Change of boundary: towards a mathematical foundation of global gravity models

Abstract

Change of boundary is a method that iteratively downward continues data from a star-shaped boundary to a regular surface, such as the sphere or the ellipsoid of revolution, and then solves a boundary value problem using the downward continued data on the changed boundary. Although the method belongs to concepts involved in the development of recent high-degree Earth’s gravity models, it is still unclear whether the iterations converge to the true solution for boundaries as complex as, for instance, the Earth’s surface. In this paper, we revisit the method and show that convergence in terms of the Cesaro limit can be achieved under the assumption that the operator performing the iterations is non-expansive. The validity of the assumption is, however, still not proved. Therefore, we examine the hypothesis numerically using boundaries of various complexity. We start with a simple synthetic topography defined by a Legendre polynomial and move to more realistic finite-degree shapes of the asteroids Bennu and Eros. The numerical experiments indicate that the assumption is valid as long as the boundary deviates not too far from a sphere and the truncation degree of the gravity model is not too high (the experiments with the synthetic topography and Bennu). Otherwise, the hypothesis seems to be false (the Eros case). Finding an analytical condition to separate between shapes for which the change of boundary method converges/diverges remains an open issue.