Nonlinear Processes in Geophysics
www.nonlin-processes-geophys.net
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14
4
2007
10.5194/npg-14-465-2007
http://www.nonlin-processes-geophys.net/14/465/2007/
http://www.nonlin-processes-geophys.net/14/465/2007/npg-14-465-2007.html
http://www.nonlin-processes-geophys.net/14/465/2007/npg-14-465-2007.pdf
465
502
2007-08-02
Scaling and multifractal fields in the solid earth and topography
S. Lovejoy
lovejoy@physics.mcgill.ca
D. Schertzer
Physics, McGill University, 3600 University st., Montreal, Que. H3A 2T8, Canada
GEOTOP_UQAM/McGill Center, Montreal, Qc, Canada
CEREVE, Ecole Nationale des Ponts et Chaussées, 6–8, avenue Blaise Pascal, Cité Descartes, 77455 Marne-La-Vallee Cedex, France
Meteo France, 1 Quai Branly, 75007 Paris, France
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.
<br><br>
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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