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Engenharia Agrícola

Print version ISSN 0100-6916On-line version ISSN 1809-4430

Eng. Agríc. vol.40 no.1 Jaboticabal Jan./Feb. 2020  Epub Feb 17, 2020 

Scientific Papers

Topography, Photogrammetry and Remote Sensing


Luciana P. C. Guedes1  *

Raquel T. Bach1 

Miguel A. Uribe-Opazo1 

1Western Paraná State University/ Cascavel - PR, Brasil


Spatial variability description of soil chemical properties by thematic maps depends substantially on suitable geostatistical models. One of the parameters composing a geostatistical model is nugget effect. This study aimed to evaluate the simultaneous influence of nugget effect and sampling design on geostatistical model estimation and estimation of soil chemical properties at unsampled sites, considering simulated data. Our results will be used as scientific basis for spatial variability analyses of soil chemical properties in agricultural areas. Given the simulation results and agricultural data, we concluded that the high nugget effect values obtained here reduced spatial estimation efficiency. Moreover, a systematic sampling design promoted the least accurate estimates of geostatistical model and at non-sampled sites. Despite that, these nugget effect estimates should be kept in the analysis. However, further studies will be needed to investigate which factors are responsible for such high nugget effect values.

KEYWORDS Geostatistics; Spatial Dependence Index (SDI); Relative Nugget Effect (RNE); Spatial Estimation


Spatial analysis of a georeferenced variable using geostatistical models enables measuring the spatial dependence degree among samples within a determined area, thus describing its spatial dependence structure (Guedes et al., 2018). The spatial dependence analysis, mainly of soil chemical properties in farmlands, allows to know their values in subregions (management zones) within the area of interest, which, in turn, enables the application of agricultural inputs at specific points (Gazolla-Neto et al., 2016).

The spatial dependence structure of a certain georeferenced variable should be described considering a stochastic process, whose data are expressed by Z(s1), Z(s2), …, Z(sn), which are known in n sites si (i = 1, …, n), being that si = (xi, yi)T is a two-dimensional vector. The georeferenced variable can be expressed by a Gaussian spatial linear model: Z(si) = µ(si) + ε(si) (Uribe-Opazo et al., 2012), in which µ(si) = µ is the deterministic term, µ is a constant, and ε(si) represents the stochastic term with mean zero, i.e., E[ε(si)] = 0; and variation between points in space separated by an Euclidean distance hij = ||h||, so that h = si - sj (i, j = 1, …, n) is determined by a covariance function: C(hij) = cov[ε(si), ε(sj)] = σij, which depends only on h. Moreover, a covariance matrix is found from the covariance function, as follows: Σ = [(σij)] (Dalposso et al., 2018; Guedes et al., 2011; 2018).

One of the functions used to describe a spatial dependence structure is referred to as a semivariance function γ(hij). It measures dissimilarity between values sampled at sites separated by a distance hij, for stationary and isotropic processes, and is associated with the covariance function by γ(hij) = C(0) – C(hij) (Uribe-Opazo et al., 2012).

In the literature, different theoretical models have been developed to define spatial dependence structure (semivariance function), and several methods have been used to estimate these models (Cressie, 2015; Monego et al., 2015; Cortés-D et al., 2016). The model estimated for semivariance function has the following parameters: range (a), partial sill (C1), nugget effect (C0), and sill (C0+ C1). Range is the longest distance between sampling sites spatially correlated within an area. Sill is the semivariance value when distance equals range, corresponding to the variance of a georeferenced variable (Kestring et al., 2015).

Semivariance function has a minimum distance (hmin), within which its semivariance value (γ(hij)) is calculated. When it is high, the phenomenon has high variability over a small range of distances. In these situations, the semivariance function has a parameter called nugget effect (Peng & Wu, 2014; Cressie, 2015; Genton & Kleiber, 2015. Small-scale variability may be associated with features of the studied process and/ or measurement errors. It may also occur due to data heterogeneity or sampling size and scheme (BOSSEW et al., 2014; Seidel & Oliveira, 2014; Lark & Marchant, 2018; Wadoux et al., 2019).

Vallejos & Osorio (2014), Cressie (2015), Bassani et al. (2018) suggested a relationship between nugget effect and (a) features related to spatial prediction, (b) a structure that describes spatial dependence and (c) sample design. Chipeta et al. (2016) described that when nugget effect is not zero, a sampling design with closer pairs of sampling points should be considered to improve geostatistical model estimation and spatial prediction.

However, there is a gap about the implications of nugget effect and sample design on geostatistical model estimation and spatial prediction simultaneously (expressed by thematic maps). This is especially true if the evaluation considers the simultaneous influence of nugget effect and sampling design.

Thus, the goals of this study were: 1) to evaluate the influence of nugget effect on geostatistical model estimation and estimation at non-sampled sites of a georeferenced variable, using Monte Carlo simulated data and considering different sample designs (random, systematic, and lattice plus close pairs); 2) to analyze the spatial variability of the following soil chemical properties: carbon, calcium, magnesium, and pH, considering the sampling data of an area whose sampling design was a lattice plus close pairs.


Simulated datasets originated from multivariate stochastic processes, assuming stationary variables, with an isotropic Gaussian linear model (Uribe-Opazo et al., 2012). Sampling designs with 100 sampling points arranged in a regular area with a maximum limit of coordinates equal to 100 m were considered. Three sample designs were simulated: systematic (lattice) 10 × 10 grid, random and lattice 19 × 19 grid added by 19 randomly chosen nearby points (lattice plus close pairs). Sample size and the latter sampling design were chosen due to the design used in the study area.

Twelve trials were considered for each sample design, totaling 36 sets of simulations. For each set of simulation were considered 100 simulations, totaling 3600 simulations for every trials. Each simulation attempt was made considering an exponential model for semivariate function, with fixed parameters for a practical range equal to 60 m and a sill of 10 m. Nugget effect was the parameter that varied between trials. Thus, 12 trials were performed considering the following nugget effect values: 0, 1, 2, 2.5, 3, 4, 5, 6, 7, 7.5, 8, and 9.

Semivariance function parameters were estimated for each simulation by maximum likelihood and respective asymptotic standard errors were calculated (De Bastiani et al., 2018). Moreover, the following measures that quantify the spatial dependence intensity were estimated: relative nugget effect (RNE) (Equation 1) and spatial dependence index (SDI) (Equation 2).

RNE=(C0C0+C1)100 (1)
SDI=MF.(C1C0+C1).(aq.MD).100 (2)

in which: MF is the model factor and reflects the spatial dependence strength of each model (for exponential, spherical, and gaussian models, MF values are 0.317, 0.375, and 0.504 respectively); C0 is the nugget effect; C0 + C1 is the sill; a is the practical range; and q.MD is the fraction (q) of the maximum distance (MD) between sample points. In this study, we assumed q as 50% of the maximum distance. RNE value was proposed by Cambardella et al. (1994), which describes the proportion of sill represented by the nugget effect. Yet, SDI was proposed by Seidel & Oliveira (2014) and includes a greater amount of information in its calculation compared to the RNE (nugget effect, sill, practical range, and semivariance function model).

Scale of values of RNE and SDI indexes are different from each other, as well as their interpretation. While RNE ranges from 0 to 100% for all semivariance function models, SDI depends on semivariance function model. According to Seidel & Oliveira (2014), for exponential, spherical, and Gaussian functions, SDI varies respectively from 0 to 31.7%, from 0 to 37.5%, and from 0 to 50.4%. Regardless of the model describing semivariance function, the closer the SDI to its maximum value, the greater the spatial dependence of the variable under study.

Considering the estimated geostatistical models, the values of the georeferenced variables at unsampled sites were estimated using the kriging method. Spatial estimation mean variance or kriging variance ( σ02¯ ) was calculated since it is an estimation efficiency measure, wherein the lower its value, the better the spatial estimation efficiency (Cressie, 2015; Kleijnen & Mehdad, 2016).

Simulations using nugget effect values from 0 to 8 were compared to those with nugget effect equal to 9. This comparison encompassed the following measures: the sum of squared difference between spatial estimates and accuracy measure, known as overall accuracy (OA) (De Bastiani et al., 2012). Comparison methods were chosen to compare georeferenced spatial estimates with those generated with a lower degree of spatial dependence.

We also analyzed the spatial variability of a set of real data from a commercial grain area of 167.35 ha. The area is located in the city of Cascavel, western Paraná State, Brazil. The geographical coordinates are the following: 24.95° S latitude, 53.37° W longitude, and 650 m above sea level. Local soil is classified as Oxisol, with clayey texture and deep layers of good water storage capacity, porosity, and permeability (De Bastiani et al., 2012). Local climate is very wet and classified as mesothermal, Cfa (Köeppen), with an average annual temperature of 21°C (Kestring et al., 2015).

A lattice plus close pairs sampling was performed, with a maximum distance of 141 m between sampling sites. In some sites, random samples were taken at shorter distances: 75 and 50 m between sites, resulting in a total of 102 sampling points. All samples were georeferenced and located with the aid of a signal reception device of Global Positioning System (GPS) GEOEXPLORE 3, in the UTM spatial coordinate system.

Soybeans have been grown under no-till system since 1994. We used data from the 2010/2011 crop season, which are related to the following soil chemical properties: carbon (C – g dm−3), calcium (Ca – cmol dm−3), magnesium (Mg – cmol dm−3), and pH. The dataset of this study was acquired by routine chemical analysis, collecting a soil sample at each marked point and five subsamples of the 0.0 to 0.2 m depth range, close to the marked points, being mixed and placed in plastic bags of about 500 g, thus composing a representative sample of the portion. These samples were sent to the Laboratory of Soil Analysis of the Central Cooperative for Agricultural Research (COODETEC) for routine chemical analyses.

The best model was fit to the semivariance function for each variable under study, according to cross-validation criteria (Lu et al. 2012; Robinson et al., 2013). Asymptotic standard errors, RNE (Equation 1) and SDI (Equation 2), were calculated for each model. Moreover, using the kriging method, thematic maps of spatial variability of variables were created for the area under study. Simulated data sets and statistical and geostatistical analyses were performed by R software (R Development Core Team, 2018), using the geoR module (Ribeiro Junior & Diggle, 2001).

Results and Discussion

Simulated Data Analysis

Nugget effect estimation was the mostly influenced parameter by sample design changes in all simulations. On average, the worst results were obtained in systematic sampling, where the estimated values were more distant from the nominal value of this parameter (Table 1). Moreover, the standard error estimates of C0 were higher (Figure 1-a) compared to those of random design and lattice plus close pairs (Figure 1-b and 1-c).

TABLE 1 Descriptive statistics for estimates of nugget effect (C0). 

C0 nominal value Systematic Random Lattice Plus Close Pairs
Mean CV (%) Mean CV (%) Mean CV (%)
0 0.297 225.670 0.068 210.730 0.254 202.170
1 5.873 69.060 0.761 74.390 0.770 125.620
2 4.531 82.690 1.600 53.850 1.396 99.400
2.5 4.472 79.270 2.028 49.120 1.724 91.300
3 4.225 78.970 2.455 46.200 2.108 82.880
4 3.546 82.080 3.368 41.270 2.820 73.150
5 2.747 91.110 4.202 41.970 3.525 67.370
6 2.162 98.960 4.961 42.380 4.369 61.430
7 1.581 113.460 5.887 40.130 5.026 59.940
7.5 1.303 125.200 6.159 41.120 5.335 60.950
8 1.058 137.860 6.589 41.850 5.606 62.170
9 0.645 167.020 7.443 44.580 6.662 53.050

CV: coefficient of variation (%).

FIGURE 1 Box plot Graphs: estimates of standard deviation (or error) (SD) of nugget effect for each sampling design: (a) systematic, (b) random, and (c) lattice plus close pairs. 

Nugget effect was overestimated in simulations using lower values of this parameter, but underestimated when higher values were used. The best estimates were achieved in random design as it was one of the sampling designs. Nugget effect estimates were on average close to the nominal value, showing less variability (Table 1) and lower standard errors (Figure 1-b).

Lattice plus close pairs was considered the second best design in terms of nugget effect estimation, as it also showed on average nugget effect estimates closer to the nominal value (Table 1) and low standard error estimates (Figure 1-c). But unlike the previous one, this model showed a greater variability of estimates compared to those of the random design.

All sampling designs in this study generated similar range and sill estimates. In this sense, relevant differences in C0 estimates influenced directly RNE and inversely SDI index calculations. When considering RNE and SDI indexes and all simulations, the lowest spatial dependence was observed in systematic sampling if compared with random design and lattice plus close pairs. Estimations of RNE and SDI indexes closer to the nominal values were obtained by random design.

These results corroborate the conclusion of Kestring et al. (2015), Zhao et al. (2016), and Bussel et al. (2016), who claimed who stated about sample design and size influences on geostatistical model estimation.

Table 2 also presents a descriptive summary of average kriging variance ( σ02¯ ) for a georeferenced variable at non-sampled sites, with different nugget effect values and sampling designs. Average kriging variance results were similar for all sampling designs and increased as nugget effect was raised.

TABLE 2 Descriptive statistics for average kriging variance ( σ02¯ ) and the sum of squared difference (SSD) between spatial estimations. CV is the coefficient of variation (%). 

Nominal value of (C0) Estimated Measures Systematic Random Lattice Plus Close Pairs
Mean CV (%) Mean CV (%) Mean CV (%)
0 σ02¯ 2.593 23.490 2.900 15.300 2.647 19.960
SSD 11090 31.770 20110 43.160 20960 45.890
1 σ02¯ 3.251 25.220 3.893 15.760 3.462 21.890
SSD 9533 32.980 17560 45.540 18890 47.630
2 σ02¯ 3.941 25.270 4.786 15.860 4.288 22.340
SSD 7999 35.230 15140 48.170 16750 49.710
2.5 σ02¯ 4.304 24.990 5.208 15.870 4.694 22.180
SSD 7230 36.820 13990 49.620 15700 50.790
3 σ02¯ 4.683 24.320 5.616 15.870 5.112 21.690
SSD 6452 38.190 12890 51.170 14610 52.080
4 σ02¯ 5.434 23.090 6.339 18.790 5.898 20.750
SSD 4970 42.08 10770 54.710 12580 54.440
5 σ02¯ 6.161 22.030 6.996 21.400 6.571 22.520
SSD 3629 48.870 8778 58.950 10660 56.970
6 σ02¯ 6.771 26.010 7.656 21.270 7.041 28.610
SSD 2320 61.810 6978 63.240 8844 60.290
7 σ02¯ 7.135 31.270 7.784 31.640 7.674 28.090
SSD 1300 87.090 5326 68.010 7192 63.290
7.5 σ02¯ 7.335 32.810 7.978 33.400 7.652 33.050
SSD 905.700 109.580 4593 71.870 6373 65.920
8 σ02¯ 7.160 40.890 7.580 45.270 7.328 42.580
SSD 595.900 145.294 3925 73.030 5668 67.430
9 σ02¯ 5.541 77.450 5.384 88.200 5.384 88.180

According to Cressie (2015), georeferenced variables can be decomposed into two random terms: a second-order stationary process and a white-noise measurement process. In this case, when interpolation is done at an unsampled point, variance estimate exceeds stationary variance by the amount of white-noise measurement variance (corresponding to nugget effect variation) (Burgess & Webster, 2019). Therefore, the nugget effect and kriging variation are directly related. Average kriging variance shows how efficient a spatial estimation of unsampled sites was, thus, the smaller the spatial estimation the more efficient it is. These results (Table 2) showed that the higher the nugget effect (i.e., the lower the spatial dependence degree of a georeferenced variable), the lower the efficiency of spatial estimation by kriging.

Kriging spatial estimation results in unsampled sites for simulations with C0 between 0 and 8 were compared to those obtained for simulation with C0 equal to 9 (Table 2 and Figure 2) by the sum of squared difference (SSD) (Table 2). In all sample designs, as the nugget effect increased, SSD between spatial estimates decreased.

This result indicates that the closer the nugget effect values are in a geostatistical model for spatial estimation, the more similar their results are in terms of estimation, regardless of the sampling design. The geostatistical model used for comparison (C0 = 9) represents a georeferenced variable with pure nugget effect, that is, without spatial dependence. Thus, the results of SSD (Table 2) showed that the stronger the spatial dependence of a georeferenced variable (lower C0 value), the greater the dissimilarity thereof with a georeferenced variable without spatial dependence, as far as spatial prediction is concerned.

Kriging equations depend on semivariance function, especially for nugget effect. Higher values of this parameter imply higher kriging weight for distant samples, which, in turn, produces a smoother thematic map (Bassani et al., 2018).

With C0 changes, the systematic design showed less sensitivity to spatial estimation (lowest SSD (Table 2), underestimating nugget effect (Table 1)).

These results evidenced the importance of accurately modelling the nugget effect, considering its association with kriging estimation and sampling design. For high nugget effect values (concerning its sill), it is therefore recommended to consider a sample design with many short-distance sites (Chipeta et al., 2016), to minimize uncertainties in semivariance function parameters and kriging estimates (Wadoux et al., 2019).

Figure 2 shows the box plot graphs for each sampling design, with similarity measure OA by comparing spatial estimations, considering estimates from data simulated using nugget effect equals to 9 (as reference for mapping) and spatial estimation from data simulated using the other values of nugget effect (as a model maps).

FIGURE 2 Box plot Graphs of overall accuracy ( OA ) for each sampling design: (a) systematic, (b) random, and (c) lattice plus close pairs. Dotted lines indicate ranges of minimum values of high similarity. 

In both random (Figure 2-b) and lattice plus close pairs (Figures 2-c) designs, the increased nugget effect in the simulations generated a high percentage of simulations with low similarity between spatial estimates (OA < 0.85, De Bastiani et al., 2012). In considering C0 = {3, 4, 5, 6, 7.5} (moderate spatial dependence; Cambardella et al., 1994) and C0 = {0, 1, 2, 2.5} (strong spatial dependence; Cambardella et al., 1994), the spatial estimates of these sampling designs had low similarity in 100% of simulations with OA values.

For systematic sampling (Figures 2-a), the findings described above are valid for all simulations considering the nugget effect values that describe the georeferenced variable as having weak spatial dependence; and most nugget effect values that consider the georeferenced variable as having moderate spatial dependence.

The systematic sampling showed the highest measurements of accuracy compared with the others, especially for C0 above 5. For C0 = 8, which belongs to the same classification as spatial dependence intensity for the data simulated with nugget effect equal to 9, simulations increased (20%) with OA ≥ 0.85, which indicates high similarity between spatial estimations carried out at non-sampled sites (De Bastiani et al., 2012).

Based on the results, the systematic design had lower sensitivity to the spatial estimation process, with changes in nugget effect value, when compared with the random sampling and lattice plus close pairs. This is due to the poor quality of nugget effect estimation in systematic sampling (Table 1 and Figure 1), which hence produced a low quality in kriging estimation.


Descriptive statistics for carbon (C), calcium (Ca), magnesium (Mg), and pH are given in Table 3. In analyzing these values, we can note that all parameters presented data with low dispersion and homogeneity. According to cross-validation criteria, the Gaussian model was the best-estimated model of the semivariance function for C, Ca, and Mg. As for pH, the best-estimated model was the exponential one. The spatial dependence radius for C, Ca, Mg, and pH were 254.90, 639.90, 685.47, and 300 m, respectively.

TABLE 3 Means and standard deviations (in brackets) of carbon, calcium, magnesium, and pH. Best-estimated model of the semivariance function and respective estimated parameter, estimation standard error (in brackets), spatial dependence index (SDI), and relative nugget effect (RNE). 

Parameter Mean Model Nugget (C0) Sill (C1) Range (a) SDI RNE
Carbon (C) 26.93 Gaussian 5.512 4.994 254.900 6.92% 52.47%
(3.25) (1.063) (1.993) (64.207)
Calcium (Ca) 5.20 Gaussian 1.432 0.5859 639.900 10.60% 70.96%
(1.37) (0.214) (0.422) (269.636)
Magnesium (Mg) 2.25 Gaussian 0.401 0.076 685.474 6.25% 84.03%
(0.69) (0.059) (0.066) (416.130)
pH 5.10 Exponential 0.099 0.053 300.000 8.26% 64.87%
(0.39) (0.027) (0.033) (264.550)

All parameters showed nugget effect values higher than sill ones. If we associate this with simulation results, we could notice that these high nugget effect values for all parameters generate loss of efficiency in spatial estimation by kriging. However, as reported by Webster & Oliver (2007), such values should not be disregarded because the model must be correctly estimated and incorporate the estimated nugget effect value. Furthermore, in analyzing spatial dependence degree, the parameters C, Ca, and pH are classified as with moderate (25% < RNE ≤ 75%) while Mg as with weak (RNE > 75%) (Cambardella et al., 1994).

Based on quartiles, the same analysis was performed for SDI since it makes up the strictest index to evaluate degree of spatial dependence. This approach was used to rank C, Ca, and Mg within a weak spatial dependence and pH within a moderate degree (Seidel & Oliveira, 2014). When comparing the two indices, we found a difference for C and Ca, which is related to the range value and the estimated model.

Figure 3 displays the thematic maps of estimates for each factor. We observed that the thematic maps for carb2on (Figure 3-a) and pH (Figure 3-d) had less smoothing as for distribution of estimates in the area, whereas calcium (Figure 3-b) and magnesium (Figure 3-c) had more smoothed maps.

FIGURE 3 Thematic maps of: (a) carbon (C), (b) calcium (Ca), (c) magnesium (Mg), and (d) pH. 

Calcium and magnesium showed weak spatial dependence and nugget effect values higher than those of sill (Table 3). Such finding emphasizes the influence of nugget effect on the spatial estimation of georeferenced parameters in non-sampled sites, and as a result, a greater thematic-map smoothing for high nugget effect values.

According to Webster & Oliver (2007) and Hofmann et al. (2010), increasing values of nugget effect provide an improved distribution of weights in spatial estimation, which hence generates smoother thematic maps. The nugget effect can be decreased by shortening gaps between samples, i.e., increasing sample density (Kestring et al., 2015); however, it is usually unfeasible for soil properties due to the costly requirements.

The spatial variability map for carbon (Figure 3-a) showed that lower carbon contents are located midwestern, while high levels are mainly within the northern region. The thematic maps for calcium (Figure 3-b) and pH (Figure 3-d) presented reduced levels mainly within the southern region. Figure (3-c) presents the thematic map for magnesium, in which can be seen that the northern and southern regions have the lowest magnesium contents of the studied area.


The behavior of near-origin semivariograms described by the nugget effect strongly affects spatial estimation by kriging of georeferenced parameters at unsampled sites. Nugget effect has negative influence on stability of this type of modeling, that is, the higher the nugget effect, the lower the kriging efficiency of spatial prediction. Nugget effect was also directly related to smoothed kriging estimates.

Systematic sampling exhibited the least accurate nugget effect estimation, the worst efficiency of spatial estimation, and the lowest degree of sensitivity to spatial changes as the nugget effect changes. Given the changes in nugget effect values, the best parameter estimates of the geostatistical model and estimates at unsampled sites occurred with the use of random design, followed by the lattice plus close pairs.

High nugget effect value was observed for all soil chemical properties. Low carbon contents were found in the midwestern region, while in the northern they were high. Calcium and pH presented the lowest values in the southern region, and the lowest magnesium levels were observed in the northern and southern regions.


The authors are grateful for the partial financial support by the Araucária Foundation of Paraná State (FA - Fundação Araucária), the Coordination of Higher-Education Personnel Improvement (CAPES – Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) - Finance Code n° 001, and the National Council for Scientific and Technological Development (CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico).


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Received: June 16, 2016; Accepted: October 01, 2019

*Corresponding author. Western Paraná State University/ Cascavel - PR, Brasil. E-mail:

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