Numerical Solution of the Fixed Altimetry-Gravimetry BVP Using the Direct BEM Formulation

Abstract: A direct formulation of the boundary element method (BEM) for the Laplace equation leads to an integral relationship that represents a superposition of the single-layer and the double-layer potential. Such boundary integral equation (BIE) can be applied to the altimetry-gravimetry boundary value problem of the third kind (AGBVP-III) that is formulated for mixed boundary conditions (BC), i.e., (i) the Dirichlet BC at oceans/seas where the disturbing potential can be evaluated on a mean sea surface (MSS) using the global geopotential models (GGMs), and (ii) the oblique derivative BC on lands in the form of surface gravity disturbances that can be obtained from terrestrial gravimetric measurements located by means of precise satellite positioning. Their 3D positions together with MSS models available from altimetry determine the Earth’s surface as a fixed boundary. In our approach the collocation technique with linear basis functions is used to discretize BIEs and transform them to a linear system of equations. The input surface gravity disturbances at collocation points on lands are simulated from GGMs. Large-scale computations performed on parallel computers result in numerical solutions directly on the complicated Earth’s surface. The surface gravity disturbances at oceans/seas and the disturbing potential on lands are the numerical results. They are compared with values evaluated directly from GGM using spherical harmonics. The differences indicate a discrepancy between the normal and actual gravity field. An additional constant is added to the Dirichlet BC in order to find an agreement between the gravity disturbances at oceans/seas computed by our approach and the ones induced by the prescribed Dirichlet BC