Multifractal analysis of the pore- and solid-phases in binary two-dimensional images of natural porous structures

Abstract

We use multifractal analysis (MFA) to investigate how the Rényi dimensions of the solid mass and the pore space in porous structures are related to each other. To our knowledge, there is no investigation about the relationship of Rényi or generalized dimensions of two phases of the same structure.Images of three different natural porous structures covering three orders of magnitude were investigated: a microscopic soil structure, a soil void system visible without magnification and a mineral dendrite. Image size was always 1024 × 1024 pixels and box sizes were chosen as powers of 2. MFA was carried out according to the method of moments, i.e., the probability distribution was estimated for moments ranging from − 10 b q b 10 and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. A meaningful interval of box sizes was determined by estimating the characteristic length of the pore space and taking the next higher power of 2 value as the smallest box size, whereas the greatest box size was determined by optimizing the coefficients of determination of the log/log fits for all q. The optimized box size range spans from 32 to 1024 pixels for all images. Good generalized dimension (Dq) spectra were obtained for this box size range, which are capable of characterizing heterogeneous spatial porous structure. They are alike for all images and phases which the exception of the solid mass of the soil void system, which shows a rather flat D q behavior. A closer examination reveals that similar patterns of structure gain similar spectra of generalized dimensions. The capacity dimension for q = 0 is close to the Euclidian dimension 2 for all investigated images and phases.

Comparison of gliding box and box-counting methods in river network analysis

“…We have applied the configuration entropy, following the work of Tarquis et al (2006) and Dathe et al (2006), to select the minimum area that should be used, in order to obtain good results in the D q estimations but realizing that the number of boxes measured and the number of points for the regression analysis was low.…”

“…The minimum box size (a min ) is obtained here based on configuration entropy analysis, which is easy to implement and has already been used for several types of images Dathe et al, 2006). Configuration entropy analysis studies the effect of scale in any measure (a scalar quantity that leads to a positive distribution) defined in a plane.…”

“…He assumed the box size to be much greater than the smallest elementary cell in the system and much smaller than the system size. Merits and limitations of the MFA have been discussed by Chhabra et al (1989), Aharony (1989Aharony ( , 1990, Beghdadi et al (1993), Andraud et al (1994), Meakin (1998), Bird et al (2006), Dathe et al (2006) and Perrier et al (2006).…”

Multifractal Characterization of Pore Size Distributions of Peat Soil

“…In soil science, the multifractal concept has been applied to analyze the pore size distribution and particles of soil [6][7][8], the image of a thin slice of soil [9], the structure of clay [10], the matrix pore structure of a porous medium [11], soil surface pressure [12] and soil spatial variability [13]. Muller and McCauley [4] were successful in applying the fractal method to characterize the properties of the fluid flow of sedimentary rocks.…”

Comparison of the multifractal characteristics of heavy metals in soils within two areas of contrasting economic activities in China

“…anthropogenically derived heavy metal concentrations in this case) from background concentrations (i.e. geologically derived heavy metal concentrations; Cheng, 2001), and multifractal spectra that highlight non-linear characteristics and identify anomalous behaviour that reflects the characteristics of some multifractal sets (Gonçalves, 2000;Albanese et al, 2007;Guillén et al, 2011), such as the presence of porous structures and spatial variations in soil properties (Caniego et al, 2005;Dathe et al, 2006). This means that multifractal techniques can be useful tools for the analysis of heavy metal pollution within soils (e.g.…”