Massdensity Green's functions for the gravitational gradient tensor at different heights
Abstract
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 GOCEbased gravitational gradients in terms of the Earth's massdensity 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 (ϑ, ϕ) 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 (ψ, α) 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 massdensity anomalies. They additionally provide a tool for evaluating the omission error of geophysically modelled gravitational gradients and its amplification when the bandwidthlimited GOCEbased gravitational gradients are interpreted at different heights above the Earth's surface. We show that the omission error of the bandwidthlimited massdensity Green's functions for gradiometric data at the GOCE satellite's altitude does not exceed 1 per cent in amplitude when compared to the fullspectrum Green's functions. However, when evaluating the bandwidthlimited Green's functions at lower altitudes, their omission errors are significantly amplified. In this case, we show that the shortwavelength content of the forwardmodelled gravitational gradients generated by an a priori density structure of the Earth must be filtered out such that the omission error of the GOCEbased gravitational gradients (i.e. the signal that has not been modelled from the GOCE data) is equal to the omission error of the forwardmodelled gravitational gradients. Only after performing such filtering can the GOCEbased gravitational gradients at low altitudes be interpreted in terms of the Earth's density structure. Both the closed and sphericalharmonic 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 verticalvertical Green's function and (ii) linearly increases with increasing computationpoint height. The spectral forms of the gravitational gradients are further used to calculate the sensitivity of the GOCEbased gravitational gradients to the depth of density anomalies, expressed in terms of the harmonic degree of the internal massdensity anomaly. We show that the largest gravitational gradient response is obtained for shallow mass anomalies, and is further amplified as the harmonic degree increases.
 Publication:

Geophysical Journal International
 Pub Date:
 March 2014
 DOI:
 10.1093/gji/ggt495
 Bibcode:
 2014GeoJI.196.1455M
 Keywords:

 Inverse theory;
 Satellite geodesy;
 Gravity anomalies and Earth structure