Fractal density modeling of crustal heterogeneity from the KTB deep hole

Abstract

Fractal or multifractal concepts have significantly enlightened our understanding of crustal heterogeneity. Much attention has focused on 1/f scaling natures of physicochemical heterogeneity of Earth crust from fractal increment perspective. In this study, fractal density model from fractal clustering point of view is used to characterize the scaling behaviors of heterogeneous sources recorded at German Continental Deep Drilling Program (KTB) main hole, and of special contribution is the local and global multifractal analysis revisited by using Haar wavelet transform (HWT). Fractal density modeling of mass accumulation generalizes the unit of rock density from integer (e.g., g/cm3) to real numbers (e.g., g/cmα), so that crustal heterogeneities with respect to source accumulation are quantified by singularity strength of fractal density in α‐dimensional space. From that perspective, we found that the bulk densities of metamorphic rocks exhibit fractal properties but have a weak multifractality, decreasing with the depth. The multiscaling natures of chemical logs also have been evidenced, and the observed distinct fractal laws for mineral contents are related to their different geochemical behaviors within complex lithological context. Accordingly, scaling distributions of mineral contents have been recognized as a main contributor to the multifractal natures of heterogeneous density for low‐porosity crystalline rocks. This finally allows us to use de Wijs cascade process to explain the mechanism of fractal density. In practice, the proposed local singularity analysis based on HWT is suggested as an attractive high‐pass filtering to amplify weak signatures of well logs as well as to delineate microlithological changes.

Fractal Dimensions of Cell Wall in Growing Cotton Fibers

“…where P o is the specific pore volume of immature cotton fibers having the primary cell wall only, D v is the fractal dimension; E = 3 is the Euclidean dimension. Similarly, the dependence of the apparent density on the scale factor, F, is the following [7,14]:…”

“…Previously, the fractal theory was applied to analyze the porous structure of various materials [7][8][9][10]. In particular, for cellulose it was found that the process of nitrogen sorption by pores of various sizes can be described by a fractal with dimension from 2.1 to 2.5 [11].…”

Mixed multifractal densities for quasi-Ahlfors vector-valued measures

“…Indeed, such densities have been applied for many time in the understanding of many phenomena, such as climate factors distributions in different regions, crashes propagation and the effect on markets, etc. See [1,7,22,24,39,42]. These applications have been tackled by using the single multifractal analysis and densities.…”

Cyclicity and Persistence of Earth's Evolution Over Time: Wavelet and Fractal Analysis

“…Wavelet analysis is a powerful tool for investigating the multiscale nature and scaling behavior of hierarchical systems (Mallat, ), thus providing an appropriate framework for both the local Hurst detection (LHD) and local singularity analysis (LSA) of fractal signal records (Chen & Cheng, ). Notably, within the context of fractals, the Hurst index characterizes the persistence or intermittency of time series, while the singularity index is for the clustering or accumulation of mass or energy.…”

“…Wavelet transform (WT) is a popular method for time series analysis (e.g., trend, cycle, and correlation) of diverse geological or geophysical records (Dashtian et al, 2015;Kumar & Foufoula-Georgiou, 1997), including various isotope signals (Prokoph et al, 2004;Prokoph & Puetz, 2015). In addition, WT advances the scaling analysis of hierarchical structures and transient phenomena (Chen & Cheng, 2017). In this study, we use WT to obtain the local fractal sequence of U-Pb age distribution and δ 18 O time series based on the fractal modeling of large data sets complied from the literatures.…”