Local Least Squares Spectral Filtering and Combination by Harmonic Functions on the Sphere

Least squares spectral combination is a well-known technique in physical geodesy. The established technique either suffers from the assumption of no correlations of errors between degrees or from a global optimisation of the variance or mean square error of the estimator. Today Earth gravitational models are available together with their full covariance matrices to rather high degrees, extra information that should be properly taken care of. Here we derive the local least squares spectral filter for a stochastic function on the sphere based on the spectral representation of the observable and its error covariance matrix. Second, the spectral combination of two erroneous harmonic series is derived based on their full covariance matrices. In both cases the transition from spectral representation of an estimator to an integral representation is demonstrated. Practical examples are given. Taking advantage of the full covariance matrices in the spectral combination implies a huge computational burden in determining the least squares filters and combinations for high-degree spherical harmonic series. A reasonable compromise between accuracy of estimator and workload could be to consider only one weight parameter/degree, yielding the optimum filtering and combination of Laplace series.