Interpolation and optimal linear prediction

This paper is concerned with the interpolation of spatially distributed observations of a quantitative phenomenon, sometimes referred to as kriging. This activity can be understood as a prediction procedure for values of random functions under stationarity assumptions in a polynomial linear regression context. After a heuristic and an exact derivation of the best linear unbiased prediction procedure (and the variance of prediction error) if the covariance function relating covariance between two possible observations to their mutual distance is known, follows the introduction of weaker assumptions admitting the definition of the variance only for increments of a certain order by a pseudoco–variance function. A particular related case is the so–called semivariogram for increments of order one. The prediction procedure turns out to be similar to that in the previous situation. The weaker assumptions allow an unbiased estimation of the unknown pseudocovahance function of polynomial form under restrictions imposed by Fourier transformation. Extension from point–wise observations or predictions to area or volume averages is touched upon.