Multifractal Filtering and Decomposition of Geochemical and Geophysical Anomalous Fields

Spatially superimposed multiple processes such as multiplicative cascade processes often generate multifractal measures possessing so-called self-similarity or self-affinity that can be described by power-law type of functions within certain scale ranges. The multifractalities can be estimated by applying multifractal modeling to the measures reflecting the characteristics of the physical processes such as the element concentration values analyzed in rock and soil samples and caused by the underlying mineralization processes and the other geological processes. The local and regional geological processes may result in geochemical patterns with distinct multifractalities as well as variable scaling ranges. Separation of these multifractal measures on the basis of both the distinct multifractalities and the scaling ranges will be significant both for theoretical studies of multifractal modeling and its applications. Multifractal scaling breaks have been observed from various multifractal patterns. This paper introduces a technique for separating multifractal measures on the basis of scaling breaks. It has been demonstrated that the method is effective for decomposing geochemical and geophysical anomalies required for mineral exploration. The method involves a power-law model representing the relationships between the power spectrum values (S) and the "area" of the set on the frequency plain with power spectrum values above S, \{Wx, Wy: > S \}, where Wx, and Wy representing the wave numbers in horizontal and vertical directions, respectively. It may be anticipated that the power-law relationship shows multi-scaling properties over multiple scale ranges. The scaling breaks bounding the multiple ranges of power spectra can be identified on log-log plots of A(> S) vs. S. Each such scale range then can be used to define a filter. Taking a simple case with two filters as an example, assuming two ranges of power spectrum can be identified by fitting two different power-law relationships with exponents b1 and b2, respectively, then the threshold S0 obtained from these two power-law relations can be used to form the two sets \{Wx, Wy: S > S0 \} and \{Wx, Wy: S0 < S \}, which can be further used to define two filters G1(Wx, Wy) = 1 if Wx, Wy belongs \{Wx, Wy: S > S0\} and otherwise G1(Wx, Wy) = 0. The other filter can be G2(Wx, Wy) = 1 - G1(Wx, Wy). Inverse Fourier transformation can be applied with these filters to convert back to space domain to yield two decomposed measures. The decomposed components can be nonfractal, fractal, or multifractal quantities with less variability in comparison with the bulk measure. Several datasets containing the concentration values of several trace elements in soil, lake sediment, stream sediment and rock samples collected from mineral districts were employed for demonstrating the application of the method for identification of mineralization associated anomalies from background fields.