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Green's function solution to spherical gradiometric boundary-value problems
Abstract. Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {$ Γ_{rr} $}, {$ Γ_{r} $θ,$ Γ_{r} $λ} and {$ Γ_{θθ} $−$ Γ_{λλ} $,$ Γ_{θλ} $}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {$ Γ_{r} $θ,$ Γ_{r} $λ} and {$ Γ_{θθ} $−$ Γ_{λλ} $,$ Γ_{θλ} $}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component $ Γ_{rr} $. The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution.
Green's function solution to spherical gradiometric boundary-value problems
Abstract. Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {$ Γ_{rr} $}, {$ Γ_{r} $θ,$ Γ_{r} $λ} and {$ Γ_{θθ} $−$ Γ_{λλ} $,$ Γ_{θλ} $}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {$ Γ_{r} $θ,$ Γ_{r} $λ} and {$ Γ_{θθ} $−$ Γ_{λλ} $,$ Γ_{θλ} $}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component $ Γ_{rr} $. The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution.
Green's function solution to spherical gradiometric boundary-value problems
Martinec, Z. (author)
Journal of Geodesy ; 77
2003
Article (Journal)
English
BKL:
38.73
Geodäsie
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