Surface fractal characteristics of preferential flow patterns in field soils: evaluation and effect of image processing

Abstract

In the last few decades, preferential flow has become recognized as a process of great practical significance for the transport of water and contaminants in field soils. Dyes are frequently used to visualize preferential flow pathways, and fractal geometry is increasingly applied to the characterization of these pathways via image analysis, leading to the determination of `mass' and `surface' fractal dimensions. Recent work by the authors has shown the first of these dimensions to be strongly dependent on operator choices (related to image resolution, thresholding algorithm, and fractal dimension definition), and to tend asymptotically to 2.0 for decreasing pixel size. A similar analysis is carried out in the present article in the case of the surface fractal dimension of the same stained preferential flow pathway, observed in an orchard soil. The results indicate that when the box-counting, information, and correlation dimensions of the stain pattern are evaluated via non-linear regression, they vary anywhere between 1.31 and 1.64, depending on choices made at different stages in the evaluation. Among the parameters subject to choice, image resolution does not appear to exert a significant influence on dimension estimates. A similar lack of dependency on image resolution is found in the case of a textbook surface fractal, the quadratic von Koch island. These parallel observations suggest that the observed stain pattern exhibits characteristics similar to those of a surface fractal. The high statistical significance ( R>0.99) associated with each dimension estimate lends further credence to the fractality of the stain pattern. However, when proper attention is given to the fact that the theoretical definition of the surface `fractal' dimension, in any one of its embodiments, involves the passage to a limit, the fractal character of the stain pattern appears more doubtful. Depending on the relative weight given to the available pieces of evidence, one may conclude that the stain pattern is or is not a surface fractal. However, this conundrum may or may not have practical significance. Indeed, whether or not the stain pattern is a surface fractal, the averaging method proposed in the present article to calculate surface dimensions yields relatively robust estimates, in the sense that they are independent of image resolution. These dimensions, even if they are not `fractal', may eventually play an important role in future dynamical theories of preferential flow in field soils.