An optimal regional averaging method with error estimates and a test using tropical Pacific SST data

Abstract

This paper provides a systematic procedure for computing the regional average of climate data in a subregion of the earth surface using the covariance function written in terms of empirical orthogonal functions (EOFs). The method is optimal in the sense of minimum mean square error (mse) and gives an mse estimate of the averaging results. The random measurement error is also included in the total mse. Since the EOFs can account for spatial inhomogeneities, the method can be more accurate than those that assume a homogeneous covariance matrix. This study shows how to further improve the accuracy of optimal averaging (OA) by improving the accuracy of the eigenvalues of the covariance function through an extrapolation method. The accuracy of the authors' procedure is tested using cross-validation techniques, which simulate past sampling conditions on the recent, well-sampled tropical Pacific SST and use the EOFs independent to the month being tested. The true sampling error of the cross-validated tests is computed with respect to the 1°x 1°data for various sampling conditions. The theoretical sampling error is computed from the authors' derived formula and compared to the true error from the cross-validation tests. The authors' numerical results show that (i) the extrapolation method can sometimes improve the accuracy of the eigenvalues by 10%, (ii) the optimal averaging consistently yields smaller mse than the arithmetic averaging, and (iii) the theoretical formula for evaluating the OA error gives estimates that compare well with the true error.