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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, 337–350, 2007
https://doi.org/10.5194/npg-14-337-2007
© 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, 337–350, 2007
https://doi.org/10.5194/npg-14-337-2007
© Author(s) 2007. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  06 Jul 2007

06 Jul 2007

Anisotropic scaling of remotely sensed drainage basins: the differential anisotropy scaling technique

A. Beaulieu1,*, H. Gaonac'h1, and S. Lovejoy2 A. Beaulieu et al.
  • 1Université du Québec à Montréal – Centre GEOTOP-UQÀM-McGill, C.P.8888 succursale Centre-Ville, Montréal, Québec, H3C 3P8, Canada
  • 2McGill University, Department of Physics, 3600 University, Montreal, Québec, H3A 2T8, Canada
  • *now at: Centre for Topographic Information in Sherbrooke, Natural Resources Canada, 2144 King Street West, suite 010, Sherbrooke, Qc, J1J 2E8, Canada

Abstract. We investigate the statistical properties of dendritic drainage areas from diverse geological environments (Deception Canyon, Utah and the Loess Plateau, China) using narrow band visible ASTER satellite images. We show that from 240 m to 7680 m, the isotropic (angle integrated) energy spectra E(k) of all the fields closely follow a power law form: E(k)∝k−β where k is a wave number and β a scale invariant exponent. In spite of this good isotropic scaling, images with very similar β's and similar isotropic multifractal exponents have distinct textures; we suggest that the differences are primarily due to anisotropy, which is nevertheless scaling. We develop the new "Differential Anisotropy Scaling" technique to characterize this scale-by-scale (differential) anisotropy and we test it on simulated anisotropic scaling fields. The method gives useful characterizations of the scale by scale anisotropy irrespective of whether or not the analyzed field is scaling. When the anisotropy is not too strong, the parameters can be interpreted as scale invariant anisotropy exponents. Viewed as a method of estimating these exponents, it has the advantage of relying on two linear regressions rather than on complex higher dimensional nonlinear ones. When applied to dendritic drainage basins we find that they have distinct anisotropies characterized by differential anisotropy stretching and rotation parameters as well as by a distinct absolute anisotropy at the reference scale of 960 m. Our new method allows us to statistically distinguish, not only between two geologically different drainage basins (the China Loess Plateau and Utah Deception Canyon), but also between different regions of the same China drainage system.

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