A Modified Discrete Karhunen-Loeve Random Field Estimator
Based upon the modified Karhunen-Loeve model of the gravity disturbance vector proposed by us recently, a corresponding Karhunen-Loeve random field estimator is developed in this paper for a two-dimensional grid of gravity data in a local finite region.The new eigenvectors obey the "required" orthogonality relations on the chosen grid provided the KL expansion is now separated into odd and even integers. The closed-form solution of the estimator isthen obtained under the "diagonal" assumption for the gain coefficients. It is shown that this assumption, without which nosolution is possible, enables the estimator to reproduce the data atgrid points when the noise in the data is absent and the number of terms in the Karhunen-Loeve expansion are equal to the number of grid points.
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.
Statistical models for the vertical deflection from Gravity-Anomaly Models
Statistical models for vertical deflections in the form of power spectral densities and autocorrelation functions are derived from theoretical gravity-anomaly models by means of the Vening Meinesz equations. Details are given for homogeneous and isotropic gravity-anomaly models described by exponential and Bessel autocorrelation functions. The use of these models for evaluation of mean-squared output errors in an inertial navigation system is described.