The surface fractal dimension of the soil–pore interface as measured by image analysis

There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (DL) within a plain has been measured, the surface fractal dimension DS is obtained by DS=DL+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of DS. In contrast to our expectations for a mathematical fractal, we found decreasing values for DS with increasing magnification. The values drift from DS=2.91 for a resolution of 2.44 μm/pixel to DS=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.