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SPECIAL SECTION: FRACTALS

Limitations in Determining Multifractal Spectra from Pore–Solid Soil Aggregate Images

^a Dep. of Crop and Soil Sciences, Michigan State Univ., East Lansing, MI 48824-1325
^b Dpto. de Matemática Aplicada E.T.S.I. Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
^c APS/CARS-CAT, The Univ. of Chicago, c/o Argonne National Lab., 9700 S. Cass Ave., Bldg. 434/A004, Argonne, IL 60439
^* Corresponding author (kravche1{at}msu.edu).

Received 15 January 2008.

Multifractal methods have the potential to be useful tools for characterizing spatial distributions of soil pores from microtomographic images of undisturbed soil cores and soil aggregates. The objective of this study was to examine the limitations of multifractal analyses in binary (void and solid) soil images and to explore conditions under which multifractal spectra can be obtained. Multifractal characteristics of binary soil images are bounded within certain limiting values corresponding to nonfractal scaling. In this study, we first addressed the theoretical limitations of multifractal analysis of binary images and examined the nonfractal scaling boundaries in multifractal calculations by the method of moments. Then we developed boundary conditions for multifractal calculations by the direct method. Results revealed that fractal scaling is potentially possible only across a relatively narrow range of cell sizes restricted by the nonfractal scaling boundaries. Moreover, the range of cell sizes where fractal scaling is potentially possible varies with pore size. That is, in multifractal calculations it changes continuously with changes in the q value. For the soil aggregates examined in this study, this range varied from two to eight pixels for low q values to 128 pixels for high q values. The varying range makes calculations of true multifractal spectra for binary soil image data impossible. These results are consistent with a general theoretical notion that binary soil images are not multifractal in a strict mathematical sense. We suggest, however, that application of multifractal formalism can generate "pseudo-multifractal spectra" that might still be useful for summarizing pore distribution information and for comparing pore data among different agricultural management regimes and soil types.

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