Gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method

Abstract

AbstractThe paper concerns a novel concept based on the iterative approach applied for solving the nonlinear satellite-fixed geodetic boundary value problem (NSFGBVP) using the finite element method (FEM). We formulate the NSFGBVP that consists of the Laplace equation defined in the 3D bounded domain outside the Earth, the nonlinear boundary condition (BC) prescribed on the disretized Earth’s surface, and the Dirichlet BC given on a spherical boundary placed approximately at the altitude of GOCE satellite mission and additional four side boundaries. The iterative approach is based on determining the direction of actual gravity vector together with the value of the disturbing potential. Such a concept leads to the first iteration where the oblique derivative boundary value problem is solved, and the last iteration represents the approximation of the actual disturbing potential and the direction of gravity vector. As a numerical method for our approach, we implement the FEM with triangular prisms. High-resolution numerical experiments deal with the local gravity field modelling in the Andean, Himalayan and Alpine mountain range, where we focus on the contribution to the disturbing potential solution by solving the nonlinear geodetic boundary value problem in comparison with the solution to the oblique derivative geodetic boundary value problem. The obtained results showed the maximal contribution of the presented approach 0.0571 [$${\text{m}}^{{\text{2}}} {\text{s}}^{{{\text{ - 2}}}}$$ m 2 s - 2 ] (Andes), 0.0702 [$${\text{m}}^{{\text{2}}} {\text{s}}^{{{\text{ - 2}}}}$$ m 2 s - 2 ] (Himalayas), 0.0066 [$${\text{m}}^{{\text{2}}} {\text{s}}^{{{\text{ - 2}}}}$$ m 2 s - 2 ] (Alps) in the disturbing potential, that is located in the areas of the highest values of the deflection of vertical.