Fractal modelling, characterization and simulation of particle-size distributions in soil

Abstract

A modelling of particle-size distribution in soil (PSD) by means of the fractal mass distribution is presented. The model is based on a new interpretation of the invariance of PSD with respect to the scale.It is shown that the modelized PSD can be mathematically determined from soil textural data. Combining some well-founded theoretical results from fractal geometry, the model allows us to simulate the PSD of a given soil and its characterization by means of the entropy dimension. The scaling behaviour of mass-size (or particlenumber) distribution empirically shown by different authors, is obtained from the model as a theoretical result.The model allows the testing of the degree of self-similarity of soil PSDs and can be used for predicting soil properties related to the PSD, as well as the characterization of soil textures.

Land-Use Conversion Changes the Multifractal Features of Particle-Size Distribution on the Loess Plateau of China

“…At every size scale ε, a number N (ε) = 2 k of cells are considered, and their respective measures, μ i (ε), are supplied by the available data. For PSDs, μ i (ε) in each subinterval of sizes represented the relative volume of soil particles of characteristic size in the subinterval [20,32]. …”

On soil textural classifications and soil-texture-based estimations

“…and so on (i, j, k = 1, 2, 3), is scale invariant. The set of textural data, together with the entropy self-similarity assumption, unequivocally determines the PSD (Martín and Taguas, 1998). Based on the theorem of Elton (1987), the mass of soil with size particles within an interval J may be computed using the IFS as follows: (a) take any starting value x 0 in I and (b) choose, at random, an integer number i of the index set 1, 2, 3 with probability p i and denote with x 1 the value ϕ i (x 0 ).…”

“…2.2 Reconstruction of the particle size distributions from data on textural fraction content The reconstruction of the particle size distribution (PSD) is based on the assumption that entropy as the measure of heterogeneity of these distributions is preserved across the support scales (Martín and Taguas, 1998). Assuming that the texture interval is divided into k textural size ranges and the respective textural fraction contents p 1 , p 2 , ..., p k , 1 ≤ i ≤ k, and k i=1 p i = 1, the Shannon information entropy (IE) (Shannon, 1948) is defined by…”

“…Martín and Taguas (1998) proposed to use the hypothesis of self-similarity and iterated function formalism to generate the particle size distribution from small number of textural fractions. In applications of this technique, they used sand, silt, and clay fraction contents with size boundaries defined by the USDA textural classification.…”

Reply to referee #3

“…2.2 Reconstruction of the particle size distributions from data on textural fraction content 5 The reconstruction of the particle size distribution (PSD) is based on the assumption that the entropy as the measure of heterogeneity of these distributions is preserved across the support scales (Martín and Taguas, 1998). Assuming the texture interval divided into k textural size ranges and that the respective textural fraction contents p 1 , p 2 , .…”