Abstract
It is already almost two decades since fractals [1] obtained their name and were developed from merely curious mathematical constructions into an important instrumentation for practically all natural and engineering sciences [2, 3, 4]. These advances were brought about by the efforts to model and characterize complex structures and processes which either cannot be approximated using Euclidean objects or such an approximation proves inefficient. To facilitate the discussion, consider an example taken from the remote sensing practice, see Fig. 1, where a laser profiler measured profile of cultivated soil [5, 6] is shown.
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Mandelbrot, B.B. (1979) Fractals, W.H. Freeman and Company, San Francisco; Mandelbrot, B.B. (1982) The Fractal Geometry of Nature, W.H. Freeman and Company, San Francisco.
Feder, J. (1988) Fractals, Plenum, New York.
Schroder, M.R. (1991) Fractals, Chaos, Power Laws, W. H. Freeman and Company, New York.
Barabási, A.-L. and Stanley, H.E. (1995) Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge.
Wegmüller, U., Mätzler, C., Active and passive microwave signature catalog (2-12 GHz), Inst. Appl. Phys. Univ. Bern, Bern, Switzerland, Tech. Rep. (1993).
Wegmüller, U., Mätzler, C., Hüppi, R., and Schanda, E. (1994) Active and passive microwave signature catalog on bare soil (2-12 GHz), IEEE Trans, on Geosci. Rem. Sen. 32, 698–702.
Bak, P., Tang, C., and Wiesenfeld, K. (1987) Self-organized criticality: An explanation of 1/f noise, Phys. Rev. Lett 59, 381–384; Bak, P., Tang, C., and Wiesenfeld, K. (1988) Self-organized criticality, Phys. Rev. A 38, 364-374.
Berry, M.V. (1979) DifTractals, J. Phys. A: Math. Gen. 12, 781–797.
Panchev, S. (1971) Random Functions and Turbulence Pergamon Prees, Oxford; Monin, A.S. and Yaglom, A.M. (1971) [itStatistical Fluid Mechanics} MIT Press, Boston.
Priestley, M.A. (1981) Spectral Analysis and Time Series Academic Press, London.
Yordanov, O.I. and Ivanova, K. (1995) Description of surface roughness as an approximate self-affine random structure, Surface Science 331-333 1043–1049.
Orey, S. (1970) Gaussian sample functions and the Hausdorff dimension of level crossings Z. Wahrsch'theorie verw. Geb. 15, 249–256.
Falconer, K.J. (1985) The Geometry of Fractal Sets Cambridge University Press, Cambridge.
Sayles, R.S. and Thomas, T.R. (1978) Surface topography as a nonstationary random process, Nature (London) 271, 431–434; Berry M.V. and Hannay, J.R. (1978) Nature (London) 273, 573 (1978); Klinkenberg, B. and Goodchild, M.F. (1992) The fractal properties of topography: A comparison of methods, Earth Surface Processes and Landforms 17, 217-234.
Mandelbrot, B.B., Passoja, D.E., and Paullay, A.J. (1984) Fractal character of fracture sufaces of metals, Nature (London) 308, 721; Zhenyi M., Langford, S.C., Dickinson, J.T., Engelhard, M.H., and Baer, D.R. (1991) Fractal character of crack propagation in epoxy and apocy composites as revealed by photon emission during fracture, J. Mater. Res. 6, 183-195.
Bell, T.H. Jr (1979) Mesoscale sea floor roughness, Deep-Sea Res. 26A, 65–76.
Fuks, I.M., (1983) Radiophys. Quantum Electron. 26, 865 (1983).
Kolmogorov, A.N. (1941) The local structure of turbulence in incompressible viscous fluid, C.R. Acad. Sci. USSR 30, 299 (1941); Chen, S., Doolen, G.D., Kraichnan, R.H., and She Z.-S. (1993) On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids A 5, 458.
Phillips, O.M. (1958) On the dynamics of unsteady gravity waves of finite amplitude, J. Fluid Mech. 4, 226; Zakharov, V.E. and Filonenko, N.N. (1967) Weak Tubulence of Capillary Waves, J. Appl Mech. Tech. Phys. 8, 62-67.
Pierson, W.J. Jr., and Moskowitz, L. (1964) A proposed spectral form for fully developed wind seas based on the similarity theory of S.A. Kitaigorodskii, J. Geophys. Res. 69, 5181; Cuissard A., Baufays C., Sobieski P. (1986) Sea surface description requirements for electromagnetic scattering calculations, J. Geophys. Res. 91, 2477-2492; Elgar, S. and Mayer-Kress, G. (1989) Observations of the fractal dimension of deep-and shallow-water ocean surface gravity waves, Physica D 37, 104-108; Stiassnie, M.Y., Agnon, Y., and Shemer, L. (1991) Fractal dimensions of random water surfaces, Physica D 47, 341-352.
Mandelbrot, B.B. and Van Ness, J.W. (1968) Fractional Brownian Motious, Fractional Noises and Applications, SIAM Review 10, 422–436; Llosa, J. and Masoliver, J. (1990) Fractal dimension for Gaussian colored processes, Phys. Rev. A 42, 5011-5014.
Voss, R.F. (1992) Phys. Rev. Lett. 68, 3805; Stanley, H.E. (1992) Physica A 186, 1.
Yordanov, O.I. and Nickolaev, N.I. (1994) Self-affinity of time series with finite domain power-law power spectrum, Phys. Rev. E 49 R2517–R2520.
Yordanov, O.I. and N. Nickolaev, N.I. (1996) Approximate, saturated and blurred self-affinity of random processes with finite domain power-law power spectrum, Physica D (accepted for publication).
N. P. Greis and H.S. Greenside, Phys. Rev. A 44 (1991) 2324.
Hasselman, K., et al., (1973) Herausgegeben vorn Deutsch. Hydrograph. Institut., Reihe A 12 95; Cuissard A., Baufays, C. and Sobieski, P. (1994) Fully and non-fully developed sea models for microwave remote sensing applications, Rem. Sens. Environment 48 25-38.
Rasigni G., F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria (1983) Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits, Phys. Rev. B, 27, 819–830.
G. K. Batchelor, Proc. Camb. Philos. Soc. 47 (1951) 359.
D. Lohse and A. Müler-Groeling, Phys. Rev. Lett. 74 (1995) 1747.
T. Hwa and M. Kardar, Phys. Rev. A 45 (1992) 7002.
N. C. Pesheva, J. G. Brankov and E. Canessa (1996) Layer features of the lattice gas model for self-organized criticality, Phys. Rev. E 53, 2099–2103.
O. I. Yordanov and A. Guissard, (1996) Approximate self-affine model for cultivated soil roughness Physica A, submitted for publication.
Press, W.H., Flaunery, B.P., eukolsky, S.A., and Vetterling, W.T. (1988) Numerical Recipes, Cambridge University Press, Cambridge.
Persival, D.B. (1991) Characterization of frequency stability: frequency-domain estimation of stability measures, Proc. of IEEE 79, 961–972.
Bender, C. M. and Orszag, S. A. (1978) Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York.
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Yordanov, O.I., Ivanova, K., Michalev, M.A. (1997). Approximate Self-Affinity: Methodology and Remote Sensing Applications. In: Groll, H., Nedkov, I. (eds) Microwave Physics and Techniques. NATO ASI Series, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5540-3_19
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