Abstract
Particle-size distribution (PSD) is a fundamental soil property usually reported as discrete clay, silt, and sand percentages. Models and methods to effectively generate a continuous PSD from such poor descriptions using another property would be extremely useful to predict and understand in fragmented distributions, which are ubiquitous in nature. Power laws for soil PSDs imply scale invariance (or selfsimilarity), a property which has proven useful in PSD description. This work is based on two novel ideas in modeling PSDs: (1) the concept of selfsimilarity in PSDs; and (2) mathematical tools to calculate fractal distributions for specific soil PSDs using few actual texture data. Based on these ideas, a random, multiplicative cascade model was developed that relies on a regularity of scale invariance called ‘log-selfsimilarity.’ The model allows the estimation of intermediate particle size values from common texture data. Using equivalent inputs, this new modeling approach was checked using soil data and shown to provide greatly improved results in comparison to the selfsimilar model for soil PSD data. The Kolmogorov-Smirnov D-statistic for the log-selfsimilar model was smaller than the selfsimilar model in 92.94% of cases. The average error was 0.74 times that of the selfsimilar model. The proposed method allows measurement of a heterogeneity index, H, defined using Hölder exponents, which facilitates quantitative characterization of soil textural classes. The average H value ranged from 0.381 for silt texture to 0.838 for sandy loam texture, with a variance of <0.034 for all textural classes. The index can also be used to distinguish textures within the same textural class. These results strongly suggest that the model and its parameters might be useful in estimating other soil physical properties and in developing new soil PSD pedotransfer functions. This modeling approach, along with its potential applications, might be extended to fine-grained mineral and material studies.
Similar content being viewed by others
References
Anderson, A.N., McBratney, A.B., and Crawford, J.W. (1998) Applications of fractals to soil studies. Advances in Agronomy, 63, 1–76.
Arya, L.M. and Paris, J.F. (1981) A physicoempirical model to predict the soil moisture characteristic from particle-size distribution and bulk density data. Soil Science Society of America Journal, 45, 1023–1030.
Caniego, F.J., Martín, M.A., and San José, F. (2001) Singularity features of pore-size soil distribution: singularity strength analysis and entropy spectrum. Fractals, 9, 305–316.
Chhabra, A. and Jensen, R.V. (1989) Direct determination of the α singularity spectrum. Physical Review Letters, 62, 1327–1330.
DeGroot, M.H. and Schervish, M.J. (2002) Probability and Statistics (3rd edition). Addison-Wesley, New York.
Everstz, C.J.G. and Mandelbrot, B.B. (1992) Multifractal measures. Pp. 921–953 in: Chaos and Fractals (H. Peitgen, H. Jürgens, and D. Saupe, editors). Springer, Berlin.
Falconer, K.J. (1994) The multifractal spectrum of statistically self-similar measures. Journal of Theoretical Probability 7, 681–702.
Falconer, K.J. (1997) Techniques in Fractal Geometry. Wiley & Sons, Chichester, UK.
Haverkamp, R. and Parlange, J.Y. (1986) Predicting the water-retention curve from particle-size distribution: I. Sandy soils without organic matter. Soil Science, 142, 325–339.
Kolmogorov, A.N. (1992) On the logarithmic normal distribution of particle sizes under grinding. Pp. 281–284 in: Selected Works of A.N. Kolmogorov (A.N. Shiryayev, editor). Vol. II, Kluwer Academic Publishers, Dordrecht, The Netherlands.
Mandelbrot, B.B. (1982) The Fractal Geometry of Nature. W. H. Freeman & Co., New York.
Martin, M.A. and Taguas, F.J. (1998) Fractal modelling, characterization and simulation of particle-size distributions in soil. Proceedings of the Royal Society of London Series A, 454, 1457–1468.
Martín, M.A., Rey, J.M., and Taguas, F.J. (2001) An entropy-based parametrization of soil texture via fractal modelling of particle-size distribution. Proceedings of the Royal Society of London Series A, 457, 937–947.
Martin, M.A., Pachepsky, Y.A., Rey, J.M., Taguas, J., and Rawls, W.J. (2005) Balanced entropy index to characterize soil texture for soil water retention estimation. Soil Science, 170, 759–766.
Montero, E. and Martín, M.A. (2003) Hölder spectrum of dry grain volume-size distributions in soil. Geoderma, 112, 197–204.
Soil Taxonomy (1975) A Basic System of Soil Classification for Making and Interpreting Soil Surveys. Agriculture handbook No. 436, Soil Survey Staff, Soil Conservation Service, U.S. Department of Agriculture.
Taguas, F.J. (1995) Modelización fractal de la distribución del tamaño de las partículas en el suelo. PhD thesis, Universidad Politécnica de Madrid, Spain.
Taguas, F.J., Martín, M.A., and Perfect, E. (1999) Simulation and testing of self-similar structures for soil particle-size distributions using iterated function systems. Geoderma, 88, 191–203.
Turcotte, D.L. (1986) Fractals and fragmentation. Journal of Geophysical Research, 91, 1921–1926.
Turcotte, D.L. (1992) Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge, UK.
Tyler, S.W. and Wheatcraft, S.W. (1989) Application of fractal mathematics to soil water retention estimation. Soil Science Society of America Journal, 53, 987–996.
Tyler, S.W. and Wheatcraft, S.W. (1992) Fractal scaling of soil particle-size distributions: analysis and limitations. Soil Science Society of America Journal, 56, 362–369.
Vrscay, E.R. (1991) Moment and collage methods for the inverse problem of fractal construction with iterated function systems. Pp. 443–461 in: Fractals in the Fundamental and Applied Sciences (H.-O. Peitgen, J.M. Henriques, and L.F. Penedo, editors). North Holland, The Netherlands.
Wu, Q., Borkovec, M., and Sticher, H. (1993) On partice-size distributions in soils. Soil Science Society of America Journal, 57, 883–890.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Martín, M.Á., García-Gutiérrez, C. Log selfsimilarity of continuous soil Particle-size distributions estimated using random multiplicative cascades. Clays Clay Miner. 56, 389–395 (2008). https://doi.org/10.1346/CCMN.2008.0560308
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1346/CCMN.2008.0560308