Quantification of soil structure based on Minkowski functions

The structure of soils and other geologic media is a complex three-dimensional object. Most of the physical material properties including mechanical and hydraulic characteristics are immediately linked to the structure given by the pore space and its spatial distribution. It is an old dream and still a formidable challenge to relate structural features of porous media to their functional properties. Using tomographic techniques, soil structure can be directly observed at a range of spatial scales. In this paper we present a scale-invariant concept to quantify complex structures based on a limited set of meaningful morphological functions. They are based on d+1 Minkowski functionals as defined for d-dimensional bodies. These basic quantities are determined as a function of pore size or aggregate size obtained by filter procedures using mathematical morphology. The resulting Minkowski functions provide valuable information on the size of pores and aggregates, the pore surface area and the pore topology having the potential to be linked to physical properties. The theoretical background and the related algorithms are presented and the approach is demonstrated for the pore structure of an arable soil and the pore structure of a sand both obtained by X-ray micro-tomography. We also analyze the fundamental problem of limited resolution which is critical for any attempt to quantify structural features at any scale using samples of different size recorded at different resolutions. The results demonstrate that objects smaller than 5 voxels are critical for quantitative analysis.