Mathematical analysis of optimal averaging and gridding for climate data

This thesis focuses on a rigorous mathematical analysis of a suite of mathematical methods of optimal averaging and gridding developed in the last three decades for climate data. The optimization is in the sense of minimum mean square errors (MSE). The first result of this thesis is the proof of correlation factor being less than or equal to one. The second result includes several estimates of MSE for the optimal spatial average. The original method and the proofs here extensively use Empirical Orthogonal Functions (EOFs), which are eigenfunctions of a subclass of Hilbert-Schmidt operators associated with the covariance matrix of an inhomogeneous climate field. Error estimators and their approximations are analytically computed for the special case of a homogeneous field on the unit circle and are compared to an error estimator using uniform sampling. The third result includes several numerical approximations to MSE under various kinds of conditions, such as a given number of truncated EOFs. This thesis has proven that this approximation converges to the exact MSE for an increasing number of EOFs. To demonstrate this behavior, a numerical example with actual precipitation observations is given.