2. Fuzziness performance index (FPI); and

3. Modified partition entropy index (MPE; Fridgen et al., 2004). To evaluate the quality of the cluster process, two analyses could be conducted:

The average comparison test (analysis of variation, ANOVA; Bazzi et al., 2015)

Smoothness index (SI; Schenatto et al., 2016), which is a measure of the smoothness of the contour curves.

Although a careful definition of MZs requires a lot of steps and analysis about data. The goal of this study was to analyze viability of MZs definition using stable and non-stable variables and use methodology to define best number of zones for each case. This would be very interesting for producers who are only starting to use precision agriculture.

Study site and data collection

Field 1: Data from a 16.9 ha soybean field (Figure 1) were evaluated. It  is  located  in a  rural  area of municipality of Cascavel, Paraná state, Brazil  (24°57ʹ13ʺS and 53°34ʹ02ʺW, average elevation: 650 m). The soil is classified as distroferric red latossol (Embrapa, 1999) with 70% clay. A Trimble Geo Explorer XT 2005 Geodesic Differential Global Positioning System (GDGPS) with post-processing was used for georeferencing the research area. Soil samples (n = 87) were irregularly spaced throughout field. The following soil attributes were measure:

 

 

1. Soil texture (clay, silt, and sand)

2. Cone index (0 to 10 cm, CI_0_10; 10 to 20  cm,  CI_10_20;  and  0 to 20  cm,  CI_0_20)

3. Chemical attributes: organic C, pH, Ca, and Mg, P and K, Cu, Zn, Fe, and Mn, and H+Al; and

4. Physical attributes: bulk density. The methodology from Embrapa (2009) was used to measured these variables. These attributes were used to define management zones because they have influence with potential yield to corn and soybean crops. Numerous researches were used for these attributes to define management zones, such as, Blackmore (2000), Fraisse et al. (2001), Johnson et al. (2003), Schepers et al. (2004), Taylor et al. (2007) and Li et al. (2007).

Field 2 (Figure 2) was approximately 35 ha in size and was nearly level (0 to 2% slope), with an average elevation of 1437 m. The area was cultivated with corn and is located in the rural area of the municipality of Wiggins, Colorado, USA (40°19ʹ59ʺN and 104°01ʹ50ʺW). The dominant soil types were Valentine fine sand (mixed, mesic, Typic Ustipsamment) and Dwyer fine sand (mixed, mesic, Ustic Torripsamment) series (USDA, 1986). The following soil attributes were measure: soil texture (clay, silt, and sand); ammonium; cation exchange capacity (CEC); nitrate; organic matter (OM); and chemical attributes (P, pH, K, Zn).

 

 

Delineation of management zones

Clustering methods are appropriate for dividing data into groups with homogeneous characteristics. Of these, the Fuzzy C-Means is used most often (Li et al., 2007; Fu et al., 2010; Zhang et al., 2013; Li et al., 2013; Moral et al., 2010). This algorithm uses a weighting exponent to control the degree of sharing between classes (Bezdek, 1981), allowing individuals to exhibit partial adhesion in each of the classes, which is important when dealing with the continuous variability of natural phenomena (Burrough, 1989).

Usually only those attributes that do not vary appreciably over time as chemical attributes are used to create MZs that are intended for use for many years (Doerge, 1996). However, for special conditions, MZs are use for nutrients and lime application in the following year. Chemical attributes may be included as MZ attribute variables. Two approaches were apply to delineate MZs: all variables approach (all_Var) and stable variables approach (sta_Var). In the first approach, variables with a significant spatial cross-correlation with the yield were select to delineate MZs. The selected variables may or may not be stable over time. In sta_Var, only those variables that were stable over time (Table 1) and spatially cross-correlated with the yield (Doerge 1996) were consider for delineation MZs (Table 1).

 

 

 

Descriptive statistics

The descriptive statistics for selected attributes of Field 1 are indicated that all variables, except for slope, were it isnormally distributed (Table 2). Most variables showed symmetric and mesokurtic distributions. The CV was low for elevation, pH, bulk density, and clay; medium for Fe, C, silt, Ca, cone index (0 to 10 cm, CI_0_10; 10 to 20 cm, CI_10_20; and 0 to 20 cm, CI_0_20), Mg, Mn, and H+Al; high for the yield and Cu; and very high for K, sand, P, the slope, and Zn. In case of Field 2 (Table 3), clay, silt, sand, OM, pH, and Zn were normally distributed. The CV was low for the pH and sand; medium for yield; high for Zn, OM, CEC, ammonium, and K; and very high for P, nitrate, clay, and silt.

 

 

 

Spatial autocorrelation and cross-correlation

The attributes analyzed values for Field 1 were show at Table 4. Variables Cu, clay, elevation, CI_10_20, CI_0_20, sand, slope, P, C, Zn, and bulk density had significant positive autocorrelations. Variables CI_10_20, elevation, CI_0_20, bulk density, slope, sand, clay, silt, pH, P, and H+Al were significantly cross-correlated with yield. Bazzi et al. (2013) found a high correlation between soybean yield with soil sand percentage, generating agricultural area MZs with this layer. Similarly, Schenatto et al. (2016), Peralta and Costa (2013) and Gavioli et al. (2016) generated MZs from elevation data and CI in experimental soybean and cornfields.

 

 

The values for Field 2 were show in Table 5. Variable yield, P, nitrate, Zn, CEC, OM, K, clay, sand, silt, and ammonium exhibited significant autocorrelations. Variables OM, CEC, sand, NH3, clay, K, silt, and Zn were significantly cross-correlated with the yield. Other studies yielded similar results: for example, Bansod and Pandey (2013) and Jaynes et al. (2003) identified high correlation between CEC and yield.

 

 

Management zones

Figure 3 shows the MZs as a function of zone number (from two to five) and approach (sta_Var and all_Var) for Field 1 (Cascavel) and Field 2 (Wiggins). In all cases, this can be see when the number of zones increased, the amount of fragments of the area is bigger.

 

 

For Field 1, data layers selected to delineate MZs using sta_Var (approach) were elevation and slope, whereas MZs based on all_Var based on CI_10_20. CI_10_20 was not selected as sta_Var because it was not stable over time. However, because it showed highest spatial correlation with yield and was cross-correlated with elevation and slope, it was included in all_Var selection. In this sense, in the lower areas, soybean yield was lower probably because rainfall was higher than average during cultivation, which may have resulted in excessive water in soil (Fausey, 1999). For Field 2, MZs based on sta_Var approach were delineated using sand, whereas in all_Var approach, OM used it. As in Field 1 case, OM was not select as sta_Var because it was not stable over time. In both approaches, selected correlated layers (sand and OM) were negatively spatially correlate with yield. In addition, sand and OM were correlate to CEC, meaning that this variable used to define MZs if sample and analysis costs were prohibitive.

The average yields differed significantly between two MZs (Table 6) in both fields. With three, four, and five zones only for Field 2, the average yields differed significantly. From these results and those reported by Zhang et al. (2013), it can be seen that the division of the area into two MZs was adequate for producing zones with different yields.

 

 

The VR of an MZ indicates whether MZ  delineation is more efficient than having no MZs on field. In all cases (Table 6 and Figure 4), VR values were positive, indicating that in all cases, total variance reduced, as expected. In case of Field 1, VR remained reasonably constant with MZs number.  However, in Field 2 case, VR increased with MZ number. The VR results confirm ANOVA results because it turns out  that divisions  within  each  zone  did  not  show  much  variance  from  average, indicating regions of similar productive potential. These results corroborates those obtained by Xiang et al. (2007) and indicated that when divisions are satisfactory, VR tends to increase despite an increase in MZs number because there is smaller variation within each zone.

 

 

ANOVA shows for Field 1, that best division was two MZs (for sta_Var and all_Var) and up to five MZs for Field 2 (sta_Var and all_Var) (Table 6). The FPI, MPE, and VR indices (Figure 4) were also evaluated; smaller FPI and MPE, shows better clustering test for both fields, considering FPI and MPE. Best efficiency was obtain with two MZs, but best option with VR was two (sta_Var) and three (all_Var) MZs for Field 1 and five MZs for Field 2 (sta_Var and all_Var).

Two approaches comparison (sta_Var and all_Var) for two MZs (selected as best number of MZs) showed that sta_Var provides better VR. That means there is no meaning of using all variables (time stable or not) in defining MZs. The sta_Var MZs (with two MZs) presented a reduction in variance from 21 (Field 1) to 50% (Field 2). Because the MZ definition through Fuzzy c-means does not use variable yield only if several years’ yields could find other important and not redundant variable, the MZs would be same.

 

 

 

The methodology proposed for MZs define has practical value and allowed MZs definition with good quality. This results may be confirmed on the analysis of the variance reduction and Anova. For divisions in two zones (best division by MPE and FPI indexes), the variance reduction and Anova showed have different yield potential in each zone when the fields were divided. The stable variables approach performed better than all variables in all cases because they showed better results on the analysis from variance reduction and Anova, indicating that stable variables have influence in the yield capacity and can be used to define management zones that must be used for a long time. The stable variables approach also served as the source of recommendation and soil analysis and pre-plant or in-season fertilizer recommendation for precision agriculture.

The authors have not declared any conflict of interests.

The authors are grateful to State University of Western Paraná (UNIOESTE), Technological Federal University of Paraná (UTFPR), Araucária Foundation (Fundação Araucária), Coordination for Improvement of Higher Education  Personnel  (CAPES),  and  National Council for Scientific and Technological Development (CNPq) for support received and for Agassis Linhares, agronomist engineer, for assignment of research area.