Mathematical and cartographic modelling of the deformation of the earth's surface in Europe using GNSS data
Authors: Tadyeyev Oleksandr 
Published: January 2025

the article is devoted to a scientific and methodological study of the problem of evaluation and graphically representation of deformations of the earth's surface using GNSS data.

The purpose of the study: testing the methodology of mathematical and cartographic modeling of horizontal deformations of the earth's surface in projection onto the geosphere using the example of the territory of Europe with the prospect of expressing the nonlinear deformation patterns.

Research result. The results of observations at GNSS stations of the EPN network during 2008–2014 were used as input data for testing of the methodology. The station coordinates were selected from the JPL Comb database of the SOPAC archive. The choice of territory and input data is justified by previously conducted studies on their basis to determine the optimal model surfaces for the evaluation of deformations at the regional level, as well as by studies of the prospects for evaluation of nonlinear deformations.

An algorithm for evaluation the deformations of the earth's surface in the projection onto the geosphere was used. The generalization of empirical discrete values of station coordinates and their displacements on the geosphere is implemented by basis functions in the form of series of spherical functions of different degrees and orders using the least squares method. The mean square errors of approximation of the explicit expression of the series of spherical functions were used as the criterion of optimality for selection of the final results.Based on this indicator, the corresponding optimal empirical formulas of the functional model were generated with subsequent evaluation of the tensor and deformation characteristics. The following deformation characteristics were evaluated: dilation, extreme expansions (stretching and compression), shear, azimuth of the main axis of deformation, rigid rotation of part of the earth's surface as an absolutely solid body.

The described algorithm for evaluation of deformations was applied under conditions of different degrees of generalization (detailing) of input data. This was achieved by the abstraction method at the stage of selecting the regions of homogeneous deformations on a probabilistic-statistical basis through the use of the sliding dispersion algorithm. Different degrees of generalization were determined by changing of the radius of averaging the dispersions of the GNSS station displacements. First of all, the value of the radius = 2800 km was empirically determined, at which the studied territory was not divided into homogeneous regions. Thus, complete smoothing of the empirical data was achieved. The characteristics calculated under this condition revealed the trend - the general pattern of deformation of the earth's surface in Europe. The value of radius = 2100 km conditionally determined a high degree of generalization of the input data and ensured the division of the territory into five regions of homogeneous deformations. The average degree of generalization of the input data was achieved at a radius value of = 1400 km. Under this condition, eight regions of homogeneous deformations were identified. For each selected region, the deformation characteristics of the corresponding part of the earth's surface were calculated according to the algorithm described above. The vast majority of the results obtained in this way showed the nonlinear patterns of the deformation of the earth's surface in Europe.

The characteristics calculated in relation to the selected regions are presented in the form of thematic schematic maps of the deformation of the earth's surface. The resulting cartographic products provided the systematization of the territory of Europe into nominally stable and potentially dangerous from a tectonic point of view components under conditions of different degrees of generalization of GNSS data.

Scientific novelty: thematic schematic maps of horizontal deformations of the earth's surface in Europe in the projection onto the geosphere with different degrees of generalization (detailing) of the input data have been created.

Keywords: GNSS, geosphere, spherical function, deformation, thematic map.