Abstrakt
Four different forms of the tensor Green's function for the gravitational gradient tensor, derived in this article, give a theoretical basis for geophysical interpretations of the GOCE-based gravitational gradients in terms of the Earth's mass-density structure. The first form is an invariant expression of the tensor Green's function that can be used to evaluate numerically the gravitational gradients in different coordinate systems (e.g.
Cartesian). The second form expresses the gravitational gradients in spherical coordinates (theta, phi) with the origin at the north pole as a series of tensor spherical harmonics.
This form is convenient to apply when the GOCE data are represented in terms of the gravitational potential as a scalar spherical harmonic series, such as the GOCO03S satellite gravity model. The third form expresses gravitational gradients in spherical coordinates (epsilon, alpha) with the pole at the computation point.
The fourth form then expresses the corresponding isotropic kernels in a closed form. The last two forms are used to analyse the sensitivity of the gravitational gradients with respect to lateral distribution of the Earth's mass-density anomalies.
They additionally provide a tool for evaluating the omission error of geophysically modelled gravitational gradients and its amplification when the bandwidth-limited GOCE-based gravitational gradients are interpreted at different heights above the Earth's surface. We show that the omission error of the bandwidth-limited mass-density Green's functions for gradiometric data at the GOCE satellite's altitude does not exceed 1 per cent in amplitude when compared to the full-spectrum Green's functions.