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

Abstract

Advantages and disadvantages of least squares collocation (LSC) and kriging
have 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 covariance
parameters estimation for spatial LSC interpolation, via a kind of cross-validation, called
hold-out (HO) validation. Two covariance models are applied in order to reveal also those
differences that come solely from the covariance model.
Typical covariance models have a few variable parameters, the selection of which requires
analysis of the actual data distribution. Properly chosen covariance parameters
result in accurate and reliable predictions. The correlation length (CL), also known as
the correlation distance in the Gauss-Markov covariance functions, the variance (C0) and
a 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 present
in the kriging-related investigations. Therefore the covariance parameter N has the same
scale as the data and can be analyzed in relation to the data errors, spatial data resolution
and prediction errors.
The vector of the optimal three covariance parameters is sometimes determined approximately
for the purposes of modeling with limited accuracy requirements. This is
done e.g. by the fitting of analytical model to the empirical covariance values. The more
demanding predictions need precise estimation of the covariance parameters vector and
the researchers solve this problem via least squares methods or maximum likelihood (ML)
inference. Nevertheless, both least squares and ML produce an error of the parameters
and it is often large. The reliability of LSC or kriging using parameters with an error
of e.g. a quarter of the parameter value is usually not discussed.