On the effect of covariance function estimation on the accuracy of kriging predictors

The kriging procedure gives an optimal linear predictor of a spatial process at a point x(o), given observations of the process at other locations x(1),..., x(n), taking into account the spatial dependence of the observations. The kriging predictor is optimal if the weights are calculated from the correct underlying covariance structure. In practice, this covariance structure is unknown and is estimated from the data. An important, but not very well understood, problem in kriging theory is the effect on the accuracy of the kriging predictor of substituting the optimal weights by weights derived from the estimated covariance structure. We show that the effect of estimation is negligible asymptotically if the joint Gaussian distributions of the process at x(o),..., x(n) under the true and the estimated covariance are contiguous almost surely. We consider a number of commonly used parametric covariance models where this can indeed be achieved.