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A mathematician.^[ref]

Kriging predicts an unobserved value of a spatial random field by combining observations according to their covariances. Nearest-neighbor interpolation uses the closest sample; kriging also asks how strongly each sample is related to the target and how much information the samples share with one another. Two nearby, highly correlated samples need not provide twice the information of one. The different forms of kriging depend on whether the mean is known, an unknown constant, or an unknown regression drift.

Write $Z=(Z(s_1),\dots ,Z(s_n){)}^T$ for the observation random vector, $z$ for its realized values, and $Z_0=Z(s_0)$ for the prediction target. Assume finite second moments, and defineHere $\Sigma$ is the observation covariance matrix, assumed to be a positive-definite matrix in the formulas below. A covariance function supplies its entries and the target covariance vector $c$. For a second-order stationary random field, the covariogram $C(h)$ depends on displacement, and the semivariogram satisfies $\gamma (h)=C(0)-C(h)$. The intrinsically stationary random field formulation permits kriging using a semivariogram even when a stationary covariogram is unavailable; the finite covariance formulation below is a convenient important case.

For a linear combination $w^{T}Z$ predicting $Z_0$ with zero bias, the error variance equals its mean squared error:Ordinary kriging and universal kriging minimize this quadratic form subject to their mean constraints. Their optimality is among the specified linear predictors with zero bias under the chosen statistical model; covariance information alone does not establish optimality among arbitrary nonlinear predictors. The Gaussian random field interpretation supplies additional conclusions in simple kriging.

The reported error variance assumes a chosen covariance function, its statistical parameters, observation locations, and a mean structure. It omits uncertainty in fitted covariance parameters and does not validate the trend. A repeated-sampling mean squared error need not equal the error variance conditional on each realized data vector.^[ref]

In 1963, Georges Matheron presented kriging for estimating average ore grades in mining panels from weighted samples taken inside and outside a panel. This setting in geostatistics explains why prediction targets can be averages over regions as well as values at points.^[ref]

By Georges Matheron. Economic Geology 58(8), 1246–1266; published 1 December 1963. DOI 10.2113/gsecongeo.58.8.1246.

Cited passages: p. 1246, abstract and introduction, on mining estimation. Primary text was inspected through a scan mirror; the mirror is not the author or publisher.