Weibull-Based Tephra Decay Models

Inspired by Bonadonna & Costa (2013; https://doi.org/10.1007/s00445-013-0742-1).

About the model

• Estimate eruption plume height, and plume stability
• An approach to Mass eruption rate (MER)
• Understanding vent location and dispersal axis

1. Overview

This repository applies the Weibull (Rosin–Rammler) functional form to describe the spatial decay of three tephra-fall properties:

• Deposit thickness
• Maximum lithic size (ML)
• Median grain size (Mdϕ)

All three variables share the same mathematical structure, differing only in parameter sets.

2. Weibull Parameters

Each Weibull curve is defined by three parameters:

• θ : scaling constant (controls magnitude)
• λ : characteristic decay length scale (km)
• n : shape parameter (controls curvature and tail behavior)

These parameters reflect eruption intensity, column height, wind field, and particle settling dynamics.

3. Contour Types and Distance Proxy

Field measurements are extracted from three contour types:

• Isopachs → thickness (cm)
• Isopleths → maximum lithic size (cm)
• Isogrades → median grain size (ϕ)

Because contours enclose areas, the distance variable is defined as:

$$ x = \sqrt{A} $$

where A is the area enclosed by the contour. This provides a shape-independent proxy for radial distance.

4. Deposit Shapes and Eruption Styles

Different eruption styles produce distinct contour geometries:

• Plinian / Subplinian → elongated isopachs due to strong wind advection
• Vulcanian / small explosive → compact, near-circular isopachs
• Phreatomagmatic → irregular or lobate patterns due to variable column height
• Coarse clast isopleths → typically more circular (less wind influence)
• Mdϕ isogrades → smooth, intermediate shapes reflecting integrated fining trends

Using sqrt(A) normalizes these geometric differences into a single decay coordinate.

5. Weibull Equations

5.1 Thickness

$$ T(x) = \theta_{th} \left( \frac{x}{\lambda_{th}} \right)^{n_{th}-2} \exp\left[-\left( \frac{x}{\lambda_{th}} \right)^{n_{th}} \right] $$

5.2 Maximum Lithic Size

$$ ML(x) = \theta_{ML} \left( \frac{x}{\lambda_{ML}} \right)^{n_{ML}-2} \exp\left[-\left( \frac{x}{\lambda_{ML}} \right)^{n_{ML}} \right] $$

5.3 Median Grain Size

$$ Md\phi(x) = \theta_{Md\phi} \left( \frac{x}{\lambda_{Md\phi}} \right)^{n_{Md\phi}-2} \exp\left[-\left( \frac{x}{\lambda_{Md\phi}} \right)^{n_{Md\phi}} \right] $$

where Y is thickness, ML, or Mdϕ.

6. Interpretation

• λ increases with eruption size and dispersal power.
• n controls whether the curve behaves more like a power law (low n) or exponential (high n).
• θ scales the overall magnitude of the deposit property.

Rayleigh Case (n = 2)

When the Weibull shape parameter equals n = 2, the decay law reduces to the Rayleigh form. In this case, the general Weibull equation becomes:

$$ Y(x) = \theta \exp\left[-\left(\frac{x}{\lambda}\right)^2\right] $$

This applies to all three variables (thickness, maximum lithic size, and median grain size), since they share the same Weibull structure.

Implication

A Rayleigh-type decay indicates smooth, Gaussian-like thinning consistent with a well-mixed umbrella cloud. This behavior is typical of Plinian or strong Subplinian eruptions, where turbulent mixing dominates over wind distortion and ballistic settling.

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