Optimal Estimation From Data Regularly Sampled On A Sphere With Applications In Geodesy

The size of the variance-covariance matrix of the data, used to obtain minimum variance estimators for collocation, is as large as the number of observations in the data set. For some arrangements of the data, such as the usual 'equal angle' (or 'regular') grid, the matrix presents a very strong Toeplitz-circulant structure that can be exploited to reduce computing in setting-up and inverting the matrix. This reduction can be quite drastic. This report discusses such structure and presents an algorithm for implementing collocation efficiently. Three applications are considered: (a) the spherical harmonic analysis of point data; (b) the same analysis using area means; (c) the estimate of the disturbing potential from gravity anomalies. The harmonic analysis is optimal for noisy data as well; with noiseless data it provides harmonic coefficients with minimum aliasing. (Author).