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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 14, issue 4
Nonlin. Processes Geophys., 14, 465–502, 2007
https://doi.org/10.5194/npg-14-465-2007
© Author(s) 2007. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

Special issue: New developments in scaling, scale and nonlinearity in the...

Nonlin. Processes Geophys., 14, 465–502, 2007
https://doi.org/10.5194/npg-14-465-2007
© Author(s) 2007. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  02 Aug 2007

02 Aug 2007

Scaling and multifractal fields in the solid earth and topography

S. Lovejoy1,2 and D. Schertzer3,4 S. Lovejoy and D. Schertzer
  • 1Physics, McGill University, 3600 University st., Montreal, Que. H3A 2T8, Canada
  • 2GEOTOP_UQAM/McGill Center, Montreal, Qc, Canada
  • 3CEREVE, Ecole Nationale des Ponts et Chaussées, 6–8, avenue Blaise Pascal, Cité Descartes, 77455 Marne-La-Vallee Cedex, France
  • 4Meteo France, 1 Quai Branly, 75007 Paris, France

Abstract. 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.

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