Abstract
Although scale invariance is a basic geodynamic symmetry, there is no reason to expect it to be isotropic; that the corresponding fractals or multifractal fields be self-similar. The analysis of various geophysical fields has indeed shown that although they are typically scaling multifractals, they are indeed anisotropic involving differential rotation, stratification and more complex scale changes. The framework for the analysis and simulation of such fields is generalized scale invariance (GSI), which comprises two elements — the generator and the unit ball. Although the generator is more fundamental, the shape of the possibly highly anisotropie unit ball will also affect the texture and morphology of the corresponding fields. While full nonlinear GSI is difficult to examine, linear approximations are always valid over finite ranges of scale; we will therefore focus on the latter. Building on earlier work using linear GSI and second order (convex/elliptical) balls, we establish a method of constructing higher order families necessary for modeling qualitatively different morphologies, especially non-convex ones. We illustrate the method using fourth order polynomials and multifractal simulations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chigirinskaya, Y., D. Schertzer, S. Lovejoy, A. Lazarev, and A. Ordanovich, Unified multifractal atmospheric dynamics tested in the tropics Part 1: horizontal scaling and self-organized criticality, Nonlinear Processes in Geophysics (1), 105–114, 1994.
Davis, A., S. Lovejoy, and D. Schertzer, Supercomputer simulation of radiative transfer inside multifractal cloud models, in I.R.S. 92, edited by S. Keevallik and O. Kärner, 112–115, 1992.
Falco, T., F. Francis, S. Lovejoy, D. Schertzer, B. Kerman, and M. Drinkwater, in IEEE Transactions on Geosciences and Remote Sensing, 34, 906–914, 1996.
Fox, C.G., and D. Hayes, Quantitative methods for analyzing the roughness of the seafloor, Reviews of Geophysics (23), 1–48, 1985.
Frisch, U.P., P.L. Sulem, and M. Nelkin, A Simple Dynamical Model of Intermittency in Fully Developed Turbulence, Journal of Fluid Mechanics, 87, 719–24, 1978.
Gabriel, P., S. Lovejoy, D. Schertzer, and G.L. Austin, Multifractal Analysis of resolution dependence in satellite imagery, Journal of Geophysical Research, 15, 1373–1376, 1988.
Gupta, V.K., and E. Waymire, A Statistics I Analysis of Mesoscale Rainfall as a Random Cascade, Journal of Applied Meteorology, 32, 251–267, 1993.
Hooge, C., Earthquakes as a Space-Time multifractal Process, M. Sc. thesis, McGill University, Montreal ( Qeubec ), Canada, 1993
Hooge, C., S. Lovejoy, D. Schertzer, S. Pecknold, J F. Malouin, and F Schmitt, Multi-fractal Phase Transitions: The Origin of Self-Organized Criticality in Earthquakes, Non-Linear Processes in Geophysics, 1 (2), 191–197, 1994.
Lavallée, D., S. Lovejoy, D. Schertzer, and P. Ladoy, Nonlinear variability and landscape topography: analysis and simulation, in Fractals in geography, edited by L. De Cola, and N. Lam, pp. 171–205, Prentice-Hall, 1993.
Lazarev, A., D. Schertzer, S. Lovejoy, and Y. Chigirinskaya, Unified multifractal atmospheric dynamics tested in the tropics Part II: vertical scaling and generalized scale invariance, Nonlinear Processes in Geophysics (1), 115–123, 1994.
Lewis, G., S. Lovejoy, and D. Schertzer, The scale invariant generator technique for parameter estimates in generalized scale invariant systems, Computers in Geoscience (submitted ), 1996.
Lovejoy, S., D. Lavallée, D. Schertzer, and P. Ladoy The £1 /2 law and multifractal topography: theory and analysis, Nonlinear Processes in Geophysics (2), 16–22, 1995.
Lovejoy, S., and D. Schertzer, Generalized Scale Invariance in the Atmosphere and Fractal Models of Rain, Water Resources Research, 21 (8), 1233–1250, 1985.
Lovejoy, S., and D. Schertzer, Our multifractal atmosphere: A unique laboratory for non-linear dynamics, Physics in Canada, 46, 62, 1990.
Lovejoy, S., and D. Schertzer, How bright is the coast of Brittany?, in Fractals in Geo-science and Remote Sensing, edited by G. Wilkinson, I. Kanellopoulos, and J. Mgier,. 102–151, Office for Official Publications of the European Communities, Luxembourg, 1995.
Lovejoy, S., D. Schertzer, and K. Pflug, Generalized Scale Invariance and differentially rotation in cloud radiances, Physica A, 185, 121–127, 1992.
Lovejoy, S., D. Schertzer, and A.A. Tsonis, Functional Box-Counting and Multiple Dimensions in rain, Science, 235, 1036–1038, 1987.
Mandelbrot, B.B., Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, Journal of Fluid Mechanics, 62, 331–350, 1974.
Meneveau, C., K.R. Sreenivasan, P. Kailasnah, and M.S. Fan, Joint multifractal measures: theory and applications to turbulence, Physical Review A, 41, 894, 1990.
Novikov, E.A., and R. Stewart, Intermittency of turbulence and spectrum of fluctuations in energy-dissipation, Izv. Akad. Nauk. SSSR. Ser. Geofiz., 3, 408–412, 1964.
Pecknold, S., S. Lovejoy, D. Schertzer, C. Hooge, and J.-F. Malouin, The Simulation of Universal Multifractals, in Cellular Automata: Prospects in Astrophysical Applications, edited by A. Lejeune, and J. Perdang, pp. 228–67, World Scientific, Singapore, 1993.
Pflug, K., S. Lovejoy, and D. Schertzer, Generalized Scale Invariance, differential rotation and cloud texture, in Nonlinear dynamics of Structures, edited by R.Z. Sagdeev, U. Frisch, A.S. Moiseev, and A. Erokhin, 72–78, World Scientific, 1991.
Pflug, K., S. Lovejoy, and D. Schertzer, Differential Rotation and Cloud Texture: Analysis Using Generalized Scale Invariance, Master’s of Science Thesis thesis, McGill University, 1992.
Pflug, K., S. Lovejoy, and D. Schertzer, Generalized Scale Invariance, differential rotation and cloud texture, Journal of Atmospheric Sciences, 50, 538–553, 1993.
Pillcington, M., and J.P. Todoeschuk, Fractal Magnetization of Continental Crust, Geophysical Research Letter, 20 (7), 627–630, 1993.
Schertzer, D., and S. Lovejoy, On the Dimension of Atmospheric motions, in Turbulence and Chaotic phenomena in Fluids, IUTAM, edited by T. Tatsumi, 505–512, Elsevier Science Publishers B. V., 1984.
Schertzer, D., and S. Lovejoy, The dimension and intermittency of atmospheric dynamics, in Turbulent Shear Flows 4, edited by B. Launder, 7–33, Springer, 1985a.
Schertzer, D., and S. Lovejoy, Generalised scale invariance in turbulent phenomena, Physico-Chemical Hydrodynamics Journal, 6, 623–635, 1985b.
Schertzer, D., and S. Lovejoy, Physical modeling and Analysis of Rain and Clouds by Anysotropic Scaling of Multiplicative Processes, Journal of Geophysical Research, D 8 (8), 9693–9714, 1987.
Schertzer, D., and S. Lovejoy, Generalized Scale Invariance and multiplicative processes in the atmosphere, PAGEOPH, 130, 57–81, 1989.
Schertzer, D., and S. Lovejoy, Nonlinear geodynamical variability: multiple singularities, universality and observables, in Non-linear variability in geophysics: Scaling and Fractals, edited by D. Schertzer, and S. Lovejoy, 41–82, Kluwer, 1991a.
Schertzer, D., and S. Lovejoy, Scaling Nonlinear Variability in Geodynamics, in Nonlinear Variability in Geophysics: Scaling and Fractals, edited by D. Schertzer, and S. Lovejoy, 41–82, Kluwer Academic Publishers, Dordrecht, 1991b.
Schertzer, D., and S. Lovejoy, From scalar cascades to Lie cascades: joint multifractal analysis of rain and cloud processes, in Space/time Variability and Interdependance for various hydrological processes, edited by R.A. Feddes, 153–173, Cambridge University Press, New-York, 1995a.
Schertzer, D., and S. Lovejoy, From scalar cascades to Lie cascades: joint multifractal analysis of rain and cloud processes, in Space/time Variability and Interdependance for various hydrological processes, edited by R.A. Feddes, 153–173, Cambridge University Press, New-York, 1995a.
Schmitt, F., D. Lavallée, D. Schertzer, and S. Lovejoy Empirical Determination of Universal Multifract al Exponents in Turbulent Velocity Fields, Physical Review Letter, 68, 305–308, 1992.
Schmitt, F., D. Schertzer, and S. Lovejoy, and Y. Brunet, Estimation of universal mul- tifractal indices for atmospheric turbulent velocity fields, Fractals, 3, 568–575.
Tessier, Y., S. Lovejoy, and D. Schertzer, Universal Multifractals: theory and observations for rain and clouds, Journal of Applied Meteorology, 32 (2), 223–250, 1993a.
Tessier, Y., S. Lovejoy, D. Schertzer, D. Lavallée, and B. Kerman, Universal Multifractal Indices For The Ocean Surface At Far Red Wavelengths, Geophysical Research Letter, 20 (12), 1167–1170, 1993b.
Van Zandt, T.E., S.A. Smith, T. Tsuda, D.C. Fritts, T. Sato, S. Fukao, and S. Kato, Studies of Velocity Fluctuations in the Lower Atmosphere Using the MU Radar, Part I: Azimuthal Anisotropy, J. Atmos. Sci. (47), 39–50, 1990.
Veneziano, D., G. Moglen, and R.L. Bras, Iterate random pulse processes and their spectral properties, Water Resources (submitted ), 1996.
Yaglom, A.M., The influence on the fluctuation in energy dissipation on the shape of turbulent characteristics in the inertial interval, Soy. Phys. Dokl., 2, 26–30, 1966.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Pecknold, S., Lovejoy, S., Schertzer, D. (1997). The Morphology and Texture of Anisotropic Multifractals Using Generalized Scale Invariance. In: Molchanov, S.A., Woyczynski, W.A. (eds) Stochastic Models in Geosystems. The IMA Volumes in Mathematics and its Applications, vol 85. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8500-4_14
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8500-4_14
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8502-8
Online ISBN: 978-1-4613-8500-4
eBook Packages: Springer Book Archive