Higher-order gravitational potential gradients by tensor analysis in spherical coordinates

Previous studies mainly focused on second- and third-order gravitational potential gradients that these values are already measurable or that appropriate measurement principles are being developed. Although the higher-order (i.e. the order is larger than or equal to four) gravitational potential gradients cannot be observed now, with the development of science and technology, it will be hoped to measure these parameters in the future. Once the values of higher-order gradients are observable, they can express the higher frequency of the Earth’s gravity field and have the higher sensitivity to the shallower structures of the Earth’s subsurface, which will be applied for the global and regional gravity field modelling and geoid determination in geodesy and the mass density mapping of the subsurface and shallower structures in geophysics. In this paper, the higher-order gravitational potential gradients in spherical coordinates are focused on by tensor analysis. Firstly, the rule of the covariant derivative of a tensor is revised based on Casotto and Fantino (2009). Secondly, the general expressions for the natural components of the fourth-order up to seventh-order gravitational potential gradients are derived based on the revised rule. Specifically, we derive the expressions for physical components of the fourth-order gravitational potential derivatives as an example. Thirdly, Laplace’s equation with a uniform tesseroid using the spherical integral kernels has been applied to validate these newly derived expressions’ correctness. The expressions for the physical components of higher-order gradients up to m order can theoretically be derived based on this paper’s research results.