Efficient Approximation of the Spatial Covariance Function for Large Datasets - Analysis of Atmospheric CO2 Concentrations

Linear mixed effects models have been widely used in the spatial analysis of environmental processes. However, parameter estimation and spatial predictions involve the inversion and determinant of the n times n dimensional spatial covariance matrix of the data process, with n being the number of observations. Nowadays environmental variables are typically obtained through remote sensing and contain observations of the order of tens or hundreds of thousand on a single day, which quickly leads to bottlenecks in terms of computation speed and requirements in working memory. Therefore techniques for reducing the dimension of the problem are required. The present work analyzes approaches to approximate the spatial covariance function in a real dataset of remotely sensed carbon dioxide concentrations, obtained from the Atmospheric Infrared Sounder of NASA's 'Aqua' satellite on the 1st of May 2009. In a cross-validation case study it is shown how fixed rank kriging, stationary covariance tapering and the full-scale approximation are able to notably speed up calculations. However the loss in predictive performance caused by the approximation strongly differs. The best results were obtained for the full-scale approximation, which was able to overcome the individual weaknesses of the fixed rank kriging and the covariance tapering.