Mass-density Green's functions for the gravitational gradient tensor at different heights

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. However, when evaluating the bandwidth-limited Green's functions at lower altitudes, their omission errors are significantly amplified. In this case, we show that the short-wavelength content of the forward-modelled gravitational gradients generated by an a priori density structure of the Earth must be filtered out such that the omission error of the GOCE-based gravitational gradients (i.e. the signal that has not been modelled from the GOCE data) is equal to the omission error of the forward-modelled gravitational gradients. Only after performing such filtering can the GOCE-based gravitational gradients at low altitudes be interpreted in terms of the Earth's density structure. Both the closed and spherical-harmonic forms of the Green's functions allow a direct interpretation in terms of the minimum lateral extent of the Earth that needs to be considered in a regional model constrained by gravitational gradients if the full information provided by the Green's functions is to be retained. We show that this extent (i) is smallest for the vertical-vertical Green's function and (ii) linearly increases with increasing computation-point height. The spectral forms of the gravitational gradients are further used to calculate the sensitivity of the GOCE-based gravitational gradients to the depth of density anomalies, expressed in terms of the harmonic degree of the internal mass-density anomaly. We show that the largest gravitational gradient response is obtained for shallow mass anomalies, and is further amplified as the harmonic degree increases.