Investigating the Spatial Structure of Soil Hydraulic Properties in a Long-Term Field Experiment Using the BEST Methodology

Abstract

Understanding the spatial structure of soil properties at field scale and introducing this information into appropriate data analysis methods can help in detecting the effects of different soil management practices and in supporting precision agriculture applications. The objectives of this study were: (i) assessing the spatial structure of soil physical and hydraulic properties in a long-term field experiment; (ii) defining a set of spatial indicators for gaining an integrated view of the studied system. In seventy-two georeferenced locations, soil bulk density (BD), initial volumetric soil water content (θ_i) and cumulative infiltration curve as function of the time (I(t)) were measured. The soil water retention curve (θ(h)) and the hydraulic conductivity function (K(h)) were then estimated using the Beerkan Estimation of Soil Transfer parameters (BEST) methodology. The volumetric soil water contents at soil matrix (h = −10 cm), field capacity (h = −100 cm) and wilting point (h = −15,300 cm) were considered. In addition, a set of capacitive indicators—plant available water capacity (PAWCe), soil macroporosity (PMACe), air capacity (ACe) and relative field capacity (RFCe)—were computed. The data were first analyzed for overall spatial dependence and then processed through variography for structural analysis and subsequent spatial interpolation. Cross-correlation analysis allowed for assessing the spatial relationships between selected physical and hydraulic properties. On average, optimal soil physical quality conditions were recorded; only PMACe values were indicative of non-optimal conditions, whereas mean values of all the other indicators (BD, Ks, PAWCe, ACe, RFCe) fell within optimal ranges. The exponential model was found to be the best function to describe the spatial variability of all the considered variables, except ACe. A good spatial dependence was found for most of the investigated variables and only BD, ACe and Ks showed a moderate autocorrelation. Ks was confirmed to be characterized by a relatively high spatial variability, and thus, to require a more intensive spatial sampling. An inverse spatial cross-correlation was observed between BD and Ks up to a distance of 10 m; significant cross-correlations were also recorded between Ks and PMACe and ACe. This result seems to suggest the possibility to use these soil physical quality indicators as covariates in predictive multivariate approaches.

1. Introduction

A clear understanding of the impact of agronomic techniques on soil quality and crop productivity is of foremost importance for the sustainable management of cropping systems and for providing proper answers to challenges posed by climate change. Soil tillage practices are known to greatly affect soil status and quality [1,2] and its ability to capture and store precipitation or irrigation water [3]. The direct action of tillage management on soil physical properties modifies air-water capacity relationships. This in turn affects soil organic matter dynamics and turnover, carbon storage and CO₂ emissions, nutrient cycling and water resources conservation [1,2,4,5].

Modifications induced by tillage practices on soil physical and hydraulic properties are not always easy to detect, and field studies have often provided contrasting results [3]. The presence of unstable conditions, when investigations are carried out in the short- and medium-term period, and also the spatial variability of the soil properties over the field experimental area may cause difficulty in discerning the effects. Spatial heterogeneity is suggested as a primary source of error [6]. When spatial variability occurs at a scale smaller than the block (and plot) size, the effect of experimental treatments on the response of primary variables may be confounded [7,8,9]. Several studies have reported that neglecting the spatial dependence can cause misinterpretation errors, and consequently, improper management decisions [10,11].

In order to assess the spatial structure of the investigated variables, specific approaches for data sampling and analysis have to be undertaken. At first instance, a more intensive sampling is required in comparison to traditional approaches. A number of samples ranging from 50 to 100 is appropriate for accurate spatial analysis [12,13,14]. Sample sizes below 50 units do not meet the basic assumption for the application of spatial analysis [15,16].

In addition, multiple, or repeated, measurements performed over the same experimental unit/plot may give rise to correlations among observations and/or residuals that need to be appropriately considered in statistical analysis [7]. In this specific case, not properly accounting for residual autocorrelation over space may cause erroneous conclusions about treatment significance [10]. Modeling the variance-covariance matrix of the residuals and incorporating this information into linear mixed effects models (LMMs) may represent a rigorous approach alternative to classical analysis of variance and standard linear models. The information about the spatial structure of a variable of interest can support farmers in delineating homogeneous areas for promoting the application of precision agriculture strategies.

The need for denser sampling in spatial studies is, however, hindered by the characteristics of conventional methods for determination of soil physical and hydraulic properties, since these methods are expensive and time consuming, and often require high technical skills. Simple, quick and accurate methodologies are envisaged and desirable [17]. A simple and rapid field method, initially developed by Haverkamp et al. [18], allowed for the simultaneous characterization of both hydraulic conductivity function and water retention curve. Lassabatère et al. [19] developed the “Beerkan Estimation of Soil Transfer parameters (BEST)” procedure in order to simplify soil hydraulic characterization [20]. The method can be conveniently applied for performing intensive samplings with a less demanding experimental effort [21], thus dealing with the requirements of spatial analysis.

Another open issue in the field of spatial analysis is that of selecting a suitable number of effective indicators, and above all, to learn how to interpret them in an integrated way. In the past, several efforts have been directed toward reaching this aim, and not always with satisfying results [22,23]. Today, it is possible to collect a large amount of literature regarding a number of old and newly introduced indicators without any attempt at presenting an integrated reading of the results provided. More specifically, the Moran index has a long-dated history, during which the method by which to use it and the related interpretation of the results have significantly improved [24]. In addition to the Moran index, even the nugget-to-sill ratio [25,26,27] has evolved to include the range and the model itself within the computation [27]. The cross-correlogram is a very powerful tool to quantitatively compare different spatial maps, thus reducing the subjectivity of visual inspection [28,29]. However, this index is often neglected in spatial analysis studies due to the difficulty in computing and interpreting the results. After the refinements of the concepts underlying those indicators, the revision of the definitions and the overcoming of misconceptions, the issue of misinterpretation or contradiction among results provided by different indicators has become much rarer. Therefore, an integrated reading of the outcomes of many indices together is not only possible, but desirable. In the present study, a set of indicators were applied to a group of variables with different degrees of spatial correlation in order to provide a systematic and integrated approach and “read between the lines” of the specific spatial features of different variables.

The general objective of this study was to assess the spatial structure of soil physical and hydraulic properties at a long-term field experimental site. Secondly, a set of spatial indicators for gaining an integrated view of the studied system were suggested and used. The BEST-procedure was adopted for the soil hydraulic characterization. The data, collected in georeferenced locations, were first analyzed for overall spatial dependence and then processed through variography for structural analysis and subsequent spatial interpolation. The quantitative assessment of the spatial relationships between selected physical and hydraulic properties was finally achieved using cross-correlation analysis. This study will also represent a preliminary step toward the inclusion of the information on the spatial structure of the investigated variables into linear mixed effects models considering residual autocorrelation. Moreover, information about spatial structure can support farmers in delineating management zones for fostering the application of precision agriculture strategies.

2. Materials and Methods

2.1. Study Area and Experimental Trial

The field experiment was performed in the autumn of 2021 at the Council for Agricultural Research and Economics (CREA-AA) in Southern Italy (41°27′03″ N, 15°30′06″ E). The climate is classified as accentuated thermomediterranean [30]. Long-term annual average rainfall is 550 mm and it is mainly concentrated in the winter period [9]. The soil is clay of alluvial origin classified as fine, mesic, Typic Chromoxerert [31].

Data were collected within a long-term field experiment, started in 2002, aimed at investigating the effects of two soil management strategies (minimum tillage, MT, and no-tillage, NT) on durum wheat (Triticum turgidum subsp. durum Desf.). The experimental design was a randomized complete block design (RCBD) with three replicates and unit plot sizes of 500 m². Further details on the experimental trial and plot management are reported in Castellini et al. [32] and Stellacci et al. [5].

Measurements of soil physical and hydraulic properties were carried out in seventy-two geo-referenced locations. The number of observations was defined to gain a minimum number of pairs for each distance class in order to assess spatial structures underlying the physical process taking place in the study area [14].

2.2. Laboratory and Field Measurements

The Beerkan Estimation of Soil Transfer Parameter (BEST) methodology by Lassabatère et al. [19] was applied for each of the selected seventy-two sampling locations. In particular, at each sampling point, the soil bulk density (BD), the initial volumetric soil water content (θ_i) and the cumulative infiltration curve as function of the time (I(t)) were determined. Then, the soil water retention curve and the hydraulic conductivity function (θ(h) and K(h) relationships, respectively) were estimated using the BEST methodology. The values of the volumetric soil water content corresponding to specific reference values of the matrix potential, i.e., at soil matrix, θ₁₀ (h = −10 cm), at field capacity, θ₁₀₀ (h = −100 cm) and at wilting point, θ₁₅₃₀₀ (h = −15,300 cm), were taken into account and analyzed to investigate the spatial variation of soil physical properties. In addition, starting from the estimated soil water retention curve, a set of four capacitive indicators accounting for the proportion between water and air into the soil was computed. These indicators were: plant available water capacity (PAWCe), soil macroporosity index (PMACe), air capacity (ACe) and relative field capacity (RFCe) [33]. More information on these indicators is reported in the following sections.

In detail, an infiltration experiment was carried out at each sampling point using a metal ring (15 cm inner diameter) and the cumulative infiltration as a function of the time (I(t)) was measured. Fifteen water volumes of 200 mL each were poured from a height of about 3 cm and the BEST-steady algorithm [34] was applied to estimate the soil water retention curve and the hydraulic conductivity function (θ(h) and K(h)). Undisturbed soil cores (0.10 m in height by 0.05 m in diameter) were collected in each sampling point at a 0 to 0.10 m depth close to the ring to quantify the soil water content at the beginning of infiltration experiments, θ_i, and the soil dry bulk density (BD). Saturated soil water content, θ_s, was estimated from BD assuming a particle density of 2.65 g cm⁻³. Disturbed soil samples were also collected at each sampling point to assess the mean particle size distribution (PSD) at 0–0.10 m depth. After checking that quasi-steady flow conditions were always reached, BD, θ_i, PSD and I(t) were used to run BEST-steady. Therefore, although the average values of the input variables are usually used for running BEST, in this investigation, only the mean PSD was used, while the single geo-referenced observations of BD, θ_i and I(t) were considered.

2.3. Soil Physical Indicators

The soil saturated hydraulic conductivity (Ks) indicates the soil water infiltration capacity and is a key property affecting soil water storage, and the transport of water and solutes in the soil [35,36]. Optimal Ks values for agriculture soils, ranging within 0.005–0.05 mm s⁻¹, favor a rapid infiltration and redistribution of plant available water [35,36]. Dry bulk density (BD) is an indicator of soil quality, correlated with soil compaction, related to soil porosity and moisture content. It is also related to thermal properties (soil volumetric heat capacity, thermal conductivity), in this way affecting chemical and biological variables. Optimal BD values range in an interval between 0.9–1.2 g cm⁻³. BD and Ks have been widely used to explore the impact of soil management from the point of view of environmental sustainability (among others, Cavalcante et al. [37]), and have been recently included in specific protocols for the assessment of sustainable soil management [38].

Plant available water capacity (PAWC) (cm³ cm⁻³) is computed as the difference between the water contents at field capacity (at h = −100 cm) and at permanent wilting point (at h = −15,300 cm) [32], and thus, represents the water available for crop growth. For the present investigation, the following PAWC limits were considered [36]: PAWC ≥ 0.20 ideal; 0.15 < PAWC < 0.20 good; 0.10 < PAWC < 0.15 limited; PAWC < 0.10 poor. Air capacity (AC) (cm³ cm⁻³) is computed as the difference between the water contents at saturation and at field capacity; providing information about the soil ability to store and transmit air [36]. Optimal AC values for clay soil fall in the interval 0.10–0.26 cm³ cm⁻³, while higher or lower values indicate poor soil aeration conditions [32,33]. The relative field capacity (RFC) was computed as the ratio between the water contents at field capacity and at water saturation [36]. Optimal values for RFC range within the interval 0.6–0.7; lower or higher values indicate “water limited” or “aeration limited” conditions, respectively [36]. Macroporosity (PMAC) (cm³ cm⁻³) gives the volume of large (macro) pores (i.e., >0.3 mm equivalent pore diameter), which indirectly indicates the soil’s ability to quickly drain excess water and facilitate root proliferation. PMAC ≥ 0.05–0.10 cm³ cm⁻³ indicates optimal conditions, while PMAC ≤ 0.04 cm³ cm⁻³ indicates soil compaction [39,40,41].

The subscript e (i.e., PAWCe, ACe, RFCe, PMACe) is used for indicating variables estimated by BEST. Since indicators of soil quality were both estimated by BEST and directly measured (BD and Ks), they assume a different meaning. Consequently, capacitive indicators (i.e., PAWCe, ACe, etc.) could provide evidence of better or worse relative soil quality, according to the different soil use or to the relative distance within the plot.

2.4. Statistical Methods

2.4.1. Exploratory and Correlation Data Analysis

Basic statistics were computed for the investigated variables: BD, ACe, PMACe, PAWCe, RFCe, Ks, θ_i, θ₁₀, θ₁₀₀ and θ₁₅₃₀₀. Finally, the hypothesis of normality was checked using Shapiro-Wilk test.

Pearson correlation coefficients were computed in order to investigate the relationships among studied variables. Moreover, the assessment of the overall spatial correlation of the considered variables was performed by means of Moran statistics [42,43]. Details regarding the Moran I index can be found in the following papers [32,42,43].

2.4.2. Geostatistical Analysis

The spatial variability of a variable can be assessed by means of the geostatistical analysis with the aim of predicting variable values at unvisited locations and producing maps. The main stages of geostatistical analysis concern structural analysis (variography), weight assessment and, finally, interpolation (kriging).

Variography

The experimental variogram and the variogram model are two different functions used to describe the features of geo-referenced data. The experimental variogram is a discrete function and represents the half of the average squared difference between points separated by the distance h (lag). The variogram model is a mathematical model parametric, a continuous and conditionally negative definite [22] (Figure 1). The variogram model is derived from the experimental variogram by means of a procedure called model fitting. In Figure 1, some examples of theoretical variogram models are reported.

The parameters of the variogram model are the partial sill (σ²), indicating the structured component of the variance, the nugget, indicating the random uncorrelated component, and the range (α). The range is defined as the distance beyond which the spatial correlation becomes negligible (Figure 1). For asymptotic models, such as Gaussian and exponential, the true range is replaced by the effective (or practical) range that is defined as the distance at which the semivariance value achieves 95% of the sill [45,46].

In this study, variogram models were used to model the experimental variograms of the investigated physical and hydraulic variables. The goodness-of-fit was evaluated through the leave-one-out cross-validation and Pearson correlation coefficients (r) between predicted and observed data were computed [22].

Spatial Interpolation

Ordinary kriging (OK) was selected as well suited to interpolate the investigated variables. OK is a univariate predictor in which the unknown value z(x₀) of a given realization of Z(x₀) is predicted from the known values z(x_i) i = 1, 2, …,N, at the support points x_i. The OK predictor is expressed as described in the following papers [26,29,47].

2.4.3. Cross-Correlation

Cross-correlations were computed for comparing the maps of the interpolated values of the studied variables. The cross-correlation is effective in highlighting the strength and the extent of the spatial correlation between a couple of variables [28,29,48]. The analytical formulation is the following:

$${\rho }_{A, B}\left(h\right)=\frac{E\left[z_{i,jA},z_{i^{\prime },j^{\prime }B}\right]-m_A·m_B}{s_A·s_B}$$

(1)

where $z_{i,jA}$ and $z_{i^{\prime },j^{\prime }B}$ represent the values at locations $\left(i-i\prime \right)$ and $\left(j-j\prime \right)$ of the two maps separated by the h distance, respectively; $h=\sqrt{{\left(i-i\prime \right)}^2+{\left(j-j\prime \right)}^2}$ represents the distance between the two locations, E denotes the mathematical expectation, $m_A$ and $m_B$ represent the populations means and $s_A$ and $s_B$ represent the populations standard deviations. If patterns are completely similar, apart from a constant, ${\rho }_{A, B}\left(1\right)$ should be equal to 1. To estimate ${\rho }_{A, B}\left(h\right)$ from the available data the following equation can be used:

$$r_{A, B}\left(h\right)=\frac{{{\displaystyle \sum }}_{i,j=1}^{N\left(h\right)}{z}_{i,jA},z_{i^{\prime },j^{\prime }B}-{\widehat{m}}_A·{\widehat{m}}_B}{{\widehat{s}}_A·{\widehat{s}}_B}$$

(2)

To compute $r_{A, B}\left(h\right)$, the procedure is as follows: from both maps, all the couples whose locations are separated by the distance h are collected. Indices ${\widehat{m}}_A$ and ${\widehat{m}}_B$ and ${\widehat{s}}_A$ and ${\widehat{s}}_B$ represent the mean and the standard deviation of mapped $z_{i,jA}$ and $z_{i^{\prime },j^{\prime }B}$, respectively. N(h) is the total number of these pairs.

The cross-correlation coefficients obtained at specified lag distances (0 m, 5 m, 10 m, 15 m, 20 m) were reported in order to provide quantitative information on the spatial relationships of soil physical and hydrological variables under study.

3. Results

3.1. Exploratory Data Analysis

The basic statistics highlighted a departure from the normal distribution of all the variables under study, excluding bulk density (BD), according to the Shapiro-Wilk test. Nonetheless, as shown in

Table 1. Summary statistics for the variables under study.

Variable	Unit	Mean	St.Dev	Median	Min	Max	Skewness	Kurtosis
Ks	mm s−1	0.043	0.055	0.016	0.001	0.270	1.864	3.630
θi	cm3 cm−3	0.347	0.029	0.351	0.247	0.399	−1.548	2.989
θ10	cm3 cm−3	0.599	0.028	0.606	0.526	0.65	−0.586	−0.357
θ100	cm3 cm−3	0.426	0.041	0.434	0.33	0.487	−0.621	−0.655
θ15300	cm3 cm−3	0.147	0.016	0.151	0.112	0.174	−0.440	−0.745
BD	g cm−3	1.008	0.051	1.002	0.904	1.129	0.286	−0.681
PMACe	cm3 cm−3	0.020	0.017	0.012	0.005	0.092	1.811	3.211
ACe	cm3 cm−3	0.193	0.034	0.183	0.141	0.291	0.817	−0.277
PAWCe	cm3 cm−3	0.279	0.025	0.285	0.218	0.315	−0.706	−0.593
RFCe	-	0.687	0.057	0.699	0.537	0.761	−0.823	−0.463

, skewness was almost always smaller than 1, indicating that distributions of considered variables were nearly symmetrical; therefore, values were not transformed into Gaussian.

3.2. Soil Physical Indicators

Measured values of physical and hydraulic variables were compared with the optimal ranges reported in the literature, as summarized in Section 2.3. Overall, observed values showed good average results as only PMACe values were indicative of non-optimal conditions, whereas mean values of all the other indicators (BD, Ks, PAWCe, ACe, RFCe) fell within optimal ranges (Table 1).

In more detail, BD showed optimal soil density with the entire range of variation of the measurements (0.904–1.129 g cm⁻³) falling in the interval 0.9–1.2 g cm⁻³ (Table 1). These results were also in agreement with those provided by ACe that always showed values in the interval 0.10–0.26 cm³ cm⁻³, except for two observations (0.2718 and 0.2917 cm³ cm⁻³). Regarding RFCe, although the values recorded at some locations were outside the optimal range (0.6–0.7), these observations were very close to the critical limits with average values of 0.73 and 0.58, respectively; in addition, only two values were lower than 0.55 and only four values were greater than 0.75. Optimal conditions were also recorded for PAWCe at all the sampling locations with observed values always greater than 0.20 cm³ cm⁻³; in particular, about 80% of the observations showed PAWCe values greater than 0.25 cm³ cm⁻³. In these locations, as expected, the lowest values of PMACe were also recorded with values lower than 0.04 cm³ cm⁻³. Finally, good hydrodynamic soil properties were observed. The average value of Ks (0.043 mm s⁻¹) fell within the limits defined in the literature [35] and the observed values lower than the critical limit (<0.005 mm s⁻¹) were quite close to this critical threshold, with only three locations showing values of saturated hydraulic conductivity smaller than 0.002 mm s⁻¹.

3.3. Correlation and Preliminary Spatial Analysis

Interesting relationships were observed among the soil variables (

Table 2. Pearson’s correlation matrix for the variables under study.

Variable	Ks	θi	θ10	θ100	θ15300	BD	PMACe	ACe	PAWCe	RFCe
Ks	1
θi	0.015	1
θ10	−0.545	0.522	1
θ100	−0.752	0.447	0.902	1
θ15300	−0.698	0.478	0.945	0.990	1
BD	0.187	−0.523	−0.839	−0.603	−0.683	1
PMACe	0.812	−0.273	−0.592	−0.860	−0.798	0.168	1
ACe	0.804	−0.235	−0.523	−0.810	−0.743	0.083	0.992	1
PAWCe	−0.764	0.447	0.869	0.995	0.976	−0.551	−0.895	−0.850	1
RFCe	−0.816	0.317	0.669	0.909	0.855	−0.270	−0.990	−0.972	0.937	1

). Significant correlations were observed between Ks and PMACe and ACe (r = 0.81 and 0.80, p < 0.0001). A negative correlation was instead observed between Ks and the two capacitive indicators RFCe and PAWCe (−0.816, −0.764). Strong positive correlations were found between RFCe with PAWCe, θ₁₀₀ and θ₁₅₃₀₀ (r = 0.937, 0.908, 0.855, p < 0.0001), while negative correlations were found with Ks, PMACe and ACe (r = −0.81, −0.99, −0.97, p < 0.0001). PAWCe was strongly negatively related to PMACe and ACe (r = −0.896, −0.85, p < 0.0001), while there was a high positive correlation with θ₁₀, θ₁₀₀, θ₁₅₃₀₀ and RFCe (r = 0.87, 0.99, 0.976, 0.937, p < 0.0001). Finally, BD showed significant inverse correlations with values of volumetric soil water content at −10 cm, −100 cm and −15,300 cm, namely −0.84, −0.6, −0.68, and at the beginning of the experiment (θ_i), of −0.52.

Moran index, indicating the overall spatial dependence, ranged from values of 0.16 (p = 0.04) and 0.196 (p = 0.023) for BD and Ks, indicating a poor spatial correlation, to values of 0.4063, 0.4329, 0.4360, 0.4364 and 0.480 (with p values < 0.001), for θ₁₀, PAWCe, θ₁₀₀, θ₁₅₃₀₀, θ_i, respectively, indicating a good spatial dependence.

3.4. Variography and Spatial Interpolation

Theoretical nested models, consisting of a nugget effect and a spatial covariance function, were used to model the experimental variograms of the investigated variables. The test for the anisotropy was performed and it was found not significant; then, isotropic models were selected for all the studied variables. Results, reported in

Table 3. Model selected, variogram parameters, nugget-to-sill ratio and cross-validation outcomes.

	Ks	θi	θ10	θ100	θ15300	BD	PMACe	ACe	PAWCe	RFCe
Nugget	0.00228	0	5.24 × 10−05	0.00039	3.7 × 10−05	0.0013	0.00021	0.00080	0.00018	0.0016
Partial sill	0.00254	0.0014	0.0009	0.0018	2.8 × 10−04	0.0015	0.00056	0.00096	0.00074	0.009
Nugget-to-sill ratio	0.4725	0	0.055	0.17	0.1156	0.46	0.27	0.45	0.20	0.15
Range	107.547	18	11.83	19.01	14.34	7.73	9.77	34.74	23.29	113.97
Model	Exp	Exp	Exp	Exp	Exp	Exp	Exp	Gau	Exp	Exp
Pred. vs. obs. (r)	0.3841	0.7	0.58	0.62	0.62	0.23	0.54	0.54	0.62	0.57

, showed that the exponential model was found to be the best theoretical function according to the considered error metrics, being chosen for all the variables except air capacity (ACe). The nugget-to-sill ratio was computed to assess the spatial structure of the variables [25], with ratios lower than 25% indicating strong spatial dependence, ratios between 25 and 75% indicating moderate spatial dependence and ratios greater than 75% indicating weak spatial dependence. The analysis of the observed values of nugget-to-sill ratio indicated a significant spatial structure for most of the variables investigated with values lower or close to 25% (Table 3). Greater values were recorded for BD (0.46) and Ks (0.4725), confirming the results of the Moran I index that showed a poor overall spatial structure for these variables, and for ACe (0.45). The lowest values were recorded for volumetric soil water content at the beginning of the experiment (θ_i) and of the soil matrix (θ₁₀).

Cross-validation outcomes showed that the correlation between predicted and observed values (pred. vs. obs.) was lower than 0.6 for Ks, BD, PMACe and ACe. All the other variables showed a pred. vs. obs. correlation larger than or closer to (RFCe and θ₁₀) 0.6 and were then sufficient. The best variable was θ_i, with a correlation between predicted and observed values equal to 0.7 (Table 3).

From the visual inspection of the resulting maps (Figure 2 and Figure 3), a significant spatial structure emerged for all investigated variables. Initial volumetric soil water content (θ_i) showed a single zone with lower values in the southern part of the map and the larger values mainly located in the upper left portion. The BD map showed a specular behavior with respect to the previous variable, and thus, with the larger values located in the southern right zone. The maps of estimated volumetric soil water contents (θ₁₀, θ₁₀₀ and θ₁₅₃₀₀) displayed similar characteristics each other and with θ_i, with lower values on the southern right part and higher values on the top of the maps; in addition, another zone with lower values was observed on the middle left part of the three maps.

As expected, maps of PAWCe and RFCe shared some characteristics with θ₁₀₀ and θ₁₅₃₀₀, being derived from these variables (Figure 3). PMACe and ACe were similar to each other and mirrored PAWCe and RFCe, confirming the results of the correlation analysis. In the middle longitudinal region, Ks map recalled the behavior observed for PMACe and Ace, and, consequently, of the estimated volumetric soil water content maps. Furthermore, there was also observed a similarity with the values located in the right southern and the middle left portions of the maps.

To enhance these results, some variables were selected to check direct or inverse spatial behavior through cross-correlation procedures (BD, Ks, PMACe and ACe).

Table 4. Cross-correlation coefficients computed at different lags (0, 5, 10, 15, 20 m) between BD, Ks and the two capacitive indicators PMACe and ACe.

	0 m	5 m	10 m	15 m	20 m
BD vs. Ks	−0.2800	−0.1774	−0.1066	−0.0482	0.0263
PMACe vs. Ks	0.6578	0.3649	0.1952	0.1033	0.0379
ACe vs. Ks	0.6311	0.3504	0.1881	0.0995	0.0322
PMACe vs. ACe	0.7190	0.4041	0.2216	0.1276	0.0544

highlights an inverse cross-correlation between BD and Ks up to a distance of 10 m; this result was coherent with the spatial structure of BD, for which the range was lower than 10 m (7.7 m; Table 3). For this reason, all values related to distances larger than that length should be considered unreliable. The sign of the relationship between the two variables derives from the Pearson linear correlation reported in Table 2. Significant spatial cross-correlations were also recorded between Ks and the two capacitive indicators PMACe and ACe. Even for these last variables, the range of correlation was around 10 m, confirming once more the results displayed in Table 3. The strong cross-correlations observed were expected after visual inspection of the reported maps (Figure 2 and Figure 3). Finally, as anticipated, a strong cross-correlation in the same spatial range was found for the two capacitive indicators, PMACe and ACe (Table 4).

4. Discussion

The presence of spatial autocorrelation of soil variables in the experimental field can represent a major factor to be considered for revealing the impact of agronomic management on soil physical and hydraulic properties. This study provided a practical methodology for the assessment of the existence of a spatial structure of the variables investigated. These results will also represent a preliminary step for the inclusion of information regarding linear mixed effects models considering residual autocorrelation.

The assessment of soil physical and hydraulic status carried out with reference to the scientific literature showed that near optimal conditions were recorded in the experimental site; only PMACe average values were indicative of non-optimal conditions, whereas mean values of all the other indicators (BD, Ks, PAWCe, ACe, RFCe) fell within optimal ranges. However, we would like to point out that soil physical quality evaluation must be considered as indicative for the variables estimated from BEST (i.e., plant available water, air capacity) and closer to physical reality for the variables directly measured (i.e., bulk density and saturated hydraulic conductivity). In this sense, the literature references were used as benchmarks by which to discriminate soil spatial variability, or to establish comparisons between soil management strategies.

Soil physical variables and capacitive indicators showed strong and significant correlations. Interesting relationships were observed for Ks with PMACe, ACe and RFCe. Remarkably, there were also strong inverse correlations between RFCe vs. PMACe and ACe. Similar to our findings, high negative correlations between relative field capacity and air capacity were observed in a previous study under different crop residue management (r within −0.949 and −0.956) and soil management strategies (r within −0.91 and −0.97) [32]. Strong positive correlations were also observed between macroporosity and air capacity. These outcomes confirm the strong correlation between RFC and AC and PMAC and AC. Decreasing RFC values at increasing AC were expected because both indicators depend on soil water content at saturation and at field capacity [32]. For the accuracy of the information brought by these indicators, in a previous study, RFC was considered as a reference indicator for soil quality assessment, and its optimal and critical ranges were used to define AC limits for agricultural soils [33].

Moran I spatial autocorrelation statistics provides useful information about the predisposition of the variables to be spatialized. In the present study, since the sample size was not excessively large, this could highly affect Moran outcomes. Therefore, values equal to, or larger than, 0.4 can be considered optimal. Generally, lower values of computed Moran statistics can be indicative of the incapability of catching the underlying spatial structure of the data and, in parallel, of the need for a more intensive sampling.

The experimental variograms calculated in this study can be considered reliable since a number of pairs for each distance class larger than the minimum required for an effective spatial analysis was ensured. Myers [14] suggested 25 pairs. In particular, the actual number of observed pairs ranged between 105 and 338.

From Table 3 it emerges that the spatial variability of all the considered variables, excluding ACe, are described by the exponential model; thus, this model was the best suited theoretical function to describe the spatial structure of the physical processes associated with the observed data. This result can be explained by two different perspectives. The first one is that of pure modeling, and the second is that of physical modeling. According to the first perspective, an exponential model represents the simplest case of the Matèrn model; the latter is considered a part of what geostatisticians refer to as “the model” and the so-called model-based geostatistics [49] is based on the assumption that the Matèrn model can effectively mimic all the other models. By the second perspective, it should be recalled that the models reported in Figure 1 represent the average squared difference between couples of the observed values at given distances. Analyzing the functional form of the exponential model, it can be determined that values at the shortest distances are similar to each other (that is, for lag values (h) close to the origin), but very soon differences increase fast (Figure 1b). This is the usual empirical behavior of the considered variables; in fact, they are rather spatially heterogeneous due to their own physical nature, as highlighted in previous works [16,50]. The generalized attribution of the exponential model to Ks is confirmed by a large proportion of the scientific literature [51,52,53,54]. It should be noted that the range parameter assessed in each cited work always has a comparable order of magnitude.

As reported by several authors [25], the nugget-to-sill ratio has been computed as the ratio between the nugget semivariance and the total semivariance. This index, also called the Spatial Dependence Index (SDI), is largely employed to check the strength of the spatial structure [15,27], enabling the comparison of the relative size of the nugget effect of studied variables. In our study, a good overall spatial dependence was found for most of the investigated variables, except BD, ACe and Ks, which showed a moderate autocorrelation. These results resemble those found by Castellini et al. [32], for a different fine-textured soil in the same experimental area, where Ks had the weakest spatial dependence among the soil variables considered. Comparing the performances of Ks spatial modeling between the two studies, the poorest spatial structure observed in [32], with a nugget-to-sill ratio of 0.91, can be connected to the sample size (52 observations). In the present study, the larger sample size (72 observations) was able to grasp the underlying spatial processes, and this was reflected by the nugget-to-sill ratio (0.47). After the above comparison, we suggest a sample size larger than 70 observations for Ks, and overall to consider with caution the conclusions of the literature drawn from too small sample sizes. In recent studies, more intensive samplings (a number of observations larger than 90 and 200) have allowed for a significant improvement in the mapping accuracy of Ks [16,50].

Putting together the indications provided by the different indices computed, namely Moran I index, nugget-to-sill ratio, cross-validations outcomes and cross-correlation, it emerged that the sample size was probably nearly sufficient and needed to be increased for some of the variables, with particular regard to the hydrodynamic properties since, notoriously, Ks is characterized by a relatively high spatial variability [16,50]. The joint analysis of the abovementioned indices can help scholars and practitioners in setting an appropriate sample size for specific variables.

We want to highlight the positive role of cross-correlation statistics in the conducted analysis. The index is often neglected due to its complex formulation (Equation (2)). However, it enables a quantitative assessment of the behaviors emerging from the visual inspection of the prediction maps, thus making more objective their comparison [15,28]. Cross-correlations computed at lag (h) zero confirmed the expected strong relationship between the two capacitive indicators, air capacity and macroporosity, but also enabled us to highlight their spatial association with Ks. The range of spatial dependence was also coherent with the results of the variography. In addition, cross-correlation analysis disclosed the inverse spatial association between Ks and BD with the same range of spatial correlation.

As previously mentioned, repeated measurements performed over the same experimental unit may give rise to correlations that need to be considered in the statistical analysis [7]. However, classical analysis of variance and standard linear models assume independent errors [7,10]. Linear mixed effect models (LMMs), which quantify spatial correlation components and filter them from the total residual term of the model, may represent an appropriate strategy to be employed to improve the protection of statistical tests [9,55]. Previous studies have also shown that these models can enhance the understanding of factors affecting plant response, and are crucial in agronomic and environmental studies [55]. For these reasons, results obtained in this study also represent a first step towards the incorporation of information on the spatial structure of the investigated variables into more complex models (LMMs), with the aim of discerning the effects of compared treatments.

5. Conclusions

This study aimed to provide a set of suitable indicators, both estimated and directly measured in the field, which were useful in assessing the soil physical and hydraulic conditions. In addition, a set of statistical indicators, for gaining an integrated view of the spatial structure of the studied system, was proposed and applied. The results obtained represent a preliminary step for the inclusion of information on the spatial structure of soil hydraulic properties into linear mixed effects models considering residual autocorrelation.

A good overall spatial dependence was found for most of the investigated variables, except BD, ACe and Ks, which showed a moderate autocorrelation. In particular, Ks was confirmed to be characterized by a relatively high spatial variability, and thus, to require a more intensive spatial sampling. In this respect, machine leaning could provide a toolbox of methods advantageously applicable to predict this variable. In addition, the good cross-correlation observed in this study with some capacitive indicators seems to suggest the possibility to use these variables as covariates in predictive multivariate approaches.

The integrated use of spatial statistics demonstrated actual effectiveness in the interpretation of the spatial features of the single variables, as well as for couples. From a practical point of view, obtaining a large and spatially distributed set of data accounting for the physical and hydraulic properties of the soil can have positive implications for the rational management of water resources. However, although the applied methodology appears suitable to adequately investigate the spatial variability of soil physical and hydraulic properties at the plot/field scale, further development of this topic is necessary to consider more robust data sets, obtained, for example, from experimental measurements rather than modeling estimations or from coupling information derived from measured and estimated data.