Errors in estimating spherical harmonic coefficients from partially sampled GCM output

This study describes a method of calculating the mean squared error (MSE) incurred when estimating the spherical harmonic coefficients of a climatological field that is sampled at a small network of points. The method can also be applied to the coefficients of any other set of orthonormal basis functions that are defined on the sphere. It, therefore, provides a formalism that can be applied in a variety of contexts, such as in climate change detection, where inferences are attempted about fingerprint coefficients that are imperfectly estimated from observational data. By incorporating the fingerprint as part of a set of basis functions, the methodology can be used to estimate the sampling error in the fingerprint coefficient. The MSE is expressed in terms of the spherical harmonics (or other orthonormal expansion) of the empirical orthogonal functions (EOFs), the locations of the points in the network and a set of weights that are applied at these points. The weights are optimised by minimising the expected MSE. The method is applied to a number of network configurations using monthly-mean screen temperature and 500 mb height simulated by the Canadian Climate Centre 2nd generation general circulation model in an ensemble of six 10-year simulations. In comparison with uniform weighting, optimal weighting can reduce the MSE by an order of magnitude or more for some spherical harmonic coefficients and some network configurations. Also, the MSEs vary seasonally for each network. In particular, the relative MSE of low order spherical harmonic coefficients is found to be larger in DJF than in JJA. We demonstrate how MSEs improve with increasing network density and identify graphically, the coefficients that can be estimated reliably with each network configuration.