The Kriging Estimator

Let $Z({/boma x})$ be the random function under study. $Z({/boma x})$ is defined on a point support and is second-order stationary, with expectation,

a constant $m$ which is generally unknown; it is supposed that the covariance

or the variogram

exists (stationarity of second order resp. intrinsic hypotheses).

The estimation of the mean value

over a domain $V(/boma{x}_{o})$ with center $/boma{x}_{o}$ is required.

The experimental data to be used consist of a set of discrete grade values $z_{i}$, $i=1,/ldots,n$. These grades are either defined on point or quasi-point supports, or else they are the mean grades $z_i({/boma x_i})$ defined on the supports $v_{i}$ centered on the points ${/boma x_i}$; the $n$ supports may differ in size and shape of each other. Note that under the hypothesis of stationarity the expectation of each of these data is $m$: $E(Z_{i})=m / /forall i$.

The linear estimator $Z_{V}$ considered is a linear combination of the $n$ data values $Z_{i}$, namely

The $n$ weights $/lambda_{i}$, are calculated as to ensure that the estimator is unbiased and that the estimation variance is minimal (the estimator $/hat{Z}_{V}$ is then said to be optimal).