In this article, we use the theory of meshfree approximation to generalize the simple kriging methods by kernel-based probabilities. The main idea is that the new kriging estimations are modeled by the Gaussian fields indexed by bounded linear functionals defined on Sobolev spaces. Moreover, the covariances of the Gaussian fields at the observed functionals can be computed by the given covariance kernels with respect to the related functionals, for example, Gaussian kernels evaluated at points and gradients. This guarantees that the generalized kriging estimations can be obtained by the same techniques of the simple kriging methods and the generalized kriging estimations can cover many kinds of the complex observed information. By the generalized kriging methods, we can model the geostatistics with the additional observations of gradients at the uncertain locations.
Ye, Q. (2017). Generalizations of simple kriging methods in spatial data analysis. In Lecture Notes in Computational Science and Engineering (Vol. 115, pp. 145–166). Springer Verlag. https://doi.org/10.1007/978-3-319-51954-8_8