MULTIFRACTAL ANALYSIS OF SOIL RESISTANCE TO PENETRATION IN DIFFERENT PEDOFORMS

Soils are highly variable across landscapes, which can be assessed and characterized according to scale, as well as fractal and multifractal concepts of scale. Thus, the objective of this study was to analyze the multifractality of the penetration resistance (PR) of vertical profiles from different slope forms (concave and convex). The experimental plot incorporated 44.75 ha, and the PR was measured at 70 sampling points in the 0-0.6 m layer, distributed in concave (Type A: 38 sampling points) and convex pedoforms (Type B: 32 sampling points). Data analysis was performed using the PR value (every 0.01 m depth) for each of the sampling points (PRmean), and their respective maximum (PRmaximun) and minimum (PRminimum) values. Multifractal analysis was performed to assess the changes in the structure, heterogeneity, and uniformity of the vertical profiles according to the scale, characterizing the partition function, generalized dimension, and singularity spectrum. The multifractal parameters of the generalized dimension and singularity spectrum demonstrated greater homogeneity and uniformity in the vertical PR profiles of pedoform B (convex) compared to those of pedoform A (concave). The minimum PR values in pedoform A (PRminimum) showed the greatest scale heterogeneity, indicating that in terms of soil management, it is more relevant to monitor the minimum values than the maximum values. The fractal analysis allowed us to describe the heterogeneity of the data on scales not evaluated by conventional analysis methods, with high potential for use in precision agriculture and the delimitation of specific management zones.


INTRODUCTION
Knowledge of soil resistance to penetration (PR, MPa) allows inferences concerning the state of soil compaction throughout the landscape and the determination of management alternatives with the least possible impact on soil attributes, especially with regard to increasing density and decreasing soil porosity (SOUZA et al., 2015;BURR-HERSEY et al., 2017;SEIDEL et al., 2018). Thus, the understanding of PR along the landscape is a key factor for the development of management alternatives with less impact on natural soil characteristics (SIQUEIRA et al., 2013), in addition to favoring the development of crops (SEIDEL et al., 2018).
According to Goovaerts (1998), variability is composed of defined or intrinsic variations and random fluctuations or noise, which need to be understood individually to facilitate a better understanding of the processes and dynamics of soil properties throughout the landscape. In this way, knowledge of landscape forms allows us to describe the variability of soil attributes and their relationships with crops at different scales. Logsdon, Perfect and Tarquis (2008) highlighted that the distinction between intrinsic variability and noise depends upon the measurement scale, and that an increase in the observation scale almost always has a noise structure. Therefore, it is necessary to understand the scales of variation across landscapes. In this sense, Leão et al. (2011) studied the spatial variability of soil attributes, including the PR of different slope forms, and found the greatest differences for the concave slope compared to the convex slope. In addition, Siqueira et al. (2018) indicated that relief forms can be used to predict and improve the estimation of soil attributes. When studying vertical PR profiles using multifractal analysis, Leiva et al. (2019) found that the heterogeneity and variability of this attribute is influenced by its position in the landscape. Therefore, it is necessary to understand how PR varies across the landscape in the context of the occurrence of different patterns on the horizontal and vertical scales, especially over small distances (KRAVCHENKO;PACHEPSKY, 2004).
In this sense, multifractal analysis aims to determine variations in different compartments according to the scale (VIDAL-VÁZQUEZ et al., 2013;LEIVA et al., 2019). In this way, it is possible to conduct an in-depth study of PR profiles with high variability through the study of different compartments, which differ and present scale heterogeneity (SIQUEIRA et al., 2013). Roisin (2007) evaluated PR using multifractal analysis to understand the state of soil structure. Siqueira et al. (2013) studied the multifractality and intrinsic variability of vertical PR profiles in soils with a high water content. Wilson et al. (2016) studied the influence of water deficit on the multifractal structure of vertical PR profiles. Overall, it is apparent that few studies have been dedicated to the study of the fractal and multifractal geometry of the PR data of different slope forms.
Thus, the hypothesis of this study was that PR measurements along the landscape vary according to the shape of the slope. The objective of this study was to analyze the multifractality of PR in the vertical profiles of different slope forms (concave and convex) in an area cultivated with soybean under no-tillage conditions.

MATERIALS AND METHODS
The study area incorporates approximately 44.75 ha and is located in the municipality of Mata Roma (Maranhão,Brazil;3º 70'80.88" S and 43º 18'71.27" W). The climate is Aw (hot and humid tropical), with an average annual temperature above 27 °C and average annual rainfall of 1,835 mm, with concentrated rains from January to June. The soil in the study area is anoxisol (SANTOS et al., 2018), and its main physical and chemical characteristics are shown in Table 1.
In the study area, soybean (Glicine max L.) and corn (Zea mays L.) have been cultivated in rotation since 2007, under a no-tillage system. Subsoiling, up to a depth of 0.32 m, is completed every five years. In the 2015/16 agricultural year, 70 sampling points were marked in the study area following the planting lines, with a regular spacing of 100 × 35 m ( Figure 1). The points were georeferenced using GPS with post-processed differential correction (DGPS). The area has a smooth wavy slope with two slope units: concave (Type A: slope between 101.8-103.8 m; 38 sampling points) and convex (Type B: slope between 103.8-105.8 m; 32 sampling points) ( Figure 1).
The resistance of the soil to penetration (PR, MPa) was determined on 04/22/2016 using an impact penetrometer, with a cone angle of 30°. The PR was evaluated in the 0-0.6 m depth layer. Values were calculated for each 0.01 m.
For the purpose of verifying the greatest differences in the PR readings across the 70 sampling points and pedoforms (A and B), we used the average (PR mean ), maximum (PR maximum ), and minimum (PR minimum ) PR values of the vertical profiles at 0.01 m intervals, which were submitted to multifractal analysis. The PR readings were recorded when the soil was at field capacity (0.171 m 3 m -3 for the 70 sampling points, 0.172 m 3 m -3 for the Type A pedoform, and 0.172 m 3 m -3 for the Type B pedoform), which was determined using a Richards chamber.
Data were analyzed using descriptive statistics in the R 3.3.1 software (R CORE TEAM, 2018). The following measurements were determined: mean, variance, standard deviation, coefficient of variation, asymmetry, kurtosis, and D (maximum deviation in relation to normal distribution, using the Kolmogorov-Smirnov test with an error probability of 0.01).
The multifractal analysis was performed for each vertical PR profile using the moment method with consideration for the scale, allowing for the determination of the partition function, generalized dimension, and singularity spectrum, according to the procedures described by Evertsz and Mandelbrot (1992). The algorithms used in this study are shown in Table 2.
The partition function was built for successive segments in 2 k , k = 0 to k = 6, and moments in the range of q = +5 to q = -5 (EVERTSZ; MANDELBROT, 1992), generalizing on a δ scale to a number of segments N(δ) = 2 k of characteristic size, δ = L × 2 -k , which corresponded to the length of the vertical PR profiles (0.6 m). The probability of the mass function, Pi (δ) (Equation 1a), describes the value of the segment [Ni (δ)] and the sum of the measure across the transept [N t ], with q defined as -∞ < q < ∞. This enabled the evaluation of the segment sizes, n (δ) in normalized scale µi (δ), according to Equation 1b, which allowed the plotting of χ (q, δ) versus δ. Then, the exponent of the mass function [τ (q) -Equation 1c] was determined.
The scale function (τ q ) is related to the generalized fractal dimension (HENTSCHEL; PROCACCIA, 1983), and was defined in Equation 2a. The generalized dimension (Dq) or Rényi dimension of order q was also determined using Equation 2b. In this case, D1 was undetermined because the denominator value was equal to zero. However, Equation 2c was used for q = 1, which enabled the determination of the moment dimensions q = 0, q = 1, and q = 2 (called the capacity dimension (D 0 ), entropy dimension or Shannon's entropy (D 1 ), and correlation dimension (D 2 ), respectively) (HENTSCHEL; PROCACCIA, 1983).
The singularity spectrum was determined using Equations 3a and 3b. Here, it is necessary to consider that in multifractal measures, the number [N δ (α)] of cells of size δ with a singularity or Hölder    exponent equal to α increase or decrease with δ. In this case, it obeys a power law: N (α) αδ -f (α) , which describes exponent f (α) as a continuous function of α. The graph depicting the typical singularity spectrum has a descending concave parabola shape, where the α values increase as a function of the heterogeneity of the measurements. The scaling of the exponents τ q and f (α) can be obtained through the transformation of Legendre; however, the present study uses Chhabra and Jensen (1989) (Equations 3a and 3b).

RESULTS AND DISCUSSION
There was an increase in the PR close to the 0.3 m soil depth, with the highest average PR value of the Type B pedoform being 3.052 MPa, while that of the Type A pedoform was 3.718 MPa. The highest average PR value for the 70 sampling points (pedoform A and B) was 3.345 MPa. According to Cortez et al. (2018), Seidel et al. (2018), and Leiva et al. (2019), high PR values are related to soil management, and crops in no-tillage systems need management strategies that consider the state of soil compaction (BURR-HERSEY et al., 2017). The Type A slope unit (concave; Figures 1 and 2b) showed the highest average PR value (3.718 MPa) and highest variation of the standard deviation along the profile, indicating the high vertical variability of the data. Leão et al. (2011) highlighted that soil attributes have greater variability in concave slope units.
The statistical analysis of the PR mean , PR maximum , and PR minimum for the vertical PR profiles (  (Table  3).

Equations
Partition function The lowest value for the coefficient of variation (CV) was described for the Type B pedoform (CV = 37.021%), followed by the 70 sampling points (CV = 41.168%) and Type A pedoform (CV = 45.648%). The differences in the CV values demonstrate the need for a separate study of PR measurements, since the difference between the lowest and highest mean CV values is approximately 8%, which corroborates the results of Roisin (2007) and Leiva et al. (2019). It is important to highlight that PR measurements have a high intrinsic variability, as PR is influenced by changes along the soil profile, particularly those related to clay content, moisture, density, and porosity (SIQUEIRA ET AL., 2013; BURR-HERSEY et al., 2017;SEIDEL et al., 2018), thus being a variable with high variability, resulting in high CV values.
According to the Kolmogorov-Smirnov data normality test (p < 0.01), the PR maximum for the 70 sampling points and Type B pedoform showed a normal frequency distribution, while the other data showed a lognormal frequency distribution.
The graph was constructed considering segments 2 k , k = 0 to k = 6, with determination coefficients (R 2 ) greater than 0.9. The PR maximum presented different adjustment values for the partition function graphs of pedoforms A (R 2 = 0.999; Figure 3e) and B (R 2 = 0.996; Figure 3h). The partition function graphs of PR mean were adjusted with R 2 = 0.999 (Figures 3a, d, and g), while those of PR minimum were adjusted with R 2 = 0.998 (Figures 3c,  f, and i). Visually, the PR minimum graphs for the 70 sampling points (Figure 3c) and pedoform A ( Figure  3f) show greater data variability in the range of q = 5 to q = -5. Morató et al. (2017), studying the multifractality of physical soil variables, presented a graph of the partition function with behavior similar to that found in this study for the PR minimum . The variations in the moments reflected the data variability, since PR minimum had the highest CV values (%; Table 3). In this way, the partition functions for the PR (Figure 3) under soybean cultivation in a no-tillage system were characterized on multiple scales, that is, multifractal scales obeying a power law, as described by Banerjee et al.   The smallest difference values between D -5 -D 5 are 0.340 (PR mean ) and 0.333 (PR maximum and PR minimum ) in pedoform B. According to Dafonte-Dafonte et al. (2015), the D -5 -D 5 difference is often used as an index of multifractality, meaning that pedoform B has different degrees of multifractality. The results indicate that the greater the adjustment of the partition function data, the greater the difference between D -5 -D 5 (Figure 3 and Table 4), indicating that the differences in the PR values at the different depths have variations that are explained and related to the soil management in the study area. Therefore, studies that facilitate the understanding of the multifractal dynamics of soil attributes are important, as they add new parameters to the decision-making process. However, it is necessary to consider that soils are not ideal fractals (KRAVCHENKO; PACHEPSKY, 2004), meaning that the decisionmaking process must consider the soil as a dynamic system and use different modeling tools.
The capacity dimension values (D 0 ; Table 4 and Figures 4a, b, and c) of the studied data are constant (D 0 = 1,000). D 0 provides global or average system information, and allows us to confirm the homogeneity of the data series being studied (PR mean , PR maximum , and PR minimum ). The information dimension or Shannon's entropy (D 1 ) quantifies the degree of distribution disorder, and must be in the range 0 < D 1 < 1. Therefore, D 1 values close to 1 characterize a system uniformly distributed on all scales, while values close to 0 reflect a subset of scales in which irregularities are concentrated (VILLAS-BOAS; CRESTANA; POSADAS, 2014; Leiva et al., 2019). The PR data series describes uniformly distributed systems, with data ranging from 0.642 (PR minimum -70 points) to 0.975 (PR meanpedoform B). It is worth mentioning that the PR in pedoform A presented the lowest D 1 values compared to those of the other data. The correlation dimension (D 2 ) computes the correlation of size intervals (L); that is, it is a geometric measure that describes the complexity of interval segments where the segments are correlated, so the larger the dimension, the larger the D 2 value. The highest D 2 values were described in pedoform B (PR mean = 0.962; PR maximum = 0.955; and PR minimum = 0.955), while pedoform A showed the lowest D 2 values (PR mean = 0.948; PR maximum = 0.942; and PR minimum = 0.550).
The D 0 , D 1 , and D 2 values showed differences within the data series, which allowed us to characterize that the systems under study are multifractal. This was done in accordance with Banerjee et al. (2011), who stated that the dimensions must follow the trend of D 0 > D 1 > D 2 for an attribute to be multifractal. Therefore, the data series represents multifractal systems, and the data allowed us to describe changes in the dimension values of the different slope forms, showing that the PR samples are affected by the landscape changes.
Pedoform A showed the greatest amplitude (α -5 -α5 ), with a PR mean of 0.603, PR maximum of 0.66, and PR minimum of 1.067. The difference between α -5α 5 is an indicator of heterogeneity, with the greatest heterogeneity in this case associated with the PR minimum (Figure 4f). It is important to note that the PR minimum for 70 points and pedoform A describe greater heterogeneity/entropy, with α -5 -α 5 values of 1.061 and 1.067, respectively. The α -5 -α 5 value for pedoform B was 0.538, which confirms the lower heterogeneity or entropy of the system in this pedoform, as the smaller the difference between α -5α 5 , the more homogeneous the system (VILLAS-BOAS; CRESTANA; POSADAS, 2014). Therefore, the singularity spectrum quantifies the variability of the PR measurements in the vertical profiles, as described by Siqueira et al. (2013). The multifractal parameters of the PR data series showed that the data variability of the different pedoforms was distinct, where the Type A (concave) pedoform was more heterogeneous than the Type B (convex) pedoform. According to Artur et al. (2014), the pedoform influences the intensity and direction of the water flow in the soil, thus justifying the greater scale heterogeneity of low PR values. Rassol, Gaikwad and Talat (2014) studied the influence of the slope forms on the spatial variability of the physical and chemical attributes of the soil, and found that hillside areas have greater variability compared to other slope forms. In terms of soil management, the presence of greater scale heterogeneity related to the PR minimum values demonstrates that monitoring the soil compaction status must consider not only the PR maximum values but also the dynamics of the PR minimum values, as these values are more susceptible to long-term compaction. This corroborates the results of SEIDEL et al. (2018), who described that the proper management of the compaction state favors crop development and an increased crop yield.
On the other hand, the multifractal analysis of all vertical PR profiles (70 points) demonstrated that the PR minimum values in pedoform A underestimated the average values for the 70 sampling points ( Figure  4f). According to the singularity spectrum, the

CONCLUSIONS
The multifractal analysis allowed us to identify differences in the scale variability for the PR measurements in the concave and convex pedoforms, demonstrating its potential for the determination of specific management zones.
The concave pedoform (Type A) had differences regarding the measurement values along the PR profile that would not be detected by conventional analysis methods.
The PR minimum of the different slope forms showed the greatest multifractality of the data, presenting greater scale variation.
The generalized dimension and singularity spectrum proved to have the greatest scale heterogeneity for the concave slope (Type A), which demonstrates that multifractal analysis is efficient in ascertaining the differences in vertical PR profiles along a landscape, allowing the delimitation of management zones considering scale variability.