On the condition number anomaly of Gaussian correlation matrices

Spatial correlation matrices appear in a large variety of applications. For example, they are an essential component of spatial Gaussian processes, also known as spatial linear models or Kriging estimators, which are powerful and well-established tools for a multitude of engineering applications such as the design and analysis of computer experiments, geostatistical problems and meteorological tasks.In radial basis function interpolation, Gaussian correlation matrices arise frequently as interpolation matrices from the Gaussian radial kernel function. In the field of data assimilation in numerical weather prediction, such matrices arise as background error covariances.Over the past thirty years, it was observed by several authors from several fields that the Gaussian correlation model is exceptionally prone to suffer from ill-conditioning, but a quantitative theoretical explanation for this anomaly was lacking. In this paper, a proof for the special position of the Gaussian correlation matrix is given. The theoretical findings are illustrated by numerical experiment.