Fractal Surfaces: Measurement and Applications in the Earth Sciences

Abstract

Earth scientists have measured fractal dimensions of surfaces by different techniques, including the divider, box, triangle, slit-island, power spectral, variogram and distribution methods. We review these seven measurement techniques, finding that fractal dimensions may vary systematically with measurement method. We discuss possible reasons for these differences, and point to common problems shared by all of the methods, including the remainder problem, curve-fitting, orientation of the measurement plane, size and direction of the sample. Fractal measurements have been applied to many problems in the earth sciences, at a wide range of spatial scales. These include map data of topography; fault traces and fracture networks; fracture surfaces of natural rocks, both in the field and at laboratory scales; metal surfaces; porous aggregate geometry; flow and transport through heterogeneous systems; and various microscopic surface phenomena associated with adsorption, aggregation, erosion and chemical dissolution. We review these applications and discuss the usefulness and limitations of fractal analysis to these types of problems in the earth sciences.

Understanding Fractal Analysis? The Case of Fractal Linguistics

“…Thus the use of specifi c language by a group establishes an identity for this group with a specialist domain of knowledge and expertise. (2) Those with a solid knowledge base in other disciplines and an understanding of mathematics or physics [3,[16][17][18] ; those that bridge the gap between (1) and (2), that is, those that develop tools for fractal analyses and aim at describing these clearly [19] . Category 3 should primarily consist of category 1 researchers that aim to make the theoretical basis more accessible to applied research.…”

“…Roughness is a measure of the average variation about a mean and is not related to scale or changes in scale of measurement. D quantifi es the variation in length, area or volume with changes in the size of the measuring scale [16] .…”

“…There exists disagreement among physicists and mathematicians about what constitutes a dimension and what measures can be included in this term. Apart from the Hausdorff dimension other so-called fractal dimensions such as the Minkowski and the Kolmogorov dimension are not really dimensions in a mathematical sense [3,7,16,41] . However, in the literature authors often refer to these two measures as dimensions possibly because of trying to simplify what is really quite a complex area in mathematics.…”

Application of Multifractal and Joint Multifractal Analysis in Examining Soil Spatial Variation: A Review

“…Therefore, multifractal behavior implies that a statistically self-similar measure can be represented as a combination of interwoven fractal dimensions with corresponding scaling exponents. The multifractal parameters are generally independent of the size of the studied objects [41] and do not assume any specific distribution in the data [42]. Multifractal analysis can transform irregular data into a compact form and amplify small differences among the variables [43,44].…”

Slow Crack Propagation and Slip Correlations

“…One main evidence of the fluctuating front propagation is the roughness of fracture surfaces. Numerous works have shown the self-affine properties of crack surfaces (BROWN and SCHOLZ, 1985;BOUCHAUD et al, 1990;MA˚LØY et al, 1992;COX and WANG, 1993;SCHMITTBUHL et al, 1995b;BOUCHAUD, 1997). The out-plane roughness exponent (f % 0:80) is found to be very robust over different materials, different fracture modes and a broad range of length scales (BROWN and SCHOLZ, 1985;POWER et al, 1987;SCHMITTBUHL et al, 1993;SCHOLZ, 1990).…”