1 Introduction

The variability of soils, both spatially and temporally, has long been recognized in soil science (Jenny 1941; Lin 2006; Simonson 1959; Wang and Shi 2018). Techniques that capture soil spatial and temporal variability have recently evolved to focus the attention of soil scientists on soil attributes related to hydrological models (Qiao et al. 2018; She et al. 2017). According to Libohova et al. (2018), the success of predictions when modeling hydrological processes depends on the accurate representation of the spatial and temporal variability of soil hydrological attributes and the main external factors, such as climate, land use and management, and soil water dynamics.

Saturated soil hydraulic conductivity (KSat) is a key soil attribute for understanding the soil water movement phenomenon and is therefore one of the main inputs used in hydrological models. Several studies have reported that KSat values are affected by the variation of pore space geometry (Baiamonte et al. 2017), topography (Wang et al. 2013), land use (Price et al. 2010; Salemi et al. 2013; Pinto et al. 2019), and scale (Picciafuoco et al. 2019). It has also been well documented in the literature that Ksat is highly variable in space (Wang et al. 2013; Papanicolaou et al. 2015; She et al. 2017). Therefore, due to its high variability as well as to the effort required for KSat data sampling, representativeness of the spatial variability of KSat at the watershed scale is often difficult to obtain (Reichardt and Timm 2020).

Geostatistical tools have been used to quantify and map the spatial variability of soil hydrology attributes aimed at supporting hydrological modeling at the watershed scale. The classical estimator of Matheron is the most used semivariogram estimator to characterize and quantify the spatial variability structure of soil attributes. However, due to its high sensitivity to the existence of outliers, and its requirement of a normal distribution in order to estimate the underlying process (Lark 2000), its use is difficult in field conditions, especially for attributes related to soil water dynamics (Lebrenz and Bárdossy 2017). In order to overcome these drawbacks, data transformation (logarithm, exponential, root fourth, etc.) from the original scale of measurement has been adopted (Song et al. 2019; Wallin and Bolin 2015). However, Goovaerts (1997) indicated that data transformation is not ideal if the aim is prediction, since in general those who ultimately use the predictions, such as land managers, hydrological modelers, and environmental scientists and so on, require values on the original scale of measurement which involves a back-transformation. Therefore, several robust estimators, such as that of Cressie and Hawkins (1980), have been proposed to minimize the effects of the presence of outliers or for use on datasets with non-normal distributions for calculating experimental semivariance values.

The determination of KSat can be costly, and much effort is required to obtain representative spatial variability at the watershed scale. Therefore, multiple regression models (namely pedotransfer functions (PTFs)) have been developed to estimate KSat from easily available soil attributes from databases. For instance, Boadu (2000) developed regression-based models for estimating the KSat of compacted soils using fractal dimension, entropy, porosity, percent of fine soil particles, and the bulk density as predictors. However, some values (e.g., fractal dimension, entropy, and soil porosity) are not readily available from soil databases.

Another alternative is to use geostatistical tools such as the cross-semivariogram to estimate KSat from an auxiliary variable of simple determination and/or low cost, since both variables are well correlated. In this sense, robust estimators can be applied for the calculation of the cross-semivariogram between KSat and other soil (such as porosity, textural fractions, bulk density, etc.), topographical (elevation, slope, aspect, etc.), and environmental (climate, land use, etc.) attributes when such data are available.

Adhikary et al. (2017) mentioned that the ordinary cokriging method in general reduces the variance of the prediction error and improves estimations compared with univariate ordinary kriging method, when the auxiliary variable is well correlated with the primary one. Wang and Shi (2018) used robust cross-semivariograms in the cokriging process to estimate soil particles (clay, silt, and sand) in a large watershed, applying some of the estimators proposed by Lark (2003).

The spatial variability of KSat in watersheds is recognized as a complex phenomenon, due mainly to the natural and anthropogenic processes that influence soil water dynamics, predominantly at the surface layer (Becker et al. 2018). In this sense, an auxiliary variable (or covariate) should consider some characteristics of the soil or the environment that are quantifiable and related to the occurrence or the impediment of the target phenomenon. Acquiring these auxiliary data should be operationally more viable compared with the acquisition of the target variable. Nowadays, topographical attributes are widely available and can be readily obtained through the geographic information system (GIS) using digital elevation models (DEMs) and digital terrain analysis techniques (She et al. 2017). Additionally, if soil hydrology attributes such as KSat can have their estimates optimized using land features as covariates obtained by GIS, costs and time can be saved (She et al. 2017).

The development of the Ellert Creek Watershed (ECW) is regionally strategic as it is the main watercourse that flows directly to the Pelotas River, supplying drinking water for the population of Pelotas, a city with some 329,000 inhabitants (Beskow et al. 2016). The ECW is a headwater watershed of the Pelotas River Watershed (PRW), where local events are possibly reflected in the hydrological behavior of the downstream PRW. Furthermore, the ECW has suffered from the impacts of human activities, being vulnerable to water erosion and floods, causing soil and nutrient losses resulting in economic and social damage.

Soil hydrology attributes (e.g., KSat) are key to understanding the processes of the hydrological cycle and are critical information for the application of hydrological models aimed at supporting decisions on water resource management (Beskow et al. 2016; She et al. 2017). Among them, KSat is a sensitive model input for the application of distributed hydrological models (Wang et al. 2013; Hu et al. 2015). Therefore, the objectives of this study were to (i) provide a reference for the selection of soil hydrology and environmental attributes that can be used as covariates for KSat estimations at the watershed scale using the non-linear Spearman correlation coefficient and multivariate analysis, (ii) assess the performance of ordinary kriging for mapping the spatial distribution of KSat using Matheron and Cressie and Hawkins estimators, and (iii) assess the performance of the ordinary cokriging method to map KSat using the Cressie and Hawkins cross-semivariogram estimator and each selected attribute as an auxiliary variable.

2 Materials and Methods

2.1 Description of the Study Site

The study was carried out in a headwater sub-basin of the Pelotas River Watershed, known as the ECW, located in the municipality of Canguçu, Southern Rio Grande do Sul (RS) state, approximately 50 km northeast of municipality of Pelotas (Fig. 1). The ECW has an area of approximately 0.7 km2, and the altitude varies from 310 to 419 m. According to the Köppen climate classification, ECW is Cfa type, a mesothermal climate indicating wet subtropical conditions, and characterized by an annual average temperature of 18 °C, with hot summers and cold winters (Kuinchtner and Buriol 2001). The precipitation is well distributed throughout the year, and its mean annual value is around 1350 mm (IBGE 1986). The regional relief varies from undulating to strong undulating, with a predominance of native forest or sparse shrub, and shallow soils.

Fig. 1
figure 1

Location, topography, hydrography, and sampling points of the Ellert Creek Watershed. a South America, b Rio Grande do Sul State, and c Ellert Creek Watershed

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The watershed comprises only one soil class, the Neossolos (Brazilian Soil Classification System) or Entisols in soil taxonomy, identified using the soil map developed by Embrapa (1999), and updated by the latest version of the Brazilian Soil Classification System (Embrapa 2006). The ECW was selected due to its economic and social importance to the municipality of Canguçu, an area greatly dominated by family farming systems. The ECW main watercourse flows directly to the Pelotas River, located in the southern part of the Rio Grande do Sul state, with a total area of approximately 940 km2, including the municipalities of Pelotas, Morro Redondo, Arroio do Padre, and Canguçu.

2.2 Sample Grid

A sample grid of 106 sampling points spaced 50 m in the west by 75 m in the south direction was established at the beginning of field work. Subsequently, in order to better model the spatial variability structure of soil hydrology attributes, 78 soil samples were collected in a specific area of the ECW watershed spaced 25 m at both directions, totaling 184 sampling points (Fig. 1). This specific landscape was selected due to its diverse features, including different land uses, soil textural classes, and features of the topographical terrain. The geographical location of each point in the ECW area was recorded using GPS navigation equipment. ArcGIS software was used to establish the sample grid and to obtain the UTM coordinates of each point (Environmental Systems Research Institute, Redlands, CA).

2.3 Soil and Topographic Attributes

Undisturbed soil samples were collected at all 184 sampling points from the 0–0.20-m layer using metallic cylinders with a 5.0-cm height and a 4.8-cm internal diameter. It is important to highlight that the samples were collected from the topsoil layer (0–20 cm), the area of the watershed most affected by different land management systems (Alvarez and Steinbach 2009).

The following soil hydrology attributes were determined: soil bulk density (BD) (Blake and Hartge 1986), soil total porosity (TP), macroporosity (Mac), and microporosity (Mic) (Klute 1986). Saturated soil hydraulic conductivity (KSat) was measured at each point using the constant head method (Klute and Dirksen 1986).

The main land uses identified in the ECW were forest, silviculture, annual cropping, and pasture. Forest here included native forest sites which accounted for around 10% of the ECW’s area; areas of silviculture were characterized by the occurrence of Pinus sp., Eucalyptus sp., and Acacia sp., equivalent to 12% of the ECW. Annual cropping areas included the cultivation of Glycine max, Zea mays, and Nicotiana tabacum which occupied the largest area of the ECW (71%). Pastures were mainly characterized by grazing areas, accounting for 7% of the total area of the ECW.

2.4 Exploratory Analysis

All data sets were submitted to basic and descriptive statistics, calculating position (arithmetic average and median) and dispersion (standard deviation, variance, and coefficient of variation) measures as well as skewness and kurtosis coefficients, which are related to the characteristics of the data distribution. The Kolmogorov-Smirnov (K-S) test at 5% significance was applied to verify normality of all data sets. Additionally, boxplots were constructed to evaluate the relationship between KSat and the identified land uses. The Spearman coefficient of correlation between KSat and the other soil hydrology attributes and environmental features was also calculated.

2.5 Cluster Analysis

Before the multivariate statistical analyses, each data set was standardized (mean = 0 and standard deviation = 1), with the aim of transforming the original data to the same order of magnitude. Because the variables present different orders of magnitude, the use of the standardized variables requires that the variance and covariance matrixes are the same as the correlation matrix. Cluster analysis was applied to construct dendrograms with the aim of grouping the standardized variables into homogeneous groups (Everitt and Hothorn 2011), using the Mojena’s (1977) criterion for cutoff. The dissimilarity between the standardized variables is a distance measure. The hierarchical cluster technique uses a Euclidean distance (metric) to separate a set of objects (variables) into groups according to chosen criteria. The group-average method was used as an aggregation criterion. It is a weighted method which calculates the distance between two clusters as the arithmetic average distance from observations in one cluster to observations in another cluster. This method tends to combine clusters that have small variances and may produce new clusters that have the same variance (Ramos et al. 2007). The number of groups was defined based on the criterion of the square root of the number of variables (Bitencourt et al. 2016).

2.6 Geostatistical Analysis

The semivariograms used for geostatistical analysis may explain the spatial variability of KSat, and are calculated from the set (Zu (xi), i = 1, 2,…, N), usually by the Matheron classical estimator (Webster and Oliver 2007), defined by Eq. (1):

$$ {\gamma}_{Z_u}^M(h)=\frac{1}{2N(h)}\sum \limits_{i=1}^{N(h)}{\left[{z}_u\left({x}_i\right)-{z}_u\left({x}_i+h\right)\right]}^2\kern0.5em $$
(1)

where \( {\gamma}_{Z_u}^M \)(h) is the semivariance value using the Matheron estimator, zu(xi) and zu(xi + h) are values of Zu at locations xi and xi + h, respectively, and N(h) is the number of pairs [zu(xi), zu(xi + h)] separated by the lag distance h.

Lark (2000) studied several robust estimators for experimental semivariograms, which deal better with the presence of outliers in the datasets. One of these estimators was developed by Cressie and Hawkins (1980) and is presented in Eq. (2):

$$ {\gamma}_{Z_u}^{CH}(h)=\frac{1}{2N(h)}\sum \limits_{i=1}^{N(h)}\frac{\left\{{\left[{\left|{z}_u\left({x}_i\right)-{z}_u\left({x}_i+h\right)\right|}^{\frac{1}{2}}\right]}^4\right\}}{0.457+\frac{0.494}{N(h)}} $$
(2)

where \( {\gamma}_{Z_u}^{CH} \)(h) is the semivariance value using the Cressie-Hawkins estimator. This estimator was designed to calculate the semivariogram of a primary process with differences normally distributed even in the presence of contaminants of a secondary process.

Cressie and Hawkins (1980) showed that the square root of the absolute difference had a normal distribution with no significant bias. Therefore, the effect of outliers is damped by the square root of the absolute differences and then increasing that values to the fourth power to resize the semivariogram scale. The denominator of Eq. (2) is a correction factor based on the assumption that the underlying process has the differences normally distributed over all lags (Lark 2000). The breakdown point of this estimator is zero, and for this reason, it may present some difficulty by not establishing any limits on the effect of outliers. In this case, a very large outlier could make the estimate non-representative (Genton 1998). However, in practice, it is an estimator that smooths the discrepant values and does not set absolute limits on its effect.

The classical cross-semivariogram of Matheron is defined by Eq. (3), being an extension of Eq. (1), where Zu and Zv are the target and the auxiliary variables, respectively.

$$ {\gamma}_{Z_u,{Z}_v}^M(h)=\frac{1}{2N(h)}\sum \limits_{i=1}^{N(h)}{\left[{z}_u\left({x}_i\right)-{z}_u\left({x}_i+h\right)\right]}^2.{\left[{z}_v\left({x}_i\right)-{z}_v\left({x}_i+h\right)\right]}^2\kern0.5em $$
(3)

where \( {\gamma}_{Z_{u,}{Z}_v}^M(h) \)is the cross-semivariance value between the variables Zu and Zv, the latter obtained by means of the Matheron estimator.

For the Cressie and Hawkins estimator, the cross-semivariogram follows the same rule described above, where the covariate assists in robust semivariance estimates, according to Eq. (4):

$$ {\gamma}_{Z_u,{Z}_v}^{CH}(h)=\frac{1}{2N(h)}\sum \limits_{i=1}^{N(h)}\frac{\left\{{\left[{\left|{z}_u\left({x}_i\right)-{z}_u\left({x}_i+h\right)\right|}^{\frac{1}{2}}\right]}^4\right\}\left\{{\left[{\left|{z}_v\left({x}_i\right)-{z}_v\left({x}_i+h\right)\right|}^{\frac{1}{2}}\right]}^4\right\}}{0.457+\frac{0.494}{N(h)}} $$
(4)

where \( {\gamma}_{Z_{u,}{Z}_v}^{CH}(h) \)is the cross-semivariance value between Zu and Zv variables, the latter obtained by means of the Cressie-Hawkins estimator.

2.7 Cross-Validation

The cross-validation technique was used to verify the quality of the geostatistical results, comparing estimated data by kriging and cokriging with KSat measurements. The statistical parameters used to evaluate the performance of the cross-validation technique were the mean error (ME, Eq. (5)), the square root of the mean squared error (RMSE, Eq. (6), and the coefficient of determination (r2).

$$ ME=\frac{1}{n}\sum \limits_{i=1}^n\left({e}_i-{m}_i\right) $$
(5)
$$ RMSE=\sqrt{\frac{1}{n}}\sum \limits_{i=1}^n{\left({e}_i-{m}_i\right)}^2 $$
(6)

where n = number of sampling points, ei = estimated value of KSat, and mi = observed KSat value. The ME is an indicator of the accuracy of the estimation, showing the tendency of the interpolator to overestimate the values if positive or to underestimate if negative. RMSE quantifies the dispersion of the measured and estimated values around the 1:1 line.

3 Results

3.1 Exploratory Analyses

The summary statistics of soil saturated hydraulic conductivity (KSat) and the other six associated attributes (BD, TP, Mac, Mic, Elev, and Land Use) for the 179 points (five soil samples were discarded during laboratory procedures) are presented in Table 1. The frequency distribution of KSat and TKSat (normalized KSat data set using the fourth root transformation) is presented in Fig. 2. According to the classification of Wilding and Drees (1983), the coefficients of variation (CV) for BD, TP, Elev, and Land Use can be considered low (CV ≤ 15%), whereas the Mac and Mic CV values were considered moderate (15% < CV ≤ 35%).

Table 1 Exploratory analysis of the data sampled in the study watershed
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Fig. 2
figure 2

a Histogram of the saturated soil hydraulic conductivity (KSat) values and b histogram of transformed data by the fourth root indicating a Gaussian behavior. TKSat: KSat normalized by fourth root [KSat(1/4)]

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The crossbar of the boxplot graphics (Fig. 3) shows the median for each dataset for different land uses. The length of the box reflects the interquartile range, and the vertical tails (or bars) of the boxplot are marked by the extremes (either the highest or lowest observed data); however, such do not qualify as outliers. Outliers are defined as data ≥ 1.5 times greater (upper fence) or ≥ 1.5 times lower (lower fence) than the interquartile range.

Fig. 3
figure 3

Boxplot of KSat for the different land uses observed in SEW

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3.2 Cluster Analysis

Table 2 presents the correlation matrix between attributes, showing that between KSat and the associated attributes (BD, TP, Macro, Micro, Elev, and Land Use), all linear correlations were significant at the 1% probability level, with the exception of elevation.

Table 2 Spearman correlation matrix for the attributes studied
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The dendrogram plotted from topographic and soil hydrology attributes and land use in the ECW are shown in Fig. 4. The cutoff was defined at 70% taking into account the statistical stopping rule suggested by Mojena (1977).

Fig. 4
figure 4

Dendrogram of all hydrology attributes, elevation, and land use, constructed from 179 data points sampled at SEW. The graph also shows the cutoff of 70% (red line)

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3.3 Geostatistical Analysis

In this study, isotropic experimental semivariograms are estimated for KSat, assuming an identical spatial correlation in all directions and neglecting the influence of anisotropy on the semivariogram parameters. Isotropy is assumed for methodological simplicity. Isotropy is a feature of a natural process or data in which directional influence is considered insignificant and spatial dependence (autocorrelation) changes only with the distance between two lags (Johnston et al. 2001).

In Table 3, the fitted parameters of the semivariogram and cross-variogram models and their respective degree of spatial dependence (DSD) values are presented.

Table 3 Parameters of univariate semivariograms and cross-semivariograms adjusted by theoretical models
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The DSD calculated from the univariate semivariograms of KSat was 43.2% for the transformed data, applying the classical Matheron estimator \( \left[{\gamma}_{Z_u}^M(h)\right] \). On the other hand, a DSD of 22% was obtained for the untransformed KSat data using the robust estimator of Cressie and Hawkins \( \left[{\gamma}_{Z_u,{Z}_v}^{CH}\right] \), indicating that these data have a moderate and weak spatial dependence (Cambardella et al. 1994). Following the same classification, the cross-semivariograms presented moderate spatial dependence (KSat × Macro = 33.2%; KSat × TP = 34.8%; KSat × BD = 36.2%), except for the KSat × LU (land use) cross-semivariogram (2.1%), which presented a weak spatial dependence. The ranges for the univariate semivariograms of KSat were 168.5 m for \( {\gamma}_{Z_u}^M(h) \) and 178.8 m for \( {\gamma}_{Z_u}^{CH}(h) \). For \( {\gamma}_{Z_u,{Z}_V}^{CH}(h) \), the range values were 152.9, 213.1, 161.8, and 212.7 m.

The maps developed by kriging methods using the parameters of \( {\gamma}_{Z_u}^M(h) \) (TKSat) and \( {\gamma}_{Z_u}^{CH}(h) \) (RKSat) are presented in Fig. 5. Furthermore, the cross-validation statistical precision (ME, RMSE, and r2) between the observed and estimated KSat values using univariate kriging and cokriging are shown in Table 4.

Fig. 5
figure 5

Spatial distribution of KSat in the Sanga Ellert Watershed using classic ordinary kriging (a), robust ordinary kriging (b), and robust ordinary cokriging (c–f)

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Table 4 Summarized statistics of the cross-validation for observed vs. interpolated data of KSat
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Figure 6 presents the scattered points around the 1:1 line from the cross-validation process, which was applied for each of the studied combinations, showing the accuracy of the geostatistical procedures applied in this study.

Fig. 6
figure 6

Cross-validation and linear adjustment of observed vs. interpolated data of KSat using classic ordinary kriging (a), robust ordinary kriging (b), and robust ordinary cokriging (c–f)

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4 Discussion

4.1 Exploratory Analyses

The average KSat and the median corresponded to 0.81 m h−1 and 0.52 m h−1, respectively. This contrast shows the great variability of this soil hydrology attribute, also demonstrated by the coefficient of variation (99.4%), being classified as having a high variability (CV > 35%) according to Wilding and Drees (1983). Godoy et al. (2019) and She et al. (2014) found a CV for KSat of 122.0 and 50.8% in studies conducted in central-western Brazil and in the Loess Plateau region (China), respectively. Even when the data sets were transformed to logarithmic (ln), She et al. (2017) obtained a CV above 35%.

All datasets followed a normal distribution as determined by applying the K-S test. KSat data presented a non-normal distribution, agreeing with results widely reported in the literature (Ahuja et al. 2010; Godoy et al. 2018; She et al. 2017). Therefore, the original KSat data was fourth root transformed (Elsenbeer et al. 1992) in order to use Matheron’s classical semivariance estimator (Fig. 2).

The boxplot analysis (Fig. 3) of KSat in the different types of land uses shows a decrease in the saturated soil hydraulic conductivity following the areas of forest, silviculture, annual cropping, and pasture. Forest soils are associated with a higher infiltration capacity (Archer et al. 2013; Kurnianto et al. 2018; Wood and King 1977) and a lower generation of surface runoff (Germer et al. 2010; Kurnianto et al. 2018; Liu et al. 2018) than soils under other types of vegetation. This occurs in nature due to preferential flows in the soil profile as a result of soil biota activity and higher organic matter content, as reported by Pinto et al. (2015), Ma et al. (2017), and Pinto et al. (2019).

The median KSat values for forest, silviculture, annual cropping, and pasture were 1.18, 0.65, 0.49, and 0.26 m h−1, respectively (Table 1). Annual cropping showed a greater number of outliers, suggesting a greater variability of KSat in this area of land use. Despite this variability being linked to the different farming activities practiced in the watershed, we did not investigate changes in Ksat patterns over the time within the same management system, which would lead to another line of investigation. In the agricultural areas of the watershed, different farming systems with differing levels of mechanization, and technology such as conventional tillage, no-tillage and minimum tillage were observed, increasing the variability of Ksat.

Several studies have shown that land use systems affect KSat values and therefore their spatial variation in a watershed (Price et al. 2010; Salemi et al. 2013; Pinto et al. 2019). Therefore, if properly quantified, it is possible to improve KSat estimates in non-sampled points when used as a covariate in a cokriging interpolation.

4.2 Cluster Analyses

The value of the Spearman correlation coefficient between KSat vs. Land Use was low (0.21), although it was significant at a probability level of 1%. A correlation of 0.5 was observed between KSat vs. Mac and KSat vs. BD; however, in the case of KSat vs. BD, the correlation was negative. The highest correlation value was found between KSat and TP (0.72). The values of Mac and TP were directly related with KSat, which is corroborated by several other studies, where values of hydraulic conductivity are reported to increase with the increase of total porosity. This is mainly due to macroporosity since it is a preferred path of water flow in the soil profile, especially in response to gravitational force (gravitational potential) (Kurnianto et al. 2018).

A negative correlation was observed between KSat and Micro (− 0.51). According to Godoy et al. (2019), this would be an inconsistent pattern considering that the KSat of the soil has no physical relationship with soil micropores. This result indicates that a simple linear correlation matrix is not enough to explain the correlations between KSat and other attributes, and thus, multivariate statistical tools should be considered for the assortment of spatial covariates and then a more efficient cokriging process.

Based on the dendrogram and the cluster analyses, two clusters were formed: the first was composed by KSat, TP, Mac, and Land Use, and the second formed by the BD, Micro, and Elev. In cluster 1, the most pronounced similarity occurred between Mac and TP and then between these two and KSat, followed by Land Use. The relationship between Mac and TP is justified by the fact that macroporosity is a straight component of total soil porosity. Soil porosity, mainly macroporosity, is the main water path in the soil profile, and in this way, the approximation of these attributes with KSat becomes clear (Godoy et al. 2019). Some studies (Kurnianto et al. 2018; Liu et al. 2018) suggest that KSat is affected by land use, mainly in the superficial soil layers. This is due to the physical modifications in the superficial layers by soil management, altering the dynamics of the water in the soil.

4.3 Geostatistical Analyses

The variographic analysis (Table 3) demonstrates the spatial dependence of univariate semivariograms fitted using the spherical theoretical model. The cross-semivariograms were also fitted by the spherical theoretical model, except for KSat × LU, which had the experimental semivariogram fitted by the Gaussian theoretical model.

The values of the semivariogram parameter [\( {\gamma}_{Z_u}^{CH}(h)\Big] \) contribution [C] and nugget effect [C0 + C] are integers and positives, except when BD was used as a covariate (KSat × BD). The negative values for these parameters of [\( {\gamma}_{Z_u,{Z}_v}^{CH}(h)\Big] \)KSat × BD occur due to the inversely proportional correlation between the attributes KSat and BD, demonstrated by the Spearman matrix. This is widely diffused in the literature, where it is assumed that changes in BD mainly affect the volume and the relationship of the macropores of a soil, thus influencing the flow of soil water (Godoy et al. 2019).

Although the highest range of values were observed for \( {\gamma}_{Z_u,{Z}_v}^{CH}(h) \)for KSat × TP and KSat × BD, it is not possible to state categorically that cross-semivariograms significantly increase their spatial range when compared with univariate semivariograms, mainly in this case, where for this parameter, \( {\gamma}_{Z_u}^M(h) \) and \( {\gamma}_{Z_u}^{CH}(h) \) of KSat presented better performance than \( {\gamma}_{Z_u,{Z}_v}^{CH}(h) \), when the covariates were Mac and LU. This may be an indication that for a more realistic assessment of the efficiency of each method, the analysis of parameters of cross-validation could be essential.

The kriging maps (TKSat and RKSat) exhibit distinct patterns (Fig. 5), where it is possible to observe that for TKSat, high values occupy a more extensive area of the ECW, whereas for RKSat, the highest values are restricted to areas near the drainage network and with a greater forest presence. These results are in accordance with (Kurnianto et al. 2018).

The map generated by the parameters of \( {\gamma}_{Z_u,{Z}_v}^{CH}(h) \) called KSat × LU presents a random pattern very different from the others, indicating a more coherent saturated flow accumulation when compared with each other. This phenomenon can be explained by the low performance of the \( {\gamma}_{Z_u,{Z}_v}^{CH}(h) \) parameters when using land use as the covariate, indicating that this environmental attribute is not suitable for cokriging interpolation in the ECW.

Maps generated from KSat × Macro, KSat × TP, and KSat × BD are more detailed than those generated by univariate ordinary kriging. Maps generated from KSat × Macro are close to the original data in relation to the maximum and minimum values and amplitude. Additionally, maps generated by univariate kriging had less spatial detail (more uniform) than those generated by cokriging (KSat × Mac; KSat × TP; KSat × BD). This spatial detail is clearer in some specific areas of the watershed, such as seen in the middle area. A greater amplitude of the high and low values distributed over the ECW area is also observed in the three KSat maps which presented better performance in the cokriging process. This suggests that the smoothing effect is more pronounced in kriging than in cokriging. Furthermore, this demonstrates that cokriging, when used the correct covariates, can better describe the variability of KSat than ordinary univariate kriging.

The RMSE values (Table 4), which indicate the data dispersion behavior, presented a better absolute performance of the KSat × Mac interpolation, with a value closer to zero. In contrast, the KSat × LU cokriging and the univariate interpolations (TKSat and RKSat) presented the worst results. On the other hand, the values of ME, which identify if the model underestimated (negative values) or overestimated (positive values) the target variable, were quite satisfactory, presenting low and negative values with a slight underestimating behavior, except for the maps generated by univariate kriging TKSat and cokriging “KSat × LU.” Only the TKSat interpolation overestimated the observed data, showing the highest ME value.

The coefficient of determination (r2) had a very high performance demonstrating a good fit for almost all of the estimated values in the interpolations. Especially in reference to KSat × Mac (r2 = 0.91), this statistic indicated an excellent linear fit in the cross-validation cokriging data (Fig. 6). The lowest performance can be attributed to cokriging where the covariate of land use was used. In addition to the extremely random standard of the map generated by cokriging, the KSat × Land Use cross-validation (Fig. 6e) showed a poor fit between the estimated and observed values. On the other hand, the results obtained by KSat × Mac (Fig. 6c) during cokriging proved to be the most efficient mapping method, with a high performance in statistical indicators and cross-validation.

Saturated soil hydraulic conductivity is the most important soil hydrology attribute for soil management, water yield and baseflow behavior in watersheds, sediment transport, and for the basic information of initial hydrogeology exploration. Several applications require a good map of KSat, such as for identifying areas with higher potential for groundwater recharge (Pinto et al. 2016; Alvarenga et al. 2012), to characterize the effect of land use on the watershed streamflow behavior by means of a distributed hydrological model (Alvarenga et al. 2016) or based on measurements in the field (Price 2012; Muñoz-Villers and McDonnell 2013; Pinto et al. 2019), to understand the water balance in watersheds (Mello et al. 2019; Salemi et al. 2013; Fleischbein et al. 2006) and for use as a support for storm hydraulic structure design (Chappell et al. 2017).

5 Conclusions

Among univariate geostatistical techniques, the use of the robust Cressie and Hawkins estimator on untransformed data presented a better performance than using the classical estimator of Matheron on transformed data. The cluster multivariate method presented a clearer separation of the covariates for the KSat cokriging process, which can be considered as the most efficient method for selection of covariates compared to classic techniques.

The cokriging technique demonstrated a greater performance than univariate techniques when robust cokriging was applied using macroporosity and bulk density as covariates, demonstrated by the quality of the KSat map increased estimates. Land use covariate, although showing an influence on KSat, was not successful in subsequent geostatistical multivariate processes for the studied watershed.

The KSat robust cokriging map can be used to improve land planning in this headwater watershed, with the aim of increasing the water infiltrability and preserving the uses and management strategies that have shown a positive impact on the hydrology of the watershed.