Estimation of the Spatial Distribution of Soil Chemicals Using Pseudo-Cross-Variograms

In using cokriging to study soil spatial variability, a key step is to determine cross-variograms. A recently developed approach was utilized to compute pseudo-cross-variograms, from which cross-variograms can be formulated. The approach does not require a large number of common locations where data are available for all variables used in the cokriging modeling and estimation processes. In this study, with only one-thirteenth of the original data for NO3 and Ca, valid cross-variograms, each with the electrical conductivity (EC), were obtained by using pseudo-cross-variograms. Based on the cross-variograms, cokriging with EC improved the estimation of NO3 and Ca significantly. Cokriging yielded a smaller mean squared error (MSE) and kriging variance, and a higher correlation between estimates and measurements. Using 20 points of NO, and 130 points of EC, cokriging provided a similar distribution pattern for NO3 as that generated with 100 points of NO.,. Cokriging with EC reduced MSE and the mean kriging variance of the estimated Ca up to 78 and 85%, respectively, compared with kriging. variograms or cross-covariances of two random variables (or more). Standard approaches for modeling cross-variograms are usually based on sums or differences of the two random variables, which are measured at the same locations. One disadvantage to the standard approaches is that only common locations can be used. Estimating the cross-variograms requires a large number of locations where data is collected for both variables, a condition that is frequently not satisfied in practice. For cases in which this type of information is not available, cokriging cannot be used to improve estimation. Clark et al. (1989) presented a variation of cokriging, using pseudo-cross-variograms, which did not require that both variables be measured at the same locations. Our objective was to apply pseudo-cross-variograms and cokriging to improve estimations of the spatial distribution of soil chemicals over relatively large areas of land. S SPATIAL VARIABILITY can be a frustrating issue in the field-scale description of soil physical, chemical, and hydrological parameters. Geostatistics is a helpful tool to study the spatial distribution of soil properties (Yost et al., 1982a,b; Burgess et al., 1981; Hatfield et al., 1984; Morkoc et al., 1985; Warrick et al., 1986). Most geostatistical research in soil science has been based on determinations of the spatial correlation function, i.e., the variogram, and a variety of linear estimators such as kriging. Cokriging is an extended technique of kriging, which incorporates both spatial and intervariable correlation. Cokriging can estimate all or some variables using the same coefficient matrix. Several applications of cokriging for describing soil-water phenomena have been reported during the past 10 yr. For example, Vauclin et al. (1983) used kriging and cokriging to predict available water content and sand content. McBratney and Webster (1983) applied cokriging to interpret the topsoil silt content, using subsoil silt or subsoil sand as joint variables. Carr and Myers (1984) studied the application of cokriging to the analysis of satellite data. Yates and Warrick (1987) and Mulla (1988) estimated soil water content using a cokriging procedure in which the bare soil surface temperature and the sand content were used as auxiliary functions. Using soil-map delineations, Stein et al. (1988) pointed out that the use of cokriging resulted in an average increase in precision of about 10% in 30-yr average moisture deficit maps. Zhang et al. (1992) improved the estimation of soil texture by using cokriging with spectral properties. The key step in cokriging is to formulate crossR.Zhang, U.S. Salinity Lab. USDA-ARS, 4500 Glenwood Dr., Riverside, CA 92501; D.E. Myers, Mathematics Dep. , and A.W. Warrick, Dep. of Soil and Water Science, Univ. of Arizona, Tuscon, AZ 85721. Received 2 Aug. 1991. * Corresponding author. Published in Soil Sci. Soc. Am. J. 56:1444-1452 (1992). THEORY Pseudo-Cross-Variograms Cokriging is a method for estimating one or more variables of interest using data from several variables by incorporating not only spatial correlation but also intervariable correlation. Let Zj(jc), . , Zm(x) denote the value of the variables Z15 . , Zm at location*, and Z(x) = [Z^(x), . , Zm(x)]. If*! , . , xn are sample locations with data Zfo), . , Z(xn), then the cokriging estimator can be written in the form