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  • Author or Editor: L. Bányai x
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The exact least squares line fit with errors in both coordinates (Reed 1992) is investigated together with the approximate solution based on the formalism of the linear Gauss-Helmert model or the unified adjustment approach of the classical textbook by Mikhail (1976). The similarities and the differences are described in details. In spite of the small differences the exact solution is preferable and the calculations are simpler.This paper does not deal with the errors-in-variables (EIV) models solved by the total least squares (TLS) principle, since the exact line fit solution is used to validate this general approach, which is basically designed to solve more sophisticated nonlinear tasks.In the most general case the fit of Person’s data with York’s weights is iteratively solved starting with the arbitrary zero initial value of the slope. The test computation with different but systematically chosen weights proved that in special cases — e.g. the weighted least squares sum of the distances between the data points and the estimated line is minimised — there is no need for iterations at all.It is shown that methods described by Detrekői (1991) and Závoti (2012) are special cases of the general exact solutions.Reed (1992) derived the variances of the slope and intercept parameters without their covariance. The simple linear estimation of variance-covariance matrix of the exact solution is also demonstrated. The importance of the stochastic models coupled with exact solution is also demonstrated.

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The theoretical and practical background of the similarity transformation together with the simultaneous estimation of local geoid undulations is presented.The mean features of the traditional network adjustment on the local ellipsoids are summarized and the different Hungarian networks and known geoid solutions are shortly described as the basic data of the test computations.The eigenvalue and eigenvector decomposition revealed that the seven parameter similarity transformation cannot be applied together with the simultaneous local geoid estimation because the rotations about the X and Y axes significantly destroy the condition of the normal equations.However, the replacement of the rotations about the X, Y and Z axes by the rotation about the ellipsoidal normal of the datum point can provide a very well conditioned solution, which takes into account the special role of the datum point of the astro-geodetic network adjustment.Based on the unit weights of the input data an optimal adjustment strategy is demonstrated from a computational point of view, where the five transformation parameters can be estimated together with a very large number of local geoid undulations. The geoid has to be known in the global reference system. The geoid unknowns describe only the relative position of this known geoid with respect to the local reference system.The application of the available and the simultaneously estimated local geoid solutions proved that neglecting local geoid heights has a most significant impact on the scale parameter, while it has no significant effects on the horizontal residuals from the statistical point of view.The small scale difference (1 ppm) and the small rotation (-0.5 arc sec) about the datum point and its ellipsoidal normal of the Hungarian local system with respect to the global GPS system demonstrate the high quality of the traditional measurements as well.

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The assessment of the results achieved in our division at the field of physical geodesy is summarised. The studies at the fields of Earth rotation, Earth tides, Earth gravity and the global and regional Earth deformations related to geodesy and geodynamics are presented by citation of the most important publications and dissertations, which cover the history of our institute. The paper ends with the case study related to the newly developed full roving GPS observation strategy.

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B. Hofmann-Wellenhof, K. Legat, M. Wieser: Navigation. Principles of Positioning and Guidance. Springer-Verlag, Wien, New York, 2003, 427 pages, 99 figs; W. Schröder ed.: Zur Entstehung der solar-terrestrischen Physik/Some aspects of the earlier history of solar-terrestrial physics. Beitrage zur Geschichte der Geophysik und kosmischen Physik, Vol. 5, No. 3., Science Edition, 2004, pp. 150

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