Measure the thickness of a fall deposit at many places, contour it, and read the area enclosed by each isopach. Thickness falls off exponentially with the square root of that area, so a plot of log thickness against √A is a straight line — and the volume under an exponential surface has a closed form: V = 2T₀ / k².
The thickness half-distance is how far you go before the deposit is half as thick — a small one means a low column dropping everything nearby, a large one means a high column carrying ash far. A deposit whose points do not fall on one line usually needs two segments, and that break is real: it is the proximal fall separating from the distal ash cloud.
Bulk volume is not magma volume. Fall deposits are mostly void — divide by two or three to get dense rock equivalent, and say which one you are quoting.
Most deaths in a large ash fall are roof collapses, tens of kilometres away, in places where nothing else happened. The load is simply density × gravity × thickness — and wet ash weighs about twice what dry ash does, so the rain that follows an eruption is part of the hazard.
| condition | bulk density kg/m³ |
|---|
A collapsing column or dome runs out until it has fallen far enough to pay for the distance. The ratio of drop to run — H/L, the Heim coefficient — is roughly constant for a given kind of flow, so a line drawn at that angle from the top of the volcano marks everything the flow can reach. It is the crudest hazard map there is, and it works.
The same arithmetic answers a lahar down a valley and a rock avalanche off a cliff. Large volumes run further than their height alone allows — the ratio falls as the volume rises, which is why the biggest events are the ones that surprise people.
The Volcanic Explosivity Index is a logarithmic scale of erupted bulk volume, each step ten times the last above VEI 2. Like magnitude, one step is not one step.
| VEI | bulk volume | column | how often, somewhere |
|---|
An ash bed is also a time plane — the same layer in two sections is the same day, which is what makes tephra the best correlation tool in Quaternary geology. Log it in Measured sections, and date it in Dating a rock.