Approximate continuation of harmonic functions in geodesy: A spline based least squares approach with regularization

This paper is concerned with the approximate reconstruction of the earth’s potential field from geometric and gravimetric data. This is an ill-posed problem involving typically large amounts of data which are to be continued by a harmonic function. The standard approach in geodesy is based on spherical harmonics which are globally supported. Thus, a least squares approach for the data fitting yields a linear system of equations with a fully populated system matrix. This becomes computationally prohibitive for large amounts of data and, therefore, presents the biggest bottleneck for fast and efficient computations. Motivated by the early work [30], we propose in this paper an alternative and pose the harmonicity requirement on the continuation together with the data fitting as a minimization problem for a least squares functional with regularization involving the Laplacian. This approach enables the use of locally supported functions in the reconstruction for which we employ tensor products of cubic splines. The linear system resulting from the weighted least squares approach is therefore sparsely populated which allows for iterative solvers of complexity proportional to the total number of unknowns. We extensively study the choice of the regularization parameter balancing the data fit and the harmonicity requirement for both synthetic as well as earth potential data. We compare the results with discretizations using finite differences and finite elements for solving Laplace’s equation.