Analysis of modern models of counterfeiting surfaces for determination of heights by GNSS-leveling method
Authors:   Lviv Polytechnic National University, FEDORCHUK A. 
Published: September 2022

This paper discusses various sources of information related to the issue of determining heights by GNSS-leveling. Implementation of the GNSS leveling method requires the presence of geoid or quasi-geoid heights, which today can be obtained from the corresponding models. In recent decades, scientists from around the world have developed many global, regional and local models of geoid and quasigeoid. This has contributed to the emergence of numerous publications on GNSS leveling. The purpose of the work is to perform the analysis of modern models of reference surfaces on the basis of materials of scientific publications according to the criteria that have a significant impact on research in the field of determining the height by GNSS-leveling. Method. The paper considers 44 publications published in the period 2001–2021. Among the research in this direction, there are three types of work: 1) 13 publications on methods of constructing the models themselves; 2) 12 works on their accuracy analysis and 3) 19 works on “adjustment” of model heights. At the first stage of the research the analysis was performed according to criteria related to geoid and quasi-geoid surface modeling, including Stokes and Molodensky theory, mathematical methods of data analysis and processing, tidal systems, zero-order geoid undulation and scale levels of geoid and quasi-geoid models. In the second stage, the frequency of publications by years was analyzed and the activity of publications of existing geoid and quasigeoid models on the sample of countries, which was made on the basis of all selected works in this study, was established. At the third stage, a quantitative analysis of official published models of geoid and quasi-geoid on the frequency of publications for the study period. The ratio of the accuracy of the heights of global geoid models in relation to the magnitude of the degree and order of their calculation is established. Results. In 58 % of the analyzed works, the authors use Stokes’ theory in their research, and in 42 % – Molodensky’s theory. Among the mathematical methods of data analysis and processing, 27 % use the method of mean square collocation, 20 % use the least squares method, the “delete-restore” method and the method of modifying Stokes’ formula with least squares (or KTH-method). The Fast Fourier transform method was used in 13 % of the works. Publications on the creation of global models of the Earth mostly use the parameters of the tide free tidal system in their calculations – a total of 40 % works. No less important criterion (33 % of works) should be taken into account the parameter of undulation of the geoid of zero order (“zero degree term”). In total, 41 % of research is aimed at creating and analyzing quasi-geoid models on a regional scale. During the study period, most works were published in 2012 and 2018. Leading countries in the development of quasi-geoid models are Canada, Poland, Sweden and the United States, and global geoid models are Germany, the United States and China. For the period 2001–2021, 99 global geoid models of various degree and order were officially presented, among which GOCO, EIGEN and EGM series models are most often used for research. Also during this period, 177 quasi-geoid models were presented, the most of which were published in 2019. Based on these data, the relationship with the frequency of publications in 2008–2021 can be traced. The accuracy of global geoid models in relation to the degree and order of their calculation is systematic in the range of 0.52–1.92 m, 0.38–0.50 m, 0.23–0.38 m and 0.12–0,14 m for models 60-220, 220-260, 260-720 and 720-2190 degree and order respectively. Scientific novelty. It is shown that the analysis of modern models of reference surfaces on the basis of materials of scientific publications in the field of GNSS-leveling method allows establishing the existing advantages and disadvantages of research in this field. Practical significance. The data of such analysis can be used to solve key problems in determining the height of the GNSS-leveling method, which require further research and to find modernized solutions.