Definition of fractal topography to essential understanding of scale-invariance

Abstract

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter Hxy, a general Hurst exponent, which is analytically expressed by Hxy = log Px/log Py where Px and Py are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which . Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

Characterizing the Complexity Assembly of Pore Structure in a Coal Matrix: Principle, Methodology, and Modeling Application

“…As pointed out in our previous study (Jin, Wu, et al, 2017), the fractal dimension is a dependent variable of fractal behavior defined by fractal topography of Ω( P , F ). In an Ω, P and F are two dimensionless parameters and denominated by scaling lacunarity and scaling coverage, respectively.…”

“…However, such a task remains a challenge because of the inadequate and vague understanding of scale invariance and complexity 10.1029/2020JB020110 types. Fortunately, the newly emergence of fractal topography theory (Jin et al, 2019;Jin, Wu, et al, 2017) sheds a light on this issue.…”

“…Their common feature is to quantify the fractal behavior by D obtained directly or indirectly. In fact, a single D cannot determine the fractal behavior (i.e., an assembly of scaling behaviors) because of its one‐to‐many relationship (Jin et al, 2019; Jin, Wu, et al, 2017), as aforementioned. Therefore, either for the static purposes (volumetric) or the dynamic purposes (heat and mass transfer), the first and essential step is to clearly identify the control mechanism of scale‐invariance properties.…”

“…Although the above models provide deep insight in the characterization of pore structure and the productivity estimation (Bird & Perrier, 2010; Sanderson & Nixon, 2015; Zhao et al, 1994), their validity can hardly be guaranteed, because D and R ns follow one‐to‐many relationships (Jin, Wu, et al, 2017). Therefore, it is insufficient to characterize the pore structure in coal by a single parameter D .…”

Detection of oriented fractal scaling components in anisotropic two-dimensional trajectories

“…Noise and fluctuations frequently display fractal-like scaling properties associated with long-range correlations in real-world complex systems, such as biological, geophysical, and economical systems 1 , 2 . The appearance of such fluctuations has been considered as a key marker associated with a universal principle hidden in the complex system dynamics 3 .…”