Increments for (CO)Kriging with Trend and Pseudo-Covariance Estimation

Abstract

This paper presents an integrated account of the interpolation techniques of universal kriging and cokriging by way of the solution of a prediction problem in a situation akin to linear regression. Polynomial expectations of degree k in the coordinates of a region are eliminated by consideration of increments only, followed by the introduction of a covariance structure by polynomial pseudo-covariance functions. Then the best linear unbiased predictor is derived as well as the variance of the ensuing prediction error, both interpretable by some analogy to the linear regression situation, but essentially different because of the non-uniqueness of the pseudo-covariance functions and the absence of unconditional positivity of the pseudo-covariance matrices involved. Estimation of essential coefficients of the pseudo-covariance functions is accomplished by the restricted maximum likelihood method which may be helpful in deciding about the value of k as well.