Toward more complete magnetic gradiometry with the Swarm mission

An analytical and numerical analysis of the spectral properties of the gradient tensor, initially performed by Rummel and van Gelderen (Geophys J Int 111(1):159–169, 1992) for the gravity potential, shows that when the tensor elements are grouped into sets of semi-tangential and pure-tangential parts, they produce almost identical signal content as the normal element. Moreover, simple eigenvalue relations can be derived between these sets and the spherical harmonic expansion of the potential. This theoretical development generally applies to any potential field. First, the analysis of Rummel and van Gelderen (1992) is adapted to the magnetic field case and then the elements of the magnetic gradient tensor are estimated by 2 years of Swarm data and grouped into ΓΓ⁽¹⁾={[∇∇B]rθ,[∇∇B]rφ} resp. ΓΓ⁽²⁾={[∇∇B]θθ−[∇∇B]φφ,2[∇∇B]θφ}. It is shown that the estimated combinations ΓΓ⁽¹⁾ and ΓΓ⁽²⁾ produce similar signal content as the theoretical radial gradient ΓΓ⁽⁰⁾={[∇∇B]rr}. These results demonstrate the ability of multi-satellite missions such as Swarm, which cannot directly measure the radial gradient, to retrieve similar signal content by means of the horizontal gradients. Finally, lithospheric field models are derived using the gradient combinations ΓΓ⁽¹⁾ and ΓΓ⁽²⁾ and compared with models derived from traditional vector and gradient data. The model resulting from ΓΓ⁽¹⁾ leads to a very similar, and in particular cases improved, model compared to models retrieved by using approximately three times more data, i.e., a full set of vector, North–South and East–West gradients. This demonstrates the high information content of ΓΓ⁽¹⁾.