Discrete Physical Geodesy
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Date
1987-06
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Publisher
Ohio State University. Division of Geodetic Science
Abstract
Discrete gravity information on the surface of the earth (and in space) is used for solutions of the free boundary value problem in physical geodesy. The primary tool for these solutions is a harmonic embedding with the use of a mathematical sphere. (For local applications, a mathematical ellipsoid is to be preferred.) Various discrete techniques in physical geodesy are displayed. L2-norm minimizations on an embedded sphere is found to be equivalent to a simple technique with impulses on the reflected topography. This equivalence is also valid for a non-spherical external surface. Predictions with the use of harmonic embedding are not invariant with respect to the choice of the radius of the embedded sphere and a technique for the determination of the optimal radius is outlined. L2-norm minimizations on non-spherical surfaces are given a tentative approach. Methods for renormalization of integral equations are used in order to obtain highly simplified solutions. This technique seems most promising for observations given in an equal area approach. The combination of local discrete observations and global spherical harmonics is explored by harmonic embedding. The difficulties of a mixed boundary value problem are recognized for a case where continuous data are given inside finite strictly defined surface elements. These difficulties are taken care of in the discrete case (with infinitesimal surface elements) by the use of a harmonic embedding. "Autoprediction" is used in order to design highly accurate predictors for smooth external surfaces. A technique with moving weighted averages gives exact predictions by the combined use of positive and "negative weights" for equidistant observations of low 'frequencies'. Higher frequencies are taken care of in a promising way by "the moving procedure". The fixed boundary value problem is discussed to some extent. It seems to deserve special interest for modern GPS-applications. Harmonic embedding is also here used in a discrete approach. The discrete methods are given the most sophisticated application in a relativistic geodesy. Some fundamental derivations are presented. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]