3D multifractal characterization of computed tomography images of soils under different tillage management: Linking multifractal parameters to physical properties

Geoderma 363 (2020) 114129

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3D multifractal characterization of computed tomography images of soils under different tillage management: Linking multifractal parameters to physical properties

T

⁎

Diego Soto-Gómeza,b, , Paula Pérez-Rodrígueza,b,c, Laura Vázquez Juíza,b, Marcos Paradelod, J. Eugenio López-Periagoa,b a

Soil Science and Agricultural Chemistry Group, Department of Plant Biology and Soil Science, Faculty of Sciences, University of Vigo, Ourense E-32004, Spain Hydraulics Laboratory, Campus da Auga, Facultade de Ciencias, Campus da Auga, University of Vigo, Ourense, Spain Laboratory of Hydrology and Geochemistry of Strasbourg (LHyGeS), Université de Strasbourg, Strasbourg, France d Department of Sustainable Agriculture Sciences, Rothamsted Research, Harpenden, Hertfordshire, United Kingdom b c

A R T I C LE I N FO

A B S T R A C T

Handling Editor: L.S. Morgan Cristine

Multifractal analysis of pore images obtained from X-ray computed tomography (CT) was used to characterize the scaling properties of macropores in soils with different managements and their correspondence with macroscopic physical properties related with the soil functions. We used CT images of twenty undisturbed soil columns to examine the multifractal properties of the pores identified by X-ray computed tomography (CT-Porosity). Multifractal spectra successfully describe the scaling of the pore network in all soil columns. The dimensions and scaling parameters of these spectra correlate with macroscopic magnitudes, namely, CT-Porosity, surface area of the pore walls, tortuosity, and bulk density. We also found strong correlations between the singularity spectra and the topological descriptors of the pore network skeleton: total slab voxels, number of branches per path, number of endpoints and sum of branch length, among others. These correlations show that the complexity of the CT-Porosity can be related quantitatively with physical properties, the organization of the pore skeleton and solute transport.

Keywords: Multifractal analysis Soil management Soil structure Pore network Organic farming

1. Introduction Soil pore network is an evolving and complex three-dimensional part of the soil that governs the transport of gases, water, solutes, colloids, and particles. At the same time, the transport of these substances along the soil pore network takes part in key processes and functions in terrestrial environments such as water balance, plant growth, nutrient cycles (Young and Crawford, 2004), and gas exchange, that includes release and capture of greenhouse gases (Quigley et al., 2018; Steffens et al., 2017). CT imaging is a non-destructive method used to extract the porous network, and allows to measure the shape and the size of the soil pores, as well as their connectivity, number of paths, junctions, branches and loops (Helliwell et al., 2013; Horgan, 1998; Torre et al., 2018a). The importance of the spatial organization of the soil pore network is increasingly recognized because determines phenomena close related to water flow and transport, such as entrapped air, irreducible water saturation and hysteresis (Jury et al., 2011). Biopores

usually constitute a small proportion of the soil pores, but due to hierarchical network structure (Elliott et al., 1999), have a critical effect on percolation and leaching water, solutes, and suspended particles (i.e. small solid objects dispersed in water) (Heuvelink and Webster, 2001; Larsbo et al., 2014). Furthermore, hierarchical organization of biopores influences the preferential transport because favors the occurrence of large continuous paths (Marcus et al., 2013; Paradelo et al., 2013; Rabbi et al., 2018). More recently, it was shown that biopore organization is related to local water front pressure instabilities during drainage (Soto-Gómez et al., 2017). Changes in the soil pore network across scales can be explained by their scaling relations by using fractal analysis. Fractal techniques were used in the assessment of the variability of soil properties and scaling (Pachepsky et al., 1996). Interestingly, fractal analysis can account for the rare occurrences in hierarchical networks such as very large pores that have a fundamental role on the hydraulic conductivity and preferential transport in depth (Alvarez-Benedi and Munoz-Carpena,

⁎ Corresponding author at: Soil Science and Agricultural Chemistry Group, Department of Plant Biology and Soil Science, Faculty of Sciences, University of Vigo, Ourense E-32004, Spain. E-mail address: [email protected] (D. Soto-Gómez).

https://doi.org/10.1016/j.geoderma.2019.114129 Received 24 May 2019; Received in revised form 5 December 2019; Accepted 8 December 2019 0016-7061/ © 2019 Elsevier B.V. All rights reserved.

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Table 1 Sieve analysis results. Treatment

Coarse Sand (%)

Fine Sand (%)

Silt (%)

Clay (%)

Organic Matter (%)

Conv. NT Conv. ST Org.

46.2 ± 0.5 42.9 ± 2.4 44.5 ± 0.2

26.1 ± 0.9 28.3 ± 1.7 29 ± 0.4

5.7 ± 2.9 5.3 ± 4.1 8.1 ± 0.3

10.9 ± 1.2 11 ± 0.6 9.2 ± 0.7

11.1 ± 2.6 12.5 ± 4.6 8.5 ± 0.5

root crops and vegetables in which tests of organic farming were performed (Org.): five from subplots that showed higher earthworm activity deduced by surface alteration (Org. A), and the other five were randomly taken in the plot (Org. B). We considered that in both groups of samples the types of pores are similar, and the difference lies in the number and size of pores: Org. A samples are extreme cases of samples from the organic plot, with a huge porosity that favors the transport. Another five columns were sampled from a conventional cereal-potato rotation plot with disrupted root channels by shallow tillage up to 10 cm depth, presenting a denser layer at the bottom of the columns (ST). The last five columns were sampled from a no-till (NT) treatment where roots were preserved. The sieve and organic matter analysis results (Table 1) showed that all managements were quite similar, but the Org. treatment presented the smallest amount of organic matter because of its historical use for root crops with the removal of stubble. The columns were carefully extracted the same day (January 2013) using the procedure explained in Soto-Gómez et al. (2018) and summarized below. PVC cases with a sharp bottom edge were inserted in the soil (at 2–12 cm depth) without compacting it when the soil was in a friable mechanical condition, and were sealed immediately after extraction. Finally, they all were stored in darkness at 4 °C. Data of solute (bromide) and colloidal breakthrough experiments (microspheres) and preferential flow paths were collected from previous works in which we used the same samples (see Soto-Gómez et al., 2018; 2019).

2004). The inherent complexity of pore networks is best described with multifractal analysis that accounts for the rich scaling properties of the soil structure. Multifractal analysis on two dimensional (2D) thin slices was used to describe the fractal behavior of soil pore space with several thresholdings (Bird et al., 2006; Giménez et al., 2002; Tarquis et al., 2009). Lately, multifractal analysis on 2D images of soil CT-Porosity is evolving and 3D analysis gained importance (Piñuela et al., 2010; Torre et al., 2018b). Pore networks reconstructed from CT images were used to characterize the scaling of pores in peat and to estimate the hydraulic properties (Rezanezhad et al., 2010). Recently, Wang et al. (2018) reported changes in the indicators of the multifractal spectra in 3D of soils sampled around opencast coal mining upon restoration actions, but no relation of physical magnitudes with fractal parameters or topologic measures were reported. Percolation and conductivity related properties of pore networks were derived for some mathematic fractal objects used to model the soil structure (Hunt, 2004; Perrier et al., 2010). Worth noting the studies using computer simulations on idealized porous media such as JiménezHornero et al. (2009), and, more recently, Lafond et al. (2012), who compared multifractal properties of porosity inferred from CT images with gas diffusion and physical properties. However, these relations are far from being achieved in non-homogeneous porous media. Linking multiscale descriptors of disordered natural porous objects with macroscopic properties and transport functions is in the early stage of its development. Indeed, advances will require combined (multi)fractal analysis and experimental measures of soil physical parameters. Despite that, some authors found correlations between features of the preferential flow and the multifractal spectrum, very little studies are available about the correlations between multifractal descriptors of the soil CT-Porosity and their transport properties (Posadas et al., 2009). The hypotheses of the work are two: (1) the soil management produces alterations in the porous soil network that condition their multifractal properties, and (2) multifractal characteristics are related with the transport properties and can be used to predict the transport and retention of substances through the soil. So, this work is focused on three main aspects (1) the comparison of multifractal properties of the CT-Porosities of soils under three different types of tillage management; (2) the analysis of the changes in the multifractal characteristics of 2D soil sections with depth; and (3) the search of correlations between multifractal descriptors and other soil properties, such as statistical descriptors of the pore network topology and parameters describing transport.

2.2. CT images acquisition, filtering, and treatment The samples were scanned (at field capacity) with a cone-beam Xray computed scanner apparatus (i-CAT 3D, Imaging Sciences International) with the X-ray tube set to 120 kV and 5 mA current. We obtained a 2D image sequence (stack) of each column with a voxel size of 0.24 mm using the ImageJ-Fiji software (Schindelin et al., 2012). Most of the parameters were calculated using the complete stack, but for the multifractal part we had to use a smaller region of interest since the software developed only works with cubes (this is clarified in the following section). Segmentation of 8 bits images was conducted to separate the pores from the matrix obtaining a set of binarized images. This is a critical step of the analysis. The appropriate segmentation method depends on the image quality, resolution, distribution of X-ray attenuation data, and, finally, the objective of the study (Pagenkemper et al., 2015). In our case, the best segmentation method for the separation of pores from the soil matrix was the Sauvola’s auto local thresholding (Sauvola and Pietikäinen, 2000). This procedure determines the value of each pixel individually, considering the surrounding area. We used the following settings: radius of 50 pixels, parameter 1 (k value) of 0.3 and parameter 2 (r value) of 128 (default value). The value of each pixel was determined by the following equation:

2. Materials and methods 2.1. Soil sampling and leaching experiments The sampling strategy was conducted for obtaining a variety of pore types (with different size and shape) in the same soil with small variations in texture and chemical properties. We sampled soils from two adjacent experimental parcels (Centro de Desenvolvemento Agrogandeiro, Ourense, north-western Spain, coordinates 42.099 N, −7.726 W WGS84). The texture of the soils is sandy loam according to the USDA texture classification, with pH in water of 5.9 ± 0.05. The study was done on twenty unaltered columns (100 × 84 mm, height by diameter). The first 10 columns were sampled from a soil used to plant

Pixel = (pixel > mean × (1 + k + (standarddeviation/r − 1)))

(1)

We employed the same segmentation method with all samples in order to get comparable results. In the section below we included a summary of the CT parameters extracted. The complete explanation of the procedure is explained in Soto-Gómez et al. (2018). An example of the segmentation of each soil management can be seen in the Supplementary Information (Figure S1A–H). The CT-Porosity (cm3), i.e., the volume of the sample that belongs to 2

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normally distributed values or the non-parametric Kruskal–Wallis on the non-parametric ones. Correlation between parameters was tested with the Pearson’s correlation coefficient (R). We considered a significance level of 0.05 for the statistical analysis.

pore space (in 3D), was calculated over the processed binary images, considering the number of voxels occupied by pores and roots. We decided to include the roots in the pore space after the transport experiments and samples sectioning: we observed that the roots and root pores, in spite of being partially occupied, were capable of conducting particles. The number of pores and the surface and volume of every single pore were also calculated and recorded with the ImageJ pluging BoneJ. The binary images were processed to obtain the skeleton of the pores with the Skeletonize 3D plugin (Doube et al., 2010). An example of the skeleton of each soil management can be seen in the Supplementary Information (Figure S2). The topologic parameters extracted from the pore network skeleton were: number of branches and junctions, end-point voxels (voxels where the branches end, in the matrix or in the extremes of the sample), slab voxels, and triple and quadruple points (junctions that link three or four branches respectively). The path length (LR) and Euclidean length (LE) of each path of the skeleton were obtained with the analyse skeleton plugin and used to calculate the average CT tortuosity (τ) (Wu et al., 2006)

τ=

LR LE

3. Results and discussion A summary of the CT-Porosity data is shown below. The size of the pores that were identified with the CT and image analysis ranged from 1.4·10−2 mm3 to 3.5·104 mm3. The latter one was visually identified as a large multi-branched earthworm burrow. This range covers almost six orders of magnitude in volume. The average number of individual pores was 2365 per soil column, with a range from 1285 to 3930. We could not find significant differences between the number of pores of the different tillage managements. This can be caused by the great intragroup variability that exists. However, it is observed a lower average number of pores in the Org. B samples, caused by the no-tillage and the lower presence (or absence) of earthworm activity. It is important to note that the root pores had a slightly lighter shade on the tomography (associated to a higher density) than the hollow pores (e.g. earthworm pores). However, as we mentioned before, when carrying out the transport and sectioning experiments, and comparing the results, we observed that tracers (particulate ones) circulated thought these pores. So we decided to use a thresholding method that include them in the porous network. Detailed information of the characteristics of each soil obtained through the CT analysis, as well as other soil properties such as bulk density, are shown in the Supplementary Information in Tables S1, S2, S3, and S4. All soil columns were examined in order to determine the parameters of the Rényi spectrum (in 3D), the multifractal singularity spectrum (in 3D) and the vertical distribution of the fractal dimensions of 2D slices. Management practices influenced the parameters that are displayed in Table 2. In 2D we found differences between the average fractal dimension (D). In the 3D cubes, the significant effects of the management were found in the multifractal dimension in 3D (D0), the correlation dimension (D1), the entropy dimension (D2), the aperture of the spectrum (Ap), the right to left symmetry of the spectrum (R-L), and the vertical symmetry of the spectrum (V). The consistency of the multifractal calculations was assessed by checking the relation D0 ≥ D1 ≥ D2. It is important to recall that the parameters showed in Table 2 extracted from 3D cubes are average values from the two parts (upper and lower) of the soil column. We decided to use this approach to enclose the bigger amount of soil and also to compare both halves.

(2)

Then, through the BoneJ plugin, we purified the binarized stack (through the plugin Purify) and calculated parameters describing the pore network, namely the Euler-Poincare characteristic (χ) defined as the number of isolated pores minus the number of redundant connections, and the connectivity (Ragan and Hinkle, 1975). This last parameter is equal to 1 − χ, and gives us an idea of the amount of branches and loops present in our structure. 2.3. Multifractal analysis The multifractal calculations were carried out for both 2D slices (XY, plane) and 3D cubes. The multifractal analysis in the 2D slices was done using the ImageJ-Fiji Fraclac plugging (Karperien, 2013) that provides the vertical distribution of parameters of the multifractal spectra. The porous network is quite anisotropic, and we decided to follow the direction of the water flow and perform the analysis in 2D slices along the Z dimension. This allowed us to determine the variations in the fractality at different depths, and can help us to understand the behavior of the substances through the soil. For the 3D multifractal calculations we developed a software based in the BoneJ plugin FracLac (Doube et al., 2010), but is limited to cubes. Therefore, two 3D cubes were extracted from two depths (Z axis) in the centre of the images (in the X-Y plane) trying to enclose the bigger amount of soil: the upper part (0 to −4.8 cm Z axis) and the lower part (from −4.8 to −9.6 cm, approximately). The edge of all cubes was 200 voxels (≈4.8 cm length). Thus, the 3D multifractal calculations for each column were done on those two depths. The method used for the calculations of the multifractal spectra, namely, Rényi spectrum and singularity spectrum, is summarized in the Supplementary Information. The multifractal parameters obtained from the above spectra were the fractal dimension in 3D (D0), the entropy dimension (D1), the correlation dimension (D2), the aperture of the spectrum (Ap), the right to left symmetry of the spectrum (R-L), and the vertical symmetry of the spectrum (V). We also calculated the aperture and slope of several ranges: q (−1 to 1); q (0 to 1); q (−1 to 0), and so on. Graphical meaning of those parameters can be visualized in Fig. 1A and B. Moreover, a complete example of the multifractal calculations for one cube is shown in the Supplementary Information.

3.1. Multifractal analysis of 3D CT-porosity: Rényi spectrum The Rényi spectrum of all columns draws a sigmoid (Fig. 2), the typical shape of the systems with multifractal behavior. In all cases, the Dq decreased with the value of q (Lafond et al., 2012). As have been pointed by Marinho et al. (2016), the wider (in the vertical axis) the Dq spectrum, the more heterogeneous the scaling distribution. On the one hand, we have the soils from the Conv. NT and Org. A, with amplitudes of 2.42 ± 0.12 (Fig. 2A) and 2.35 ± 0.15 (Fig. 2B), respectively. The presence of a broad range of pore root sizes in NT increases the heterogeneity in both sides of the spectrum, and even the positive part (related with the bigger pores) presents a slight drop (San José Martínez et al., 2010). In Org. A samples, the shape indicates multifractality, but the bigger pores (right part of the spectrum) do not present many changes with scale (Dq is almost constant). On the other hand, Org B. samples are significantly less heterogeneous with an amplitude of the Rényi spectrum of 1.52 ± 0.13 (Fig. 2C). Conv. ST samples present a medium value of the spectrum width (2.10 ± 0.05), and have the minimum standard error. Tillage techniques create pore populations more homogenous. This can be appreciated comparing the standard errors and by taking a look at Fig. 2D and B, the ones that

2.4. Statistical analysis Normality of all the sets of multifractal indicators and soil variables was assessed with the Kolmogorov–Smirnov test. The influence of soil management on multifractal indicators was tested either with F-test for 3

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Fig. 1. A) Representation of an example of the Rényi spectrum (q Vs Dq). This example belongs to the sample number 16 (Conv. ST). Orange arrows indicate interesting values of the generalized dimensions: D0 (Dq when q = 0), D1 (Dq when q = 1), D2 (Dq when q = 2), and Dmin (Dq when q = -5). B) Representation of an example of the Singularity spectrum (α Vs f(α)). This example belongs to the sample number 16 (Conv. ST). Orange arrows indicate some interesting values of α: α0 (α when q = 0), αmin (α when q = +5), and αmax (α when q = -5). The green line represents the total aperture of the spectrum (Ap), the yellow lines are the apertures for the right (R) and left (L) branches of the spectrum, and the vertical grey line points the vertical symmetry of the spectrum (V). V is the difference between the f(αmin) (the value of f(α) when q = +5), and f(αmax) (the value of f(α) when q = -5). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

significant differences when comparing management techniques. However, the differences are caused by the extreme values of the Org. B zone (Table 2), as happened with the parameters of the Rényi spectrum. Despite that, a comparison of some multifractal features across the management practices can be illustrative. The columns from no-tilled soil presented an average wider spectrum (3.19 ± 0.41) than other treatments, and this means that the pore structure is more complex due to the variety of pores generated by roots. It is important to note that some of these columns presented big earthworm pores, like the column n° 7. This sample has a high CTPorosity (Table S2) when compared with the samples of the same plot, but the presence of a giant pore fills the space that could be occupied by small complex populations. Moreover, this core is a good example for the understanding of the value of multifractals since de single fractal dimension in 3D (D0) is fairly big because of the CT-Porosity, but the pores are not so complex when compared with the samples of the same plot. In the NT plot, we found the only two examples (n°2 and 11) of spectrums with a positive value of R-L (horizontal symmetry) and a negative V (Vertical symmetry) (Fig. 4A). This means that the small pore populations of these two samples are more complex and common in the soil profile. That is caused by the low presence of earthworm pores. Moreover, when plotting the Aperture of the spectrum versus the Dmin (Fig. 5), the NT cores are in the zone of high pore complexity (excluding the core 7). This relation is based in the works of Ge et al. (2015), where plotted D0 in front of Dmin to determine the complexity of the network. However, we substituted D0 by the aperture, since we consider it a sensitive parameter that approximates more closely to the observed. In the organic field (Org. A), the presence of earthworm pores (in

belong to soils without tillage. This is one of the results of the tillage: the homogenization of the soil. The value of the D0 follows the decreasing order Org. A > Conv. NT > Conv. ST > Org. B, with values of 2.71 ± 0.17, 2.70 ± 0.03, 2.67 ± 0.12 and 2.34 ± 0.13, respectively. However, only the Org. B shows significant differences with the other zones. The 2.34 is far from the topological dimension, 3, which means that the pore distribution range is narrow (Wang et al., 2016). The results of this parameter are similar to the corresponding one in 2D (Fig. 3), and the differences with the Org. B zone are maintained in other generalized dimensions (D1, D2 and Dmin). Considering the tomography porosities (CT-Porosity) of the four zones (Tables S1, S2, S3 and S4), it is possible to conclude that the generalized dimensions are determined mainly by the distribution of the pores, since the Conv. NT zone presents a low CT-Porosity (23.07 ± 4.68 cm3) compared to the Org. A (43.21 ± 4.98 cm3) and the Conv. ST (40.78 ± 6.92 cm3) zones. The CT-Porosity also conditions the generalized dimensions, and that can explain the significantly different values of the Org. B, a plot with an average CT-Porosity of 15.56 ± 3.61 cm3. Those CT-Porosity values were close to the expected since the tillage is used for increasing the soil aeration and the porosity (ST), but some of those pores are occupied by the roots and can produce a decrease in the CT-Porosity under no-till conditions (NT). The OA treatment has a high CT-Porosity caused by the earthworms, but the OB presents a lower value which is caused by the absence of tillage.

3.2. Multifractal analysis of 3D CT-Porosity: Singularity spectrum Multifractal characteristics of the singularity spectrum also showed

Table 2 Mean values of the features of the multifractal spectra of 2D slices and 3D cubes that showed significant differences between treatments: D, fractal dimension in 2D; D0, generalized dimension for q = 0; D1, generalized dimension for q = 1; D2, generalized dimension for q = 2; Ap, aperture of the multifractal spectrum; R-L, horizontal symmetry of the multifractal spectrum; and V, vertical symmetry of the multifractal spectrum. Sample number

Conv. ST Conv. NT Org. A Org. B

Average profile of 2D slices

Parameters of the Rényi and the multifractal spectrum of 3D pores

D

D0

1.07 1.28 1.11 0.95

± ± ± ±

0.11 0.22 0.14 0.11

2.67 2.70 2.71 2.34

D1 ± ± ± ±

0.12 0.03 0.17 0.13

2.34 2.38 2.37 1.98

D2 ± ± ± ±

0.18 0.05 0.14 0.13

4

2.26 2.23 2.27 1.88

Ap ± ± ± ±

0.19 0.10 0.14 0.10

2.68 3.19 3.02 2.01

± ± ± ±

0.14 0.41 0.45 0.22

R-L

V

0.47 ± 0.48 0.53 ± 0.25 0.65 ± 0.24 −0.40 ± 0.20

0.42 ± 0.37 −0.10 ± 0.53 0.37 ± 0.21 −0.40 ± 0.37

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Fig. 2. Rényi spectra for all columns organized by tillage managements: A) Conv. NT; B) Org. A; C) Conv. ST; and D) Org. B.

feature that will be considered in the next section of this article. The Org. B samples are significantly different from the others (Fig. 4D). For example, the apertures of the singularity spectra are quite low (2.01 ± 0.22). The pores of this soil are caused by some earthworms and remains of roots and vegetal residues. However, the average CT-Porosity is quite low and the pores are poorly distributed. When comparing the morphological features obtained from the tomography, this zone is significantly different, with fewer junctions and branches, and a lower connectivity. The lack of complexity can be appreciated in Fig. 5. The spectra of these samples are inverse to the ones of the Org. A, with negative values for both symmetries: they have complex big pores concentrated in some parts, and simple smaller pores well distributed. A similar result has already reported by Posadas et al. (2003) but using 2D images. The exception is the number 19, a sample that shows a unique behavior with a negative R-L and a positive V: complex big pores better distributed than smaller ones. However, the vertical symmetry is too close to 0 that we can consider that the distribution of pore populations in the sample is fairly similar.

some columns) increased the width of the spectra as expected (3.02 ± 0.45). It is important to remember that the Org. A columns were sampled in zones where the presence of earthworms was more evident, while the Org. B columns were randomly taken in the same plot. Since the presence of earthworm pores in the soil is not homogeneous, we observed a great variation (like in the case of NT samples). The main difference is caused by sample n° 10, the one with the lower CT-Porosity (Table S3). Samples of the Org. A plot have, in general, complex structures according to the Dmin Vs Aperture plot (Fig. 5). The singularity spectrum of all of the samples has the same shape (Fig. 4B): positive values for the two symmetries. That means that the smaller pores (from among the obtained through CT) are more complex but less common, and the bigger pores are well distributed in the samples but they are also simpler. Conv. ST soil samples have an average aperture of 2.68 ± 0.14, a value halfway between the complex spectra of Conv. NT and Org (Fig. 4C). A samples, and the simple one of the Org. B zone, as can be appreciated in Fig. 5. Most of the cores of this plot presents a spectrum similar to the obtained with Org. A samples, with positive values for both symmetries, indicating the presence of simple (less complex) big pores well distributed in all the sample. Those pore populations were produced by earthworms, crop residues and tillage. This last factor breaks the clods, disrupts the pores created by the soil fauna, and generate big cracks. The only sample that shows a different behavior is the number 12, a core that has relatively big pores poorly distributed. It is important to consider that the parameters showed in Table 2 are average values from the two parts (upper and lower) of the sample, a

3.3. Vertical variation of fractal properties and influence of the soil management The CT-Porosity of the CT-pore network and 2D fractal dimension in the X-Y slices (D) decrease with depth. Correlation between CT-Porosity and D is reported, as it was already pointed by other authors (Hatano et al., 1992; Oleschko et al., 2000). The mutual decrease of CT-Porosity and D, and its correlation can be related to the decreasing in the pore 5

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Fig. 3. A) Average fractal dimension (in 2D) for the four treatments. B) and C) Variation of the fractal dimension of each image with the depth: Column n°7, Conv. NT treatment (B) and Column n°6, Conv. ST treatment (C). (It is important to note that the scale is different).

Conv. ST, plots where the drop is very strong, indicating that the pore populations of the lower part are quite disordered. This can be related to the decrease in the values of connectivity with depth pointed by Muñoz-Ortega et al. (2015) for uncultivated and tilled soils. In the case of the singularity spectrum, when considering the aperture, there is a decrease, in almost all samples, in the range of values of the Holder exponent (i.e., the aperture of the spectrum, for further details see Supplementary Information: Fractal and Multifractal Theory) needed to describe the pore network (Fig. 6). So, the complexity of the CT extracted pore network is reduced with depth. If we compare the reduction with the value near the surface (from 0 to −4.8 cm), the lower part (from −4.8 to −9.6) is 9.19 ± 0.06% less complex. However, the change is not so sharp in the Conv. NT and Org. A zones. The Conv. ST presents the higher variations: the number 14, a sample extremely compacted near the lower extreme, presents a reduction in the complexity of 45.98%, but the number 18, a compacted sample with a big multi-branched earthworm pore located near the bottom, presents a spectrum 71.21% wider in the lower part. Finally, the case of the Org. B is the simplest one: the reduction in complexity happens in all the samples (21.87 ± 5.54%). Considering the symmetries, both values (R-L and V) tend to 0 with the depth, i.e., in general, the spectrum tends to symmetry: R-L is reduced from 0.43 ± 0.14 to 0.19 ± 0.13 and V from 0.09 ± 0.15 to 0.06 ± 0.11. This indicates more equilibrated pore populations in the lower parts of the samples: relatively big and small pores equally distributed with similar complexity. However, we found some exceptions. The left part (big pores) of the singularity spectra of the cores 18 (Conv. ST) and 1 (Org. A) does not change with depth, while the small pores are better distributed and less complex in the upper part.

network complexity with the depth that was reported by Yang et al. (2018). Moreover, the fractal dimension peaked in some narrow depth intervals that also showed a local increase in the CT-Porosity. The vertical variation of the Rényi and singularity spectra in 3D cubes was assessed by the calculation of the difference between the features of the spectrum of the cube in the top of the column minus those features of the lower cube (these characteristics are shown in Table 2). The D0 (the fractal dimension in 3D) decreases in the lower half of the column an average of 0.12 ± 0.03 for almost all samples (Fig. 2B and 2C). This decrease in D0 also correlates with the decrease in the CT-Porosity in agreement with the 2D fractal dimension. This effect can be caused by the decrease of biological activity and human alteration with depth: the CT-Porosity (and its complexity) is mostly generated by tillage, and by soil biota, factors that have more weight in surface layers. The drop in D0 is not so strong in the cores of the Conv. NT zone (0.07 ± 0.01) since the roots (using a visual approach) seem to be well distributed in the entire sample. In the Org. B zone, the decrease is also small (0.10 ± 0.10), but this plot has a bigger standard deviation and the initial values of D0 are also lower. Moreover, the pore populations are poorly distributed in all the sample (not only in the lower part). Org. A and Conv. ST samples showed a pronounced drop in almost all samples (0.13 ± 0.07 and 0.17 ± 0.08, respectively), caused by a compacted lower half in the case of Conv. ST, and by a bigger presence of earthworm pores and vegetal rests in the upper part in the case of Org. A. Something similar happens with the values of D1, a parameter related to the disorder. Again, the Conv. NT samples have a constant value of uniformity in both halves. In Org. B the value of D1 is quite low, and the decrease is not so evident as in the case of Org. A and 6

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Fig. 4. Singularity spectrum of all the samples that were identified in this study. A) Conv. NT, B) Org. A, C) Conv. ST, and D) Org. B. The number of each sample appear in the lower part of the graphs.

lattice of 2 × 2 pixels (0.23 mm2) has 24 different configurations. However, to represent a larger pore structure we need a larger region e.g., with 2 × 2 mm2, we have ~2 1069 different configurations. That huge number can support very complex fractal structures. Dependency on the support size can explain why most of the macroscopic magnitudes derived from the CT images (volume and surface) and topological descriptors of the skeleton (slab voxels, branches and junctions per path, among others) showed good correlations with the multifractal parameters. Values of all these descriptors depend on the size and complexity of the pore network. So that the discrete representation of junctions, branches, etc. needs a larger support (i.e., a larger area with many pixels) than objects with small Hausdorff exponents, the fractal dimension of soil parts with the same Holder exponent (for further details see Supplementary Information: Fractal and Multifractal Theory), e.g., a straight pore represented by an alignment of a few pixels, or a small circular pore (one single pixel). Starting with the descriptors extracted from the Rényi spectrum (D0 and D2), they are correlated with almost all the parameters considered in Table 3. Those two multifractal characteristics are related with the distribution of the pores in the soil, so the bigger the CT-Porosity, pore volume and surface, the bigger the probability of having a better distribution of pores, with pores more spread throughout the sample, and the bigger the value of D0 and D2. And with the parameters related to the skeleton, the same happens: the higher the complexity (number of junctions, branches, slab voxels, a higher connectivity…), the bigger the values of those two parameters. Tortuosity is a characteristic of the pore network with important implications for the transport of substances throughout soil (Roberts et al., 1987). The positive correlation with the tortuosity can be explained through the CT-Porosity: this feature is extremely correlated with the tortuosity (R = 0.89), and indicates that big pores (earthworm pores, for example) increase the average tortuosity. In short, the presence of big pores tends to increase the tortuosity and improve the distribution of the CT-Porosity, making

Fig. 5. Crossplot representing average Aperture and Dmin of all the columns. The complexity of the pore network increases from the bottom left to the top right.

3.4. Correlations between multifractality and descriptors of the pore network and macroscopic soil physical properties. In the Table 3 are shown some of the correlations found between the average multifractal parameters in 3D (in this part we are not considering the fractal dimension in 2D), and some soil physical properties and topological descriptors of the pore network skeleton. First, to understand the relations of Table 3 we need to consider that large structures are represented by a large number of pixels with a large number of different pixel configurations, i.e., degrees of freedom. It can be inferred that large pore structures with many degrees of freedom can cover wide ranges of scale factors. A small 2D region represented by a 7

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Fig. 6. Plots illustrating the influence of depth in the multifractal spectrum of the soil pore network (left) and 3D representations of the columns (right). The plots show the average spectrum of the upper half of the column (orange squares) and the lower part (blue circles) from two different soil columns. In the A) (ST plot, column no. 6) the Aperture decreases from 3.4 in the upper half to 2.3, and the drop of D0 has a value of 0.3. That means a remarkable decrease of both number and diversity of pore structures in the bottom half. B) The same effect, but to a much lesser extent, was observed in the other columns (e.g., in column no. 8 in Org. A). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

more probable to find pores along the soil profile. Finally, the parameters of the Rényi spectrum are also related to the bromide dispersion coefficient, which suggests a link between the complexity across scales and transport of solutes. This can be explained by the hypothesis that complex pore networks may favor the occurrence of solute transport pathways at several scales, and that will increase the range of travel times (the time that needs the solute to cross the sample) and pore water velocities (the average velocity of the water through the soil considering the water content and the flow), which leads to a bigger dispersion. When considering the correlations with the parameters of the multifractal spectrum, the total aperture and the aperture between q = 1 and q = -1 (Ap (−1, 1)), showed similar values, but the Ap (−1, 1) seems to be more sensitive. This feature is positively correlated with the CT-Porosity, the volume and surface of pores and the skeleton parameters, as expected. If a sample has more pores, there are more

possibilities of having different pore-shapes. It is also interesting the correlation existing between the Ap, D0, D2, R-L, V and the surface area of the preferential colloidal paths described in the colloid transport experiments reported by Soto‐Gómez et al. (2019). Fig. 7A, B and C show that the complexity pointed by the broad range of scales of the singularity spectrum correlates with the surface stained by the fluorescent colloid tracer. This indicates that the soils with complex and well distributed small pores enhance de distribution of microspheres. 4. Conclusions In this article we have presented the fractal and multifractal analysis of the soil pore network using CT images of intact soil columns sampled from plots devoted to different agricultural management practices. Spatial distribution of macroporosity showed multifractal properties represented by a sigmoidal Rényi spectrum and a wide singularity 8

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Table 3 Correlation between the average values of some multifractal properties and other soil characteristics such as bulk density and CT-parameters.

3

CT-Porosity (cm ) Total pore volume (mm3) Total pore surface (mm2) Average pore volume (mm3) Average pore surface (mm2) Number of Branches Number of Junctions End-point voxels Slab Voxels Triple Points Quadruple Points Average CT tortuosity Connectivity Bulk Density (g cm−3) DBr (Dispersion) Average Stained Area (%)

D0

D2

Ap

R-L

V

Ap(-1,1)

0.707*** 0.717*** 0.811*** 0.528* 0.613** 0.752*** 0.744*** 0.627** 0.801*** 0.750*** 0.723*** 0.522** 0.720*** −0.277 0.494* 0.802***

0.797*** 0.798*** 0.806*** 0.704*** 0.714*** 0.731*** 0.786*** 0.476* 0.735*** 0.788*** 0.775*** 0.651** 0.794*** −0.450* 0.545* 0.842***

0.383 0.401 0.642** 0.040 0.283 0.535* 0.464* 0.573** 0.790*** 0.477* 0.441 0.184 0.398 0.058 0.278 0.614**

0.746*** 0.763*** 0.820*** 0.655** 0.757*** 0.680*** 0.746*** 0.406 0.771*** 0.750*** 0.734*** 0.628** 0.778*** −0.436 0.422 0.823***

0.846*** 0.861*** 0.668** 0.755*** 0.733*** 0.575** 0.700*** 0.205 0.421 0.698*** 0.696*** 0.822*** 0.735*** −0.674** 0.311 0.776***

0.677** 0.697*** 0.774*** 0.390 0.487* 0.690*** 0.659** 0.625** 0.751*** 0.669** 0.622** 0.472* 0.591** −0.199 0.308 0.807***

Probability values for the Pearson’s correlation coefficient: * P ≤ 0.05; ** P ≤ 0.01; *** P ≤ 0.001.

Fig. 7. Relation between the average stained surface by a particulate fluorescent tracer and some parameters of the multifractal spectrum: A) Aperture, B) R-L (horizontal symmetry) and C) V (vertical symmetry).

and, consequently, the pore complexity, decrease with depth. Is noteworthy that the multifractal parameters are correlated with macroscopic soil variables, such as bulk density and tortuosity, as well as with some topological descriptors of the pore network skeleton. These correlations suggest the existence of links between the spatial distribution of the pore network and the multiplicity of scale factors. Besides, the parameters studied also present correlations with the bromide dispersion and with the surface of soil stained by a particulate tracer. The above results are interesting from the point of view of the

spectrum. Characteristics defining the average shape of the Rényi and singularity spectra of soil columns were influenced by management, but only one study plot showed significant differences (Org. B). Presence of compacted layers in tilled soil, the abundance of root channels in nottilled soil and earthworm burrows in organic management produced significant changes in the multifractality. Moreover, it is possible to establish different levels of the complexity of soil pore networks considering the Dmin and the aperture of the multifractal spectrum. We also concluded that the fractal dimension (in 2D), multifractal parameters 9

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parameterization of the complexity of the porous networks of soils, and can help to link the physical properties of the soil porosity, the pore network topology and the transport. This is quite helpful when making more precise predictions regarding the fate of substances in the environment.

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Acknowledgments The authors want to acknowledge the following funding sources: D.S.G. is funded by the Pre-Doctoral Fellowship Program (FPU) of Spain’s Ministry of Education FPU14/00681, and P.P.R is funded by a post-doctoral fellowship awarded by Xunta de Galicia (Gain program ED481B-2017/31). L.V.J. was additionally funded by CIA and BV1 research contracts (FEDER, Xunta de Galicia). This work is partially founded by the INOU program K840. Authors thank the Centro de Desenvolvemento Agrogandeiro (Ourense) for allowing the sampling in their plots. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.geoderma.2019.114129. References Alvarez-Benedi, J., Munoz-Carpena, R., 2004. Computer models for characterizing the fate of chemical in soil: pesticide leaching models and their practical application. Soil-water-solute process characterization: an integrated approach. Bird, N., Díaz, M.C., Saa, A., Tarquis, A.M., 2006. Fractal and multifractal analysis of pore-scale images of soil. J. Hydrol. 322, 211–219. https://doi.org/10.1016/j. jhydrol.2005.02.039. Doube, M., Kłosowski, M.M., Arganda-Carreras, I., Cordelières, F.P., Dougherty, R.P., Jackson, J.S., Schmid, B., Hutchinson, J.R., Shefelbine, S.J., 2010. BoneJ: fee and extensible bone image analysis in ImageJ. Bone 47, 1076–1079. https://doi.org/10. 1016/j.bone.2010.08.023. Elliott, E.T., Heil, J.W., Kelly, E.F., Curtis Monger, H., 1999. Soil structural and other physical properties. Oxford University Press, New York. Ge, X., Fan, Y., Li, J., Aleem Zahid, M., 2015. Pore structure characterization and classification using multifractal theory—an application in Santanghu basin of western China. J. Pet. Sci. Eng. 127, 297–304. https://doi.org/10.1016/j.petrol.2015.01.004. Giménez, D., Karmon, J.L., Posadas, A., Shaw, R.K., 2002. Fractal dimensions of mass estimated from intact and eroded soil aggregates. Soil Tillage Res. 64, 165–172. https://doi.org/10.1016/S0167-1987(01)00253-7. Hatano, R., Booltink, H.W.G., 1992. Using fractal dimensions of stained flow patterns in a clay soil to predict bypass flow. J. Hydrol. 135, 121–131. https://doi.org/10.1016/ 0022-1694(92)90084-9. Helliwell, J.R., Sturrock, C.J., Grayling, K.M., Tracy, S.R., Flavel, R.J., Young, I.M., Whalley, W.R., Mooney, S.J., 2013. Applications of X-ray computed tomography for examining biophysical interactions and structural development in soil systems: a review. Eur. J. Soil Sci. https://doi.org/10.1111/ejss.12028. Heuvelink, G.B.M., Webster, R., 2001. Modelling soil variation: past, present, and future. Geoderma 100, 269–301. https://doi.org/10.1016/S0016-7061(01)00025-8. Horgan, G.W., 1998. Mathematical morphology for analysing soil structure from images. Eur. J. Soil Sci. 49, 161–173. https://doi.org/10.1046/j.1365-2389.1998.00160.x. Hunt, A.G., 2004. Percolative transport in fractal porous media. Chaos Soliton. Fract. 19, 309–325. https://doi.org/10.1016/S0960-0779(03)00044-4. Jiménez-Hornero, F.J., Gutiérrez de Ravé, E., Giráldez, J.V., Laguna, A.M., 2009. The influence of the geometry of idealised porous media on the simulated flow velocity: a multifractal description. Geoderma 150, 196–201. https://doi.org/10.1016/j. geoderma.2009.02.006. Jury, W.A., Or, D., Pachepsky, Y., Vereecken, H., Hopmans, J.W., Ahuja, L.R., Clothier, B.E., Bristow, K.L., Kluitenberg, G.J., Moldrup, P., Šimůnek, J., van Genuchten, Horton, Th.M.R., 2011. Kirkham’s legacy and contemporary challenges in soil physics research. Soil Sci. Soc. Am. J. 75, 1589. https://doi.org/10.2136/sssaj2011.0115. Karperien, A., 2013. FracLac for ImageJ [WWW Document]. User’s Guid. FracLac, V. 2.5. Lafond, J.A., Han, L., Allaire, S.E., Dutilleul, P., 2012. Multifractal properties of porosity as calculated from computed tomography (CT) images of a sandy soil, in relation to soil gas diffusion and linked soil physical properties. Eur. J. Soil Sci. 63, 861–873. https://doi.org/10.1111/j.1365-2389.2012.01496.x. Larsbo, M., Koestel, J., Jarvis, N., 2014. Relations between macropore network characteristics and the degree of preferential solute transport. Hydrol. Earth Syst. Sci. https://doi.org/10.5194/hess-18-5255-2014. Marcus, I.M., Wilder, H.A., Quazi, S.J., Walker, S.L., 2013. Linking microbial community structure to function in representative simulated systems. Appl. Environ. Microbiol. https://doi.org/10.1128/aem.03461-12. Marinho, M.A., Pereira, M.W.M., Vázquez, E.V., Lado, M., González, A.P., 2016. Depth distribution of soil organic carbon in an Oxisol under different land uses: stratification indices and multifractal analysis. Geoderma. https://doi.org/10.1016/j.

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