Gravity field determination using boundary element methods

Abstract

The Boundary Element Method (BEM), a numerical technique for solving boundary integral equations, is introduced to determine the earth's gravity field. After a short survey on its main principles, we apply this method to the fixed gravimetric boundary value problem (BVP), i.e. the determination of the earth's gravitational potential from measurements of the intensity of the gravity field in points on the earth's surface. We show how to linearize this nonlinear BVP using an implicit function theorem and how to transform the linearized BVP into a boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation into a linear system of equations. We discuss the major problems of this approach for setting up and solving the linear system. The BVP is numerically solved for a bounded part of the earth's surface using a high resolution reference gravity model, measured gravity values of high density, and a 50 ⋅ 50 m² digital terrain model to describe the earth's surface. We obtain a gravity field resolution of 1 ⋅ 1 km² with an accuracy of the order 10⁻³ to 10⁻⁴ in about 1 CPU-hour on a Siemens/Fujitsu SIMD vector pipeline machine using highly sophisticated numerical integration techniques and fast equation solvers. We conclude that BEM is a powerful numerical tool for solving boundary value problems and may be an alternative to classical geodetic techniques.