Two covariance models in Least Squares Collocation (LSC) tested in interpolation of local topography

Abstract

Advantages and disadvantages of least squares collocation (LSC) and kriginghave recently been discussed, especially as interdisciplinary research becomes popular.These statistical methods, based on a least squares rule, have infinite number of applications,also in the domains different than Earth sciences. The paper investigates covarianceparameters estimation for spatial LSC interpolation, via a kind of cross-validation, calledhold-out (HO) validation. Two covariance models are applied in order to reveal also thosedifferences that come solely from the covariance model.Typical covariance models have a few variable parameters, the selection of which requiresanalysis of the actual data distribution. Properly chosen covariance parametersresult in accurate and reliable predictions. The correlation length (CL), also known asthe correlation distance in the Gauss-Markov covariance functions, the variance (C0) anda priori noise parameter (N) are analyzed in this paper, using local terrain elevations.The covariance matrix is used in LSC, as analogy to the correlation matrix often presentin the kriging-related investigations. Therefore the covariance parameter N has the samescale as the data and can be analyzed in relation to the data errors, spatial data resolutionand prediction errors.The vector of the optimal three covariance parameters is sometimes determined approximatelyfor the purposes of modeling with limited accuracy requirements. This isdone e.g. by the fitting of analytical model to the empirical covariance values. The moredemanding predictions need precise estimation of the covariance parameters vector andthe researchers solve this problem via least squares methods or maximum likelihood (ML)inference. Nevertheless, both least squares and ML produce an error of the parametersand it is often large. The reliability of LSC or kriging using parameters with an errorof e.g. a quarter of the parameter value is usually not discussed.