The altimetry-gravimetry problem using orthonormal base functions
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Date
1986-12
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Ohio State University. Division of Geodetic Science
Abstract
This dissertation was undertaken in view of finding a numerical solution on a spherical earth of a mixed boundary value problem, the one of altimetry-gravimetry which is defined from gravity anomalies determined by gravimeter measurements mostly on continents and from geoid undulations known on the oceans from the methods of satellite altimetry. The disturbing potential is represented by an expression of new orthonormal base functions over the sphere. These new base functions are formed using the Gram-Schmidt orthonormalization process applied to the spherical harmonics base functions. Also the Orthonormalization process needed to be applied to mixed domains. The new orthonormal base functions are related to the integration of two associated Legendre functions. This integration is computed using newly developed recursive relations similar to the ones integrating one associated Legendre function developed by Paul (1978). Then the fast Fourier transform is used in a similar way as the spherical harmonics analysis and synthesis. The result of this solution to the "altimetry-gravimetry problem" is a set of coefficients of the new orthonormal base functions. These coefficients were retransformed into the ones of the usual spherical harmonics expansion. The spherical harmonic coefficients can then easily be analyzed and compared with existing earth's gravity field expansions. This method is a "Least-Squares method" solution but it is different than a "Least-Squares adjustment". It is stressed that the Least-Squares method i.e. minimizing the integral and not the sum of the squares of the residuals is solved using orthonormal base functions. It is the solution that has been numerically applied here but it should be emphasized it is also the solution that permits the computation of the usual spherical harmonic geopotential coefficients in the classical single boundary value problem in physical geodesy. Numerical tests show that this Least-Squares method can solve the altimetry-gravimetry problem.