Weibull Diameter Distribution
Weibull Diameter Distribution Model · Also known as: Weibull distribution, size-class distribution
The Weibull diameter distribution is a flexible three-parameter probability model used to describe the size-class distribution (proportion of trees by diameter class) in forest stands. Introduced by Bailey and Dell in 1973, the Weibull function provides an excellent fit to observed diameter distributions across diverse forest types and management histories. It is widely used in growth simulators, yield models, and forest inventory analysis because it can capture a variety of distribution shapes (right-skewed, near-normal, and even multi-modal) with just three parameters.
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When to use it
Use Weibull distributions when analyzing forest inventory data, parameterizing growth simulators, or projecting stand development. It is particularly valuable for even-aged plantations and naturally regenerated even-aged stands where diameter structure is relatively unimodal. The method works less well in uneven-aged multi-cohort forests where the diameter distribution is multi-modal or irregular. Essential when calibrating diameter-increment models or yield tables.
Strengths & limitations
- Highly flexible three-parameter model fitting most observed forest diameter distributions
- Mathematically tractable: closed-form equations for cumulative distribution, quantiles, and moments
- Simple to fit: standard maximum-likelihood and regression algorithms are stable and reliable
- Directly applicable to yield and growth models: parameterized distributions can be projected forward in time
- Well-validated across hundreds of forest types, species, regions, and management scenarios
- Low data requirements: only DBH measurements needed, no expensive instrumentation
- Assumes a single underlying distribution: fails in multi-aged stands with multiple tree cohorts
- Parameters must be re-estimated over time as the stand grows; temporal tracking is labor-intensive
- Does not capture zero-inflated or bimodal patterns common in stands with gaps or canopy layers
- Shape and scale parameters are correlated, making parameter uncertainty and confidence intervals difficult to interpret
- Fitting is sensitive to outliers (very large or very small trees); robust methods are needed
Frequently asked
What are the three Weibull parameters, and what do they control?
The location parameter (a) sets the minimum diameter; the scale parameter (b) determines the spread or width of the distribution; the shape parameter (c) controls skewness. Higher shape values produce more symmetric, bell-shaped distributions; lower values yield right-skewed distributions with a long tail of large trees. For forest stands, c typically ranges from 1 to 3.
How do I know if the Weibull fit is good enough?
Use a goodness-of-fit test such as chi-square or Kolmogorov-Smirnov to compare observed and predicted distributions. A non-significant p-value suggests the fit is acceptable. Visually inspect the histogram overlaid with the fitted Weibull curve; major discrepancies indicate a poor fit. If the fit is poor, consider alternative distributions (gamma, beta) or investigate whether your stand has multiple cohorts.
Can I fit separate Weibull distributions for different species?
Yes. If your stand is multi-species, fit separate distributions for each species or species group. This is common in tropical and mixed-species forests. Alternatively, fit a mixture of Weibull distributions (e.g., a weighted sum of two Weibull curves) if the stand has distinct structural layers or cohorts.
How do I project a diameter distribution forward in time?
Growth projection requires parameterizing a diameter-increment model (which predicts how much each diameter class grows annually) and a survival model (which predicts mortality rates). Apply these models to the diameter classes and re-fit the Weibull to the projected distribution. Most operational growth simulators handle this automatically.
Sources
- Bailey, R. L., & Dell, T. R. (1973). Quantifying diameter distributions with the Weibull function. Forest Science, 19(2), 97–104. DOI: 10.1093/forestscience/19.2.97 ↗
- Rennolls, K., Geary, D. N., & Rollinson, T. J. (1985). Characterizing diameter distributions by the use of the Weibull distribution. Forestry, 58(1), 57–66. DOI: 10.1093/forestry/58.1.57 ↗
How to cite this page
ScholarGate. (2026, June 3). Weibull Diameter Distribution Model. ScholarGate. https://scholargate.app/en/forestry/weibull-diameter-distribution
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