№ 84 (2016)

Journal “Geodesy, Cartography and Aerial Survey” –
Technical requirements for papers

Requirements for papers

1. UDC 528.721.287:537.533.35
О. IVANCHUK1*, O. TUMSKA2
1*Department of Photogrammetry and Geoinformatics, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013, tel. +38(096)4143409, e-mail: ivanchuk@inbox.ru
2Department of Photogrammetry and Geoinformatics, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013, tel. +38(050)7455711, e-mail: ol.tums@yandex.ru
DEVELOPMENT AND RESEARCH OF TECHNOLOGY FOR AUTOMATION OF THE CALIBRATION AND ACCOUNT OF DIGITAL SEM IMAGES HAVING GEOMETRIC DISTORTION OBTAINED WITH JCM 5000 (NeoScope) (JEOL, JAPAN)
Purpose. Solids microsurface digital images obtained with scanning electron microscopes (SEM) are characterized by significant geometric distortions, which must be defined and taken into account when determining the quantitative parameters of microsurface solids with high accuracy. Therefore, this problem is so important and urgent, especially during the high-tech quality control of production processes using nanotechnology, particularly in mechanical engineering, aircraft construction, and for creating space and military equipment. It is also important to perform SEM images measurements and obtain quantitative parameters of the studied microsurface in automatic mode that would allow processing of the data much faster without operator intervention. Therefore, the aim of this work was to develop and research the effective technology for automated calibration and account for the SEM images geometric distortion as well as the creation of a software package that would implement it. Methods. Digital SEM images processing and their transformations were used to automate the measurement. Results. The developed automation technology for digital SEM images measurement was tested and studied in the treatment of SEM images of the reference test object with a resolution of r = 1425 lines/mm, obtained with SEM using the JCM-5000 (NeoScope) in the range of magnifications from 1,000x to 15,000x. Accuracy and quantified characteristics of SEM images, in particular, their scales along the axes of the image, as well as their geometric distortion, are comparable to measurements made by hand. Automation of this process eliminates monotonous work, especially in the processing of SEM images in a range of relatively small increases in M=1000h-5000h, significantly reducinge the measurement time, and avoiding human errors that can occur. Scientific novelty. The developed technology for automation of the measurement of digital SEM images and to determine their geometric parameters is executed for the first time in Ukraine. The proposed technological measurement automation scheme of SEM images, and the creation of this authoring software show its efficiency and expediency. The practical significance. This technology allows you to automatically and accurately determine the true value of the SEM images increase (scale), the value of their geometric distortion. These values use determine with high accuracy of the quantitative spatial parameters of micro surface solids to increase the reliability end efficiency devices, mechanisms, and materials made from them.
Key words: scanning electron microscope (SEM); the test object; SEM digital image; geometric distortion SEM images; fractal dimension; measurement automation.

Література – 14.

Вимоги до оформлення статей збірника “Геодезія, картографія і аерофотознімання”

УДК 528.952
N. HRYTSKIV*, L. LAYKUN, L. BABIY
Department “photogrammetry and geoinformatics”, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013, *tel. +38(032)2582616, e-mail Nazar.Z.Hrytskiv@lpnu.ua
MAPPING OF AVALANCHE DANGEROUS TERRITORIES
USING GIS TECHNOLOGIES
Purpose. Avalanching is a natural phenomenon that can cause loss of life, destructions of residential and industrial buildings, inflict losses on forestry, agriculture and tourism development in mountainous areas. Monitoring and research of potential avalanche dangerous territories is an urgent task. Geoinformation modeling plays leading role for its solving. The purpose of this paper is to create raster thematic maps of avalanching hazards on the base of analysis of certain morphometric and environmental factors of avalanching in the territory of the mountain ridge Polonyna Borzhava. Methodology. Digital terrain models were obtained by vectorization of cartographic materials. This created vector models were interpolated into raster model, which became the basis for classification of geomorphometric factors of avalanching. The following factors of avalanche formation have been selected: fixed (hillside slopes, curvature and hills exposure) and variable (direction of winds). Resulting rasters were reclassified for further operations using the tools of map algebra, such as raster calculator. The classes of avalanche dangerous areas, depending on the combination of values of slope and exposure, were sorted in the resulting summation of rasters of slopes, curvature and orientation of slopes and generalization of the number of received classes. Results. A digital elevation model and raster thematic map of avalanche dangerous territories of mountain ridge Polonyna Borzhava were created. Originality. A comprehensive approach that includes integration of cartographic material, statistical meteorological data and some geomorphometric data about surface in a unified geographic information system allocates the territories with consistent snow avalanche phenomena of mountain ridge Polonyna Borzhava. Practical significance. Certain geomorphometric parameters of relief of mountain ridge Polonyna Borzhava were considered to determine areas with different degrees of avalanche hazard. Mapping of snow avalanche phenomena of Polonyna Borzhava ridge in scale of 1 : 50000 was implemented. The thematic map of relative avalanche hazard will allow professionals to focus on specific territories during the planning activity considering potential threats caused by snow avalanches.
Key words: avalanching hazard, thematic mapping, geomorphometric parameters of surface, digital elevation model, GIS-technologies, ArcGIS.

Література – 12.

UDC 528.4: 528.1: 332.3
YU. HUBAR
The Department of Cadastre of Territory of Lviv Polytechnic National University 12, S. Bandera str., Lviv, Ukraine, 79013,
tel. +38 (032) 2582631, e-mail Yurii.P.Hubar@lpnu.ua
THE USE OF UNMANNED AERIAL VEHICLES
FOR ESTIMATING MARKET REAL-ESTATE VALUE
The Purpose. The problem of applying unmanned aerial vehicles for real estate estimations has become important with the development of modern methods of distance sensing of residential areas. For several decades aerial photography has been an effective tool for performing geodetic survey, geophysical studies, and various types of monitoring. Modern technology used in creating topographic and cadastral plans is based on using digital data of aerial photography [Burshtynska H., Stankewych S., 2013]. However, the cost of airplanes and helicopters used in applying the local large-scale survey is costly .The alternative solution is applying unmanned aerial vehicles (UAVs) for real estate estimation. The Methodology. The methodology is based on the necessity of utilizing UAVs for estimation of the market real estate value. Creating large-scale plans of rural areas on the basis of data obtained from UAVs is required for projecting gain plans [Glotov V, Korduba Yu, 2011; Glotov V, Gunina A., 2014]. This is related to the land accounting and setting boundaries in a particular region. This previous sentence is duplicated in paragraph below “In Practical Significance so it is best not repeated in 2 times. UAVaerial photography has additional advantages in comparison with traditional and space photography. The Results. The investigations show the importance of UAV application for real estate estimation. The course of using UAVs is to obtain photos with certain characteristics. Aerial photography is known as a distance sensing technology and economically justified method of collecting spatial information, as the basis for creating cadastral plans and maps, three-dimensional models of the relief and the territory [Glotov V, Tserklevych A. et al., 2014]. It can also be used for estimation of the market value of real estate. Scientific Innovation. The research connected to the use of UAVs for estimation the market of the real estate value has been done for the first time. This article proves that at the time of collection and analysis of basic data in order to improve the quality of raw materials reliability, as well as for rapid and precise geometric characteristics of the studied object and selecting similar objects to the estimative that unmanned aerial devices should be used. The Practical Significance. The UAV application for the estimation of market real estate value is of high economic efficiency. As the result the determination of geometric characteristics is cheaper compared to traditional surveying methods. UAV photography has additional advantages in comparison to traditional methods, , particularly the possibility of obtaining high accuracy images (one point per centimeter) in the territory. It gives the possibility to make detailed photos of small objects and small areas when there is no value or technical possibility to do it by using other methods as in urban conditions. The other advantages of UAV photography are its mobility; high efficiency; and the environmental purity of flights.
Keywords: UAV, estimation of the real estate, distance sensing, aerial photography.
Література – 18.

UDC 528.48: 551.48
M. ZVYAHINA1, I. TREVOHO2, J. KOSTETSKA3
1 SE “CHORNOMORNDIPROEKT”
2 Department Geodesy, Lviv Polytechnic National University
3 Department Engineering Geodesy, Lviv Polytechnic National University
GEODESIC WORK OPTIMIZATION FOR DETERMINING LIKELY MARKS
OF WATER LEVEL IN THE DIFFERENT SYSTEMS OF HEIGHTS
Purpose. There is a need for State leveling network reconstruction through the determination of Reineke – Tenberg mark in BS77 for optimization of geodetic works in marine industry, which will allow determination of BS zero mark in the State Height system BS77 at each water gauge station. The possibility permits the matching of the State Leveling system of the country with leveling systems of the neighboring countries which is expected within framework of fulfilling the UELN project. Methodology and the results of the work. By projecting, construction, and reconstruction of the marine industry objects after height system BS77 implementation, several difficulties arose connected with the use of GNSS – technologies, height network, polygonometry, created in BS, as well as marks of interval water levels at gauge stations. The outstanding difficulties as to heights arose by construction of narrow hydrotechnical objects of substantial amounts, considering the marks by contractions are interval water levels. The geodesist M. S. Zvyahina investigated the problem for the first time in 2006, and not hydrologist was employed. Any data on BS77 and even the name of system BS77 itself were absent in the beginning. Therefore, in the port cities the decisions on the values differences between the systems were not considered by the Department of UAGS (all-union astornomo-geodetic society). SE “CHORNOMORNDIPROEKT” as early as 1983 there were average differences throughout GUGK USSR for the many sea ports excluding those, which had differences from UAGS. However, it became necessary to collect the data as to difference in the height marks values of matched lines, changes of values within the time for different terrains, but in BS77. Then it became obvious, that new system did have its zero. The territories of different ports were searched for stability as well as for changes of values within time for the constant subsiding. The tables for approximations for some marksvary 70 years backward and forward. The researched behavior for the outstandingly subsiding territory – is port Poti and its water gauge station. Using the data of the Ukrainian scientist [Marchenko, Yarema, 2006] on the constant rate of rising level of the Black Sea was determined by the satellite altimetry level and average sea level, through the approximation of the marks of the working and control reference marks, matched to the BS as well as to BS77 for Poti area for the selected 2005 year. The level minus 5,6 cm against level 5,4 cm was determined (Fig. 3) out of [Marchenko, Yarema, 2006]. Our research and conclusions as to two different zeros BS and BS77 was confirmed by copying from the Instructions on office processing of measured exceedance data and induction of two groups of adjustments, after which the exceedances remain measured but ready for balancing. BS was implemented when the first group of adjustments was mastered, and BS77 – when both groups were used at the same time. Thus, these two independent systems, the relations between which should be determined by adjustment and gravity affection on the Kronstadt footstock, as well as on each point of the physical surface, in order to secure the transition to BS77, if there is a need to use earlier data and GNSS-technologies. Academic novelty. We state the system error as found, which can be easily computed on the Kronstadt footstock through the determination of its quasigeoid mark in BS, which allows the quasigeoid EGG97 to be easily transformed into adopted in Ukraine as well as in other Eastern European countries system “Baltic 77”. Practical meaning. This question now is very difficult due to the present situation in geodetic management, which except minors, has at all levels not geodesists, but specialist in other fields, which could study only essentials of geodesy. Those bureaucrats cannot insist or prove anything in the field of geodesy. However, until the instructions are not updated, until the BS77 zero appears on graphics, nobody has the right in any industry to change anything in their spheres of activity. If the huge and unnecessary expenses of marine industries where charged the accounts of the said bureaucrats, they could agree to sovereignty of geodetic science. We drew tables of determination of all derived system zeros, it will take SE “CHORNOMORNDIPROEKT” 2–3 years to match all the components, however at the beginning the State Geodetic service should get the funds for reliable determination of dБС for all 27 sea ports on the seacoasts as well as for areas near seaports on the large navigatable rivers (Danube, South Bug, Dnipro). We insist that it was not good to agree as to matching of geodetic and gravity surfaces – average level and quaisgeoid on the exit point (Kronstadt footshock).
Key word: Zero of Baltic 1977, height system, unified zero of sea post, seaport zero.

Література – 13.

UDС 528.71, 528.42,528.93
V. LOZYNSKYI1*, V. NIKULISHYN2, T. ILKIV 3
1*,2 Department of Cartography and Geospatial Modeling, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013, tel. 098-71-31-398, e-mail viktor.lozynskyi@gmail.com
3 Department of Photogrammetry and Geoinformatics, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013
TECHNOLOGICAL FEATURES OF CREATION OF A LARGE-SCALE TOPOGRAPHICAL PLAN OF LVIV CITY LANDFILL USING COMBINED METHOD
Purpose. The compliance with maintenance requirements of a landfill is an important factor that have affects on its functioning. The functioning of Lviv city landfill started in 1959 and continued till 2016. According to evidences from various resolutions, regulations, and scientific publications, it was used and exploited with disabilities and did not meet environmental and sanitary standards. On May 30, 2016 a waste flow slide occurred in consequence of fire and its extinguishing. To update the topographic information about the situation at the landfill, and to correct the remediation project, the following tasks should be performed: to create a topographical plan of scale 1 : 500 with a contour interval of 0.5m, to identify technological features of combined methods using UAV TRIMBLE UX-5 and the electronic total station Leica TCR 405, and to select and take into account the peculiarities of the researched object. Methodology and results. When creating large-scale topographic plans for different kinds of objects it should be noted that each object has its own peculiarities that should be considered. In the process of the territory reconnaissance, the boundaries of the surveyed site were determined and the possibility of applying an aerial survey by UAV and remote method of tacheometry survey were considered. According to the purpose, the large-scale topographical plan of Lviv city landfill with the scale of 1 : 500 with 0.5 m relief interval with coordinate system
SC-63 and Baltic height system was created using combined methods. Additionally control of created DEM was implemented, the root-mean-square errors of the DEM were calculated before and after the use of technological operations and statistical methods. The results correspond to the requirements specified in the instructions for the topographic survey at an appropriate scale. Originality and practical significance. The developed and tested method of creating large-scale plans for the landfill enables designing organizations to solve a number of the following problems: designing new maps for storage place of solid waste, performing calculation of excavation works volume, creating working drawings for strengthening of existing dams and construction of new dams, and developing a plan for the location of the filtrate drainage system.
Key words: large-scale topographical plan, Lviv city landfill, unmanned aerial vehicle (UAV), combined survey.
Література – 27.

UDC 528.33
I. SAVCHYN*, M. DUMA
*Department of Higher Geodesy and Astronomy, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013,
tel. 097-14-37-858, e. mail: savchyn.ih@gmail.com
DNISTER PSPP CONTROL GNSS NETWORK OPTIMIZATION
Goal. The development of conceptual frameworks and proposals to optimize the geometry of Dnister PSPP control GNSS network and to identify ways to improve the accuracy of GNSS measurements. Methodology. To select optimal geometric deployment of new and to clarify the position of existing points of Dnister PSPP control GNSS network it was developed a special methodology of optimizing the geometric configuration of the network. It foresees detecting of points position at which the value of optimization criteria will be minimal. As optimization criterion it was used the determinant of covariance matrix. Results. A methodology for optimizing the geometric network configuration using mathematical modeling was devised. As a result of the in-field inspection of points as well as detailed analysis conducted and processed measurements there were highlighted three key challenging groups of points of Dnister PSPP control GNSS network: points with poor reception of satellite signal; points centered using a tripod; points damaged during construction works. In order to improve rigidity and accuracy of Dnister PSPP control GNSS network it is necessary: to exclude the application of 4 GNSS measuring points (Portal-2, Nyzhniy, OZS-1-1 and OZS-23-2); to strengthen 4 points (GZ-10, GZ-11A, GZ-11B and GZ-12) with joint satellite angular and linear measurements; to replace 4 existing points (PP-221, PP-100, Obryv and OGZ-1) and set new 4 points
(GZ-21, GZ-22, GZ-23 and GZ-24). To install the new points four areas were determined and they need monitoring. Optimization of Dnister PSPP control GNSS network using the devised methodology resulted in improved accuracy (by 8.3-10.0 %) depending on the amount of used GNSS receivers. Scientific novelty and practical significance. A new methodology of optimizing the geometric network configuration using mathematical modeling is proposed. Using this methodology Dnister PSPP control GNSS network was optimized. The methodology can also be applied to optimize other geodetic monitoring networks.
Key words: optimization; D -criterion, GNSS network, the geometric configuration of Dnister PSPP control GNSS network.

Література – 7.

UDC 528.2/.3:551.24
О. TADYEYEV
Department of Geodesy and Cartography, National University of Water and Environmental Engineering,
Soborna str., 11, 33028, Rivne, Ukraine, tel. +38(096)7488449, e-mail: oleksandrtad@gmail.com
EVALUATION OF THREE-DIMENSIONAL DEFORMATION FIELDS
OF THE EARTH BY METHODS OF PROJECTIVE DIFFERENTIAL GEOMETRY;
RIGID ROTATIONS OF THE EARTH
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.
Key words: Riemannian diffeomorphic manifolds; space metric tensor; deformation analysis; rigid rotation of the Earth; deformation of the coordinate system.

Література – 27.

UDC 528.3
O. TERESHCHUK1, I. NYSTORIAK2
1 Department of Geodesy, Cartography and Land Management, National Technical University of Chernihiv, Chernihiv, Ukraine,
e-mail: olexter1957@gmail.com
2 Department of Geodesy, Cartography and Land Management, National Technical University of Chernihiv, Chernihiv, Ukraine, vanyas31@yandex.ua
EFFICIENCY OF APPLICATION OF SATELLITE TECHNOLOGY WHEN PERFORMING LAND AND CADASTRAL WORKS IN SETTLEMENTS
Purpose. The purpose of this work is to study the effectiveness of using satellite technology in real-time kinematics mode for work performed to determine the areas of land of different size within a settlement. Methodology. To realize this рurpose we have conducted experimental research on satellite observation points and triangulation of polygonometry in Chernihiv and the region. During the research it was assumed to get the control coor¬dinates on the basic points from static observation. For basic triangulation points around Chernihiv were selected – Kyiinka (KIIN) Yatsevo (JATS) Glushets (GLUS), where three teams spend the first day of observation in “static” mode for over 4 hours. During this time, the other three teams conducted observations at points of the city polygonometry, each time starting with the hour mode “Fast-Static” and then in RTK-mode. Then rover receivers were set to receive amendments of the network System.NET. For this in the controllers there were created six projects that had a different configuration. The research resulted in modeling of objects of different shapes and sizes. For this, we used a network of urban polihonometry, and triangulation points around Chernihiv, where satellite observations were performed. Nine models of landfills were planned, the areas of which were calculated based on coordinates issued from the catalog coordinates and measured using RTK mode by satellite receivers. The program-methodical complex, developed by scientists of the Scientific and Research Institute of Geodesy and Cartography, were used for transformation and recreation of coordinates. Results. Coordinates transformation from MSC into SC-63 were completed by the key and formulas, and then transformation of the coordinates x, y in Gauss-Kruger coordinates into geodetic coordinates B, L via formulas was performed. For theoretical defining of area of the ellipsoid site we used method of numerical integration per a contour that is specified by geodetic latitude B and longitude L. In order to reduce errors in cartographic projections in the cities we used the local coordinate system, the mathematical basis of which is Gauss – Kruger projection with a displaced axial meridian. Considering the fact that the distortion in this case will be minimal, calculation of areas of intended object models was carried via the coordinates in the local system. In accordance with our program of satellites observations at the points of polygonometry and triangulation, the measurements were carried out in real time in six different configurations. Based on the analysis of these studies, we analyzed the dependence of the measured areas on modeled land plots by defining coordinates using settelite observations and evaluated the accuracy of such definitions, as well made a conclusion on the possible permissible values of distortions of land areas within the boundaries of a settlement. Scientific novelty. The data of the research again confirm the efficacy of using RTK observations and re-course to a new geodetic framework created on the base of modern measuring GNSS-technology. The feasibility of using the local coordinate system to determine the areas, takes place under certain circumstances, in particular, it is known that the former geodetic networks and networks of thickening were developed with appropriate accuracy for that time base. Clearly, the quality of former networks cannot fully ensure the accuracy of the current work. However, the results of our research of using of points of the network at the local level, such as at the city traverse network of a settlement, when determining the areas of objects with the size up to 800 hectares causes no doubts and meets the requirements of accuracy. The practical significance. Improving the accuracy of coordinate definitions is connected with the introduction of modern, uncontested satellite technology. Analysis of the study confirms the feasibility of using a local coordinate system while performing work at the local level with the area of up to 1000 hectares object. If it is necessary to define areas of objects larger than 1000 hectares with the accuracy of 50 m2, the satellite methods of measurements should be used, that ensure error of coordinates within 0.005–0.020 m.
Key words: GPS; GNSS; reference station; SGN; networks of thickening; RTK-technology, VRS.

Література – 15.

Технічні вимоги до оформлення статей
збірника “Геодезія, картографія і аерофотознімання”

UDC 528.33:551.24
М. FYS, А. SOHOR, М. YURKIV
Department of Cartography and Geospatial Modeling, Lviv Polytechnic National University, 12, S. Bandery str., Lviv, Ukraine, 79013, e-mail: Mariana.I.Yurkiv@lpnu.ua
ELABORATION OF EQUIPOTENTIAL SURFACES OF PLANETS USING BIORTHOGONAL EXPANSIONS
Purpose. Using known and fixed Earth potential, presented asthe biorthogonal expansion, to culculate the geoid surface, which describes the actual shape of the planet. The external gravitational field is generally described by the series of spherical functions. Since the geoid is determined with the help of such functions, a question arises converning the identity to define the shape, moreover its several points does not belong to the region of convergence. Methodology and results. We consider representation of potential by convergent series everywhere, which makes it possible to find the geoid without specifying the location of points on the surface, although the geoid heights calculation is carried out by various relations. According to the known function of the mass distribution of the Earth, represented by the second degree polynomial, internal and external potential of elliptical planet are defined and the equipotential surfaces are found. Calculated values via these formulas and their degree of coincidence was analyzed. Defined in two ways surfaces do not coincide with each other because the difference in the values of the radius-vector amouts up to ten meters. So, when applying biorthogonal expansions of higher orders in constructing equipotential surfaces based on information about the external gravitational field it is necessary to take into account characteristics of expansion. Originality. Method of determining the shape of the Earth using the biorthogonal expansions of mass distribution function is proposed. This representation is characterized by a convergence for considered series and gives the opportunity to build digital models of the geoid (volumetric or as an isolines map). Practical significance. The results of numerical experiments, described in the article, led to the conclusion about the possibility of determining the equipotential surfaces that adequately describe the physical surface of the planet not only of the second but higher orders using biorthogonal expansions only with additional investigations. Calculation of geoid heights with high accuracy opens the way to observe many regional and local geodynamic phenomena, such as the movement of tectonic plates, and high accuracy leveling using GPS technology can solve a number of geodetic problems.
Key words: potential, spherical functions, level surface, convergence of series, ellipsoid.
Література – 21.

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