Geomathematical Advances in Satellite Gravity Gradiometry (SGG)

Abstract

A promising technique of globally establishing the fine structure and the characteristics of the external Earth’s gravitational field is satellite gravity gradiometry (SGG). Satellites such as ESA’s gradiometer satellite GOCE are able to provide sufficiently large data material of homogeneous quality and accuracy. In geodesy, traditionally the external Earth’s gravitational potential and its Hesse matrix are described using orthogonal (Fourier) expansions in terms of (outer) spherical harmonics. Spherical and outer harmonics are introduced for the global modeling of (scalar / tensor) fields. We briefly recapitulate the results interconnecting spherically the potential coefficients with respect to tensor spherical harmonics at Low Earth Orbiter’s (LEO) altitude to the corresponding coefficients with respect to scalar spherical harmonics at the Earth’s surface. The relation between the known tensorial measurements g (i.e., gradiometer data) and the gravitational potential F on the Earth’s surface is expressed by a linear integral equation of the first kind. This operator equation is discussed in the framework of pseudodifferential operators as an invertible mapping between Sobolev spaces under the assumption that the data are not erroneous. In reality, however, the data g are noisy such that the Sobolev reference space for the (noisy) tensorial data g must be embedded in a larger Sobolev space. Under these conditions, we base our inversion process on the fact that the reference Sobolev subspace is dense in the larger Sobolev space and that, e.g., a smoothing spline process or a signal-to-noise procedure in multiscale framework open appropriate perspectives to approximate F (in suitable accuracy) from noisy data g.