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a University of Kentucky, Dep. of Agronomy, N-122 Ag. Science North, Lexington, KY 40502
b Dep. of Biosyst. and Agric. Eng., University of Kentucky, 218 C.E. Barnhart, Lexington, KY 40546-0276
* Corresponding author (mueller{at}uky.edu)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Abbreviations: FULL, datasets created by combining the validation and 30.5-m grid datasets • IDW, inverse distance weighted • IDW1.2, IDW interpolation with a distance exponent of 1.2 • MSE, mean squared error • PE, prediction efficiency PEvalidation, PE determined using an independent data set • PEcross-validation, PE determined with cross-validation • 
PEvalidation, difference in PE between ordinary kriging and IDW1.2 as determined using an independent validation dataset • PEcross-validation, difference in PE between ordinary kriging and IDW1.2 as determined with cross-validation • r2semivariogram, F, the coefficient of determination for the relationship between semivariogram pairs and the semivariogram model values • RangeF, range of spatial correlation as determined with the FULL datasets • RangeS, range of spatial correlation as determined with the sample dataset RSV relative structural variability • RSVF, RSV as determined with the FULL datasets • RSVS, RSV as determined with the sample datasets • RSV, RSVF – RSVS • SSFM, Site-specific fertility management
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Two of the most commonly used interpolation methods for SSFM are IDW and kriging (Kravchenko and Bullock, 1999). The IDW procedure has been used primarily because it is simple and quick. Kriging has been used because it provides best linear unbiased estimates; however, it is more complex and time-consuming. For mathematical descriptions of these methods, see Weber and Englund (1992), Hosseini et al. (1994), Gotway et al. (1996), Nalder and Wein (1998), Kollias et al. (1999), Kravchenko and Bullock (1999), Schloeder et al. (2001), and Kravchenko (2003).
Many studies have compared IDW and kriging. In some cases, the performance of kriging was generally better than IDW (Tabios and Salas, 1985; Hosseini et al., 1994; Dalthorp et al., 1999; Kravchenko and Bullock, 1999). In other studies, IDW generally out performed kriging (Weisz et al., 1995; Nalder and Wein, 1998; Weber and Englund, 1992). Often, however, the results have been mixed (Gotway et al., 1996; Kollias et al., 1999; Schloeder et al., 2001; Mueller et al., 2001; Lapen and Hayhoe, 2003).
The relevance of these studies depends on the methods used for analyses. Some have assessed predicted and measured values using independent validation data sets to compare IDW and ordinary kriging (Tabios and Salas, 1985; Laslett et al., 1987; Wartenberg et al., 1991; Weber and Englund, 1992; Weisz et al.; 1995; Gotway et al., 1996; Mueller et al., 2001; Kravchenko, 2003). The exact number of validation points needed to produce accurate results is not known. However, five validation points in one study may have been insufficient (Tabios and Salas, 1985). Cross-validation (Isaaks and Srivastava, 1989) has been used in many cases to compare differences between kriging and IDW (Tabios and Salas, 1985; Hosseini et al., 1994; Nalder and Wein, 1998; Kravchenko and Bullock, 1999; Schloeder et al., 2001; Lapen and Hayhoe, 2003). However, some have cautioned against this procedure for selecting interpolation procedures for mapping (Isaaks and Srivastava, 1989; Cressie, 1993; Goovaerts, 1997). In one instance, it was demonstrated that the use of cross-validation could lead to erroneous conclusions regarding the relative performance of two interpolators (Isaaks and Srivastava, 1989). Unfortunately, this issue has not been extensively studied.
Few have attempted to explain the underlying factors that impact the relative performance of ordinary kriging and IDW. One study investigated the impact of scale of sampling on the relative performance of ordinary kriging and IDW (Mueller et al., 2004). The performance of kriging improved relative to IDW interpolation as sampling intensity increased. The impact of scale is important because sampling intensities may vary widely. Situations have been identified where map quality has been poor at intensities 
0.09-ha (Mueller et al., 2001). Some farmers, however, are sampling with grids as large as 4 ha. Several studies have examined the relative performance of kriging and IDW within this range (i.e., 0.09 grid size 4.0 ha) (Gotway et al., 1996; Kravchenko and Bullock, 1999; Mueller et al., 2001; Kravchenko, 2003). However, many studies have been conducted at lower (i.e., >4-ha grids; Tabios and Salas, 1985; Laslett et al., 1987; Weber and Englund, 1992; Hosseini et al., 1994; Nalder and Wein, 1998; Lapen and Hayhoe, 2003) or higher intensities (i.e., <0.09-ha grids; Dalthorp et al., 1999). Unfortunately, it may be difficult to use these studies to aid with site-specific, soil fertility management decision-making.
The degree to which spatial structure is known impacts the relative performance of IDW and kriging. One study found that by better resolving the spatial structure with additional closely spaced samples, ordinary kriging generally outperformed IDW (Mueller et al., 2001). In a simulation study (Kravchenko, 2003), IDW was compared with kriging with known (i.e., semivariogram models were determined from an exhaustive dataset) and unknown (i.e., semivariogram models were determined from the sample dataset) spatial structure. As might be expected, the performance of kriging improved relative to IDW when spatial structure was known. Given the importance of spatial structure, it may be possible to use geostatistical indices to predict the relative performance of ordinary kriging and IDW.
Despite the work done in this area, little is known about the factors that impact the relative performance of IDW and kriging at sampling intensities used for site-specific management. Therefore, we considered two objectives. The first was to describe the factors that may determine differences between the performance of IDW and ordinary kriging. The second was to explore potential ways in which these differences might be predicted from statistical properties of the data. These analyses were conducted using soil fertility data from another published study (Mueller et al., 2004). Ordinary kriging and IDW interpolation were conducted. Multiple regression analyses were conducted to relate the statistical properties of the data with the relative performance of these procedures.
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MATERIAL AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Semivariance values were calculated with SAS (SAS Institute, Cary, NC) procedure VARIOGRAM for each of the datasets. Semivariogram models were fit based on visual assessment. Parameters included the nugget, partial sill, and range of spatial correlation. The total sill is the sum of the nugget and partial sill. The range of spatial correlation refers to the practical range. For exponential models, the practical range is the lag distance at which the semivariance is equal to 95% of the total sill. The relative structural variability is the ratio of the partial to total sill. For more detailed descriptions of these parameters, see Isaaks and Srivastava (1989) and Schabenberger and Pierce (2002). A summary of the statistical parameters for the FULL dataset is presented in Mueller et al. (2004).
For each of the FULL datasets, semivariogram values for individual pairs were calculated with Variowin (Pannatier, 1996). The r2 values for the relationship between the individual pairs and the model semivariograms (r2semivariogram, F) were calculated for the FULL dataset for lags between 0 and 26 m. This index described the adequacy with which the semivariogram model described the spatial variability in the field.
Interpolated grids (1 by 1 m) were calculated with ordinary kriging and IDW (Surfer Version 8, Golden Software, Golden, CO) for the 30.5- and 61.0-m grids datasets. Validation with an independent dataset and cross-validation were used to obtain predicted and measured values. Bilinear interpolation was used to obtain the predicted values for the independent dataset. Errors associated with bilinear interpolation were very small as determined by Mueller et al. (2001)(2004). As described by Mueller et al. (2004), the predicted and measured values were used to calculate various measures of map quality including mean squared error (MSE), bias, precision, and PE.
Preliminary analyses were conducted to determine appropriate interpolation parameters for subsequent analyses. For kriging and IDW, 15 sample points were used to calculate weighted spatial averages. This value was chosen because beyond 15 points, there was little change in prediction efficiency values. For all IDW interpolation, a distance exponent value of 1.2 was used. This value maximized average prediction efficiency across locations, sampling intensities, and soil properties. For all IDW interpolations, a search radius of 125 m was used because there was little change or improvement in prediction efficiency values with larger distances. For ordinary kriging, the search radius was set to be the lag distance beyond which the semivariogram model no longer represented the spatial structure of the data.
The SAS (Cary, NC) PROC REG procedure (SELECTION = Stepwise; SLENTRY = 0.3; SLSTAY = 0.2) was used to model prediction efficiency calculated with an independent validation dataset (PEvalidation). Diagnostics for assessing multicolinearity (i.e., intercept adjusted variance-inflation factors and condition indices statistics) were calculated by using the VIF and COLLINOINT model options. The INFLUENCE option of PROC REG was used to calculate the RStudent statistic for identifying potential outliers. Many of the pH and Bph points in the Kentucky and Michigan datasets were indicated as potential outliers. Careful examination of plots of the difference between modeled and actual semivariogram values for individual pairs versus lag distance revealed spatial anomalies for pH and Bph. The behavior occurred because the electrodes did not have time to adequately equilibrate. Therefore, each pH measurement was correlated with a previous analysis. Since all but the Shelby location samples were analyses in an order that corresponded to the spatial distribution of points across the fields, the correlations in the analyses were manifested as spatial trends in the data. For these reasons, Mueller et al. (2004) removed these points before the final analyses. These outliers have also been removed in this study.
The 30.5- and 61.0-m grid data were grouped into two categories based on semivariogram analysis of the FULL dataset. The first group included observations for variables with semivariograms that reached plateaus as determined by the FULL dataset analysis. The second group included observations of variables with semivariograms that did not reach plateaus as determined by the FULL dataset analysis. For both groups, separate multiple regression analyses were performed using the (i) 30.5- and 61.0-m grids, (ii) the 30.5-m grids, and (iii) the 61.0-m data.
For each of the datasets, models were developed using two sets of candidate regressors. The first set included sample intensity and the following variables from the FULL dataset:
RSV), and The second set included sample intensity and the following variables from the sample datasets. These included
For independent variables in the first and second sets, the natural logarithms, squares, cubes, and fourth powers these variables were also used as candidate variables when possible. Exceptions included sampling intensity and PEcross-validation, S. To avoid multicolinearity, not all variables were allowed in the model. The final model included no more than one untransformed variable (e.g., rangeS) or one of its transformations (e.g., natural logarithm of rangeS, the square, cube, or fourth power of rangeS) for each variable that was considered for entry in the model.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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PEValidation statistic was the difference in prediction efficiency between ordinary kriging and IDW1.2 as determined using an independent validation dataset. Positive values indicated that ordinary kriging outperformed IDW1.2. This statistic was negative for 18 of 22 sample data sets that had been modeled with only a nugget structure. While all soil properties in this study were spatially structured according to the full dataset analysis (Mueller et al., 2004), spatial structure was not apparent for two variables at the 30.5-m grid scale and 20 observations at the 61.0-m grid scale. For these datasets, prediction efficiency values were 15% lower on average with ordinary kriging compared with IDW1.2. Clearly, of the two methods in question, IDW1.2 was a better choice. Because the variables that were not spatially structured behaved differently than all others, these observations were excluded from the multiple regression analyses.
Fundamental Factors Impacting PEValidation
The PEValidation statistic generally became more positive as RSV decreased (Table 1). Recall that the RSV was the difference between the RSV determined with the analyses of the FULL datasets (RSVF) and the RSV determined with the analyses of the sample datasets (RSVS). Because RSVS values were generally underestimates of the degree of true spatially structured, RSV values were generally positive. The t-values from these models indicated that the RSV values were in some cases the most significant of the factors examined that influenced the relative performance of IDW1.2 and ordinary kriging.
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PEvalidation and spatial structure depended on sampling intensity (Table 1). Specifically, PEValidation increased as the dispersion of the semivariogram pairs decreased (i.e., as R2semivariogram, F increased) and the rangeF of spatial autocorrelation increased at the 30.5-m sampling intensity (Model 2 in Table 1). At the 61.5-m grid scale, PEValidation increased as R2semivariogram, F and the RSVF decreased (Model 3 in Table 1). The seemingly contradictory impact of spatial structure was associated with the resolution of sample semivariograms at shorter lags. At the 30.5-m grid sample intensity, spatial structure generally was adequately represented with the sample semivariograms. This was evident by the improvement in the performance of ordinary kriging with improved spatial structure of the data. However, short-range spatial variability was generally not adequately resolved with the 61.0-m grid sample datasets. So as spatial structure improved, the model parameters as determined from the sample datasets (i.e., RSVS) became increasingly inadequate. Therefore, the performance of ordinary kriging diminished as the structure of the underlying spatial process improved at the 61.5-m grid scale.
For variables with semivariograms that did not reach a plateau according to the FULL dataset analysis, little variation in PEValidation could be explained (i.e., Models 4, 5, and 6 in Table 1). The spatial structure of the data did likely impact the relative performance of ordinary kriging and IDW1.2 for these variables. Unfortunately, the exponential model semivariogram parameters were not simply a function of the spatial structure of the data. They also depended on the modeling choices made by the authors. Specifically, there were limitless combinations of model parameters that could have adequately fitted the experimental semivariograms. Models with either small range and sill values or large range and sill values could have been adequately fitted to semivariograms exhibiting intrinsic behavior. Therefore, no clear relationship with spatial structure parameters could be defined.
These results substantiate those of Mueller et al. (2001) and Kravchenko (2003) who found that the performance of ordinary kriging was better when spatial structure was more accurately defined either with closely spaced samples (Mueller et al., 2001) or with an exhaustive dataset (Kravchenko, 2003). Clearly, in this study, RSVF, the degree and range of spatial structure and the dispersion of observations about the semivariogram model (i.e., r2semivariogram, F) were significant factors in the regression models because the relative performance of ordinary kriging and IDW was governed by the degree to which the true spatial structure was known.
PEValidation as a Function of Statistics Calculated from the Sample Datasets
The PEcross-validationS statistic was a significant factor in several of the models (Table 2). Recall that PEcross-validationS was the difference in prediction efficiency between ordinary kriging and IDW1.2 as determined with cross-validation. Cross-validation clearly was useful here for comparing interpolation procedures when used in conjunction with the rangeS or RSVS statistics. Unfortunately however, without other factors in the model, the explanatory values of the models diminished markedly across locations, soil properties, and sampling intensities (i.e., r2 = 0.18). Therefore, caution should still be used when interpreting studies that have compared IDW and kriging solely with cross validation (e.g., Tabios and Salas, 1985; Hosseini et al., 1994; Nalder and Wein, 1998; Kravchenko and Bullock, 1999; Schloeder et al., 2001; Lapen and Hayhoe, 2003).
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RSV in Model 4 (Table 1), which had a negative coefficient as described in Table 1.
Separately, cross-validation and sample spatial structure parameters explained insufficient variability for predicting the relative performance of ordinary kriging and IDW. Together, they explained 43 and 61% of the variability of PEValidation (Table 2). Plots of estimated and measured prediction efficiency values (Fig. 1) demonstrated the performance of the regression models with significant factors developed with sample dataset statistics as candidate regressors.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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ACKNOWLEDGMENTS |
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Received for publication September 2, 2003.
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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  T.-L. Liu, K.-W. Juang, and D.-Y. Lee Interpolating Soil Properties Using Kriging Combined with Categorical Information of Soil Maps Soil Sci. Soc. Am. J., May 23, 2006; 70(4): 1200 - 1209. [Abstract] [Full Text] [PDF]
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