Revival: Fractals in Soil Science (1998)

Advances in Soil Science

Philippe Baveye, Jean-Yves Parlange, B.A. Stewart, B.A. Stewart

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351365680
• Edition: 1
• Topic: Technik & Maschinenbau
• Subtopic: Landwirtschaft

The application of fractals and fractal geometry in soil science has become increasingly important over the last few years. This self-contained and timely book was designed to provide detailed and comprehensive information on the current status of the application of fractal geometry in soil science, and on prospects for its future use. With a detailed and specific introductory chapter, particular attention is paid to comparing and contrasting "fractal" and "fragmentation" concepts. Some uses of fractals, such as to quantify the retention and transport properties of soils, to describe the intricate geometry of pore surfaces and macropore networks, or to elucidate the rooting patterns of various plants, are discussed. Applications of fractals in soil science are both relatively recent and in constant evolution. This book reflects accurately existing trends, by allowing sharp differences among the viewpoints expressed in contributed chapters to be presented to the reader in one self-contained volume.

Fractal Geometry, Fragmentation Processes and the Physics of Scale-Invariance: An Introduction

P. Baveye and C.W. Boast

Contents

1. I. Introduction
2. II. A Gallery of Mathematical Monsters
   1. A. Cantor's Set and the Devil's Staircase
   2. B. The Peano-Hilbert Plane-Filling Curves
   3. C. The Triadic von Koch Curve
   4. D. The Sierpinski Gasket and Carpet
   5. E. The Menger Sponge
   6. F. Bolzano-Weierstrass-Like Functions
   7. G. Random Monsters and Fractional Brownian Motion
3. III. Hausdorff Dimension and Alternatives
   1. A. Hausdorff Measure and Dimension
   2. B. Similarity Dimension
   3. C. Box-Counting Dimension
   4. D. Divider Dimension
   5. E. Pointwise Dimension and Hölder Exponent
   6. F. Correlation Dimension
   7. G. An Arsenal of Dimensions
4. IV. Fractals
   1. A. Physical Motivation
   2. B. "Definition" of Fractals
   3. C. "Natural" versus Mathematical Fractals
5. V. Is "Power Law" Equivalent to "Fractal"?
   1. A. Power Law or Paretian Distribution and Fractals
   2. B. Fragmentation Fractals
6. VI. The Physics of Fractals
   1. A. Going Beyond Fractal Dimensions
   2. B. Illustration: Diffusion-Limited Aggregation (DLA)
7. VII. Multifractal Measures
   1. A. The Binomial Fractal Measure
   2. B. Parameterization of Multifractal Measures
   3. C. Alternative Definitions of Multifractal Measures
   4. D. Beyond the Binomial Measure
8. Acknowledgments
9. References

I Introduction

Since its formal introduction in 1975, the concept of fractal has captured the imagination, and has entered into the toolbox, of many scientists in a wide range of fields. Papers discussing fractals in various contexts, including geophysics and soil science, now appear almost daily. A rich and constantly growing panoply of textbooks describes the multiple facets of the theory of fractals.

In this flurry of publications, one of the most striking features is the extreme variety that exists in the mathematical background assumed of the readership. In some articles and textbooks, little or no knowledge of mathematics is needed; these are typically publications that emphasize the graphical aspects of fractals, the "hypnotically intricate visual patterns and images" (Jones, 1993) that one may base on fractals, or the use of fractals to describe natural systems. Contrastedly, another set of papers and books is grounded in the belief that much if not all of the "beauty of fractals" is to be found in their mathematics (Falconer, 1990). These publications usually require of the readers a solid background in geometry and set theory.

Rare are the articles or books that analyze the connection between the mathematical beings defined and manipulated by the geometricians, and the "natural" fractals identified in the real world. Reluctance to embark on this analysis appears to have caused much confusion in the literature; the term "fractal", when applied to natural systems, often means different things to different people, creating unnecessary difficulties in communication. A first objective of the present chapter, therefore, is to attempt to fill this gap and to make more explicit the connection between theoretical and natural fractals. A similar approach is followed in the last section of this chapter, which deals briefly with the increasingly important multifractal measures.

Another area in which researchers in geophysics and soil science have as yet seldom ventured deals with the reasons for the fractal behavior exhibited by natural objects. From a descriptive standpoint, it is of great interest to be able to look at an object or material as a fractal and to associate with it a particular fractal dimension, or a number of characteristic parameters if the object is best described as a multifractal. In some cases, this may result in significant advances in our understanding of the range of possible responses of the object or material to various imposed stimuli. However, one would ideally also like to know why this object or material exhibits a fractal behavior. Section VI of the present chapter briefly mentions various processes that, in a number of physical systems, have been shown to display a fractal behavior. One of these processes, diffusion-limited aggregation, is described in some detail for illustrative purposes.

This chapter is not meant to be a treatise on fractal geometry. Its objective is to introduce the aspects of fractal theory that have found application...