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Fractal Characterization of Aggregate-Size Distribution: The Question of Scale Invariance

USDA-ARS, National Soil Tilth Lab., 2150 Pammel Dr., Ames, IA 50011

D. Gimenez

USDA-ARS, Hydrology Lab., Bldg. 007, Rm. 104, BARC-West, Beltsville, MD 20705

R. R. Allmaras

USDA-ARS, 439 Borlaug Hall, Univ. of Minnesota, St. Paul, MN 55108

*Corresponding author (logsdon{at}nstl.gov).

ABSTRACT

Aggregate-size distributions are often described as fractal based on the power law relationship between cumulative aggregate number and aggregate size. The slope of the natural log of cumulative number of aggregates as a function of natural log of diameter is the fractal dimension. If aggregate density, shape, and relative diameter (diameter as a fraction of the smallest diameter of the size-class) are scale invariant, then aggregate number can be calculated from mass. These three factors can be combined into a single unknown factor, G_i. The objectives of this study were to test the assumption of a scale invariant G_i and to compare the fractal dimension from calculated cumulative number of aggregates with the fractal dimension from counted cumulative number of aggregates. Numbers of aggregates were counted for each size class of two data sets. Calculations for the 48 samples of Data Set I spanned six classes ranging from sizes 1 to 32 mm, and calculations for 12 samples of Data Set 2 spanned four classes ranging from sizes 0.5 to 8 mm. The fractal dimension from counted cumulative number of aggregates was significantly smaller than the fractal dimension determined from calculated cumulative number of aggregates for Data Set 1 (2.44 vs. 2.51) but significantly larger for Data Set 2 (2.41 vs. 2.03). The G_i factor was significantly different across many of the size classes for both data sets. Since G_i consisted of subcomponents (density, shape, relative diameter), we cannot be certain which subcomponent(s) was (were) scale variant. Scale variant G_i complicates the use of fractal mathematics to describe dry soil aggregate distributions.

Received for publication April 17, 1995.

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