Error Covariance Analysis of a Karhunen-Loeve Random Field Estimator

An error covariance analysis of a two-dimensional Karhunen-Loeve random field estimator (KLE) for gridded gravity data is presented without actually using the data. Attention is focused particularly on the north-south deflection component of the disturbance vector so that the rms value of its residual error may not exceed 3 percent of the signal rms for grid spacing of not less than 5 nm, when signal-to-noise ratio in the data is varied within reasonable limits. To achieve this rather stringent goal, a discrete KLE derived entirely from a two-dimensional grid configuration is needed. The KLE for geodesy applications was developed initially by Bose. Using continuous measurements and the orthogonality relations of the Karhunen-Loeve (KL) eigenvectors in a continuous domain, continuous KL gain coefficients ßmnj and consequently the continuous KL gain vector Kmn were obtained; thus a satisfactory continuous KLE was achieved. However the discrete version of the KLE was then given by keeping the old (continuous) ßmnj, and simply computing the KL eigenvectors on the grid coordinates to evaluate the discrete KL gain vector Kmn. Clearly this is not a satisfactory discrete KLE since , ßmnj are not constants but rather depend on the measurement noise variances. So if the measurements are now going to be discrete, one must rederive ßmnj for this situation.