfit goes to 2.99 mm/yr, which is what you get for predicting no motion at all. Any single source beats that, and none of them beat it by much. Second source Switch it on. The two controls below wake up, and the article's answer is 77.4 km at −107.5. Set source 1 to 20.80 km and 25.44 while you are there, and the misfit falls to 0.17 mm/yr, below the noise in the data. Source 2 depth 77.4 km Deeper means broader. The deep source has to be broad enough to pull the rim down while leaving the centre alone, which is why it ends up three or four times further down than the shallow one. Source 2 volume rate −107.5 × 10⁶ m³/yr Negative means the source is shrinking. Push it positive and the moat vanishes instantly, because you now have two inflating sources and the sign rule applies to their sum as well.

RMS misfit: – variance reduction: – lowest predicted u_z: – sign changes: –

–

The observations are fixed and they are synthetic, built by the club from two hidden sources at 20.0 and 80.0 km plus seeded noise of 0.20 mm/yr. They peak at +9.71 mm/yr, cross zero at 24.7 km, and bottom out at −1.97 mm/yr at 42 km. That is a caricature of the published Uturuncu pattern, made deliberately so we know the right answer.

Best one source: 12.25 km, +6.62e6 m³/yr, RMS 1.29 mm/yr
Best two: 20.80 km +2.544e7, and 77.40 km −1.075e8, RMS 0.17 mm/yr
Hidden truth: 20.0 km and 80.0 km, recovered to 4.0% and 3.2%
Deep / shallow: the deep source has to be 4.2 times larger

Model 2. What would that inflation have to build?

Suppose you take the deformation at face value and read the volume change as magma arriving. How long before there is enough of it to erupt?

The division has three versions, because it depends entirely on what you think the volume is made of. Take all of it as eruptible magma and you get the alarming answer. Allow for the magma body being only a quarter melt, so four cubic metres of crystal mush must be reorganised per cubic metre of extractable liquid, and the answer grows fourfold. Compare against the rate at which magma actually arrived while the volcano was building itself, 85 km³ over about 640,000 years, and it grows by another two orders of magnitude.

Inflation rate 25.44 × 10⁶ m³/yr The default is our fitted shallow source. Sparks and colleagues put the real figure near one cubic metre per second, which is 31.6 on this scale, so drag it there and see how little changes. Target eruption 1.0 km³ One cubic kilometre is a Pinatubo-scale eruption. Drag to 85 and you are asking the ground to rebuild the entire mountain. The Altiplano-Puna ignimbrites were hundreds to thousands of times bigger than that. Melt fraction of the mush 25 % The paper images at most 25% melt in the magma body. Push it to 100 and you are assuming a tank of pure liquid, which is the assumption that makes volcano headlines. Watch the filled marker slide left as you do. Years of inflation elapsed 33 33 years is the whole observational record, from the first radar measurement in 1992 to now. Drag right and watch how far the bar has to travel. The play button walks it for you.

accumulated: – years needed, all magma: – allowing for melt fraction: – at the long-run magma supply: –

inflation vs long-run supply: – if steady since 250 ka: –

–

At the defaults: 39 years of inflation for 1 km³ if all of it is magma
Allowing 25% melt: 157 years, which is 1.6 centuries
Long-run supply: 7,529 years, which is 75 centuries
The ratio: the inflation rate is 192 times the long-run magma supply
The reductio: steady since 250 ka gives 6,360 km³, 75× the whole volcano

Why the second bench matters more than the first

Model 1 is a shape-matching exercise and it has an honest limit: getting the shape right does not tell you what the shape is made of. Our two-source fit reproduces the sombrero beautifully and then demands a deflating source at 77 kilometres depth that has no obvious physical object to be. Fialko and Pearse hit the same factor of four on the real data, judged it implausible, and went looking for a different mechanism.

Model 2 is where the trouble for the magma story actually shows up. A Mogi source measures volume change. Volume change is not mass arriving, and a compressible fluid can change the volume of a pore network by expanding where it already is. The same volume rate costs about 61 million tonnes a year as dacitic magma and 5 to 10 million tonnes a year as supercritical water and carbon dioxide, and the elastic model cannot tell the two apart.

Which is the whole argument of the paper, arrived at from the other direction. Liu and colleagues went and looked, with 1,700 earthquakes and a tomographic inversion, and found gas and brine in a chimney where fresh magma would have to be.