Keywords:X-ray computed tomography of soil Image analysis Soil pore space Soil spatial variability Long-range dependence Multifractal analysis Hurst exponent A correct statistical model of soil pore structure can be critical for understanding flow and transport processes in soils, and creating synthetic soil pore spaces for hypothetical and model testing, and evaluating similarity of pore spaces of different soils. Advanced visualization techniques such as X-ray computed tomography (CT) offer new opportunities of exploring heterogeneity of soil properties at horizon or aggregate scales. Simple fractal models such as fractional Brownian motion that have been proposed to capture the complex behavior of soil spatial variation at field scale rarely simulate irregularity patterns displayed by spatial series of soil properties. The objective of this work was to use CT data to test the hypothesis that soil pore structure at the horizon scale may be represented by multifractal models. X-ray CT scans of twelve, water-saturated, 20-cm long soil columns with diameters of 7.5 cm were analyzed. A reconstruction algorithm was applied to convert the X-ray CT data into a stack of 1480 grayscale digital images with a voxel resolution of 110 microns and a cross-sectional size of 690 x 690 pixels. The images were binarized and the spatial series of the percentage of void space vs. depth was analyzed to evaluate the applicability of the multifractal model. The series of depth-dependent macroporosity values exhibited a welldefined multifractal structure that was revealed by singularity and Rényi spectra. The long-range dependencies in these series were parameterized by the Hurst exponent. Values of the Hurst exponent close to one were observed indicating the strong persistence in variations of porosity with depth. The multifractal modeling of soil macropore structure can be an efficient method for parameterizing and simulating the vertical spatial heterogeneity of soil pore space.

Abstract. Saturated hydraulic conductivity (Ksat) is an important soil parameter that highly depends on soil's particle size distribution (PSD). The nature of this dependency is explored in this work in two ways, (1) by using the information entropy as a heterogeneity parameter of the PSD and (2) using descriptions of PSD in forms of textural triplets, different than the usual description in terms of the triplet of sand, silt, and clay contents. The power of this parameter, as a descriptor of ln⁡Ksat, was tested on a database larger than 19 000 soils. Bootstrap analysis yielded coefficients of determination of up to 0.977 for ln⁡Ksat using a triplet that combines very coarse, coarse, medium, and fine sand as coarse particles; very fine sand, and silt as intermediate particles; and clay as fine particles. The power of the correlation was analysed for different textural classes and different triplets using a bootstrap approach. Also, it is noteworthy that soils with finer textures had worse correlations, as their hydraulic properties are not solely dependent on soil PSD. This heterogeneity parameter can lead to new descriptions of soil PSD, other than the usual clay, silt, and sand, that can describe better different soil physical properties, that are texture-dependent.

We investigated eleven particle-size distribution (PSD) models to determine the appropriate models for describing the PSDs of 16349 Chinese soil samples. These data are based on three soil texture classification schemes, including one ISSS (International Society of Soil Science) scheme with four data points and two Katschinski's schemes with five and six data points, respectively. The adjusted coefficient of determination r 2, Akaike's information criterion (AIC), and geometric mean error ratio (GMER) were used to evaluate the model performance. The soil data were converted to the USDA (United States Department of Agriculture) standard using PSD models and the fractal concept. The performance of PSD models was affected by soil texture and classification of fraction schemes. The performance of PSD models also varied with clay content of soils. The Anderson, Fredlund, modified logistic growth, Skaggs, and Weilbull models were the best.

We have developed a new type of fragmentation algorithm that was inspired by a theoretical question raised by A.N. Kolmogorov—and still unanswered after 60 yr—regarding the characteristics of fragment size distributions when the size of the fragments, rβ, exhibits a power‐law dependence on the size of the original material, r, with 0 ≤ β ≤ 1. Our fragmentation algorithm uses β and N (which denotes the number of particles produced in the fragmentation) as input parameters and was used for various simulations performed with N values of 2, 3, and 4 and β values from 0 to 1. Simulations with β = 0 resulted in lognormal distributions according to the Kolmogorov–Smirnov goodness‐of‐fit test at a confidence level of 95%. On the other hand, simulations with fractional values of β > 0 gave highly heterogeneous distributions exhibiting multifractal characteristics. Rényi dimensions (Dq) and Hölder exponents [α(q)] at q = 0 and 1 were defined with coefficients of determination R2 > 0.95 in 78.3% of samples. The resulting α(0), box dimension D0, and entropy dimension D1 = α(1) values spanned the ranges 0.55 to 1.82, 0.52 to 1, and 0.48 to 0.94, respectively, and were thus suggestive of a multifractal nature in the simulated fragment size distributions. The multifractal characteristics of the simulated distributions were consistent with similar analyses performed on actual soil particle size distributions. These results suggest that the new algorithm can be useful for modeling natural fragmentation processes.

Abstract. The notion of representative elementary area (REA) developed to address heterogeneity and scale problems in quantitative soil pedology comes from the notion of representative elementary volume of fluid dynamics in porous media. The REA allows the identification of the minimum area of a soil block section that is required to represent the pedofeature of interest based on its distribution in soil space. In this paper eight samples were imaged with two different techniques: the confocal microscope and the conventional film camera. These techniques provided information about pore sizes between 3.62 µm and 161.98 µm, and between 39.72 µm and 1776.34 µm, respectively. Sixteen of the resulting digital images were then analyzed to investigate the representative elementary area of the multifractal patterns of the spatial distribution of voids related to the micro and macroporosity by means of the entropy dimension. Our results permit the location of the REA region over the domain of the microstructures rendered by the analysis of the microscope images. They also suggest that this region seemingly spans scales of the macrostructures as revealed by the analysis of the camera pictures.