Unbiased estimation of the fractal dimension of soil aggregate size distributions

The fractal dimension has been used to characterize the size distribution of soil aggregates produced by tillage. The fractal dimension is normally estimated from the slope of the cumulative number-size distribution plotted on a log-log scale, D1. It can also be estimated non-linearly from the cumulative number-size distribution, Dn1. In both methods, the fitting is done over the entire range of scales for which data are available. However, fractal models indicate deviations from ideal behavior during the initial stages of fragmentation. Inclusion of data from the early stages of fragmentation may result in biased estimates of D1 and Dn1. We used the derivative of the cumulative number-size distribution plotted on a log-log scale after the fourth stage of fragmentation as an unbiased estimator of the fractal dimension, Dd. Values of D1 Dn1 and Dd were compared for soil aggregate size distributions resulting from the fragmentation of dry clods collected from seven cropping/ tillage treatments and crushed at five specific energy levels. Estimates of the fractal dimension ranged from 1.179 to 2.803. The range in D1 was much wider than the ranges in Dn1 or Dd. No significant relationship existed between D1 and Dd. The relationship between Dn1 and D1 had an R2 of only 0.183 (P<0.05), and was significantly different from 1:1. In contrast, a highly significant 1:1 relationship existed between Dn1 and Dd (R2 = 0.965, P<0.01). These relations indicate estimates of D1 are biased towards data from the early stages of fragmentation. Since both Dn1 and Dd provided unbiased estimates of the fractal dimension, either could be used in future studies. Values of Dn1 and Dd were used to predict scale-invariant probabilities of failure for the different cropping/tillage treatments. The calculations indicated minimally tilled clods are more likely to fail than conventionally tilled clods when crushed to the same energy level. This result may be explained by differences in clod shape observed between tillage treatments.