A Finite Element Model Of The Earth's Anomalous Gravitational Potential
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Date
1988-06
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Ohio State University. Division of Geodetic Science
Abstract
This study addresses certain theoretical and numerical issues regarding the application of the finite element method to geodetic boundary value problems. It follows, to a large extent, the effort of Meissl (1981). The quasi-harmonic equation, controlled by boundary conditions on a potential and its normal derivative, is recast into a weak formulation which is seen to be sufficiently general to treat a number of different geodetic boundary value problems involving gravity anomalies, geoid heights and radial components of the gravity disturbance vector, on either a spherical or ellipsoidal boundary. Discretizing the weak formulation by expanding the potential into a series of piecewise-continuous. Hermite tri-cubic basis functions leads to a symmetric. positive-definite system of linear equations to be solved for the finite element nodal parameters. A spectral analysis of the spherical harmonic expansion of the potential permits a measure of the error of omission of the finite element model, expressed as a function of the width and altitude the of finite element. A finite element model of the anomalous gravitational potential was calculated from the numerical solution of the Laplace equation constrained by gravity anomalies on a bounding sphere. The spherical harmonic expansion of Rapp (1978), truncated to even degrees ranging between 2 and 16, served as reference fields. The geoid heights of the finite element model ranged between -40 and 28 meters with an RMS value of 10.4 meters, compared to a range of -101 to 78 meters and an RMS geoid height of 28.9 meters for the degree 16 reference field, which had the lowest recovery error. Finite element modeled gravity anomalies ranged from -32 to 31 milligals with an RMS value of 7.6 milligals. The reference field had gravity anomalies between -48 and 37 milligals and RMS gravity anomaly of 13 milligals. The low degree basis functions inherent to the finite element method enable rapid synthesis calculations of geodetic signals. In a point-by-point mode, five signals (geoid heights, gravity anomalies and components of the gravity disturbance vector) can be computed approximated 50 times faster with a finite element model of equivalent resolution, than when using a 180 degree spherical harmonic expansion. The calculation of signals at the nodes of a one degree grid is approximately 30 percent faster than the accelerated spherical harmonic procedures.