Tesseral harmonic coefficients and station coordinates from satellite observations by collocation
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Date
1974-12
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Publisher
Ohio State University. Division of Geodetic Science
Abstract
The usual formulation of the least squares collocation method requires the inversion of matrices with dimensions equal to the number of observations. Such a procedure poses difficult numerical problems when large numbers of observations are involved. It is shown in this report how this difficulty can be overcome. The size of the matrices to be inverted can be reduced to the number of unknown quantities, i.e. the amount of numerical work will be the same as for the corresponding adjustment problem. A comparison between the adjustment and the collocation solution shows that the basic structure of the two approaches is very similar and that existing algorithms for the adjustment solution can easily be implemented to include the collocation solution. Furthermore, the results of the adjustment solution can be used to compute the collocation solution without additional matrix inversions by a Neumann series expansion. If a further reduction of matrix dimensions is necessary a method similar to matrix partitioning can be used to determine tesseral harmonic coefficients and station coordinates separately. A similar partitioning into two groups of unknowns is achieved with an iterative procedure. It is shown that the iteration converges to a solution which is identical to that obtained by the direct approach. Numerical questions as the treatment of orbital parameters, the determination of the speed of convergence, the stability of different solutions, and the choice of a covariance function are discussed. The determination of tesseral harmonic coefficients from a combination of satellite and gravimetric data is treated in some detail and the results are related to previous work. The proposed method uses a very general model and can be applied to problems other than the one treated in this report.