Optimal gravity reduction when anomalous potential is known

Knowledge of the potential difference (altitude above sea level Hγ) or spatial position (ellipsoidal height H) at the measurement point leads to two main types of free-air gravity reduction: anomalies Δg and disturbances δg.This gives rise to geodetic boundary problems of determining the external anomalous potential T or its transformants (on the Earth's boundary surface S and beyond it), which are solved, in particular, using a family of functions orthogonal on a geometrically regular surface maximally close to the boundary surface (a sphere Ω or an oblate ellipsoid); in a more general case, the apparatus of integral equations should be used (for example, with respect to the density of a simple layer φ, which explains the external anomalous field).When the anomalous potential is found from the solution of one boundary value problem, it is possible to transform it to a gravity reduction corresponding to another boundary value problem.In modern conditions, a situation is theoretically and practically possible when both the normal Hγ and geodetic H heights are known, thus, the anomalous potential T itself at a point on the Earth's surface is considered known, the accuracy of its calculation in a linear measure is limited by the accuracy of knowledge of the heights (the first centimeters).Further, calculations of the elements of the anomalous field can be formally performed based on the solution of the first boundary value problem, but an increase in the order of the derivative of the anomalous potential leads to a loss of accuracy during the next differentiation.Therefore, as initial data, it is better to have such a derivative whose order is as close as possible to the desired value, so that the relevance of gravity measurements does not decrease.As a result of such measurements complex, it is possible to calculate separately the Δg and δg.It is generally believed that calculations using δg lead to a more accurate result due to the known boundary surface (although using the potential difference instead of the spatial position is more justified from a physical point of view).But which type of gravity reduction would be optimal?Solution of the geodetic boundary value problem for determining the anomalous potential T at an external point P has a very simple form in the spherical approximation if we introduce a special gravity reduction Πg with the corresponding boundary condition on the Earth's surface S:The integration kernel 2/r is convenient because it does not contain the natural logarithm.It is of interest to derive a more accurate boundary condition taking into account Earth's flattening.Also, in spherical approximation, one can obtain an integral equation (*), which is significantly simpler than the integral equations (**) and (***):The simplicity of such a gravity reduction was first noted by L.P.Pellinen and O.M.Ostach in their article on some topographic gravity anomalies. See: Stud Geophys Geod 18, 319–328 (1974), https://doi.org/10.1007/BF0162718