Kwang-Yul Kim
Climate System Research Program
Texas A&M University
The Karhunen-Loève basis functions (more frequently referred to as principal components or empirical orthogonal functions) of the noise response of the climate system are of interest in developing and improving detection and prediction techniques of climate change. These functions are difficult to estimate from observations because observational records are not long enough, sampling errors contaminate observations, and insufficient spatial coverage tends to bias the estimates. In this study, the Karhunen-Loève functions were modeled using a simple coupled climate model which reasonably reproduces the second-moment statistics of noise fluctuations of our climate system. Comparisons with estimates from observations provide some insights into and benchmark tests of the sensitivity of the Karhunen-Loève basis functions to sampling errors and the suitability of the modeled Karhunen-Loève basis functions in the studies of detecting and predicting climatic change.
Spatial and temporal sub-sampling errors in estimating empirical orthogonal functions (EOFs) and the corresponding eigenvalues are also studied. The sub-sampling errors are measured in terms of the minimum difference between an estimated eigenvalue and an exact one, and the minimum Euclidean norm of distance between an estimated eigenfunction and a linear space spanned by a certain number of exact eigenfunctions. This linear space represents a set of exact EOFs which admix into an estimated EOF. The extent of modal mixing, or variance split, is determined by the proximity of adjacent eigenvalues and the degree of sub-sampling error, and is estimated using one of the theorems of a posteriori error estimates for the eigenvalues and the eigenfunctions. Such a representation of the EOF errors allows a systematic approach to the sampling problems of EOFs. This methodology is applied to the surface air temperature field, in which errors in computing the EOFs and the corresponding eigenvalues are measured in terms of the sampling insufficiency spatially and temporally. To this end, different configurations of spatial sampling points, and different lengths of the temperature record have been used. This study addresses the spatial and temporal ``sampling theorem'' for the EOFs and the eigenvalues of the global surface temperature field.