Numerical Aspects of the Universal Kriging Method for Hydrological Applications

TLDR: In this paper, a QR factorization of the Universal Kriging matrix (M) is proposed to reduce the condition number of M (cond(M)) by using a set of functions derived from the eigenvectors of the variogram matrix (Γ).

Abstract: Many hydrological variables usually show the presence of spatial drifts Most often they are accounted for by either universal or residual kriging and usually assuming low order polynomials The Universal Kriging matrix (M), which includes in it the values of the polynomials at data locations (matrix F), in some cases may have a too large condition number and can even be nearly singular due to the fact that some columns are close to be linearly dependent These problems are usually caused by a combination of pathological data locations and an inadequate choice of the coordinate system As suggested by others, we have found that an appropriate scaling can alleviate the problem by significantly reducing the condition number of M (cond(M)) This scaling, however, does not affect the linear independence of this matrix We show that a QR factorization of matrix F leads to a significant improvement on cond(M) An alternative to drift polynomials consists on using a set of functions derived from the eigenvectors of the variogram matrix (Γ) Although their potential usefulness as interpolating functions remains to be ascertained, they are optimal from the point of view of optimizing cond(M) In fact we are able to provide a rigorous proof for an upper bound of cond(M) Applications of the theoretical developments to hydraulic head data from an alluvial aquifer are also presented