Green's functions in physical geodesy
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Date
1965-06
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Publisher
Ohio State University. Division of Geodetic Science
Abstract
Green's Functions in Physical Geodesy: Of the two main approaches to boundary value problems--the use of Green's functions and of integral equations--the latter approach has been chosen almost exclusively for the geodetic boundary value problem. The present report shows how Green's functions may be used for re-deriving known formulas and also for obtaining new results. Formulas for the third boundary value problem for the sphere are developed and then specialized to Stokes' problem. As a limiting case, the boundary value problem for the plane is briefly considered. Then a formula for the variation of Green's function with the boundary surface is developed and applied to the problem of Molodensky. The Computation of the External Gravity Field and the Geodetic Boundary Value Problem: In the usual way of computing the external gravity field, the earth is considered as a level surface although, strictly speaking, the free-air gravity anomalies refer to the non-level physical surface of the earth. The main purpose of the present paper is to give formulas and, as an appendix, some estimates for the effect of topographic height on these computations. An application of Green's identities yields direct, but complicated, formulas for the effect of the disturbing potential T and the gravity anomaly Δg outside the earth. A simpler solution for T is obtained through the use of a fictitious surface layer, a coating, on the earth's physical surface. A third method, which seems to be optimal for practical computations, is a free-air reduction to sea level. The accurate performance of this reduction is a problem of downward continuation of Δg, which may be handled by iterative solution of a simpler integral equation. After this reduction, however, the conventional spherical formulas can be applier. In addition, the paper presents connections between the determination of the external gravity field from surface data, which is related to the conventional boundary value problems of potential theory, and the determination of the earth's physical surface itself, which is specifically geodetic boundary value problem.
Description
Prepared for Air Force Cambridge Research Laboratories, Office of Aerospace Research, United States Air Force, Bedford, Massachusetts: Contract No. AF 19(628)-2771, Project No. 7600, Task No. 760002