An Optimal Method to Estimate the Spherical Harmonic Components of the Surface Air Temperature

This paper describes a method that minimizes the mean squared error (MSE) in estimating the spherical harmonic components of the surface air temperature field. The ratio of the MSE to the variance of the spherical harmonic component is expressed in terms of the length scale Xo, and the positions and weights of the measurement stations. The weights are optimized by the condition of minimizing the sampling error. To present an analytical example, we assume the homogeneous statistics of the temperature anomaly field, and take the low frequency approximation (i.e. ignoring the time dependence). The spectra of the temperature anomaly are the coefficients of a Fourier-Legendre series of the covariance function, and they are analytically derived from a linear noise forced energy balance climate model. Consequently, the MSE, the percentage sampling error, and the signal-noise ratio are computed for a given network of stations. Our results show that: (i) the sampling errors computed from both optimal weights and uniform weights increase with respect to the order of the spherical harmonic component; (ii) the sampling errors computed from optimal weights are significantly smaller than those from uniform weights for sufficiently dense networks. With about 60 reasonably positioned stations for sampling the spherical harmonic components T , , Tlo and TII, one can get the sampling error below 10 per cent when the optimal weights are applied. An experiment with 210 stations produces the sampling errors of less than 10 per cent for the spherical harmonic components from Too up to TS4.