World Library  


Add to Book Shelf
Flag as Inappropriate
Email this Book

Scaling and Multifractal Fields in the Solid Earth and Topography : Volume 14, Issue 4 (02/08/2007)

By Lovejoy, S.

Click here to view

Book Id: WPLBN0003988730
Format Type: PDF Article :
File Size: Pages 38
Reproduction Date: 2015

Title: Scaling and Multifractal Fields in the Solid Earth and Topography : Volume 14, Issue 4 (02/08/2007)  
Author: Lovejoy, S.
Volume: Vol. 14, Issue 4
Language: English
Subject: Science, Nonlinear, Processes
Collections: Periodicals: Journal and Magazine Collection, Copernicus GmbH
Historic
Publication Date:
2007
Publisher: Copernicus Gmbh, Göttingen, Germany
Member Page: Copernicus Publications

Citation

APA MLA Chicago

Schertzer, D., & Lovejoy, S. (2007). Scaling and Multifractal Fields in the Solid Earth and Topography : Volume 14, Issue 4 (02/08/2007). Retrieved from http://hawaiilibrary.net/


Description
Description: Physics, McGill University, 3600 University st., Montreal, Que. H3A 2T8, Canada. Starting about thirty years ago, new ideas in nonlinear dynamics, particularly fractals and scaling, provoked an explosive growth of research both in modeling and in experimentally characterizing geosystems over wide ranges of scale. In this review we focus on scaling advances in solid earth geophysics including the topography. To reduce the review to manageable proportions, we restrict our attention to scaling fields, i.e. to the discussion of intensive quantities such as ore concentrations, rock densities, susceptibilities, and magnetic and gravitational fields.

We discuss the growing body of evidence showing that geofields are scaling (have power law dependencies on spatial scale, resolution), over wide ranges of both horizontal and vertical scale. Focusing on the cases where both horizontal and vertical statistics have both been estimated from proximate data, we argue that the exponents are systematically different, reflecting lithospheric stratification which – while very strong at small scales – becomes less and less pronounced at larger and larger scales, but in a scaling manner. We then discuss the necessity for treating the fields as multifractals rather than monofractals, the latter being too restrictive a framework. We discuss the consequences of multifractality for geostatistics, we then discuss cascade processes in which the same dynamical mechanism repeats scale after scale over a range. Using the binomial model first proposed by de Wijs (1951) as an example, we discuss the issues of microcanonical versus canonical conservation, algebraic (Pareto) versus long tailed (e.g. lognormal) distributions, multifractal universality, conservative and nonconservative multifractal processes, codimension versus dimension formalisms. We compare and contrast different scaling models (fractional Brownian motion, fractional Levy motion, continuous (in scale) cascades), showing that they are all based on fractional integrations of noises built up from singularity basis functions. We show how anisotropic (including stratified) models can be produced simply by replacing the usual distance function by an anisotropic scale function, hence by replacing isotropic singularities by anisotropic ones.


Summary
Scaling and multifractal fields in the solid earth and topography

Excerpt
Agterberg, F.: Geomathematics, 596 pp., Elsevier, 1974.; Agterberg, F.: New Applications of the Model of de Wisj in regional geochemistry, Math. Geol., 39, 1–25, doi:10.1007/s11004-006-96063-7, 2007.; Ahrens, L. H.: A fundamental law of geochemistry, Nature, 172, 1148–1152, 1953.; Aitchison, J. and Brown, J. A. C.: The lognormal distribution, with special reference to its uses in economics, 176 pp., Cambridge University Press, 1957.; Aviles, C. A., Scholz, C. H., and Boatwright, J.: Fractal Analysis Applied to Characteristic Segments of San Andreas Fault, J. Geophys. Res., 92, 331–344, 1987.; Bacry, A., Arneodo, A., Frisch, U., Gagne, Y., and Hopfinger, E.: Wavelet analysis of fully developed turbulence data and measurement of scaling exponents, in: Turbulence and coherent structures, edited by: Lessieur, M. and Metais, O., pp. 703–718, Kluwer, 1989.; Blakely, R. J.: Potential Theory in Gravity and Magnetic Applications, 441 pp., Cambridge University Press, 1995.; Brax, P. and Pechanski, R.: Levy stable law description on intermittent behaviour and quark-gluon phase transitions, Phys. Lett. B, 253, 225–230, 1991.; Bahr, K.: The route to fractals in magnetotelluric exploration of the crust, in: Fractal behaviour of the earth system, edited by: Dimri, V. P., Springer, Heidelberg, 2005.; Bak, P., Tang, C., and Weiessenfeld, K.: Self-Organized Criticality: An explanation of 1/f noise, Phys. Rev. Lett., 59, 381–384, 1987.; Bak, P., Tang, C., and Weiessenfeld, K.: Self-Organized Criticality, Phys. Rev. Lett., A 38, 364–374, 1988.; Balmino, G.: The spectra of the topography of the Earth, Venus and Mars, Geophys. Res. Lett., 20(11), 1063–1066, 1993.; Balmino, G., Lambeck, K., and Kaula, W.: A spherical harmonic analysis of the Earth's topography, J. Geophys. Res., 78(2), 478–481, 1973.; Bansal, A. R. and Dimri, V. P.: Self-affine gravity covariance model for the Bay of Bengal, Geophys. J. Inter., 161, 21–30, 2005.; Barton, C. C.: Fractal analysis of scaling and spatial clustering of fractures, in: Fractals in the Earth Sciences, edited by: Barton, C. C. and La Pointe, P. R., pp. 141–178, Plenum Press, New York, 1995.; Barton, C. C. and Scholz, C. H.: The fractal size and spatial distribution of hydrocarbon accumulations: Implications for resource assessment and exploration strategy, in: Fractals in Petroleum Geology and Earth Processes, edited by: Barton, C. C., and La Pointe, P. R., pp. 13–34, Plenum Press, New York, 1995.; Bean, C. and McCloskey, J.: Power-law random behaviour of seismic reflectivity in boreholes and its relationship to crustal deformation models, Earth Planet. Sci. Lett., 21, 2641–2644, 1993.; Bell, T. H.: Statistical features of sea floor topography, Deep Sea Res., 22, 883–891, 1975.; Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F., and Succi, S.: Extended self-similarity in turbulent flows, Phys. Rev. E., 48, R29–R32, 1993.; Berkson, J. M. and Matthews, J. E.: Statistical properties of seafloor roughness, in: Acoustics and the Sea-Bed, edited by: Pace, N. G., p. 215–223, Bath University Press, Bath, England, 1983.; Bourlon, E., Mareschal J.-C., Gaonac'h H., Lovejoy S., and Schertzer, D.: Anisotropic Scaling Rock Density and the Earth's Gravity Field, paper presented at Nonlinear VAriability in Geophysics, 4 July 1998, Roscoff, France, 1998.; Burridge, L. and Knopoff, L.: Model and theoretical seismicity, Bull. Seismol. Soc. Am., 57, 341–371, 1967.; Cahalan, R.: Bounded cascade clouds: albedo and effective thickness, Nonlin. Processes Geophys., 1, 156–167, 1994.; Cargill, S. M., Root, D. H., and Bailey, E. H.: Estimating unstable resources from historical industrial data, Econ. Geol., 76, 1081–1095, 1981.; Carlson, J. M., Langer, J. S., and Shaw, B. E.: Dynamics of earthquake faults, Rev. Mod. Phys., 66, 6

 

Click To View

Additional Books


  • Entropy-information Perspective to Radio... (by )
  • Polar Spacecraft Observations of the Tur... (by )
  • Distribution of Petrophysical Properties... (by )
  • Nonlinear Dynamics of Turbulent Waves in... (by )
  • Coupling Coefficients and Kinetic Equati... (by )
  • Model Error Estimation in Ensemble Data ... (by )
  • Combining Inflation-free and Iterative E... (by )
  • The Turnstile Mechanism Across the Kuros... (by )
  • Prediction of Magnetic Storm Events Usin... (by )
  • Crossing a Narrow-in-altitude Turbulent ... (by )
  • Excitation of Kinetic Alfvén Turbulence ... (by )
  • Universal Multifractal Martian Topograph... (by )
Scroll Left
Scroll Right

 



Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.