Spatial analysis includes an expanding array of methods which address different spatial issues, ranging from remote sensing to spatial error uncertainty. Each of these methods focuses on geographically raw data correlated by statistical methods. In general, spatial interpolation and stochastic Kriging, in particular, will be addressed in this chapter. Ordinary Kriging (OK) foundations are presented in the first section which encompasses eight sub-sections (in accordance with the eight myGeoffice© options). Section two introduces Kriging with Trend (KT but sometimes known as Universal Kriging) including five sub-sections: Geocomputation of KT, estimation mapping, the cross-validation procedure, validation using an extra dataset and KT versus OK comparison. Finally, Indicator Kriging (IK) is explored in section three together with nine sub-sections: First and second cutoff definition, first and second probabilistic interpolation maps, construction of the conditional cumulative distribution function, entropy of Shannon, E-type spatial estimation (including misclassification risks and economic classification), morphologic geostatistics and probabilistic interval mapping.
TopOrdinary Kriging
The difference between inverse distance weighted (IDW) and Ordinary Kriging (OK) derives from the mathematical process that minimizes the minimum squared error variance of the estimation. The OK equations system can be obtained from the following two algebraic expressions stressed in Figure 1.
Figure 1. wj are the weights to be assigned to the ith observation and Ψ represents the Lagrange multiplier or slack value required for forcing total weights to equal one when the partial derivatives system is resolved (set to minimize the estimation variance). γ(xi,xj) is the variogram value between sample i and j while γ(xi,x0) equals the variogram value between i sample and x0 site.
Analogous to Thiessen and B-Splines, Kriging is an exact interpolator in the sense that the sample and estimation are equal. This happens because the variogram value between k sample and x0 estimation, γ(xk,x0), equals zero since the estimated site is the sample itself. Therefore, its Kriging variance equals zero. By resolving the Kriging matrix system presented in Figure 2, the wk weight is one while all other weights (including the Lagrangean dummy parameter) become zero (the k row of matrix B changes to zero, if the nugget-effect is considered zero).
Figure 2. The Kriging matrix system
Under this elegant matrix layout, each interpolated value is calculated as the sum of weighted known points whose weights are calculated from the (n+1) simultaneous linear equations set: A×W=B or W=A-1×B. The statistical distance (it incorporates anisotropic distance, direction, spatial autocorrelation and clustering information) between n samples and distances from each sample to the grid estimation point is used to compute the model variance reproduced on A (among samples) and B matrices (between each sample and the estimated location). While A-1 underlies the declustering factor, B represents the structural distance between the estimation and all samples. In addition, the product of A-1 by B adjusts the raw inverse statistical distance weights of B matrix to account for possible redundancies between samples (Isaaks & Srivastava, 1989). As expected, if no spatial autocorrelation is found among the available samples, the Kriging estimator equals the sample average.
With a zero nugget-effect, the diagonal of A matrix equals zero while Kriging weights decrease as the dataset location gets farther from the estimated one. Hence, major software computation uses the covariogram model to setup A and B matrices to avoid the diagonal zeros of A. After all weights are calculated, the estimation of the regular grid surface can be determined with a great deal of calculation. In fact, the computation intensity to solve OK weights is proportional to the cube of the number of observations. Regarding the uncertainty layout, this is closely related to OK variance and given by Figure 3.
Figure 3. C00 equals the variance of the estimated point value (as the variable becomes more erratic, this term increases in magnitude), Cij is the covariance between ith and jth sample (as samples get closer together, a clustering issue, the average covariance increases), wi and wj denote the OK weights while Ci0 represents the covariance between the ith sample and the unknown value being estimated (uncertainty decreases as samples are closer to the value being estimated).