Fourth order Taylor–Kármán structured covariance tensor for gravity gradient predictions by means of the Hankel transformation

Abstract

Vector-valued stochastic processes have been used for data processing, prediction, filtering, collocation, even network design and analysis in Geodesy, Physical Geodesy, and navigation etc. Recently, LEO satellite missions, such as CHAMP, GRACE and GOCE, provide a number of measurements to study the gravitational field of the Earth. When the gravity gradients, i.e. the second derivatives of the gravitational potential, are used for prediction and filtering by Kolmogorov–Wiener/Gauss–Markov concept, we will need the fourth-order covariance/correlation matrices of the gravity gradient signals. With the assumptions of homogeneous and isotropic field and the random functions of “potential type”, the paper aims at the development of the fourth order tensor-valued Taylor–Karman structured covariance/correlation matrices. The characteristic functions of these tensor-valued covariance/correlation matrices, namely the lateral and longitudinal components, will be derived for n-dimensional spaces, here specified for \(n=3\) dimensions in the paper. A special part is devoted to the Hankel transformation for gravity gradients and their variance-covariance in order to guarantee consistency well-known from problems in using Fourier transformations. We use the variance-covariance function of type (i) isotropic, (ii) homogeneous and (iii) potential as prior information for fitting the discrete data of variances and covariances estimated from observations.