Minimum Estimation Variance

The estimation variance $E(Z_{V}-/hat{Z}_{V})^{2}$ can be expanded as follows:

with

The terms $m^2$ are eliminated and, thus, we have

The estimation variance can, thus, be expressed as a quadratic form in $/lambda_i, /lambda_j$, and is to be minimized subject to the non-bias condition $/sum /lambda_{i}=1$. The optimal weights are obtained from standard Lagrangian techniques by setting to zero each of the $n+1$ partial derivatives (in respect to $/lambda_i$ and to the Lagrange parameter $/mu$) of

This procedure provides a system of $n+1$ linear equations in $n+1$ unknowns (the $n$ weights $/lambda_i$, and the Lagrange parameter $/mu$) which is called the ``kriging system'':

The minimum estimation variance, or ``kriging variance'', can then be written as

This kriging system can also be expressed in terms of the semi-variogram function $/gamma({/boma h})$, particularly when the random function $Z({/boma x})$ is intrinsic only and the covariance function $C({/boma h})$ is not defined: