Multifractal modeling of 2D geochemical landscape based on Wavelet Leaders

As one field in nonlinear sciences, It has been demonstrated that the concepts and models relevant to multifractal theory are useful not only for characterizing the fundamental properties of non-linearity of the mineralization processes, the singular distribution of mineral deposits and ore element concentrations in mineral districts, but also for singularity analysis and anomaly delineation. This paper compares the box-counting moment based multifractal method with a newly developed wavelet leaders moment based multifractal method. The former is based on the technique characterized by the form of integrals or sums concerned with probability measures usually used in dynamical systems, while the latter by increments with formulated multiscale functions used in turbulence signals. Both deterministic and random multiplicative cascade synthetic images and geochemical landscapes created form Gejiu mineral district of southern China were chosen to demonstrate the fact that by convention both two methods assumed a biased estimation for scaling singularity exponents and spectra and exhibit large uncertainties for various datasets. The selective and filtering behavior caused from wavelet leaders directly lead to lots of smaller values compared to the measures basis, and thus, by moment generating process which offers us an unstable estimation. As for the characteristic of wavelet transform, we have the reasons to believe that the similar behavior could existed in those signals which owned the properties such as small sample sizes, short scaling range, low fluctuation and high frequency superimposition, and so on, whereas for those signals owned large sample size and abruptly changes, the wavelet leaders method could be an efficient tools for multifractal analysis. In addition the introduction of wavelet leaders based multifractal method could complement by the original scaling system and estimation procedures in the geological field aimed at interpreting geological phenomenon.