A Comparison of Bjerhammar's Methods and Collocation in Physical Geodesy.
Abstract
In 1963 A. Bjerhammar solved the geodetic boundary value problem by applying Poisson's integral equation for a finite set of observed free-air gravity anomalies. Due to the relation between the number of observations (m) and the number of chosen unknowns (N) different solutions are obtained: non-singular (m = N), least squares (m > N) and minimum norm solutions (m < N). In the special case N approaches infinity, it is shown that the Bjerhammar solution with Poisson's kernel and a solution by collocation with the corresponding kernel are identical. Bjerhammar's method is generalized by using other kernel functions, and each minimum norm solution is shown to correspond to one specific set of degree variances in collocation. The impulse approaches (reflexive prediction, Dirac method) of Bjerhammar are presented. In the theoretical case with a continuous coverage of observations at the surface of the earth, it is shown that both the Dirac method and collocation give a unique solution for any choice of positive degree variances of the kernel functions, whenever the solutions exist. However, the intermediate solutions for Delta g star and X at the Bjerhammar sphere do not exist in general. If collocation is applied by solving the Wiener-Hopf integral equation, a convergent solution is proved outside a sphere. However, inside the bounding sphere of the earth the convergence is still not proved.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 01, 1978
- Accession Number
- ADA063194
Entities
People
- Lars Sjoeberg
Organizations
- Ohio State University