BG Biogeosciences BG Biogeosciences 1726-4189 Copernicus Publications Göttingen, Germany 10.5194/bg-7-3799-2010 Fractal Metrology for biogeosystems analysis Torres-Argüelles V. 1 Oleschko K. 2 Tarquis A. M. 3 Korvin G. 4 Gaona C. 2 Parrot J.-F. 5 Ventura-Ramos E. 1 Universidad Autónoma de Querétaro, Querétaro, Mexico Centro de Geociencias, Universidad Nacional Autónoma de México (UNAM), Blvd. Juriquilla 3001, 76220, Querétaro, Mexico Dept. of Applied Mathematics to Agricultural Engineering (E.T.S.I.A.) C.E.I.G.R.A.M., Technical University of Madrid (U.P.M.), Ciudad Universitaria, 28040, Madrid, Spain King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Instituto de Geografía, UNAM, Ciudad Universitaria, México, Distrito Federal, Mexico 24 11 2010 7 11 3799 3815 Copyright: © 2010 V. Torres-Argüelles et al. 2010 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://bg.copernicus.org/articles/7/3799/2010/bg-7-3799-2010.html The full text article is available as a PDF file from https://bg.copernicus.org/articles/7/3799/2010/bg-7-3799-2010.pdf

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 (<i>Chernozem, Solonetz</i> and <i>"Chocolate" Clay</i>) 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 <i>Fractal Metrology</i> (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.

 References Alvarez-Ramirez, J., Echeverria, J. C., and Rodriguez, E.: Performance of a high-dimensional <i>R</i>/<i>S</i> method for Hurst exponent estimation, Physica A, 387, 6452–6462, 2008. Avnir, D. and Farin, D.: Molecular fractal surfaces, Nature, 308, 261–263, 1984. Bakucz, P. and Krüger-Sehm, R.: A new wavelet filtering for analysis of fractal engineering surfaces, Wear, 266, 539–542, 2009. Barton, C. C. and La Pointe, P. R. (Eds.): Fractals in Petroleum Geology and Earth Processes, Springer New York, 342 pp., 1995. Bassler, K. E., Gunaratne, G. H., and McCauley, J. L.: Markov processes, Hurst exponents, and nonlinear diffusion equations: With application to finance, Physica A, 369, 343–353, 2006. Behery, G. M.: Simulation of fractal dimension evaluations, Int. J. Model. Simul., 26, 91–97, 2006. Beiró, M. G., Alvarez-Hamelin, J. I., and Busch, J. R.: A low complexity visualization tool that helps to perform complex systems analysis, New J. Phys., 10, 125003, https://doi.org/10.1088/1367-2630/10/12/125003, 2008. Bigerelle, M., Gautier, A., and Iost, A.: Roughness characteristic length scales of micro-machined surfaces: A multi-scale modeling, Sensor Actuat. B-Chem., 126, 126–137, 2007. Borri-Brunetto, M., Carpinteri A., and Chiaia B.: Scaling phenomena due to fractal contact in concrete and rock fractures, Int. J. Fracture, 95, 221–238, 1999. Brewer, R.: Fabric and Mineral Analysis of Soils, John Willey and Sons, New York, 470 pp., 1964. Brink, L.: The Nobel Prize in Physics 2008, Presentation Speech, <a href="http://nobelprize.org/nobel_prizes/physics/laureates/2008/presentation-speech.html">http://nobelprize.org/nobel_prizes/physics/laureates/2008/presentation-speech.html</a>, 2008. Chandrasekaran, S. and Sundararajan, S.: Effect of microfabrication processes on surface roughness parameters of silicon surfaces, Surf. Coat. Tech., 188–189, 581–587, 2004. Chang, K. J., Ray, G., Bradbury, D., Nakagawa, O. S., Oh, S.-Y., Bartelink, D., Chang, E., Stine, B., Maung, T., Divecha, R., Boning, D., and Chung, J.: Using a Statistical Metrology Framework to Identify Systematic and Random Sources of Die- and Wafer-level ILD Thickness Variation in CMP Processes, IEDM Tech. Digest, 499–502, 1995. Christensen, K. and Moloney, N. R.: Complexity and Criticality, Imperial College Press Advanced Physics Texts, Vol. 1, London, 392 pp., 2005. Cox, M. G., Rossi, G. B., Harris, P. M., and Forbes, A.: A probabilistic approach to the analysis of measurement processes, Metrologia, 45, 493–502, 2008. Cox, S.: Information Technology: the global key to precision agriculture and sustainability, Comput. Electron. Agr., 36, 93–111, 2002. Curtis, M. and Farago, F.: Handbook of Dimensional Measurement, fourth edition, Industrial Press Inc., New York, USA, 608 pp., 2007. Dathe, A., Eins, S., Niemeyer, J., and Gerold, G.: The surface fractal dimension of the soil-pore interface as measured by image analysis, Geoderma, 103, 203–229, 2001. Denisov, S. I.: Fractal dimension of random processes, Chaos Soliton. Fract., 9, 1491–1496, 1998. Diehl, R. J. and Holm, P.: Roughness measurements – Have the necessities changed? J. Sound Vib., 293, 777–783, 2006. Dimri, V. P. and Prakash, M. R.: Scaling of power spectrum of extinction events in the fossil record, Earth Planet. Sc. Lett., 186, 363–370, 2001. Dodds, P. S.: Optimal form of branching supply and collection networks, Phys. Rev. Lett., 104, 048702, https://doi.org/10.1103/PhysRevLett.104.048702, 2010. Eberhardt, K., Hagwood, C., Kacker, R., Levenson, M., Liu, H. K., Vangel, M., Yen, J., and Zhang, N. F.: 5. 6 Bayesian Metrology, in: Report of activities of the statistical engineering division, NIST, edited by: Eberhardt, K., available at: <a href="http://www.itl.nist.gov/div898/pubs/ar/ar2000/node40.html">http://www.itl.nist.gov/div898/pubs/ar/ar2000/node40.html</a>, 2001. Ellis, C.: The sampling properties of Hurst exponent estimates, Physica A, 375, 159–173, 2007. El-Sonbaty, I. A., Khashaba, U. A., Selmy, A. I., and Ali, A. I.: Prediction of surface roughness profiles for milled surfaces using an artificial neural network and fractal geometry approach, J. Mater. Process. Tech., 200, 271–278, 2008. Falconer, K.: Techniques in Fractal Geometry, John Wiley & Sons, 274 pp., 1997. Feagin, R. A.: Relationship of second order lacunarity, Brownian motion, Hurst exponent, and pattern organization, Physica A, 328, 315–321, 2003. Feagin, R. A., Wu, X. B., and Feagin, T.: Edge effects in lacunarity analysis, Ecol. Model., 201, 262–268, 2007. Fleming, R. W., Dror, R. O., and Adelson, E. H.: Real-world illumination and the perception of surface reflectance properties, J. Vision, 3, 347–368, 2003. Fransson, J. H. M., Talamelli, A., Brandt, L., and Cusso, C.: Delaying transition to turbulence by a passive mechanism, Phys. Rev. Lett., 96(6), 064501, https://doi.org/10.1103/PhysRevLett.96.064501, 2006. Gallant, J. C., Moore, I. D., and Hutchinson, M. F.: Estimating fractal dimension of profiles: a comparison of methods, Math. Geol., 26, 455–481, 1994. Germer, T. A.: Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry, Phys. Rev. Lett., 85, 349–352, 2000. Gomez, S., Hale, K., Burrows, J., and Griffiths, B.: Measurements of surface defects on optical components, Meas. Sci. Technol., 9, 607–616, 1998. Hardy, G. H.: Weierstrass's non-diferentiable function, T. Am. Math. Soc., 17, 301–325, 1916. Heilmann, R. K., Chen, C. G., Konkola, P. T., and Schattenburg, M. L.: Dimensional Metrology for Nanometer-Scale Science and Engineering: towards sub-nanometer accurate encoders, Nanotechnology, 15, 504–511, 2004. Hopp, T. H.: Computational Metrology, Manufacturing Review 6, American Society of Mechanical Engineers, New York, 295–304, 1993. Hurst, H. E.: Long-term storage capacity of reservoirs, T. Am. Soc. Civ. Eng., 116, 770–808, 1951. International Organization for Standardization (ISO): International Vocabulary of Basic and General Terms in Metrology, 3rd edition, Geneva, Switzerland, 55 pp., 2004. Jensen, J. E.: Computationally modeling the effects of surface roughness on soft X-ray multilayer reflectors, M.S. thesis, Department of Physics and Astronomy, Brigham Young University, 164 pp., 2006. Joint Committee for Guides in Metrology (JCGM): International Vocabulary of Metrology – Basic and general concepts and associated terms, 3rd edition, 90 pp., 2008. Joint Committee for Guides in Metrology (JCGM): Evaluation of measurement data – An introduction to the "Guide to the expression of uncertainty in measurement" and related documents, 1st edition, 20 pp., 2009. Jones, C. L. and Jelinek, H. F.: Wavelet Packet Fractal Analysis of Neuronal Morphology, Methods, 24, 347–358, 2001. Katz, J. S.: Indicators for complex innovation systems, Res. Policy, 35, 893–909, 2006. Kim, T. W., Bhushan, B., and Cho, Y. J.: The contact behavior of elastic/plastic non-Gaussian rough surfaces, Tribol. Lett., 22, 1–13, 2006. Klinkenberg, B.: A review of methods used to determine the fractal dimension of linear features, Math. Geol., 26, 23–46, 1994. Kogan, J.: Introduction to clustering large and high dimensional data, Cambridge University Press, New York, 222 pp., 2007. Korvin, G.: Fractal models in the earth sciences, Elsevier, Amsterdam, 396 pp., 1992. Korvin, G.: Is the optical image of a non-Lambertian fractal surface fractal?, IEEE Geosci. Remote S., 2, 380–383, 2005. Levitz, P.: Toolbox for 3-D imaging and modeling of porous media: Relationship with transport properties, Cement Concrete Res., 37, 351–359, 2007. Lojkowski, W., Turan, R., Proykova, A., and Daniszewska, A. (Eds.): Eight Nanoforum Report: Nanometrology, available at: <a href="http://www.innovationsgesellschaft.ch/images/fremde_publikationen/NANOMETROLOGY_Report.pdf">http://www.innovationsgesellschaft.ch/images/fremde_publikationen/NANOMETROLOGY_Report.pdf</a>, 2006. Lu, X. and Guan, J.: A new approach to building histogram for selectivity estimation in query processing optimization, Comput. Math. Appl., 57, 1037–1047, 2009. Majumdar, A. and Bhushan, B.: Fractal model of elastic-plastic contact between rough surfaces, ASME J. Tribol., 113, 1–11, 1991. Malamud, B. D. and Turcote, D. L.: Self-affine time series: measures of weak and strong persistence, J. Stat. Plan. Infer., 80, 173–196, 1999. Mandelbrot, B. B.: The fractal geometry of nature, edited by: Freeman, W. H. and Company, New York, USA, 1982. Mandelbrot, B. B.: Gaussian Self-Affinity and fractals, Springer-Verlag, New York, 654 pp., 2002. Mandelbrot, B. B. and Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422–437, 1968. Mandelbrot, B. B. and Wallis, J. R.: Noah, Joseph and Operational Hydrology, Water Resour. Res., 4, 909–918, 1968. Mandelbrot, B. B., Passoja, D. E., and Paulay, A. J.: Fractal character of fracture surfaces of metals, Nature, 308, 721–722, 1984. Marschal, A., Andrieux, T., Compagon, P. A., and Fabre, H.: Chemical metrology – QUID?, Accredit. Qual. Assur., 7, 42–49, 2002. MINITAB for Windows: MAVV. Home. WIN 1221.03737., available at: <a href="http://www.minitab.com">http://www.minitab.com</a>, 1998. Monroe, D.: Rough skin is good, Phys. Rev. Focus, available at: <a href="http://focus.aps.org/story/v17/st6">http://focus.aps.org/story/v17/st6</a>, 2006. NIST Bayesian metrology, National Institute of Standards and Technology: Manufacturing Engineering Laboratory, <a href="http://www.itl.nist.gov/div898/bayesian/overview.html">http://www.itl.nist.gov/div898/bayesian/overview.html</a>, access: 9&nbsp;November 2010, 2001. NIST/SEMATECH: e-Handbook of statistical methods, Probability Density Function, <a href="http://www.itl.nist.gov/div898/handbook/">http://www.itl.nist.gov/div898/handbook/</a>, 2006. Oleshko, K., Vadiunina, A. F., Zilaeva, V. A., and Truchin, V. I.: Influencia del campo magnético sobre las propiedades del suelo y plantas, Pochvovedenie, 7, 91–100, 1980 (in Russian). Oleschko, K.: Effect of electric current on clay minerals in "chocolate" clays, Universidad Estatal de Moscú M.V. Lomonósov, Facultad de Suelos, Boletín de la Universidad de Moscú, serie Pochvovedenie (Edafología), No. 2, 1981 (in Russian). Oleschko, K., Figueroa, B., Miranda, M. E., Vuelvas, M. A., and Solleiro R. E.: Mass fractal dimensions and some selected physical properties of contrasting soils and sediments of México, Soil Till. Res., 55, 43–61, 2000. Oleschko, K., Korvin, G., Balankin, A. S., Khachaturov, R. V., Flores, Figueroa, L., Urrutia, J., and Brambila, F.: Fractal Scattering of Microwaves from Soils, Phys. Rev. Lett., 89, 188501, https://doi.org/10.1103/PhysRevLett.89.188501, 2002. Oleschko, K., Korvin, G., Figueroa, B., Vuelvas, M. A., Balankin, A., Flores, L., and Carreon, D.: Fractal radar scattering from soil, Phys. Rev. E, 67, 041403–1:041403–13, https://doi.org/10.1103/PhysRevE.67.041403, 2003. Oleschko, K., Parrot, J.-F., Ronquillo, G., Shoba, S., Stoops, G., and Marcelino, V.: Weathering: toward a fractal quantifying, Math. Geol., 36(5), 607–627, 2004. Oleschko, K., Parrot, J.-F., Korvin, G., Esteves, M., Vauclim, M., Torres-Argüelles, V., Gaona-Salado, C., and Cherkasov, S.: Fractal Image Informatics: from SEM to DEM, Proceedings of 4th International Conference: GIS in Geology and Geosciences, Vista of new approaches for the Geoinformatics, AIP, 109, 2509–2517, 2008. Oleschko, K., Korvin, G., Flores, L., Brambila, F., Gaona, C., Parrot, J.-F., Ronquillo, G., and Zamora, S.: Probability density function: A tool for simultaneous monitoring of pore/solid roughness and moisture content, Geoderma, https://doi.org/10.1016/j.geoderma.2009.10.015, in press, 2010. Orbach, R.: Dynamics of fractal networks, Science, 231, 814–819, 1986. Palisade Corporation: Guide to Using @RISK: Risk Analysis and Simulation Add-In for Microsoft® Excel Version 5.5, 2005. Parfenova, E. I. and Yarilova, E. A.: Guide on Micromorphological Studies in Soil Science, Nauka, Moscow, 198 pp., 1977 (in Russian). Partis, L., Croan, D., King, B., and Emslie, K. R.: Biometrology and biomeasurement: overview of current technologies and priorities for a biological measurement program in Australia, Australian Government Analytical Laboratories, Pymble, NSW, Australia, AGAL Public Interest Program Public, Interest Report Series Number, 2002-1, 2002. Paschotta, R.: Encyclopedia of laser physics and technology, RP Photonics, Wiley-VCH, Berlin, 844 pp., 2009. Pawley, J. B. (Ed.): Handbook of biological confocal microscopy, 3rd edition, Springer Science + Business media, New York, 985 pp., 2006. Pelletier, J. D. and Turcotte, D. L.: Self-affine time series: 11. Applications and models, Adv. Geophys., 40, 91–166, 1999. Pendleton, D. E., Dathe, A., and Baveye, P.: Influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media, Phys. Rev. E, 72, 041306, https://doi.org/10.1103/PhysRevE.72.041306, 2005. Pentland, A. P.: Fractal based description of natural scenes, IEEE T. Pattern Anal., PAMI-6, 661–674, 1984. Plowman, D. A., Solansky, S., Beck, T. E., Baker, L., Kulkarni, M., and Villarreal Travis, D.: The role of leadership in emergent, self-organization, The Leadership Quarterly, 18, 341–356, 2007. Puente, C.: A universe of projections: may Plato be right?, Chaos Soliton. Fract., 19, 241–253, 2004. Rehman, S. and Siddiqi, A. H.: Wavelet based Hurst exponent and fractal dimensional analysis of Saudi climatic dynamics, Chaos Soliton. Fract., 40, 1081–1090, 2009. Rodriguez-Iturbe, I. and Rinaldo, A.: Fractal river basins: chance and self-organization, Cambridge University Press, New York, 564 pp., 1997. Scipioni, A., Rischette, P., Bonhomme, G., and Devynck, P.: Characterization of self-similarity properties of turbulence in magnetized plasmas, Phys. Plasmas, 15, 112303, https://doi.org/10.1063/1.3006075, 2008. Seffens, W.: Order from chaos, Techsighting Software, Science, 285, 5431, https://doi.org/10.1126/science.285.5431.1228a, 1999. SML, Surface Metrology Laboratory: Departament of Mechanical Engineering, Worcester Polytechnic Institute, 2008. Stanley, H. E.: Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 336 pp., 1971. Statistical Package for the Social Sciences (SPSS), 13.0 for Windows, Copyright © SPSS Inc., released on 6 December 2004. Strauss, O.: Quasi-continuous histograms, Fuzzy Set. Syst., 160, 2442–2465, 2009. Su, Z.-Y. and Wu, T.: Music walk, fractal geometry in music, Physica A, 380, 418–428, 2007. Sun, W., Xu, G., Gong, P., and Liang, S.: Fractal analysis of remotely sensed images: a review of methods and applications, Int. J. Remote Sens., 27, 4963–4990, 2006. Tancrez, J.-S., Semal, P., and Chevalier, P.: Histogram based bounds and approximations for production lines, Eur. J. Oper. Res., 197, 1133–1141, 2009. Tang, D. and Marangoni A. G.: 3-D fractal dimension of fat crystal networks, Chem. Phys. Lett., 433, 248–252, 2006. Tél, T. and Gruiz, M.: Chaotic Dynamics: An Introduction Based on Classical Mechanics, Cambridge University Press, 393 pp., 2006. TruSoft, Intern. Inc.: Benoit 1.3 Software, 1999. Turcotte, D. L.: Fractals and Chaos in Geology and Geophysics, Cambridge University Press, 416 pp., 1997. Vadyunina, A. F., Sokolova, T. A., Oleshko, K., and El Lakani, A. A.: Effect of electric current on clay minerals in "Chocolate Clays", Vestnik MGU, 50–54, 1980 (Pochvovedenie, in Russian). Van Gorp, A., Bigerelle, M., Grellier, A., Iost, A., and Najjar D.: A multi-scale approach of roughness measurements: Evaluation of the relevant scale, Mat. Sci. Eng. C, 27, 1434–1438, 2007. Verhoest, N. E. C., Lievens, H., Wagner, W., Álvarez-Mozos, J., Moran, M. S., and Mattia, F.: On the soil roughness parameterization problem in soil moisture retrieval of bare surfaces from synthetic aperture radar (Review), Sensors, 8, 4213–4248, 2008. Villarrubia, J. S.: Issues in line edge and line width roughness metrology, AIP Conference proceedings: Characterization and metrology for ULSI Technology, 386–393, 2005. Wang, L., Rong, W., and Sun, L.: Elastic-plastic adhesive contact of fractal microparts surfaces with low adhesion parameters, Proceedings of the 3rd IEEE Int. Conf. on Nano/Micro Engineered and Molecular Systems, Sanya, China, 6–9 January, 592–595, 2008. Weisstein, E. W.: Fourier Transform, From Mathworld – A Wolfram Web Resources, <a href="http://mathworld.wolfram.com/FourierTransform.html">http://mathworld.wolfram.com/FourierTransform.html</a>, 2010 Willink, R.: Principles of probability and statistics for metrology, Metrologia, 43, S211–S219, 2005. Wilson, T. H.: Short notice, Fractal strain distribution and its implications for cross-section balancing: further discussion, J. Struct. Geol., 19, 129–132, 1997.