Multifractal Modeling Of 2D Geochemical Landscape Based On Wavelet Transforms

Scale-invariance has been viewed as a fundamental property of the nature for describing the geological characteristics and patterns caused from various geological processes and events. 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.As an extension to Fourier analysis, it has been demonstrated that singular or non-stationary signals such as geochemical or geological patterns under wavelet analysis shows more efficient than Fourier analysis based on the retaining both space or time position and frequency components. Wavelet analysis and multifractals are different descriptions concerned with scaling analysis, one focuses on the general similarity and the other focuses on the scale information of one point.This dissertation mainly devoted to some contents as follows:Firstly, various multifractal formalisms based on wavelet transforms are summarized including wavelet coefficients, WTMM, and WLs methods. It is demonstrated by the case study of de Wijs model and stream sediment elements concentrations that box-counting is superior to the WLs and wavelet coefficient.Secondly, a new scaling analysis method called cumulant analysis is proposed, which has been demonstrated to be an efficient method for general scaling analysis in fully developed turbulence fields and efficiently overcome the influence of some noises. Meanwhile, the lower orders moment estimation derived from cumulant analysis, compared with direct polynomial expansion using difference derivatives, shows good behavior for simple multifractals.Finally, a new filter is proposed based on the power laws existed between wavelet coefficients and scales extended from C-A, S-A models. It has been showed us that by the case of stream sediment elements concentrations that W-A model is an efficient model for anomalies identification and can effectively avoid the influence caused from pseudo-high frequency components.