Isopach Thinning Analysis & Tephra Volume Estimation

A browser-based tool for fitting isopach thinning curves and estimating tephra-fall deposit volumes

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Overview

Tephra deposits thin systematically with distance from the vent. Plotting $\sqrt{A}$ (km) against thickness $T$ (cm) on a semi-log axis reveals this thinning structure, and integrating the curve yields the total erupted volume. This tool fits four models simultaneously, reporting fitted equations, $R^2$, and volume in km³ or m³.

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Background

The x-axis uses $\sqrt{A}$ — the square root of the area enclosed by each isopach contour — rather than distance, so no assumption about isopach shape (circular, elliptical, irregular) is required (Fierstein & Nathenson 1992). The volume integral is:

$$V = \int_0^{\infty} T , dA$$

Model	Equation	Volume
Exponential (1-seg)	$T = T_0 \exp(-k\sqrt{A})$	$V = \dfrac{2T_0}{k^2}$
Exponential (2-seg) proximal	$T = T_0 \exp(-k_1\sqrt{A}), \quad \sqrt{A} \le \sqrt{A_{\rm ip}}$	$V_{\rm prox} = \dfrac{2T_0}{k_1^2}\left[1 - (k_1\sqrt{A_{\rm ip}}+1)\exp(-k_1\sqrt{A_{\rm ip}})\right]$
Exponential (2-seg) distal	$T = T_1 \exp(-k_2\sqrt{A}), \quad \sqrt{A} > \sqrt{A_{\rm ip}}$	$V_{\rm total} = \dfrac{2T_0}{k_1^2} + 2T_0\left[\dfrac{k_2\sqrt{A_{\rm ip}}+1}{k_2^2} - \dfrac{k_1\sqrt{A_{\rm ip}}+1}{k_1^2}\right]\exp(-k_1\sqrt{A_{\rm ip}})$
Power law	$T = T_{\rm pl},(\sqrt{A})^{-m}$	$V = \dfrac{2T_{\rm pl}}{2-m}\left[C^{2-m}-B^{2-m}\right]$
Weibull	$T = \theta\exp\left[-(\sqrt{A}/\lambda)^n\right]$	$V = \dfrac{2\theta\lambda^2}{n},\Gamma\left(\dfrac{2}{n}\right)$

$R^2$ is computed on $\ln T$ residuals. Power-law integration requires finite limits $B$ (proximal, default 0.3 km) and $C$ (distal, default 1 000 km).

Symbol glossary

Symbol	Description
$T$	Deposit thickness (cm)
$A$	Area enclosed by an isopach contour (km²)
$\sqrt{A}$	Square root of isopach area; used as the distance proxy (km)
$T_0$	Extrapolated thickness at $\sqrt{A} = 0$ for the proximal segment (cm)
$k$	Exponential decay rate of the 1-segment fit (km⁻¹); steeper $k$ = faster thinning
$k_1$	Exponential decay rate of the proximal segment in the 2-segment fit (km⁻¹)
$k_2$	Exponential decay rate of the distal segment in the 2-segment fit (km⁻¹); typically $k_2 < k_1$
$T_1$	Extrapolated thickness at $\sqrt{A} = 0$ for the distal segment (cm)
$\sqrt{A_{\rm ip}}$	Break-in-slope: value of $\sqrt{A}$ where the two exponential segments intersect (km)
$V_{\rm prox}$	Volume contributed by the proximal segment alone, from $\sqrt{A} = 0$ to $\sqrt{A_{\rm ip}}$ (km³)
$V_{\rm total}$	Total deposit volume integrating both proximal and distal segments (km³)
$T_{\rm pl}$	Power-law scaling coefficient (cm)
$m$	Power-law exponent (dimensionless); controls rate of thinning with $\sqrt{A}$
$B$	Proximal integration limit for the power-law model (km); approximates vent radius
$C$	Distal integration limit for the power-law model (km); approximates maximum dispersal
$\theta$	Weibull thickness scale; maximum thickness extrapolated to $\sqrt{A} = 0$ (cm)
$\lambda$	Weibull characteristic dispersal length (km); distance at which thickness drops to $\theta/e$ when $n = 1$
$n$	Weibull shape parameter (dimensionless); $n = 1$ recovers the exponential, $n < 1$ gives a heavier distal tail
$\Gamma$	Gamma function: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t},dt$
$R^2$	Coefficient of determination computed on $\ln T$ residuals; measures goodness of fit on the log scale

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Entering your data

Up to 10 isopach pairs can be entered manually:

Field	Unit	Requirement
T	cm	> 0
A	km²	> 0

Row order does not matter — data are sorted by $\sqrt{A}$ internally. Four presets are available for immediate exploration: Mt. St. Helens 1980, Ruapehu 1996, Hatepe 186 A.D., and Quizapu 1932.

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References

Bonadonna, C. & Costa, A. (2012). Estimating the volume of tephra deposits: A new simple strategy. Geology, 40, 415–418. https://doi.org/10.1130/G32769.1

Bonadonna, C. & Houghton, B. F. (2005). Total grain-size distribution and volume of tephra-fall deposits. Bulletin of Volcanology, 67, 441–456. https://doi.org/10.1007/s00445-004-0386-2

Fierstein, J. & Nathenson, M. (1992). Another look at the calculation of fallout tephra volumes. Bulletin of Volcanology, 54, 156–167. https://doi.org/10.1007/BF00278005

Pyle, D. M. (1990). New estimates for the volume of the Minoan eruption. Thera and the Aegean world III, 2, 113-121.

Pyle, D. M. (1989). The thickness, volume and grainsize of tephra fall deposits. Bulletin of Volcanology, 51, 1–15. https://doi.org/10.1007/BF01439773

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License

Released under the MIT License. Free to use, modify, and redistribute with attribution.

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