Mathematical Geosciences

, Volume 40, Issue 5, pp 533–573 | Cite as

Anisotropic Scaling Models of Rock Density and the Earth’s Surface Gravity Field

  • S. LovejoyEmail author
  • H. Gaonac’h
  • D. Schertzer
Special Issue


In this paper we consider an anisotropic scaling approach to understanding rock density and surface gravity which naturally accounts for wide range variability and anomalies at all scales. This approach is empirically justified by the growing body of evidence that geophysical fields including topography and density are scaling over wide range ranges. Theoretically it is justified, since scale invariance is a (geo)dynamical symmetry principle which is expected to hold in the absence of symmetry breaking mechanisms. Unfortunately, to date most scaling approaches have been self-similar, i.e., they have assumed not only scale invariant but also isotropic dynamics. In contrast, most nonscaling approaches recognize the anisotropy (e.g., the strata), but implicitly assume that the latter is independent of scale. In this paper, we argue that the dynamics are scaling but highly anisotropic, i.e., with scale dependent differential anisotropy.

By using empirical density statistics in the crust and a statistical theory of high Prandtl number convection in the mantle, we argue that \(P(\underline{K},k_{z})\approx(|K/k_{s}|^{H_{z}}+|k_{z}/k_{s}|)^{-s/H_{z}}\) is a reasonable model for the 3-D spectrum (K is the horizontal wavevector and K is its modulus, k z is a vertical wavenumber), (s,H z ) are fundamental exponents which we estimate as (5.3,3), (3,3) in the crust and mantle, respectively. We theoretically derive expressions for the corresponding surface gravity spectrum. For scales smaller than ≈100 km, the anisotropic crust model of the density (with flat top and bottom) using empirically determined vertical and horizontal density spectra is sufficient to explain the (Bouguer) g z spectra. However, the crust thickness is highly variable and the crust-mantle density contrast is very large. By considering isostatic equilibrium, and using global gravity and topography data, we show that this thickness variability is the dominant contribution to the surface g z spectrum over the range ≈100–1000 km. Using estimates of mantle properties (viscosity, thermal conductivity, thermal expansion coefficient, etc.), we show that the mantle contribution to the mean spectrum is strongest at ≈1000 km and is comparable to the variable crust thickness contribution. Overall, we produce a model which is compatible with both the observed (horizontal and vertical) density heterogeneity and surface gravity anomaly statistics over a range of meters to several thousand kilometers.


Geogravity Geopotential theory Fractals Multifractals Scaling 


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Copyright information

© International Association for Mathematical Geology 2008

Authors and Affiliations

  1. 1.PhysicsMcGill UniversityMontrealCanada
  2. 2.GEOTOPUQAMMontrealCanada
  3. 3.Université Paris-Est, ENPC/CEREVEMarne-la-Vallee Cedex 2France

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