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The study of soil structure, i.e., the pores, is of vital importance in different fields of science and technology. Total pore volume (porosity), pore surface, pore connectivity and pore size distribution are some (probably the most important) of the geometric measurements of pore space. The technology of X-ray computed tomography allows us to obtain 3D images of the inside of a soil sample enabling study of the pores without disturbing the samples. In this work we performed a set of geometrical measures, some of them from mathematical morphology, to assess and quantify any possible difference that tillage may have caused on the soil. We compared samples from tilled soil with samples from a soil with natural vegetation taken in a very close area. Our results show that the main differences between these two groups of samples are total surface area and pore connectivity per unit pore volume.

There is order in chaos. This order appears, for instance, at: (i) the fractality of the chaotic attractors, (ii) the universality of the period doubling bifurcations and (iii) the visibility patterns of the time series that are generated by some unimodal maps. In this paper, we focus on the geometric structure of time series that can be deduced from a visibility mapping. This visibility mapping associates to each point (t, x(t)) of the time series to its horizontal visibility basin, i.e. the number of points of the time series that can be seen horizontally from each respective point. We apply this study to the class of unimodal maps that give rise to equivalent bifurcation diagrams. We use the classical logistic map to illustrate the main results of this paper: there are visibility patterns in each cascade of the bifurcation diagram, converging at the onset of chaos. The visibility pattern of a periodic time series is generated from elementary blocks of visibility in a recursive way. This rule of recurrence applies to all periodic-doubling cascades. Within a particular window, as the growth parameter r varies and each period doubles, these blocks are recurrently embedded forming the visibility pattern for each period. In the limit, at the onset of chaos, an infinite pattern of visibility appears containing all visibility patterns of the periodic time series of the cascade. We have seen that these visibility patterns have specific properties: (i) the size of the elementary blocks depends on the period of the time series, (ii) certain time series sharing the same periodicity can have different elementary blocks and, therefore, different visibility patterns, (iii) since the 2-period and 3-period windows are unique, the respective elementary blocks, {2} and {2 3}, are also unique and thus, their visibility patterns. We explore the visibility patterns of other low-periodic time series and also enumerate all elementary blocks of each of their periodic windows. All of these visibility patterns are reflected in the corresponding accumulation points at the onset of chaos, where periodicity is lost. Finally, we discuss the application of these results in the field of non-linear dynamics.

Soil structure controls important physical and biological processes in soil–plant–microbial systems. Those processes are dominated by the geometry of soil pore structure, and a correct model of this geometry is critical for understanding them. Soil tomography has been shown to provide rich three‐dimensional digital information on soil pore geometry. Recently, mathematical morphological techniques have been proposed as powerful tools to analyze and quantify the geometrical features of porous media. Minkowski functionals and morphological functions built over Minkowski functionals provide computationally efficient means to measure four fundamental geometrical features of three‐dimensional geometrical objects, that is, volume, boundary surface, mean boundary surface curvature, and connectivity. We used the threshold and the dilation and erosion of three‐dimensional images to generate morphological functions and explore the evolution of Minkowski functionals as the threshold and as the degree of dilation and erosion changes. We analyzed the three‐dimensional geometry of soil pore space with X‐ray computed tomography (CT) of intact soil columns from a Spanish Mediterranean vineyard by using two different management practices (conventional tillage versus permanent cover crop of resident vegetation). Our results suggested that morphological functions built over Minkowski functionals provide promising tools to characterize soil macropore structure and that the evolution of morphological features with dilation and erosion is more informative as an indicator of structure than moving threshold for both soil managements studied.

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