where C is the covariance matrix among the n points 0i used in the estimation, i.e. the semi-variances corresponding to the distances separating the various pair of points, as read on the variogram model; w is the vector of weights to be estimated (with the constraint that the sum of weights must be 1); and d is a vector containing the covariances between the various points 0i and the grid node to be estimated. This is where a variogram model becomes essential; it provides the weighting function for the entire map and is used to construct matrix C and vector d for each grid node to be estimated. X is a Lagrange parameter (as in Section 4.4) introduced to minimize the w variance of the estimates under the constraint X wi = 1 (unbiasedness condition). The solution to this linear system is obtained by matrix inversion (Section 2.8):
Vector d plays a role similar to the weights in inverse-distance weighting since the covariances in vector d decrease with distance. Using covariances, the weights are statistical in nature instead of geometrical.
Kriging takes into account the grouping of observed points 0- on the map. When two points 0- are close to each other, the value of the corresponding coefficient Cj in matrix C is high; this contributes to lowering their respective weights w-. In this way, the redundancy of information introduced by dense groups of sampling sites is taken into account.
When anisotropy is present, kriging can use two, four, or more variogram models computed for different geographic directions and combine their estimates when calculating the covariances in matrix C and vector d. In the same way, when estimation is performed for sampling sites in a volume, a separate variogram can be used to describe the vertical spatial variation. Kriging is the best interpolation method for data that are not on a regular grid or display anisotropy. The price to pay is increased mathematical complexity during interpolation.
Among the interpolation methods, kriging is the only one that provides a measure of the error variance for each value estimated at a grid node. For each grid node, the error variance, called ordinary krlglng variance (sOK), is calculated as follows (Isaaks & Srivastava, 1989), using vectors w and d from eq. 13.21:
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