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United States Department of Agriculture

Agricultural Research Service

Title: Fractal Techniques to Assess Spatial Variability in Soils

Authors
item Kravchenko, Alexandra - MICHIGAN STATE U.
item Pachepsky, Yakov

Submitted to: Soil-Water-Solute Process Characterization: An Integrated Approach
Publication Type: Book / Chapter
Publication Acceptance Date: January 8, 2004
Publication Date: January 12, 2004
Citation: Kravchenko, A.N., Pachepsky, Y.A. 2004. Fractal techniques to assess spatial variability in soils. Soil-Water-Solute Process Characterization: An Integrated Approach. pp. 617-638

Interpretive Summary: Understanding soil variability becomes imperative as the management tools become more sophisticated and soil uses become more multifaceted. A correct model of variability is necessary to interpret and use results of soil sampling. Fractal variability models have become popular in soil studies because they are designed to mimic and parameterize irregular objects that have similar features at different scales. Such similarity is common in soils which are naturally hierarchically organized. The objective of this book chapter is to introduce and demonstrate applications of fractal models to the variability that can customarily be found in data from grid sampling or from GIS. We begin with monofractal models for spatial variability in which consecutive variations have a dependence on each other. Examples of spatial variability in soil pH, organic carbon content, clay content, and phosphorus illustrate applications of such models. Then multifractal models are introduced that are suitable for systems exhibiting more variability as the observation scale becomes finer. Finally, fractal models for simulating spatial variability are presented as means to fill missing data or sparsely sampled area with the most probable values, to design sampling strategies, and to evaluate new soil sensors. The ability of fractal models to better simulate rare occurrences in soils is emphasized since rare occurrences, such as large pores, preferential pathways, and localized bacteria habitats, often define soil behavior at scales coarser than observational ones. Fractal models present the opportunity to quantify the spatial variability for further comparison and use of the variability parameters.

Technical Abstract: Quantifying soil variability is imperative because of its significance. As other soil properties, variability changes with scale of soil sampling or description. Recently fractal geometry has become an important source of scaling laws in soil science. The objective of this chapter is to present applications of fractal and multifractal models to soil spatial variability in "classical" sense, i.e. variations in soil properties measured with the same support. Two example data sets are used representing typical grid sampling, and typical GIS output. Monofractal models and parameters, including fractal dimension and Hurst exponent, are introduced based on the physical model of the fractal Brownian motion. Using semivariograms to estimate parameters of a fractal scaling is demonstrated. Soils may have more complex scaling behavior which can be simulated with multifractal models. We present different techniques to esimate parameters of multifractal models and caution about relying on a single technique. We emphasize the importance and suitability of fractal models for simulations of spatial variability, and demonstrate superiority of multifractal models in simulating variability of soil phosphorus and organic matter data. One important feature of fractal models is their ability to better simulate rare occurrences, such as large pores, preferential pathways and localized bacteria habitats, which define soil behavior at scales coarser than observational ones. Fractal models provide quantification of soil spatial variability for further comparison and use of the variability parameters.

Last Modified: 8/22/2014
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