Aim. Analysis of the current state of solving the problem the estimating of earth deformation fields based on continuum mechanics, improving traditional methods the estimating of horizontal deformations, definition an alternative approach and justification of ways and the algorithm of solving the problem based on it. Methodology and results. The analysis showed some shortcomings of traditional solving of the problem. In order to minimize their impact the improved method of mathematical and cartographic modeling of linear deformations is proposed. The essence of improvements consists in the necessity of a priori statistical test of conditions of the locally homogeneous linear model and forming finite elements of the surface on its results in the particular implementation of deformation fields. The finish result of geodetic data processing is a synthetic inventory map of decisions. Some results of the method approbation in Europe are shown. An improved method provides reliable indicators of deformations of the surface where the linear homogeneous hypothesis is confirmed. But do not allow to fully estimating the deformation of the surface of the study area as a whole. In order to avoid this shortcoming, an alternative approach to the solving of the problem is proposed. Prospects of the solving with the geometric point of view based on the projective-differential geometry are substantiated. To search for ways of solving the problem was elected the theory of the surfaces mapping. According to the hypothesis that distortions of the initial surface in the transition to the mapping surface are caused by the geodynamic factor, this approach allows to generate the mapping tensor (deformation tensor) and submit distortions by different numerical characteristics. A tensor defines the function that implements the mapping. Its components are partial derivatives of the function of deformed surface coordinates from her initial coordinates. The theory of mapping does not limit the class of such functions, but only imposes on them homeomorphism conditions. This allows you to transfer deformations by nonlinear functional models. Depending on types of geodetic data defined the ways of solving the problem of deformation fields estimating. Data types define geodetic reference frame surfaces with corresponding to them coordinate systems. The choice of coordinate systems was associated with types of parameterizations and mapping of surfaces. Mathematical solving of the problem on the plane in a rectangular system (quasiconformal mapping of Riemann surfaces parameterization) and also on the geosphere and ellipsoid of revolution in corresponding curvilinear coordinate systems (mapping of surfaces with isometric parameterization) are obtained. Prospects of using the theory of mapping into estimating of spatial earth’s deformations in the geocentric coordinate system are substantiated also. Originality. Solving of tasks the estimating of earth’s deformation fields been achieved by methods of the projective-differential geometry on an alternative theoretical basis - the theory of the surfaces mapping. Practical significance. The chosen alternative approach has greater potential capabilities compared with traditional which is based on the linear homogeneous model of continuum mechanics. The obtained solving makes it possible to estimate the deformation fields within the framework of any empirical nonlinear functional models only on homeomorphism conditions of surfaces mapping. On this basis, a general algorithm for solving the problem is generated.
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