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Soil Physics

Brittle Fracture of Soil Aggregates

Weibull Models and Methods of Parameter Estimation

^a Dep. of Agroecology, Danish Institute of Agricultural Sciences, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark
^b Dep. of Earth and Planetary Sciences, Univ. of Tennessee, 1412 Circle Dr., Knoxville, TN 37996

^* Corresponding author (lars.munkholm{at}agrsci.dk)

Brittle fracture of soil aggregates is usually analyzed with the Weibull "weakest-link" model. Failure is expressed in terms of a probability distribution function (pdf) of aggregate strengths. Traditionally a two-parameter Weibull model is fitted to double log-transformed data with the Weibull parameters ( and ß) estimated using linear regression. The main objective of this study was to compare the goodness-of-fit for a three-parameter versus a two-parameter Weibull model. In addition, we compared three common methods of parameter estimation: linear regression, nonlinear regression, and maximum likelihood. The different models and methods of estimation were evaluated using previously published and unpublished aggregate rupture energy data from three contrasting soil types (Bygholm sandy loam, Maury silt loam, and Karnak silty clay). Overall, the goodness-of-fit was not markedly improved by using a three-parameter as compared with a two-parameter Weibull model. The choice of model had a significant effect on the parameter estimates. The three-parameter model produced lower estimates of ß than the two-parameter model. The data were always best fitted using nonlinear regression. Nonlinear regression also resulted in a greater power of distinction between management treatments and aggregate sizes for on the Maury soil. We recommend fitting aggregate rupture data to a two-parameter Weibull model and estimating the model parameters using nonlinear regression.

Abbreviations: AIC, Akaike's information criterion • ANOVA, analysis of variance • cdf, cumulative probability density function • D, Kolmogorov-Smirnov statistic • E, rupture energy • E₀, the location parameter—the value of E where the probability of failure is estimated to be zero • F, Fisher's F statistic • L, likelihood function • LIN, linear regression • LIN-2, linear regression, two-parameter Weibull model • ML, maximum likelihood • ML-2, maximum likelihood, two-parameter Weibull model • ML-3, maximum likelihood, three-parameter Weibull model • n, number of fits or samples • NLIN, nonlinear regression • NLIN-2, nonlinear regression, two-parameter Weibull model • NLIN-3, nonlinear regression, three-parameter Weibull model • P, probability • pdf, probability density function • R², coefficient of determination • RSS, residual sums of squares

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