A modified number-based method for estimating fragmentation fractal dimensions of soils

Fractal theory has been applied to the characterization of particle- and aggregate-size distributions in soils. We used a number-based method for estimating fragmentation fractal dimensions from these distributions. This method has several inconsistencies. The objectives of our study were to: (i) propose a modified number-based method, (ii) evaluate the modified method using published data on particle- and aggregate-size distributions, and (iii) apply the modified method to a large particle-size distribution data base to analyze the validity of fractal scaling. Assuming scale-invariant fragmentation to be a valid model of particle-size distribution within size ranges of fractions, we derived a formula expressing the characteristic grain size as a function of the fractal dimension and limits of the grain size range. Parameters of Turcotte's fractal fragmentation model were found by minimizing the sum of squares of differences between measured and calculated masses of grain fractions. Comparison berween original and modified number-based methods showed that the modified method generally resulted in lower fragmentation fractal dimensions than the original method. The modified method was applied to a data set of particle-size distributions of 2600 soil samples. In 80% of samples, the fractal scaling was not applicable across the whole range of particle size between 0.002 and 1 mm, since errors of the fractal fragmentation model were statistically significantly larger than measurement errors, and estimates of the fractal dimension were larger than 3. It appears that models more sophisticated than scale-invariant fragmentation are required to simulate soil particle-size distributions.