First-order derivatives of principal and main invariants of gravity gradient tensor of the tesseroid and spherical shell

The invariants of gravity (or gravitational) gradient tensor can be applied as the additional internal parameters for the gravity gradient tensor, which have been widely used in the recovery of the global gravity field models in geodesy, interpretation of geophysical properties in geophysics, and gravity matching in navigation and positioning. In this contribution, we provide the general formulae of the first-order derivatives of principal and main invariants of gravity gradient tensor (FPIGGT and FMIGGT), where their physical meaning is the change rate of the invariants of gravity gradient tensor, and their expressions consist entirely of gravity gradient tensor and gravitational curvatures (i.e. the third-order derivatives of gravitational potential). Taking the mass bodies (i.e. tesseroid and spherical shell) in spatial domain as examples, the expressions for the FPIGGT and FMIGGT are derived, respectively. The classic numerical experiments with the summation of gravitational effects of tesseroids discretizing the entire spherical shell are performed to investigate the influences of the geocentric distance and latitude using different grid resolutions on the FPIGGT and principal invariants of gravity gradient tensor (PIGGT). Numerical experiments confirm the occurred very-near-area problem of the FPIGGT and PIGGT. The FPIGGT and PIGGT of the tesseroid using the Cartesian integral kernels can avoid the polar-singularity problem. Meanwhile, the finer the grid resolution, the smaller the relative approximation errors of the FPIGGT. The grid resolution lower than (or including) 1 × 1 at the satellite height of 260 km provides satisfactory results with the relative approximation errors of the FPIGGT and PIGGT in Log scale less than zero. The proposed first-order derivatives of principal and main invariants of gravity gradient tensor will provide additional knowledge of the gravity field for geodesy, geophysics, and related geoscience applications.