Weibull form: Y(x) = θ · (x/λ)^(n−2) · exp[−(x/λ)^n]
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λ increases with eruption size & dispersal power
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n controls curve shape: low n → power-law-like · high n → exponential-like
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θ scales the overall magnitude of the deposit property
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Plume height: Ht = 5.01 · λML0.55 (R²=0.93, Eq. 7)
Rayleigh Case (n = 2) — When n = 2 the general law reduces to the Rayleigh form:
Y(x) = θ · exp[−(x/λ)²].
Applies equally to thickness, max lithic size and Mdϕ, as all share the same Weibull structure.
A Rayleigh decay indicates smooth, Gaussian-like thinning consistent with a well-mixed umbrella cloud —
typical of Plinian or strong Subplinian eruptions where turbulent mixing dominates over wind distortion and ballistic settling.
★ marks are shown in the table where n ≈ 2.
Plot I — Deposit Thinning
Thickness (cm) vs √Area (km)
Plot II — Median Grain Size
Mdϕ (cm) vs √Area (km)
Plot III — Max Lithic Size & Plume Height Estimator
Max. lithic size ML (cm) vs √Area (km)
Literature Examples — click to activate / deactivate
Eruption
VEI
λth
θth
nth
λML
θML
nML
Ht (km)
Style
λ in km · θ in cm · ★ n ≈ 2 (Rayleigh case) · Source: Tables 1 & 4, Bonadonna & Costa (2013)
Plume Height Estimator
Enter max lithic clast size (cm) at each √isopleth area (km).
Fits Weibull curve → derives λML → estimates Ht via Eq. 7.
Minimum 3 points required.
cm · range 1–20
range 0.5–2.0
Fitted Parameters & Result
λML (dispersal scale)—
θML (magnitude scale)—
n (shape)—
Plume height Ht—
Ht ± 25 % range—
★ n ≈ 2 — Rayleigh case: smooth Gaussian-like thinning. Well-mixed umbrella cloud.
Consistent with Plinian / strong Subplinian dynamics.