Scaling and multifractal fields in the solid earth and topography
type: Article
item type: journalArticle
is part of: Nonlinear Processes in Geophysics
  type: Journal
  volume: 14
  number: 4
authors: Person
  surname: Lovejoy
  given name: S.
authors: Person
  surname: Schertzer
  given name: D.
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 ba…
date: 2007
pages: 465–502