The datum choice problem in Geodesy
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This contribution presents a theoretical review of the datum problem in Geodesy, inside the framework of the Gauss-Markov model. In this review, it is mathematically demonstrated that a solution with constraints is biased when the rank of the design matrix is not full, i.e., . In other words, there is a rank defect in the design matrix due to the datum deficiency in the geodetic network, which generally occurs at least in terms of the origin. The minimum inner constraints are mathematically demonstrated through the conditions of no net translation (NNT), no net rotation (NNR), and no net scale (NNS). Furthermore, we provide a step-by-step S-transformation to change or update the datum from a minimally constrained solution to another. In particular, the cases of geodetic deformation analysis, where different observation epochs must be at the same datum, and the geodetic network densification based on pre-existing reference points are discussed. A simulated trilateration network demonstrates several possibilities for defining the datum matrix. Finally, comparisons were made between five different types of constraints in a levelling network: absolute constraint; weight constraint; NNT condition for all network points; NNT condition only for reference points and weighted or generalized NNT condition according to Kotsakis (2013). The results obtained by Monte-Carlo simulations demonstrate that the generalized NNT condition provides the best datum choice in the densification of geodetic networks problem.
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