Fractal geometry applied to soil and related hierarchical systems (original) (raw)

Fractals and Dynamic Systems in Geoscience

Journal of Structural Geology, 1996

Fractal packing and highly irregular shaped particles increase the mechanical properties of rocks and building materials. This suggests that fractal methods are good tools for modeling particle mixes with efficient properties like maximum strength and maximum surface area or minimum porosity and minimum permeability. However gradings and packings are calculated by "Euclidean" disk models and sphere models. Surprisingly even the simplest models are far more complex than they appear. The fractal "Appolonian packing model" is proposed as the most universal two-dimensional packing mode!. However the inhomogeneity of gradings and the irregularity of natural grain shapes and surfaces are not reflected by these models. Consequently calculations are often far from empirical observations and experimental results. A thorough quantification of packings and gradings is important for many reasons and still a matter of intense investigation and controversial discussion. This study concentrates on fractal models for densely packed non-cohesive rocks, crushed mineral assemblages, concrete and asphalt mixtures. A summary of fractal grain size distributions with linear cumulative curves on log-log plots is presented for these mixtures. It is shown that fractal two-dimensional and three-dimensional models for dense packings reflect different physical processes of material mixing or geological deposition. The results from shear-box experiments on materials with distinct grain size distributions show a remarkable increase of the mechanical strength from non-fractal to fractal mixtures. It is suggested that fractal techniques need more systematical application and correlation with results from material testing experiments in engineering geology. The purpose of future work should lead towards the computability of dense packings of angular particles in three dimensions.

Fractal Metrology for biogeosystems analysis

The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes of this complex system. In the present research, we studied the aggregation process as self-organizing and operating near a critical point. The structural pattern is extracted from the digital images of three soils (Chernozem, Solonetz and “Chocolate” Clay) and compared in terms of roughness of the gray-intensity distribution quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them measured in terms of standard deviation. Some of the applied methods are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc.) while the others have been recently developed by our Group. The combination of these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM). We show the usefulness of FM for complex systems analysis through a case study of the soil’s physical and chemical degradation applying the selected toolbox to describe and compare the structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.

Fractal Behaviour of Soil Physical and Hydraulic Properties

Physical and hydraulic properties of soils show spatial and temporal variability at different scales. The measurement and understanding of soil properties is essential to describe dominant hydrologic processes/parameters at various scales. Due to self - similar (or scale invariant) properties of fractals and their representation by a single parameter, fractal dimension, they have potential as a descriptive tool for scaling up various parameters. The objectives of this study was to illustrate fractal depiction of spatial behaviour of soil physical and hydraulic properties on a field scale, and to describe their anisotropic feature and possible impact on fractal dimension. The variogram plotted on a log - log scale was used to estimate fractal dimension of data collected from soil core samples and in-situ field measurements. The results indicate that soil properties show fractal behaviour and cannot capture anisotropic variability of soil properties on a field scale. There is a furthe...

Statistical properties of ecological and geologic fractals

Ecological Modelling, 1996

To use fractal models for ecological and geologic data, the statistical properties of fractals need to be clarified. No sampling or estimation theory for fractals currently exists. Several concrete steps in this direction are taken here. First, the information fractal dimension is proposed as a new measure that is relatively robust with respect to sampling error and can handle intensive data. The information fractal is tested with field data and is shown to be capable of delineating stratified structures and defining the scale of heterogeneity in the data. Comparison to semivariance analysis reveals the superiority of the fractal model for sample data that are nonisotropic and nonstationary. It is argued that approaches using regression to estimate fractal dimensions of spatial patterns are statistically invalid, and alternatives are proposed. Sampling of natural objects with transects (e.g., wells) is explored. For nonisotropic media (or maps), random placement of transects is shown to give an unreliable estimate of pattern. For transects taken perpendicular to a directional pattern (i.e., strata), it is shown that the mean of multiple estimates of the multiscale fractal dimensional profile does converge to the true value. Other sampling issues are addressed.

Fractal approach in characterization of spatial pattern of soil properties

EURASIAN JOURNAL OF SOIL SCIENCE (EJSS), 2017

The objective of the study was to characterize spatial pattern of soil properties (CaCO3, soil organic carbon, P2O5, K2O, and clay content) using fractal concept. Total of 141 topsoil samples (0-30 cm) were collected on 1850 ha in karst polje (Petrovo polje, Croatia) and analyzed for listed soil properties. The semi-variogram method was used to estimate fractal dimension (D) value which was performed from both of isotropic and anisotropic perspective. The D value of soil properties ranged between 1.76 to 1.97, showing a domination of the short-range variations. The SOC and K2O fractal D values 1.79 and 1.76 respectively, exhibited a spatial continuity at the entire analysed range of the scale. The D value for P2O5 (1.97) showed a nearly total absence of the spatial structure at all scales. The CaCO3 and clay content indicated a multifractal behavior mainly attributed to effects of alluviation, differences in geology and its spatial changes and transitions. The results of anisotropic analysis of soil properties pattern have showed strong relations with directions and partial self-similarity over limited ranges of scales defined by scale-break. Finally, our results showed that fractal analysis can be used as a appropriate tool for the characterization of spatial pattern irregularities of soil properties and detection of soil forming factors that cause it.

Spatial Pattern of Biological Soil Crust with Fractal Geometry

EGUGA, 2015

Soil surface characteristics are subjected to changes driven by several interactions between water, air, biotic and abiotic components. One of the examples of such interactions is provided through biological soil crusts (BSC) in arid and semi-arid environments. BSC are communities composed of cyanobacteria, fungi, mosses, lichens, algae and liverworts covering the soil surface and play an important role in ecosystem functioning. The characteristics and formation of these BSC influence the soil hydrological balance, control the mass of eroded sediment, increase stability of soil surface, and influence plant productivity through the modification of nitrogen and carbon cycle. This study focus on characterize the spatial arrangements of the BSC based on image analysis and fractal concepts. To this end, RGB images of different types of biological soil crust where taken, each image corresponding to an area of 3.6 cm2 with a resolution of 1024x1024 pixels. For each image and channel, mass dimension and entropy were calculated. Preliminary results indicate that fractal methods are useful to describe changes associated to different types of BSC. Further research is necessary to apply these methodologies to several situations.

Linear fractal analysis of three Mexican soils in different management systems

Soil Technology, 1997

The purpose of this study was to document the fractal nature of three soils of Mexico with contrasting genesis and marked differences in morphology and to estimate the fractal dimensions of their sets of aggregates and pores. These dimensions were estimated along lines and were called linear fractal dimensions. A single, 'ideal' fractal dimensionality was detected in the three soils studied. The soil linear fractal dimensions, calculated from macro and micromorphological data, had larger values than the dimension of the Cantor fractal dust model, but were less than unity. It was shown, that the fractal structure of the soil pore space could not be described by the same dimension as that of the aggregates. The linear fractal dimensions of soils of distinct genesis, were significantly different on all scales compared, but the differences fluctuated between 0.4% and 9.1%. 0 1997 Elsevier Science B.V.

The fractal dimension of pore distribution patterns in variously-compacted soil

Soil and Tillage Research, 1998

Pore-size distribution pattern signi®cantly alters many soil properties affecting water movement and root growth. The distribution is largely in¯uenced by soil compaction but information on how to describe this effect is very limited. In this study we used the fractal dimension to characterize pore distribution patterns in variously-compacted soil. The soil used was an Orthic Luvisol (Lublin Region, Poland). The various soil compaction was obtained by wheel traf®c treatments: unwheeled (L); moderately compacted, 3 tractor passes (MC); strongly compacted, 8 tractor passes (SC). Pore distribution patterns of all pores (>0.3 mm) and water-conducting pores were analyzed with an image analyzer and the two-dimensional fractal dimension was estimated. All pores were analyzed on the drawings obtained from the polished surfaces of soil blocks 8Â9Â2 cm. To analyze the water-conducting pores, soil cores were taken in cylinders of length 20 cm and diameter 21.5 cm from the plots on which methylene blue solution was applied. The pores were analyzed on horizontal cuts at 2 cm depth intervals. Mean values of fractal dimensions for all pores (D p 2) in the horizontal plane of surface soil in L, MC and SC were 1.69, 1.42 and 1.35, respectively. In the vertical plane, the corresponding values were 1.48, 1.35 and 1.29. In L the fractal dimension re¯ected pores of different size ranging from a few tenths of a millimetre to a dozen or so millimetres with rather smooth walls. In MC the contribution of large pores decreased whereas that of medium-sized pores considerably increased forming net-like patterns. However in SC, the largest area was of massive structure with longitudinal cracks and scarce and unevenly-distributed larger pores. The D p 2 was linearly correlated with areal porosity (R0.965) and the arithmetic mean of the areas of pores (R0.914). Mean values of fractal dimensions for the blue staining patterns (D s 2) in the plough layer ranged from 1.06 to 1.12 in L and MC whereas it decreased to 0.94 in SC. The wide range of D s 2 values for 2 cm layers of upper soil in L re¯ected the high variability of pore structures in this treatment. In the subsoil, the D s 2 varied from 1.03 to 1.09 and re¯ected mostly the distribution pattern of earthworm channels. The values of fractal dimensions for roots (D r 2) re¯ected different branching and root growth in variously-compacted soil. This study showed that fractal analysis provides a relevant quanti®cation of the changes of pore and root structure in relation to soil compaction. #

information, fractal, percolation and Geo-environmental Complexities

Recent theoretical developments in landscape studies have emphasized the relationship between pattern and processes while the effects of changes in spatial scale allowed us to extrapolate the information across the scales. The scientific breakthrough in nonequilibrium thermodynamics and its new ordering principles have deepened our understanding on the complexities and dynamics of the landscape that is far from equilibrium. New insights into environmental dynamics that have emerged from landscape studies, has lead the hypotheses to test in diversity of systems and at many scales. Different landscape indices reflect operating processes at different scales. Unlike others, landscape has organized complexities. The greater the complexity of the landscape, the lesser is its predictability. The functional complexities of the landscape are evident on spatial domain which could be distinguished from pattern development. The spatial pattern of the ecosystem type is a unique new phenomenon that arises at landscape level. The problem, therefore, is to detect and quantify pattern in the spatial heterogeneity of the landscapes. The approach is to develop and test a set of indices that capture important aspects of landscape pattern. Environmental complexities at landscape level thus could be determined by scale based system approach. The present work seeks to expand the concept of scale based system approach to understand environmental complexities by quantifying landscape pattern. The mathematical metrics used on spatial domain are information theory, fractal geometry and percolation theory. Information theory helps us to analyze the spatial heterogeneity of the landscape. Entropy values are determined to evaluate the non-regularity in the distributional pattern of different habitats in the landscapes. Fractal geometry measures the degree of disorder in the shape of the landscape elements while the connectivity of the landscape patches is determined by percolation theory.