Evaluation of three-dimensional deformation fields of the earth by methods of projective differential geometry;rigid rotations of the earth

: pp. 25 – 38
Department of Geodesy and Cartography, National University of Water and Environmental Engineering

Aim. The aim is this research is the evaluation of the Earth’s rigid rotation as a component of global deformation fields in interconnection with angular distortions of the geocentric spatial coordinate system. Methodology. Solutions will be achieved by methods of  projective differential (metric) geometry based on the differential presentation of transformations of Riemannian space images in the form of its complicated diffeomorphic manifolds. Based on the essence of the  International Terrestrial Reference System (ITRS).in which the input data are given, and on a global scale the  deformation fields, as a Riemannian manifold it is defined as a the tangent of Euclidean space. To solve the problem are used the methods of description the change of the Riemannian metric in the tangent Euclidean space,  is parameterized by the Cartesian coordinate system. Results. The  basis of the methods used enabled  the results, which are the  terms of its content. The practical application has a dual interpretation. In the former , an expression of angular distortions for needs of the deformation analysis is derived from formulas for angles of the rigid Earth's rotation into projections on ITRS coordinate planes. At the same time, it is proven that these angles are indicators of the coordinate system distortion. The hypothesis of probable deformations of the spatial geocentric coordinate system is substantiated by the geophysical content of the ITRS concept. The identity of conditions of the Earth's parameterization by ITRS and of the tangent Euclidean space parameterization by the Cartesian coordinate system has been proven. On this basis, the truthfulness of the hypothesis can be verified by empirical values of angles that are defined from results of GNSS-observations. In this case of   significant  importance, they are indicators of angular distortions of the ITRS system or an  expression by deviations from the axes orthogonality in its ITRF version as measures of the oblique-angled Cartesian system into the any epoch of observation that follows. Using  methods of projective differential geometry the formulas are obtained  for the coordinate axes directions of the deformed system. Scientific novelty. It is proven that the approach for  solving the problem of the deformation analysis in geodynamics based on the Riemannian geometry it is generalizing relative to its use. On this basis, prospects for filing of deformation fields by nonlinear functional models are substantiated. Practical significance. The obtained results are designed to be used for the  evaluation of global deformation fields of the Earth and solving problems of the modern geodesy in its interconnection with geodynamics in the context of  reference frame research. All analytical expression of angular distortions is  given in general form, which is able to transfer the nonlinear deformation tendencies.
A methodology of the deformation analysis is adapted to be used as  input data for the results of the Global Navigation Satellite System (GNSS)  monitoring station coordinates, taking into account the probable ITRS angular distortions.

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