Miguel Ángel Martín

Laser diffraction and multifractal analysis for the characterization of dry soil volume-size distributions

Abstract Fractal scalings of the type N ( x > X )≈ X − D f and M ( x X )≈ X d have been studied in previous works for the cumulative number and mass distributions of soil aggregates and particles of size x greater or less, respectively, than a characteristic size X . The exponents D f (fragmentation fractal dimension) and d represent rough estimates of the scale dependence that takes place in the number or mass accumulated, with respect to length scale. A closer look at the mass of soils (aggregates and primary particles) accumulated across the different regions of the interval of sizes would lead, in general, to detect denser or rarer regions across different length scales. These heterogeneity features are present in most real distributions in nature, displaying a behavior named multifractal . The laser scattering method and the multifractal analysis offer the possibility of studying the volume distribution of soils in scales that are not often explored. The goal of this paper is the application of laser diffraction and multifractal analysis to the characterization of dry volume-size distributions in soils. For each of the 20 soil samples considered for the study, an adequate measure is constructed with data corresponding to laser diffraction analysis of those samples. Multifractal analysis techniques are applied to such measures. Volume-size distribution of soils showed, in general, suitable scaling properties that make multifractal analysis a useful mathematical tool for its characterization by means of the singularity spectrum f ( α ( q )). Samples showed great variability in their multifractal behavior. An adequate range of the parameter q is selected for the characterization of each sample, in such a way that the R 2 values of the corresponding scaling fits are greater than 0.95. In that range, different features of the singularity spectrum f ( α ( q )) have been analyzed. The singularity spectrum f ( α ( q )) reveals to be useful for the characterization of soil volume-size distributions.

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

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.

Simulation and testing of self-similar structures for soil particle-size distributions using iterated function systems

Abstract Particle-size distribution (PSD) is a fundamental soil physical property. The PSD is commonly reported in terms of the mass percentages of sand, silt and clay present. A method of generating the entire PSD from this limited description would be extremely useful for modeling purposes. We simulated soil PSDs using an iterated function system (IFS) following Martin and Taguas [ Martin, M.A., Taguas, F.J., 1998 . Fractal modeling, characterization and simulation of particle-size distribution in soil. Proc. R. Soc. Lond. A 454, 1457–1468]. By means of similarities and probabilities, an IFS determines how a fractal (self-similar) distribution reproduces its structure at different length scales. The IFS allows one to simulate intermediate distributional values for soil textural data. A total of 171 soils from SCS [ Soil Conservation Service, 1975 . Soil taxonomy: a basic system of soil classification for making and interpreting soil surveys. Agricultural Handbook no. 436. USDA-SCS, USA, pp. 486–742] were used to test the ability of different IFSs to reconstruct complete PSDs. For each soil, textural data consisting of the masses in eight different size fractions were used, and different PSDs were predicted using different combinations of three similarities. The five remaining data points were then compared with the simulated ones in terms of mean error. Those similarities that gave the, lowest mean error were identified as the best ones for each soil. Fifty-three soils had an error less than 10%, and 120 had an error less than 20%. The similarities corresponding to the sand, silt and clay fractions, i.e., IFS {0.002, 0.05}, did not, in general, produce good results. However, for soils classified as sand, silt loam, silt, clay loam, silty clay loam and silty clay, the same similarities always produced the lowest mean error, indicating the existence of a self-similar structure. This structure was not the same for all classes, although loams and clays were both best simulated by the IFS {0.002, 0.02}. It is concluded that IFSs are a powerful tool for identifying self-similarity in soil PSDs, and for reconstructing PSDs using data from a limited number of textural classes.

SINGULARITY FEATURES OF PORE-SIZE SOIL DISTRIBUTION: SINGULARITY STRENGTH ANALYSIS AND ENTROPY SPECTRUM

In this paper, several features of pore-size soil distribution are first analyzed, suggesting that they are closer to those of singular measures than to those of distributions with smooth density. In a second step, the weighted singularity strength of an experimental measure obtained by image analysis of soil samples is evaluated. The results of this analysis show the singular nature of pore-size distribution. Finally, the distribution is characterized by means of a spectrum of entropies computed on distorted measures associated with the original experimental measure.

On the role of Shannon's entropy as a measure of heterogeneity

Ž . Ibanez et al. 1998 proposed the use of Shannon’s entropy to analyze the ̃ diversity of the world pedosphere on the basis of data compiled by the F.A.O. at the scale 1:5,000,000. Here we will try to provide some mathematically founded arguments to justify the use and interpretation of Shannon’s information entropy as a measure of diversity and homogeneity. Information entropy h is computed from a discrete probability distribution 4 p : is1, 2, . . . , N via Shannon’s formula hsyÝ p log p . This quantity i i i i was originally proposed by Shannon as a measure of the average information content that is gained from observing the realization of an experiment with N possible outcomes with probabilities of occurence given by p , p , . . . , p . 1 2 N Ž . Well-known mathematical facts are that a h attains its maximum value log N Ž . only in the equiprobable case, that is p s1rN for all i’s, and b h vanishes in i Ž . the case that some p s1 and thus p s0 if i/ j . These two extreme j i Ž . Ž . situations, respectively, correspond with a the most informative case and b Ž . the least informative case, since observing the actual outcome provides a much Ž . rich information being all outcomes equally probable and b very poor information in the case that outcome j has all the chances to occur. Also, the number h depends continuously on the probabilities p so that similar distributions render i close values of h.

Hölder spectrum of dry grain volume-size distributions in soil

Abstract Fractal parameters have been applied before to describe number and mass size distributions of particles and aggregates through power scaling laws. Exponents such as the fragmentation fractal dimension, D f , represent rough estimates of the scale dependence present in the number or mass accumulated with respect to length scale. However, a deeper and more accurate study of these distributions may be done using multifractal analysis techniques when a large enough amount of data is available. The light scattering method and the multifractal analysis enable the study of these distributions in scales not very often explored. In the present work, attention is focused on the Holder exponents spectrum , α ( q ), and its capability to characterize dry grain volume-size distributions in soil. Twenty samples were analyzed by laser diffraction and Holder spectrum analyses, showing a great variability in Holder exponent results. Coefficients of determination have been considered to study and characterize scalings achieved. Holder spectrum analysis revealed to be useful in the characterization of these distributions as they showed suitable scaling properties in general, producing the best fits for the entropy dimension ( D 1 ). Thus, the combination of laser diffraction and Holder spectrum analyses is a potential tool for detecting changes in soil grain distributions due to physical or degradation processes. The similarity of these results to those obtained from mathematical multifractal measures suggests that this way of characterizing dry soil grain distributions may be used to simulate empirical data by means of mathematical algorithms based in fractal geometry (Iterated Function Systems and related ones).

AN INTRODUCTION TO FLOW AND TRANSPORT IN FRACTAL MODELS OF POROUS MEDIA: PART I

This is the second part of the special issue on fractal geometry and its applications to the modeling of flow and transport in porous media, in which 10 original research articles and one review article are included. Combining to the first part of 11 original research articles, these two issues summarized current research on fractal models applied to porous media that will help to further advance this multidisciplinary development. This whole special issue is published also to celebrate the 70th birthday of Professor Boming Yu for his distinguished researches on fractal geometry and its application to transport physics of porous media.

BALANCED ENTROPY INDEX TO CHARACTERIZE SOIL TEXTURE FOR SOIL WATER RETENTION ESTIMATION

Soil hydraulic parameters are needed in most projects on transport and fate of pollutants. Pedotransfer procedures are often used to estimate soil hydraulic properties from basic soil data available from soil surveys. Soil particle size distribution, or texture, is known to be a leading soil property affecting soil ability to retain and transmit water and solutes. A substantial effort has been put in searching for texture parameters useful for estimating soil hydraulic properties. Recently a new, entropy-based index has been proposed that characterizes the non-evenness of particle size distributions. This index called balanced entropy has a potential to reflect probable packing of soil particles. Our objective was to see whether the balanced entropy can serve along with other basic soil properties as one of variables-predictors of soil water retention. We computed the balanced entropy for 9871 soil samples in the NRCS soil characterization database and applied the data mining tools to estimate water retention from soil textural composition, organic carbon content, and bulk density. The balanced entropy was the best single predictor and the most important predictor of volumetric water contents at −33 kPa, which are notoriously difficult to estimate. Using the balanced entropy is a promising approach to improve the accuracy of estimated soil hydraulic properties.

QUANTIFYING THE RELATIONSHIP BETWEEN DRAINAGE NETWORKS AT HILLSLOPE SCALE AND PARTICLE SIZE DISTRIBUTION AT PEDON SCALE

Nowadays, translating information about hydrologic and soil properties and processes across scales has emerged as a major theme in soil science and hydrology, and suitable theories for upscaling or downscaling hydrologic and soil information are being looked forward. The recognition of low-order catchments as self-organized systems suggests the existence of a great amount of links at different scales between their elements. The objective of this work was to research in areas of homogeneous bedrock material, the relationship between the hierarchical structure of the drainage networks at hillslope scale and the heterogeneity of the particle-size distribution at pedon scale. One of the most innovative elements in this work is the choice of the parameters to quantify the organization level of the studied features. The fractal dimension has been selected to measure the hierarchical structure of the drainage networks, while the Balanced Entropy Index (BEI) has been the chosen parameter to quantify the heterogeneity of the particle-size distribution from textural data. These parameters have made it possible to establish quantifiable relationships between two features attached to different steps in the scale range. Results suggest that the bedrock lithology of the landscape constrains the architecture of the drainage networks developed on it and the particle soil distribution resulting in the fragmentation processes.

COMPUTER SIMULATION OF RANDOM PACKINGS FOR SELF-SIMILAR PARTICLE SIZE DISTRIBUTIONS IN SOIL AND GRANULAR MATERIALS: POROSITY AND PORE SIZE DISTRIBUTION

A 2D computer simulation method of random packings is applied to sets of particles generated by a self-similar uniparametric model for particle size distributions (PSDs) in granular media. The parameter p which controls the model is the proportion of mass of particles corresponding to the left half of the normalized size interval [0,1]. First the influence on the total porosity of the parameter p is analyzed and interpreted. It is shown that such parameter, and the fractal exponent of the associated power scaling, are efficient packing parameters, but this last one is not in the way predicted in a former published work addressing an analogous research in artificial granular materials. The total porosity reaches the minimum value for p = 0.6. Limited information on the pore size distribution is obtained from the packing simulations and by means of morphological analysis methods. Results show that the range of pore sizes increases for decreasing values of p showing also different shape in the volume pore size distribution. Further research including simulations with a greater number of particles and image resolution are required to obtain finer results on the hierarchical structure of pore space.