Spectral analysis of the full gravity tensor

SUMMARY The five independent components Γ xz , Γ yz , Γ zz , Γ xx ‐ Γ yy , and Γ xy , of the gravity tensor are measurable by gradiometers. When grouped into {Γ zz }, {Γ xz , Γ yz } and {Γ xx ‐ Γ yy , 2Γ xy } and expanded into an infinite series of pure‐spin spherical harmonic tensors, simple eigenvalue connections can be derived between these three sets and the spherical harmonic expansion of the gravity potential. The three eigenvalues are ( n + 1)( n + 2), ‐( n + 2)√ n ( n + 1) and √( n ‐ 1) n ( n + 1)( n + 2). This result permits an easy analytical incorporation of all measurable tensor components into a spectral signal and noise analysis of gravity quantities on a sphere. Analogous relations also exist for a 2‐D Fourier (flat earth) expansion of these three sets. An additional advantageous feature of the set {Γ xx ‐ Γ yy , 2Γ xy }, besides the simple eigenvalue relation, is its invariance with respect to small position uncertainties, e.g., of the trajectory of the satellite or airplane carrying the gradiometer. Hence a complete framework exists in terms of the eigenvectors of all operators connecting the zeroth, first, and second derivatives of the gravitational potential. At the same time the results facilitate the planning of gradiometer missions and their data analysis.