What it does
Tephra fall deposits thin away from the vent. Mapping lines of equal thickness (isopachs) and measuring the area inside each gives pairs of thickness T and area A. Plotted as ln T against √A they fall close to straight lines, and integrating the fitted curve gives the deposit’s bulk volume. This tool fits the three standard models side by side:
- Exponential (Pyle 1989), with one, two or three straight segments (Fierstein and Nathenson 1992; Bonadonna and Houghton 2005). Segments join where their lines cross and the last one runs to infinity.
- Power law T = Tpl · x−m (Bonadonna and Houghton 2005), integrated between a proximal limit B and a distal limit C that you choose.
- Weibull T = θ (x/λ)n−2 exp(−(x/λ)n) (Bonadonna and Costa 2012), with volume 2θλ²/n.
It also gives the VEI class, the mass for your deposit density and the dense-rock-equivalent (DRE) volume.
How to use it
- Paste or type one isopach per line: thickness, then area (the whole area inside the line). Open a CSV or Excel file if you have one. Pick the units.
- Choose how many exponential segments to fit. The page tries every split and keeps the one with the smallest squared error in ln T.
- Set the power-law limits B and C as √area in km, and the deposit and magma densities.
- Read the volumes, VEI and mass in the table, look at the plot (thickness on a log scale against √area), and download the report or CSV.
How the volumes are worked out
With x = √A, the volume is V = ∫ T(x) 2x dx from 0 outwards. For one exponential segment T = T₀e−kx this is V = 2T₀/k²; each segment of a multi-segment fit adds 2T₀/k² [(1 + kx₁)e−kx₁ − (1 + kx₂)e−kx₂] between its ends. The thickness half-distance bt = ln 2 / (k√π) is shown for each segment. The Weibull parameters are fitted by least squares on ln T (θ is solved exactly for each λ and n, which are searched from many starting points).
Worked example
The example’s nine synthetic isopachs, from 128 cm over 1.5 km² to 0.5 cm over 902.9 km², bend at about 9 km. With two exponential segments (split after the fourth isopach, lines crossing at √A = 8.73 km) the bulk volume is 0.0485 km³, which is VEI 3. One segment gives 0.0514 km³ and three give 0.0532 km³. The Weibull fit gives 0.0500 km³ (λ = 10.9 km, n = 1.09), and the power law between 1 and 300 km gives 0.0900 km³, showing how much it depends on its limits. At 1000 kg/m³ the two-segment volume is 4.85 × 10¹⁰ kg, or 0.0194 km³ DRE at 2500 kg/m³.
Tips
- Fewer than about six isopachs can not support more than one segment; the fits will look fine but the volume beyond the last isopach is a guess.
- When the models disagree a lot, the deposit’s distal part is poorly known. Report the range.
- If the Weibull fit “runs off to a power law”, your points lie on a straight line on log-log axes; use the power law with sensible limits.
Frequently asked questions
Is VEI only about volume? VEI also uses column height and other observations; this tool assigns the class from bulk tephra volume alone.
Is my data uploaded? No. Everything is calculated in your browser and nothing leaves your device.
Where the numbers come from
- Methods (formulas only, no data copied): Pyle (1989) exponential thinning; Fierstein and Nathenson (1992) and Bonadonna and Houghton (2005) multi-segment exponential and power-law integration; Bonadonna and Costa (2012) Weibull function; Newhall and Self (1982) VEI volume classes.
- Example isopachs: synthetic, made by fibub from a two-segment exponential with a little noise. Densities: round values, editable.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.