A New Model for Quantifying Anisotropic Scale Invariance and for Decomposition of Mixing Patterns

Abstract

A new power-law function has been derived to represent the relationship between area of the set consisting of wave numbers with spectral energy density above S (A(>S)) on the two-dimensional frequency plane and S. The power-law relation holds if the field concerned possessing isotropic scale invariance or generalized scaling invariance involves rotational and ratio-scale changing transforms. The equation is valid for dealing with common exploration geophysical and geochemical fields encountered in mineral exploration and environmental assessment. This power-law function not only provides a new model for characterizing anisotropic scaling invariance for generalized scaling field, for example, estimating the power exponent of power spectrum of generalized scale invariance measure in frequency domain, but also forms a theoretical base for the S-A filtering technique developed for decomposing a mixing field into components on the basis of distinct scaling properties in the frequency domain. It is demonstrated that the method has potential to become a general technique for image processing and pattern recognition.

Density/area power-law models for separating multi-scale anomalies of ore and toxic elements in stream sediments in Gejiu mineral district, Yunnan Province, China

“…A fractal filtering method developed on the basis of the generalized self-similarity principle (Cheng et al, 1999;Cheng, 2004) was used to separate the mixing patterns of As in the area.…”

“…where ∝ stands for "proportional to", β is the anisotropic scaling exponent, and d is a parameter representing degree of overall contraction (Cheng et al, 1999;Cheng, 2004). Since the shape of A involved in these models (C-A and S-A) at different concentrations (C) or spectral energy densities (S) can be any self-similarly shaped contour, these self-similar contours characterize the anisotropic scale invariance of the density distribution.…”

“…Since the shape of A involved in these models (C-A and S-A) at different concentrations (C) or spectral energy densities (S) can be any self-similarly shaped contour, these self-similar contours characterize the anisotropic scale invariance of the density distribution. Thus, the exponent of the power-law relation becomes an essential index that characterizes generalized self-similarity, implying that a geo-field shows diversity in the spatial domain and depicting self-similarity in a special domain such as a Fourier domain (Cheng 2004(Cheng , 2005.…”

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

“…The acquisition of remotely sensed multiple spectral images is thus a unique source of data for determining the scale invariant characteristics of the radiance fields (e.g. visible, short-wave infrared, infrared, microwave) related to the chemical composition of soil and bedrock, their humidity content, their surface temperature (e.g., Laferrière and Gaonac'h, 1999;Maître and Pinciroli, 1999;Lovejoy et al, 2001a, b;Harvey et al, 2002;Beaulieu and Gaonac'h, 2002;Gaonac'h et al, 2003;Cheng, 2004).…”

“…In the case where the anisotropy is only a function of size (but not location) the appropriate notion of scale for these structures is determined by a power law where the exponent is a matrix instead of a scalar. This matrix G, generator of the anisotropy, determines the notion of scale and hence the differential scaling according to the directions (for example see Lovejoy et al, 1992Lovejoy et al, , 2001aLovejoy et al, , b, 2005Pflug et al, 1993;Pecknold et al, 2001;Lilley et al, 2004;Cheng, 2004). But anisotropy is important for image processing and pattern recognition too.…”