Recovering the global gravitational field from satellite measurements of the full gravity gradient

The problem of recovering the earth’s potential coefficients $${\bar C_{nm}}$$ _n,m from satellite gradiometry missions is studied by the space-wise approach. The conventional procedure of solving this problem is based on constructing from the start several boundary value (BV) relations, each of them corresponding to a separate second order potential derivative. In the present paper an optimal approach is elaborated for solving the spaceborne scalar boundary value problem in which the external gravitational field is reproduced from the total magnitude Γ of the gravity gradient measured in a spacecraft. Correspondingly, instead of constructing a set of BV equations, a unique one is derived combining the components of the gravity gradient tensor and the first order potential derivatives entering the expression for Γ. This equation is solved analytically in form of a quadrature formula for $${\bar C_{nm}}$$ _n,m , depending on the observed anomaly △Γ of the gravity gradient magnitude Γ.