Regularisation and Deregularisation of Covariances and Variograms for Univariate Random Functions in One-dimensional Space

The regularisation on a segment of a stationary or intrinsic variogram model is usually expressed as an integral. By introducing an auxiliary function, it is shown that this integral is a linear combination of elementary terms. This paper shows how these terms are arranged and provides a simple and safe procedure to calculate them. An important application of this procedure is the deregularisation problem, that is, the conversion of a regularised variogram to a point variogram. This ill-posed problem is tackled using an iterative optimisation algorithm designed to produce a family of point variograms that best fit the initial regularised variogram. Subsequently, each of these point variograms can be regularised on any segment, which results in a fully coherent family of simple and cross regularised variograms.