On the relation between number-size distributions and the fractal dimension of aggregates

Number-size distributions (i.e. particle- and aggregate-size distributions) have historically been used as indicators of soil structure, and recent work has aimed to quantify this link using fractals to model the soil fabric. This interpretation of number-size distributions is evaluated, and it is shown that a number-size relation described by a power law does not in itself imply fractal structure as suggested, and a counter example is presented. Where fractal structure is assumed, it is shown that the power-law exponent, φ, describing the number-size distribution cannot be interpreted as the mass-fractal dimension, D_M, of the aggregate. If the probability of fragmentation is independent of fragment diameter, then the exponent may be identified with the boundary dimension, D_B, of the original matrix. If, however, as is likely, this probability is scale-dependent, then φ may over- or under-estimate the boundary dimension depending on whether the fragmentation probability increases or decreases with fragment size. The significance of these conclusions is discussed in terms of the interpretation of number-size distributions, and alternative methods for quantifying and interpreting soil structure are evaluated.