On Combining the Directional Solutions of the Gravitational Curvature Boundary-Value Problem

In global studies, the Earth’s gravitational field is conveniently described in terms of spherical harmonics. Four integral-based solutions to a gravitational curvature boundary-value problem can formally be formulated for the vertical-vertical-vertical, vertical-vertical-horizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal components of the third-order gravitational tensor. Each integral equation provides an independent set of spherical harmonic coefficients because each component of the third-order gravitational tensor is sensitive to gravitational changes in the different directions. In this contribution, estimations of spherical harmonic coefficients of the gravitational potential are carried out by combining four solutions of the gravitational curvature boundary-value problem using three methods, namely an arithmetic mean, a weighted mean and a conditional adjustment model. Since the third-order gradients of the gravitational potential are not yet observed by satellite sensors, we synthesise them at the satellite altitude of 250 km from a global gravitational model up to the degree 360 while adding a Gaussian noise with the standard deviation of 6.3 × 10⁻¹⁹ m⁻¹ s⁻². Results of the numerical analysis reveal that the arithmetic mean model provides the best solution in terms of the RMS fit between predicted and reference values. We explain this result by the facts that the conditions only create additional stochastic bindings between estimated parameters and that more complex numerical schemes for the error propagation are unnecessary in the presence of only a random noise.