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DOI: 10.4018/978-1-5225-3270-5.ch009

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TopThe 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.

w_{j} are the weights to be assigned to the i^{th} 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(x_{i},x_{j}) is the variogram value between sample i and j while γ(x_{i},x_{0}) equals the variogram value between i sample and x_{0} 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 x_{0} estimation, γ(x_{k},x_{0}), 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 w_{k} 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).

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.

C_{00} equals the variance of the estimated point value (as the variable becomes more erratic, this term increases in magnitude), C_{ij} is the covariance between i^{th} and j^{th} sample (as samples get closer together, a clustering issue, the average covariance increases), w_{i} and w_{j} denote the OK weights while C_{i0} represents the covariance between the i^{th} sample and the unknown value being estimated (uncertainty decreases as samples are closer to the value being estimated).

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