Soil Science Society of America Journal 67:919-927 (2003)
© 2003 Soil Science Society of America
DIVISION S-8—NUTRIENT MANAGEMENT & SOIL & PLANT ANALYSIS
Improved Prediction and Mapping of Soil Copper by Kriging with Auxiliary Data for Cation-Exchange Capacity
J. Wua,
W. A. Norvell*,b,
D. G. Hopkinsc,
D. B. Smithd,
M. G. Ulmere and
R. M. Welchb
a Dep. Crop & Soil Sci., Cornell Univ., Ithaca, NY 14853
b U.S. Plant, Soil & Nutrition Lab., USDA-ARS, Tower Rd., Ithaca NY 14853
c Dep. Soil Sci., North Dakota State Univ., Fargo, ND 58105
d U.S. Geol. Survey, Denver, CO 80225
e USDA-NRCS, Bismarck, ND 58501
* Corresponding author (wan1{at}cornell.edu)
 |
ABSTRACT
|
|---|
Measurements of Cu or other trace elements in soils are rarely available in sufficient abundance to permit accurate mapping of large areas. In contrast, information is more widely available for major soil characteristics, such as cation-exchange capacity (CEC). Using data for soils of northern North Dakota, we compared four geostatistical methods as predictors of soil Cu: ordinary kriging (OK), ordinary kriging combined with regression (OKR), ordinary cokriging (OCK), and standardized ordinary cokriging (SCK). Ordinary kriging utilized data for soil Cu only, whereas the other three methods made use of CEC, which served as a secondary variable to improve the prediction and mapping of soil Cu. Quantitative predictions of soil Cu were tested by partitioning 619 sites of Cu into a training set of 310 sites, which was used to build models, and a testing set of 309 sites, which was reserved to test predictions derived from the training set. All three of the methods utilizing CEC data improved predictions substantially in comparison to OK. A larger data set containing data for soil Cu, CEC, or both was compiled from several comparable sources to prepare maps of Cu in soils of the 18 counties of northern North Dakota. These maps, based on data for 1002 sites within northern North Dakota, were quite similar for the three kriging methods which used data for both Cu and CEC, but different from the map derived from OK. Thus, the differences among the maps for soil Cu were consistent with our conclusion that the prediction of soil Cu was substantially improved by the use of CEC as an auxiliary variable.
Abbreviations: CEC, cation-exchange capacity • CECd, (geographically) detrended CEC • Cu, soil Cu • Cud, (geographically) detrended Cu • MAD, mean absolute difference • Max, maximum • Min, minimum • OC, organic C • OCK, ordinary cokriging • OK, ordinary kriging • OKR, ordinary kriging combined with regression • SCK, standardized ordinary cokriging • SD, standard deviation
|
INTRODUCTION
|
|---|
MINOR ELEMENTS are present naturally in all soils in variable but typically small amounts. Understanding the distribution of these elements in soils would be advantageous for agronomic and environmental reasons. Some, such as Cu, are essential micronutrients, required in small amounts by plants or animals for normal nutrition and health (Welch, 1995). Soils providing inadequate amounts of micronutrients to crops are widespread, but their distribution is not well understood (Sillanpää, 1990; Welch et al., 1991). On the other hand, many minor elements can also be detrimental to plants or animals when present in the soil at elevated concentrations. Efforts to document or evaluate soil contamination are often hampered by a lack of information about normal or background levels of elements in soils.
Information concerning the distribution of minor elements in soils is relatively sparse compared with the abundance of information about major soil characteristics. This disparity seems likely to persist, because of the costs of sampling and analysis, difficulties in comparing data obtained by diverse methods, and the lack of a compelling need to collect such data over large areas. Data for minor elements in soils are most often obtained from low density geochemical surveys (Shacklette and Boerngen, 1984; Garrett, 1994), which can provide relatively little information about small scale variability, or alternatively, the data are obtained from intensive studies of small areas that provide detailed information about localized spatial relationships (Atteia et al., 1994; Wu et al., 2002), but not about the distribution of elements over wider areas. In contrast to the limited availability of information on the distribution of minor elements, information on macroelements or major soil characteristics, such as pH, CEC, texture, or organic C (OC), is much more readily available. For example, many states have detailed records from decades of soil testing for plant nutrients, soil pH, or salinity; and, the National Cooperative Soil Survey has analyzed more than 20 000 pedons of U.S. soils for major soil characteristics (National Soil Survey Center, 2002).
Knowledge of the spatial distribution of minor elements in soils would be greatly enhanced by geostatistical methods that could draw useful inferences from the more-readily available data for major soil characteristics. The potential benefits from such an approach became obvious to us during efforts to summarize data from our ongoing study of trace elements in soils of northern North Dakota, USA. Preliminary evaluation of data for total soil Cu in our study showed that it was reasonably well related to soil CEC (r = 0.66***, n = 619). Others have noted similarly good correlations (Holmgren et al., 1993; Kabata-Pendias and Pendias, 2001). Such a relationship between Cu and CEC is reasonable, particularly within a limited range of parent materials, because CEC is related to the amounts of clay-size minerals and soil organic matter. The existence of these relationships suggests that auxiliary data for CEC should be helpful in mapping Cu more accurately.
A variety of geostatistical methods are available to describe spatially dependent variation and to estimate variables at unsampled locations (Warrick et al., 1986). Ordinary kriging is the anchoring algorithm for geostatistical interpolation (Deutsch and Journel, 1998, p. 66), and this method of interpolation combines many of the desirable features of other approaches (Isaaks and Srivastava, 1989, p. 318–321). Ordinary kriging interpolates effectively among values of a single variable at unsampled locations, but OK does not directly utilize auxiliary information about the distribution of other variables.
Conventional OCK uses the existence of spatial covariability within a variable and between variables to improve the estimations of a primary variable (Journel and Huijbregts, 1978, p. 324; Isaaks and Srivastava, 1989, p. 399). For example, OCK has been used to improve estimates of water content from data on the distribution of soil texture (Vauclin et al., 1983), and also from data on surface temperature or sand content (Yates and Warrick, 1987). Similarly, Mulla (1988) used OCK and densely measured data on the distribution of soil surface temperature to help predict water, clay, and sand contents. Electrical conductivity data were used by Zhang et al. (1992) to help predict concentrations of NO-3 and Ca2+, and by Vaughan et al. (1995) to predict salinity.
A modified approach to OCK, called standard ordinary cokriging by Deutsch and Journel (1998)(p. 74) and Goovaerts (1998), was proposed by Isaaks and Srivastava (1989)(p. 409) as a means to improve estimates of the primary variable by reducing restrictions on the influence of the secondary variable. Unlike OCK, which uses separate unbiasedness constraints on the weights of the primary and secondary variables, SCK uses only a single joint constraint on their weights, thereby increasing the influence of the secondary variable on interpolated estimates of the primary variable.
Another approach for incorporating auxiliary information is to utilize statistical regression to expand the set of data before kriging (Ahmed and De Marsily, 1987), a method called ordinary kriging combined with regression. Regression analysis establishes a relationship between a primary variable and a secondary or auxiliary variable, and then this relationship is used to estimate the primary variable at locations where only the secondary variable is known. After this enlargement of the data set, OK is used to estimate values of the primary variable at unsampled locations. This approach was used by Knotter et al. (1995) and Juang and Lee (1998) who reported that OKR produced better estimates than OCK, as well as being easier to implement.
Our study has three objectives: (i) to quantitatively test predictions from four methods of kriging (OK, OKR, OCK, and SCK) using a set of data for Cu and CEC that may be partitioned into training sites and testing sites, (ii) to demonstrate that prediction and mapping of soil Cu can be improved by incorporating information on the distribution of CEC, and, (iii) to use the most successful of these approaches to prepare a map of Cu in soils of the northern 18 counties of North Dakota.
|
MATERIALS AND METHODS
|
|---|
Data for Quantitative Testing of Kriging Methods
The main source of data for this paper is a concurrent study of the geographical distribution of micronutrients and trace elements in northern North Dakota (Norvell, W.A., J. Wu, D.B. Smith, D.G. Hopkins, M. Ulmer, and R.M. Welch, unpublished data, 2002). This data set characterizes soils of northern North Dakota, an agricultural region that produces the majority of durum wheat (Triticum durum L.) grown in the USA. Study of these soils was initiated in response to concerns about the accumulation of trace elements in durum wheat and other crops. The soil data from this region of 18 counties (Fig. 1)
includes total Cu and CEC at 625 sites. In brief, soil sampling was conducted according to a stratified randomized design (Petersen and Calvin, 1986) that placed a site in two out of every four townships (i.e., a site density of 1 site per 200 km2). Within a selected township, the site was located in a randomly selected quarter section on the dominant agriculturally suited soil. To evaluate short-range spatial variation, two or three sites per county were selected as starting points for additional sampling at intervals of about 500, 1000, 1500, and 2500 m. Surface soils (0–15 cm) and subsurface soils (30–50 cm) were collected.
View larger version (34K):
[in this window]
[in a new window]
|
Fig. 1. Upper panel: Distribution of 1088 sites with measured concentrations of soil Cu or CEC, used in preparing maps of soil Cu in the 18 counties of northern North Dakota. Lower panel: Distribution of 619 sites in the training or testing data sets, used for quantitative comparison of kriging methods.
|
|
Soil samples were air-dried, crushed with wooden or ceramic utensils, and passed through a 2-mm stainless steel sieve. The concentration of soil Cu was measured by inductively coupled Argon-plasma-emission-spectrometry, following sample decomposition in a mixture of HF-HCl-HNO3 (Briggs, 1990). The detection limit for Cu by this method is 1 mg kg-1. Cation-exchange capacity was measured according to Chapman (1965) and expressed on an air-dried soil basis (cmolc kg-1).
Six hundred nineteen of the 625 sites with complete data for soil Cu and CEC were selected for quantitative comparison and testing of kriging methods. Six samples from a small area of atypically trace element-rich soils were excluded as being representative of a separate population of significantly different geochemical and spatial characteristics. The 619 sites were partitioned into two sets, using a stratified and randomized approach, to create both a training set and a testing set. To provide stratification, all sites were first sorted by their site-numbers, which had been assigned generally according to their geographic locations from west to east and from north to south within a county. Then, for every group of four sites, two were randomly selected as training sites and the other two as testing sites. This process partitioned the full set of 619 sites into a set of 310 for training and a set of 309 for testing. The overlapping geographic distributions of the training and test sites are shown in Fig. 1.
Data for Mapping of Copper in Northern North Dakota
Five sources of data were combined for the preparation of soil Cu maps for northern North Dakota (Fig. 1). The largest of these subsets was from our current study of trace elements in this region (Norvell et al., unpublished data, 2002), which included 625 sites with data for Cu and CEC, and another 36 sites with data for Cu or CEC, but not both. Additional data for Cu or CEC were available for another 341 sites from four other sources. The first of these auxiliary sources was data compiled by the (USDA) Natural Resources Conservation Service in Bismark, ND for soil characterization pedons, collected as part of the National Cooperative Soil Survey (National Soil Survey Center, 2002). A total of 216 sites with soils suited for farming were located within our study area. Among these, 162 had data available for CEC, while 54 had data for clay content and OC, but not for CEC. For the latter sites, the CEC values were estimated from the concentrations (g kg-1) of clay and OC, using a regression Eq. [1] based on the 128 sites, which had complete data for all three characteristics:
 | [1] |
A second subset of auxiliary data was extracted from the trace metal survey by Holmgren et al. (1993), which included 13 sites with data for Cu and CEC in the study area. (The CEC data from the National Soil Survey Center [2002] and from Holmgren et al. [1993] were available on an oven-dry soil basis. For compatibility with our data for air-dry soils, these auxiliary CEC data were adjusted to an air-dry soil basis using a relationship, established with our very similar soils, showing that the air dry moisture content was closely related to CEC, i.e., 1.2 g H2O cmol-1c). A third subset of auxiliary data consisted of CEC values (air-dry soil basis) for 51 soils collected by Franzen (1999) as part of a study of plant-available micronutrients. The fourth source of auxiliary data was a regional geochemical study of the Canadian prairies and adjoining areas of the USA (Garrett, 1994; R.G. Garrett, personal communication, 1997). This study included data for Cu at 61 sites from within the study area and at 86 sites from nearby locations outside of the study area. The latter were included to improve map accuracy near boundaries, but are excluded from summaries of data for the northern counties of North Dakota.
In summary, the total number of sites with useful data for mapping within the 18 counties was 1002. This included 638 locations with data for Cu and CEC, 91 locations with data for Cu only, and 273 sites with data for CEC only. Summaries of data for the 729 sites with known Cu concentrations and the 911 sites with CEC are included in Table 1.
View this table:
[in this window]
[in a new window]
|
Table 1. Summary statistics for soil Cu and cation-exchange capacity (CEC) in data sets used in comparing kriging methods or in preparing maps of soil Cu in northern North Dakota.
|
|
Geostatistical Methods
For performing geostatistical inferences, the spatial variability of attributes was modeled with the aide of semivariograms or cross-semivariograms (Journel and Huijbregts, 1978). A spherical model with nugget variance was used to fit the variograms in this study. This widely used model provided a continuous quantitative function to describe the spatial variability for use in interpolation algorithms (Journel and Huijbregts, 1978, p. 83 and 153):
 | [2] |
where C0, nugget variance; C, structural variance; and A, range.
For geostatistical inferences involving kriging, the variables are assumed to meet the requirements of second-order stationarity. Because this assumption was violated by the presence of significant spatial trend in our study area (see Fig. 2
and the discussion in Results below), we modeled the trend with a second-order polynomial function:
 | [3] |
where Z is the variable being described, X and Y are coordinates of sampling points; and a, b, c, d, e, and f are the least-squares fitted coefficients. After the modeled trend is subtracted from the original data, the residual or detrended data are more suited to geostatistical interpolation.
View larger version (14K):
[in this window]
[in a new window]
|
Fig. 2. Experimental omnidirectional semivariograms of Cu and cation-exchange capacity (CEC) prepared from original (raw) data at 619 sites.
|
|
The OK estimator can be expressed as:
 | [4] |
where z*(u0) is the value of Z to be estimated at location u0; 
(ui) is the weight associated with the measured value of Z at location ui; n is the number of the measured values used in estimation.
The OKR approach uses the same estimator as Eq. [4]. But, it first builds a regression model between a primary variable and a secondary or auxiliary variable, then uses this model to predict values of the primary variable for locations where only the values of the secondary variable are available. Finally, sites with either measured or predicted values of the primary variable are treated as equivalent in the interpolation process (Ahmed and De Marsily, 1987).
With one secondary variable, the OCK estimator for the primary variable is written:
 | [5] |
where Z1 is the primary variable and Z2 is the secondary variable; z1*(u0) is the value of Z1 to be estimated at location u0; Z1(ui) is the weight associated with the measured value of Z1 at location ui; Z2(uj) is the weight associated with the measured value of Z2 at location uj; n1 is the neighborhood of Z1 and n2 is the neighborhood of Z2 used in estimation.
The SCK estimator, using one secondary variable is similar to OCK, except that SCK rescales Z2 to the same mean as Z1, and then uses only one nonbias condition instead of two:
 | [6] |
where mz1 is the mean of variable Z1 and mz2 is the mean of variable Z2; and other symbols are the same as in Eq. [5]. The single unbiasedness constraint on Eq. [6] gives more weight to the secondary variable in SCK than in OCK.
Thorough descriptions and explanations of geostatistical methods can be found in books such as Journel and Huijbregts (1978), Isaaks and Srivastava (1989), or Goovaerts (1997).
All kriging and cokriging inferences were made using GSLIB (Deutsch and Journel, 1998). A super block search strategy was employed for all interpolations. This search strategy partitions the data into a grid network superimposed on the area being considered. Then, the search for neighboring sites to any location is limited to sites falling in nearby super blocks. To maximize the use of the spatially relevant information, we chose a search radius of double the range distance as determined from semivariograms. Within this radius, we set the maximum number of neighbor points at 36 for both primary and secondary variables.
Comparison of Kriging Methods and Maps of Soil Copper
For quantitative comparison of kriging approaches, the data for Cu from the 309 sites in the testing set were reserved for testing the accuracy of predictions. Data for Cu at the 310 sites of the training set, plus CEC data for all 619 sites were used in geostatistical inferences leading to prediction of Cu at the sites reserved for testing. Several statistical criteria were used to quantitatively compare known concentrations of Cu with the predictions based on OK, OKR, OCK, or SCK. In addition to simple descriptive statistics, we computed the mean absolute difference (MAD) and the coefficient of determination (r2p):
 | [7] |
 | [8] |
where z(ui) is the measured value of Z at location ui; and z*(ui) is the estimated value at the same location. Errors in prediction of testing site data were shown also in scatter plots.
Maps produced by the four kriging methods from the full set of data for Cu and CEC in northern North Dakota were compared by linear regression and by visual inspection of differences. Linear regression was used to estimate the extent to which the interpolated values expressed in one map were related to values expressed in another (ESRI Inc., 1994, p. 363). Maps of differences between the SCK map for Cu and those prepared by OK, OKR, and OCK were created to depict the distribution and the degree of differences. These comparisons provide a good measure of agreement between maps, but, obviously, cannot directly assess the accuracy of predicted values of Cu, as was accomplished by the quantitative comparison of training and testing data.
|
RESULTS
|
|---|
Quantitative Comparison of Soil Copper Predictions by Four Kriging Methods
Table 1 shows that the Cu concentrations varied widely from a low of 3 to a high of 37 mg kg-1 among the 619 sites selected to compare the four kriging methods. Soil CEC values varied widely also, from 3.9 to 47.2 cmolc kg-1. These data were distributed in an approximately symmetrical manner, characterized by small skewness and a median close to the mean.
Variograms computed from the raw data for Cu and CEC indicated the existence of significant geographic trend (Fig. 2). Their semivariances increased rapidly with lag distance out to about 20 km, continued to rise out to a lag distance of about 240 km, and then declined erratically. Neither semivariogram displayed a consistent sill nor met satisfactorily the requirements for second-order stationarity. Therefore, we used Eq. [3] to describe the general trends for Cu and CEC, providing the following relationships:
 | [9] |
 | [10] |
where X and Y are relative coordinates (X + 611 km and Y - 2694 km) in kilometers in Albers projection; Cu is in milligrams per kilogram (mg kg-1) of soil and CEC is in centimole of charge per kilogram (cmolc kg-1) of soil.
These fitted equations were used to remove trends from the original data. The detrended data for the 619 sites had mean values near zero, and very small skewness (Table 2). The detrended data for the subsets of 310 training sites and 309 testing sites had statistical characteristics generally similar to the complete set of 619 sites. For example, they were almost identical for SD, maximum (Max), and minimum (Min) values; and, the skewness and kurtosis were changed little, even though the numbers of sites were reduced by about one-half.
View this table:
[in this window]
[in a new window]
|
Table 2. Summary statistics of detrended data for total Cu and CEC in soils of the full set, training subset, and testing subset used in comparing four kriging methods.
|
|
Semivariograms and cross-semivariograms built from the detrended data were plotted and fitted with the spherical model (Fig. 3) . Variograms for Cu and CEC in the training set (310 sites) were similar to those for the complete data set (619 sites), but the former were slightly more variable as would be expected because the number of observation pairs available within any lag distance was reduced by about one-half. A range distance of 20 km was found for detrended Cu (Cud), detrended CEC (CECd), or Cud x CECd in all variograms. The sill values for variograms of detrended data for the full set (619) and training set (310) were similar for Cud, CECd, or Cud x CECd, respectively, while the corresponding nugget variances were close.
View larger version (34K):
[in this window]
[in a new window]
|
Fig. 3. Experimental omnidirectional semivariograms and cross-semivariograms prepared from detrended data for Cu and CEC at 619 sites of the full data set or at 310 sites of the training set. The fitted curves and parameters (C0, nugget variance; C, structural variance; and A, range) of the spherical model (Eq. [3]) were used in quantitative comparison of kriging methods.
|
|
Estimates of values by OK, OCK, and SCK at the 309 testing locations were calculated using the fitted model parameters for the training set. Because these comparisons involved point data only, the appropriate model for Cud as estimated by OKR at the testing sites was provided simply by the regression equation relating Cud to CECd at the 310 training sites:
 | [11] |
Summary statistics for detrended values of Cu estimated by the four methods for the 309 testing sites are tabulated in Table 3, along with statistics for measured values of Cud, which were excluded from predictions based on the training set and reserved solely to test predictions. Of the four approaches, the results for OK were substantially less successful. Compared with the measured data for Cud, the predictions of OK produced the largest MAD (3.01 mg kg-1), and the largest differences in the mean, median, SD, Min, and Max. Predictions of Cud by SCK were the closest to the testing set data for SD, Max, skewness, and kurtosis. In addition, SCK produced the lowest MAD (1.84 mg kg-1), differing significantly in this statistic from OK or OKR. Results from OCK were almost as good as those from SCK. Ordinary kriging combined with regression was slightly less successful than SCK or OCK, but this approach was still highly effective, and the estimated mean and median by OKR were actually the closest to those in the test set data.
View this table:
[in this window]
[in a new window]
|
Table 3. Comparison of detrended concentrations of soil Cu (mg kg-1) in the testing data set with predicted values based on four kriging methods.
|
|
Detailed comparisons of measured values of Cu in the test set with estimates from the four kriging methods are shown in Fig. 4 , wherein the contributions of general trend from Eq. [9] have been added to Cud so that actual concentrations of soil Cu are portrayed. For OK, OKR, OCK, and SCK, the coefficients of determination for predictive success, r2p, were 0.42, 0.74, 0.76, and 0.77, respectively, showing the same relative differences among kriging methods as discussed above for Cud. Deviations of predicted data from measured data of the test set are evident in Fig. 4. Results for OK were clearly much more scattered and showed bias toward underestimating large values and overestimating small values. For the three methods utilizing CEC data, OKR, OCK, and SCK, the predicted values were more closely associated with measured values along the 1:1 line.
View larger version (16K):
[in this window]
[in a new window]
|
Fig. 4. Comparison of concentrations of soil Cu at the 309 sites of the test data set with the predictions of ordinary kriging (OK), ordinary kriging combined with regression (OKR), ordinary cokriging (OCK), and standardized ordinary cokriging (SCK). The coefficient of determination, r2p, characterizes the fit of the data to the predicted relationship derived from kriging, represented here by the dashed line (1:1).
|
|
As a confirmation of the analyses presented above, we reversed the testing and training sets of data, and then repeated the geostatistical inferences described above. The results were almost identical (data not shown). In addition, we partitioned the whole data set into a training set containing 3/4 of the samples (465 sites) and a testing set containing the remaining samples (154 sites). Again, all geostatistical inferences were repeated, and again the results were almost identical (data not shown). These results indicate that for this data set the comparisons among kriging methods were robust and reproducible, that is, not accidents of partitioning.
Comparisons of Soil Copper Maps for Northern North Dakota
Maps of soil Cu were created by the four kriging approaches using all available data for locations within, or near, the 18 counties of northern North Dakota (815 sites for Cu and 911 sites for CEC). Trend surfaces were removed by Eq. [9] and [10] before regression or geostatistical interpolations, then added back to detrended values from kriging or cokriging to produce maps of the distribution of Cu concentrations in soil.
Based on the results from the first part of this study, SCK was expected to produce a map that was as good or perhaps better than the other kriging approaches. This map is shown in some detail in the upper panel of Fig. 5
. Maps produced by the other three kriging approaches differed in varying degrees, as revealed by regression analysis and the distribution of deviations from the map by SCK. The map from OCK was quite similar, accounting for 99.6% of the variation in the map by SCK, with a slope of 0.997 and an intercept of -0.01. The map produced by OKR was also similar, accounting for 97.8% of variability of the map from SCK, with a slope of 0.962 and an intercept of 0.59. The map from OK was less similar, accounting for 91.7% of variation in soil Cu in the map from SCK, with a slope of 0.928 and an intercept of 1.12.
View larger version (61K):
[in this window]
[in a new window]
|
Fig. 5. Upper panel: Soil Cu map prepared by standardized ordinary cokriging (SCK) from data for soil Cu at 815 sites and data for cation-exchange capacity (CEC) at 911 sites in northern North Dakota. Lower panel: Maps of the absolute (ABS) difference in predicted soil Cu concentrations between the map from SCK and those from ordinary kriging (SCK–OK), ordinary kriging combined with regression (SCK–OKR), and ordinary cokriging (SCK–OCK).
|
|
Another sensitive comparison among maps is provided by the absolute difference between the mapped values from SCK and those from OCK, OKR, or OK. The lower panel of Fig. 5 displays the extent and spatial distribution of these differences as separate maps of the same area. Differences between the SCK and OCK maps were the smallest, exceeding 0.5mg kg-1 in only a few areas, and exceeding 2 mg kg-1 in only one area. Differences between the SCK and OKR maps were more evident, exceeding 2 mg kg-1 in small but scattered areas across the region, and exceeding 4 mg kg-1 in a few areas. Differences between the SCK and OK maps were larger and more extensive, exceeding 6 mg kg-1 in numerous areas.
|
DISCUSSION
|
|---|
Ordinary kriging is the anchoring algorithm of geostatistics (Deutsch and Journel, 1998), but it does not make use of auxiliary information about spatially correlated variables. The potential importance of auxiliary information is well illustrated by our predictions of soil Cu at the 309 sites of the testing data set, utilizing data for soil Cu in the training set and CEC data for all sites (Table 3, Fig. 4). The incorporation of CEC data improved the agreement between predicted and measured concentrations of soil Cu, from an r2p of about 0.42 for OK to 
0.74 for OKR, OCK, or SCK. This large improvement in estimates of soil Cu was a benefit arising from the strong spatial covariability of Cu and CEC. The improvement was strengthened by the combination of relatively low sampling density (about 1 site per 200 km2) with a relatively short-range distance for spatial dependency (about 20 km, Fig. 3). For data with these characteristics, there is great potential to improve accuracy of maps by using auxiliary data for secondary variables, especially so in any area where data for the primary variable is sparse relative to the secondary variable.
Among the three methods that utilized CEC information to predict soil Cu, SCK was the most successful, although only to a very small degree. Standardized ordinary cokriging produced the highest r2p between estimated and measured values, and the smallest MAD. The results from OCK were almost identical. Improvements in predictions by SCK over OCK have been attributed by Isaaks and Srivastava (1989)(p. 416) to the requirement in SCK for only one nonbias condition in Eq. [6], which reduces the risk of unreasonable estimates (Goovaerts, 1998). However, with our data for Cu and CEC this improvement was slight.
Ordinary kriging combined with regression is an interesting and flexible approach for using auxiliary data. With our data for soil Cu and CEC, OKR was slightly less effective in predicting Cu at the testing site locations than SCK or OCK, but the differences were very small. Possible reasons for the slightly lower effectiveness of OKR include the fact that the use of regression assumes that one relationship fits data from all sites. While such an assumption may be true in a generalized sense, regression still cannot make the most effective use of localized relationships between a primary and secondary variable. In addition, OKR should be inherently less effective in using information on the spatial dependency of an auxiliary variable. On the other hand, OKR has some significant advantages over OCK or SCK. Ordinary kriging combined with regression is much less computationally intensive, and it easily accommodates nonlinear or multivariate relationships among variables, in addition to linear relationships.
The potential advantages of OKR over OCK were not important factors in our evaluation of soil Cu, but they could certainly be important in other work. For example, we are aware of two studies that found that OKR produced better results. Knotter et al. (1995) reported that OKR produced better results than OCK in estimating the depth of soft layers in soil from auxiliary data on bulk electrical conductivity. One factor that favored OKR in their study was that the relationship between depth of soft layers and electrical conductivity was exponential, so that the linear algorithms of cokriging were less effective in interpolating among measured values. Juang and Lee (1998) also reported that OKR produced better results than OCK in predicting pollutant concentrations in a metal-contaminated paddy field of about 10 ha. Here, the effectiveness of OCK may have been limited by the low number of samples available for the development of variograms
Comparisons amongst the Cu maps produced by the four kriging methods cannot prove, in any sense, which is the more valid map of soil Cu. However, these comparisons show clearly that the maps produced by methods that utilized CEC data were very much more similar to each other than to the map produced by OK. Thus, these comparisons are fully compatible with the conclusions from our quantitative evaluation of the four kriging methods as predictors of Cu concentrations in the 309 soils of the test data set. We believe that the maps utilizing auxiliary data for CEC provide a more accurate representation of the distribution of soil Cu. We believe, too, that the map produced by SCK is preferable, albeit by a very small margin, to those derived from OCK or OKR. The map produced by SCK from all available data for Cu or CEC is shown in Fig. 5 to portray the distribution of soil Cu in the 18 counties of northern North Dakota. The distribution of soil Cu in this region is, of course, linked to its geologic history and to processes of soil genesis, but these interesting associations are beyond the scope of this paper.
Finally, the advantages that accrued from the use of auxiliary data in predicting and mapping soil Cu are very likely to occur also for other trace elements. Data for trace elements in soils are usually relatively sparse, so that correlations with related characteristics, such as CEC, OC, clay content, or pH, should improve estimates of concentrations of many trace elements. Relatively large amounts of data for major soil properties have been accumulated and tabulated over many decades. These data are readily available to improve predictions of the distribution of trace elements in soils, using appropriate methods of kriging or cokriging.
|
ACKNOWLEDGMENTS
|
|---|
We thank Drs. R.G. Garrett of Geol. Survey of Canada, R.L. Chaney of USDA-ARS, and D.W. Franzen of North Dakota State Univ. for contributing data or samples for inclusion in this study.
|
NOTES
|
|---|
Mention of proprietary product or vendors does not imply approval or recommendation by the USDA.
Received for publication February 22, 2002.
|
REFERENCES
|
|---|
- Ahmed, S., and G. De Marsily. 1987. Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity. Water Resour. Res. 23:1717–1737.
- Atteia, O., J.P. Dubois, and R. Webster. 1994. Geostatistical analysis of soil contamination in the Swiss Jura. Environ. Pollut. 86:315–327.[Medline]
- Briggs, P. 1990. Elemental analysis of geological material by inductively coupled plasma-atomic emission spectrometry. p. 83–91. In B.F. Arbogast (ed.) Quality assurance manual for the Branch of Geochemistry, U.S. Geol. Survey. Open-File Rep. 90–668. Geol. Survey, U.S. Dep. Interior.
- Chapman, H.D. 1965. Cation exchange capacity. p. 891–913. In C.A. Black et al. (ed.) Methods of soil analysis. Part 2. Agron. Monogr. No. 9. ASA, Madison, WI.
- Deutsch, C.V., and A.G. Journel. 1998. GSLIB, Geostatistical software library and user's guide. 2nd ed. Oxford University Press, New York.
- ESRI Inc. 1994. ARC/INFO grid manual. ESRI Inc., Redlands, CA.
- Franzen, D.W. 1999. North Dakota survey of soil copper, pH, zinc, and boron levels. Ext. Rep. 52. Ext. Service, North Dakota State Univ., Fargo, ND.
- Garrett, R.G. 1994. The distribution of cadmium in A horizon soils in the prairies of Canada and adjoining United States. p. 73–82. In Current research 1994-B. Geol. Survey of Canada. Ottawa, Canada.
- Goovaerts, P. 1997. Geostatistics for natural resources evaluation. Oxford University Press, New York.
- Goovaerts, P. 1998. Ordinary cokriging revisited. Math. Geol. 30:21–42.
- Holmgren, G.G.S., M.W. Meyer, R.L. Chaney, and R.B. Daniels. 1993. Cadmium, lead, zinc, copper, and nickel in agricultural soils of the United States of America. J. Environ. Qual. 22:335–348.[Abstract/Free Full Text]
- Isaaks, E.H., and R.M. Srivastava. 1989. Applied geostatistics. Oxford University Press, New York.
- Kabata-Pendias, A., and H. Pendias. 2001. Trace elements in soils and plants. 3rd ed. CRC Press, Boca Raton, FL.
- Knotter, M., D.J. Brus, and J.H. Oude Voshaar. 1995. A comparison of kriging, co-kriging and kriging combined with regression for spatial interpolation of horizon depth with censored observations. Geoderma 67:227–246.[ISI]
- Journel, A.G., and Ch. Huijbregts. 1978. Mining geostatistics. Academic Press, London.
- Juang, K.W., and D.Y. Lee. 1998. A comparison of three kriging methods using auxiliary variables in heavy-metal contaminated soils. J. Environ. Qual. 27:355–363.
- Mulla, D.J. 1988. Estimating spatial patterns in water content, matric suction, and hydraulic conductivity. Soil Sci. Soc. Am. J. 52:1547–1553.[Abstract/Free Full Text]
- National Soil Survey Center. 2002. National soil characterization data [Online]. Soil Survey Division, USDA-NRCS-NSSC, Lincoln, NE. Available at http://soils.usda.gov/soil_survey/nscd/main.htm (verified 30 Dec. 2002).
- Petersen, R.G., and L.D. Calvin. 1986. Sampling. p. 33–51. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI.
- Shacklette, H.T., and J.G. Boerngen. 1984. Element concentrations in soils and other surficial materials of the conterminous United States. U.S. Geol. Surv. Prof. Pap. 1270.
- Sillanpää, M. 1990. Micronutrient assessment at the country level: An international study. FAO Soils Bulletin 63. FAO/Finnish International Development Agency, Rome, Italy.
- Vauclin, M., S.R. Vieira, G. Vachaud, and D.R. Nielsen. 1983. The use of cokriging with limited field soil observations. Soil Sci. Soc. Am. J. 47:175–184.[Abstract/Free Full Text]
- Vaughan, P.J., S.M. Lesch, D.L. Corwin, and D.G. Cone. 1995. Water content effect on soil salinity prediction: a geostatistical study using cokriging. Soil Sci. Soc. Am. J. 59:1146–1156.[Abstract/Free Full Text]
- Warrick, A.W., D.E. Myers, and D.R. Nielsen. 1986. Geostatistical methods applied to soil science. p. 53–82. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI.
- Welch, R.M. 1995. Micronutrient nutrition of plants. Crit. Rev. Plant Sci. 14:49–82.
- Welch, R.M., W.H. Allaway, W.A. House, and J. Kubota. 1991. Geographic distribution of trace element problems. p. 31–57. In J.J. Mortvedt et al. (ed.). Micronutrients in agriculture, 2nd ed. SSSA Book Ser. No. 5. SSSA, Madison, WI.
- Wu, J., W.A. Norvell, D.G. Hopkins, and R.M. Welch. 2002. Spatial variability of grain cadmium and soil characteristics in a durum wheat field. Soil Sci. Soc. Am. J. 66:268–275.[Abstract/Free Full Text]
- Yates, S.R., and A.W. Warrick. 1987. Estimating soil water content using cokriging. Soil Sci. Soc. Am. J. 51:23–30.[Abstract/Free Full Text]
- Zhang, R., D.E. Myers, and A.W. Warrick. 1992. Estimation of the spatial distribution of soil chemicals using pseudo-cross-variograms. Soil Sci. Soc. Am. J. 56:1444–1452.[Abstract/Free Full Text]
This article has been cited by other articles:
|
|
|
|
 
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]
|
|
|
TOP
ABSTRACT
INTRODUCTION
