The gravitational potential and its first- and second-order partial derivatives of an ellipsoidal tesseroid based on the Cartesian integral kernel

Gravity forward modelling is a fundamental problem in the fields of geophysics and geodesy at regional and global scales. Considering the curvature of the Earth, tesseroids are suitable to accurately simulate the theoretical gravity field. In general, the spherical tesseroid is regarded as an ideal model, but it cannot consider the oblateness of the Earth. Therefore, we define an ellipsoidal tesseroid at the local Cartesian coordinate system. Then we propose the formulas of the gravitational potential and its first- and second-order partial derivatives of the ellipsoidal tesseroid based on the Cartesian integral kernel. To enhance the practicality, we approximate the ellipsoidal tesseroid to the spherical tesseroid and derive the formulas of the gravitational potential and its partial derivatives. Moreover, we discuss the formulas of the gravity field for the model with linear variable density. The ellipsoidal tesseroid, which is selected as the fundamental mass element, can more accurately simulate the gravity and gravity gradient anomalies of the Earth. Compared with methodologies that make use of integral kernels expressed in spherical coordinate system, the formulas based on the Cartesian integral kernel are given in compact and computationally attractive form. Besides, these formulas can avoid the polar singularity of the spherical coordinate system. The numerical simulation and comparison with previous methods validate the new ellipsoidal tesseriod formulas.