THE ASSESSMENT OF THE CELESTIAL BODY INFLUENCE ON THE GEODETIC MEASUREMENTS

VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Darius POPOVAS THE ASSESSMENT OF THE CELESTIAL BODY INFLUENCE ON THE GEODETIC MEASUREMENTS SUMMARY OF DOCTORAL DISSERTATION TECHNOLOGICAL SCIENCES, MEASUREMENT ENGINEERING (10T) Vilnius 2011

Scientific Supervisor Prof Dr Habil Petras PETROŠKEVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Measurement Engineering 10T). The dissertation is being defended at the Council of Scientific Field of Measurement Engineering at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Vladas VEKTERIS (Vilnius Gediminas Technical University, Technological Sciences, Measurement Engineering 10T). Members: Assoc Prof Dr Vladislovas Česlovas AKSAMITAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Measurement Engineering 10T), Prof Dr Habil Ramutis Petras BANSEVIČIUS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering 09T), Prof Dr Habil Vytautas GINIOTIS (Vilnius Gediminas Technical University, Technological Sciences, Measurement Engineering 10T), Prof Dr Habil Kazys KAZLAUSKAS (Vilnius University, Physical Sciences, Informatics 09P). Opponents: Dr Habil Saulius ŠLIAUPA (Nature Research Center, Physical Sciences, Geology 05P), Prof Dr Habil Algimantas ZAKAREVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Measurement Engineering 10T). The dissertation will be defended at the public meeting of the Council of Scientific Field of Measurement Engineering in the Senate Hall of Vilnius Gediminas Technical University at 2 p. m. on 19 December 2011. Address: Saulėtekio al. 11, LT-10223 Vilnius, Lithuania. Tel.: +370 5 274 4952, +370 5 274 4956; fax +370 5 270 0112; e-mail: doktor@vgtu.lt The summary of the doctoral dissertation was distributed on 18 November 2011. A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saulėtekio al. 14, LT-10223 Vilnius, Lithuania). Darius Popovas, 2011

VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS Darius POPOVAS DANGAUS KŪNŲ ĮTAKOS GEODEZINIAMS MATAVIMAMS VERTINIMAS DAKTARO DISERTACIJOS SANTRAUKA TECHNOLOGIJOS MOKSLAI, MATAVIMŲ INŽINERIJA (10T) Vilnius 2011

Disertacija rengta 2006 2011 metais Vilniaus Gedimino technikos universitete. Mokslinis vadovas: prof. habil. dr. Petras PETROŠKEVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, matavimų inžinerija 10T). Disertacija ginama Vilniaus Gedimino technikos universiteto Matavimų inžinerijos mokslo krypties taryboje: Pirmininkas: prof. habil. dr. Vladas VEKTERIS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, matavimų inžinerija 10T). Nariai: doc. dr. Vladislovas Česlovas AKSAMITAUSKAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, matavimų inžinerija 10T), prof. habil. dr. Ramutis Petras BANSEVIČIUS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija 09T), prof. habil. dr. Vytautas GINIOTIS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, matavimų inžinerija 10T), prof. habil. dr. Kazys KAZLAUSKAS (Vilniaus universitetas, fiziniai mokslai, informatika 09P). Oponentai: habil. dr. Saulius ŠLIAUPA (Gamtos tyrimų centras, fiziniai mokslai, geologija 05P), prof. habil. dr. Algimantas ZAKAREVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, matavimų inžinerija 10T). Disertacija bus ginama viešame Matavimų inžinerijos mokslo krypties tarybos posėdyje 2011 m. gruodžio 19 d. 14 val. Vilniaus Gedimino technikos universiteto senato posėdžių salėje. Adresas: Saulėtekio al. 11, LT-10223 Vilnius, Lietuva. Tel.: (8 5) 274 4952, (8 5) 274 4956; faksas (8 5) 270 0112; el. paštas doktor@vgtu.lt Disertacijos santrauka išsiuntinėta 2011 m. lapkričio 18 d. Disertaciją galima peržiūrėti Vilniaus Gedimino technikos universiteto bibliotekoje (Saulėtekio al. 14, LT-10223 Vilnius, Lietuva). VGTU leidyklos Technika 1936-M mokslo literatūros knyga. Darius Popovas, 2011

Introduction Topicality of the problem. Geodetic measurements are interrelated with the gravitational field which tend to vary within the time running. The major influence makes such celestial bodies as the Sun and Moon. The gravitational pull of the moving celestial bodies alternate the field of the Earth s gravity as well as its form. The effect of the Sun and the Moon cause periodical and constant deformations of the Earth. The mentioned above deformations influence the direction of the vertical, the gravity, the equipotential surfaces of the gravity field and etc., what in its turn induces the results of geodetic measurements. To calculate the corrections of the tides there are applied the spherical function series of the tide generating potential. Modern accurate calculations on the tide potential by means of the Sun s and Moon s coordinates allow to assess precisely the variation of the gravity field to a rigid Earth. However, the Earth is not rigid and homogenic body and be able to react on to the effect of the celestials bodies in a more complex way, that is why the available models of the tides and the algorithms diverge in between more or less. The classical geodesy use simpler and unlike methods (in levelling, gravimetry) of the tide corrections which have to be revised and specified when more precise measurements are introduced. Relevance of the work. The effect of the celestial bodies has to be assessed when carrying out geodetic measurements, determining the coordinates of the benchmarks and the parameters of the Earth s gravity field. These issues are significant when solving geodetic tasks not only by the classical methods but the methods of cosmic geodesy. The data obtained by the cosmic geodesy methods are mostly applied for the solution of the tasks of geophysics and geodynamics, for compiling global and regional geodetic networks, that is why tide models to be used for the corrections of these data have to satisfy the required accuracy typical for these data. The increased accuracy of geodetic and gravimetric measurements requires more precise and detailed investigation of the effect of the celestial bodies as well as more precise its assessment. The mentioned above are significant for the geodetic works executed in Lithuania which are considered to be essential to be able to establish modern geodetic reference frame. Objective of the work. The effect of the celestial bodies on to the elements of the gravity field and the methodology of the assessment of the changes of the gravity field in the geodetic measurements. 5

Aim of the work. To investigate and assess the effect of the celestial bodies on to the elements of the gravity field and to upgrade the methodology of the assessment of the celestial bodies effect on the geodetic measurements. Tasks of the work 1. To carry out the research on the potential of the tide caused by the celestial bodies. 2. To analyse the effect of the celestial bodies on to the gravity field elements. 3. To assess and investigate the effect of the celestial bodies on to the geodetic measurements. 4. To improve the methodology of the assessment of the celestial bodies effect on to the geodetic measurements. Methods of the work. Theoretical and experimental research of Earth s gravity change applying the theory of the tide potential, spherical function series, horizontal and equatorial coordinate systems and assessing the rigidity of the Earth. Scientific novelty 1. There was analysed, assessed and investigated the effect of the celestial bodies on to the elements of the gravity field instead of the elements of the orbit making use of the known coordinates of the celestial bodies. 2. The zonal waves of the tide potential were analysed in a detailed way, their effect to the gravity field elements was evaluated. 3. The methodology of the assessment of the tide effect on to the geodetic measurements was advanced. 4. Methodology to specify the geoid surface and gravity system assessing the permanent part of the tide generating potential. 5. The effect of the celestial bodies on to the gravity field elements in the territory of Lithuania and their dispersion. Practical value. The results of the research work could be applied for the assessment of the celestial bodies effect on the geodetic, gravimetric and 6

geodetic astronomy measurements as well as when selecting the surfaces of the Earth s and geoid and the system of the gravity in cases of the assessment of the celestial bodies effect. The presented methodology of the Lunisolar effect on the height differences was applied for the geodetic vertical first class Lithuanian network measurements. Defended propositions 1. The new series of the tide potential preconditioned by the impact of the celestial bodies and expressed by the spherical functions, where the arguments are the celestial body coordinates, allows to asses the impact of the celestial bodies by using the finite equations for each member of the series. 2. The use of the coordinates of the celestial bodies allows to carry out the assessment of the long period waves. It allows the selection of the gravity system and equipotential surfaces for geoid determination. 3. The use of the improved methodology regarding the impact of the celestial bodies on the geodetic measurements allows to carry out the assessment of the permanent and variable part of the impact of the celestial bodies. 4. Presented methodology allows the reduction of the data of various geodetic measurements to be introduced into the uniform assessment system dealing with the impact of the celestial bodies. The scope of the scientific work. The research consists of the following parts, namely the introduction, four chapters and summary of the results, two supplements are included as well. The research contains 108 pages, apart from the supplements; there were used 138 numbered equations, 45 figures and 1 table. There were used 153 literature sources when working on the research. 1. Celestial bodies effect to the gravity field and geodetic measurements valuation problems The chapter deals with the review of the development of the theory of the effect of the celestial bodies, the analysis of the contemporary methods used in the assessments of the tides and the reliability of the assessments made. The scientific literature issues on the research theme concerning the effect of the celestial bodies on the Earth s gravity field and geodetic measurements are 7

analysed. The problems regarding the assessment of the effect of the celestial bodies are named and classified. In executing the assessment of the celestial bodies on the geodetic measurements, the research very often is limited only to the first member of the tide potential. Together with the increase in the accuracy of the measurements it is advisable to assess the members of the higher series, n=3; n=4 of the tide potential. The geodetic experiences indicate mostly single Love number used, which after the perfection of the accuracy of the measurements inadequately represents the elasticity of the Earth. In accordance with the methodology of the assessment of the zonal waves there could be applied different conceptions regarding the Earth s surface, geoid and gravity. The contemporary used geodetic data present the uneven assessment of the effect of the celestial bodies, which is why the general implementation of various geodetic data requires their reduction into the unanimous system for the assessment of the effect of the celestial bodies to be made. 2. Theoretical research of the effect of the celestial bodies on the gravity field The effect of the celestial bodies on the gravity field is presented by using the series of the spherical functions of the tide potential: n Gm R VT = Pn (cos z) r n= 2 r, (1) where Pn (cos z ) is the Legendre polynomials, G is a gravitational constant, m is the mass of the celestial body, r is the geocentric distance to the celestial body, R is the geocentric distance to the point of the Earth s surface, and z is the geocentric zenith distance of the celestial body. Taking into consideration only the first three members of the series it is possible to express the following: 2 3 2 GmR 3 3 4 GmR 3 1 5 3 VT = cos z cos z cos z r 2 2 + + r 2 2 4 GmR 35 4 30 2 3 cos cos 5 z z + r 8 8 8. (2) 8

The derivative of the tide potential towards the vertical direction is equal to the projection of the tide force onto the vertical. It expresses the change δg of the gravity g due to the effect of the celestial body: V δ g = T. (3) R By differentiating (2) it is received in the following way: ( 2 2 3GmR 1 3cos ) ( 3cos 5cos 3 ) GmR δ g = z + z z + 3 4 r 2r GmR 2r 5 3 2 4 ( 30 cos z 35cos z 3). (4) By applying the theorem of the sum of the spherical functions there have been written the equation for the gravity change expressed by the equatorial coordinates: When n = 2 : 2 2 ( )( ) T GmR δ g2 = 3sin δ 1 3sin Φ 1 + 3sin 2δ sin 2Φ cos t + 3 2r 2 2 3cos δ cos Φ cos 2t, (5) where δ, and t is the declination of the celestial body and hour angle, Φ is the geocentric latitude of the point. When n = 3 : δ g 2 T 3GmR 3 4 3 ( 5sin δ 3sinδ 3 ) ( 5sin Φ 3sinΦ ) = 4r 2 2 ( ) ( ) + 3 cos δ 5sin δ 1 cos Φ 5sin Φ 1 cos t + 2 2 2 5 3 3 15cos δ sinδ cos Φ sin Φ cos 2t + cos δ cos Φ cos 3t 2. (6) 9

When n = 4 : 3 4 2 ( ) ( 35sin 4 Φ 30 sin 2 Φ 3) 64 T 4GmR 1 δ g4 = 5 35sin δ 30sin δ + 3 r 3 ( ) 5 cos δ 7sin δ 3sin δ 8 3 ( ) cos Φ 7sin Φ 3sin Φ cos t + ( ) ( ) 5 cos 2 7sin 2 1 cos 2 7sin 2 1 cos 2 16 δ δ Φ Φ t + + + 35 3 3 35 4 4 cos δ sinδ cos Φ sin Φ cos 3t + cos δ cos Φ cos 4t 8 64. (7) Then the change of the gravity is equal to: T T T 2 3 4 δg = δg + δg + δg. (8) The constant part of the regional waves on the gravity which depends only on the latitude, it is possible to determine from the equation: ( 2 3 3GmR 3sin 1) ( 35sin 4 30sin 2 3) T GmR δ gzφ = Φ Φ Φ +. (9) 3 5 2r 16r This part of the gravity change in the given latitude tends not to vary in the course of time. The other part of the zonal waves depends not only on the latitude, but also on the declination of the celestial body. To assess its effect it is possible to use the mean integral values of δ functions. Thus, the effect on the gravity could be expressed by the following: δ g T zv ( ) GmR 3 2 2 = 1 sin ε 3sin 1 3 Φ 2r 2 10

( ) 3 3GmR 35 4 4 2 5 sin ε 5sin ε + 1 35sin Φ 30sin Φ + 3 16r 8, (10) where ε is the inclination of the celestial body orbit to the equator. By applying the potential of the tide there were received not only the above presented equations expressing the change of the gravity but the equations describing the deviation of the vertical, the deformations of the equipotential surface and the components of the tide force. For the assessment of the whole elements of the gravity field there were obtained the equations by applying the first three members of the tide potential as well as there were carried out the transitions from the horizontal to the equatorial coordinates. There were analysed in a detailed way the zonal waves which depend on the latitude of the point and declination of the celestial body. There were selected two forms for the assessment of the zonal waves. The first form is the one which evaluates the effect depending on the latitude, the second form is the one additionally evaluating the average integral effect of the functions depending on the declination of the celestial body. The formulas of the zonal waves were specified and they could be applied when selecting various surfaces of the Earth s and geoid as well as gravity system. The effect of the celestial body on the gravity field and its elements of the real Earth were analysed. The Love numbers dependant on the degree of tide potential was used when evaluating the elasticity of the Earth. 3. The research of the effect of the Moon and Sun on the gravity field elements The chapter presents the analysis of the Moon and Sun effect on the elements of the gravity field when the zenith distance of the celestial body is changing, submits the assessment of the effect of the constant part of the zonal waves and carries out the research on the general effect of both the celestial bodies. The equations received in the second chapter are used. Due to the effect of the celestial body when the potential of the gravity of the Earth is changing there are proceeded the deformation of the equipotential surface of the gravity field. 2 T γ 2GmR 3 2 1 ζ = cos z 3 + gr 2 2 3 γ 3GmR 5 3 3 cos z cos z 4 + gr 2 2 11

4 γ 4GmR 35 4 30 2 3 cos z cos z 5 + gr 8 8 8. (11) where γ n is the coefficient evaluating the elasticity of the Earth. In accordance with the equation (11) there was assessed the influence of the Moon and Sun on the equipotential surface of the real Earth (Fig. 1). The range of deformation changes of the equipotential surface under the influence of the Moon when n = 2 is equal to 0.4677 m, under the influence of the Sun it is 0.1801 m The values of the deformations in Lithuania may be 0.3821 m due to the influence of the Moon and 0.1338 m due to the influence of the Sun. ξ m 0.4 0.3 0.2 0.1 0.0-0.1-0.2 0 30 60 90 120 150 180 z Deformation of equipotential surface due to the Moon effect Deformation of equipotential surface due to the Moon effect in Lithuania Deformation of equipotential surface due to the Sun effect Deformation of equipotential surface due to the Sun effect in Lithuania Fig. 1. Deformation of equipotential surface as the function of zenith distance (n=2) The range of the change of the deformations of the equipotential surface due to the influence of the Moon due to the second member (n=3) of the equation (11) is 12.9 mm; due to the Sun it is 0.01 mm. The values of deformations in Lithuania could be up to 6.41 mm caused by the effect of the Moon and 0.006 mm because of the Sun. The range of the deformation of the equipotential surface due to the effect of the Moon of the equation (11) of the last member (n = 4) is 0.018 mm. The effect of the Sun is insignificant and the 10 surface deformation because of the Sun does not exceed 4x10 mm. 12

The variation of the geocentric distance of the Moon is able to change the equipotential surface up to 0.1502 m, and that of the Sun up to 0.0165 m. The variation of the geocentric distances of the points of the Earth s surface up to 20 km change equipotential surface, because of the Moon the change is up to 0.0031 m, and because of the Sun it is up to 0.0012 m. The analogues research was applied to investigate the ranges of the vertical deviations and gravity changes due to the effects of the Moon and Sun. The influence of the first three members of the tide potential was assessed on to the gravity field elements, particularly distinguishing the effect caused in the territory of Lithuania. The research was carried out to investigate common effect of both celestial bodies, and dependence on the phases of the Moon, changes of the geocentric distances to the celestial bodies and the latitude were analysed. Zonal waves of the three first members of the tide potential were analysed. There was determined the effect of the zonal waves on the gravity, deviation of the vertical and equipotential surfaces of the gravity field. There was indicated and distinguished the constant part of the zonal waves which depend only on the latitude of the point. The average integral effect of the part of the period of the length of the regional waves was determined. 4. Upgrading and implementation of the methodology of the assessment of the celestial bodies effect on the geodetic measurements Based on the theoretical investigations carried out and described in the second chapter of the work as well as carried out research of the changes of the gravity field elements presented in the third chapter, there was prepared the methodology for the assessment of the effect of the celestial bodies for the measurements of the vertical network. The change of the levelled height difference is expressed by the formula: δ = dv = dv cos( A a), (12) a where A is the azimuth of the celestial body, accounted from the direction to the North, a is the azimuth of the levelling line and d is the distance between the benchmarks. After inserting the value deviation of the vertical we receive the following formula: 3Gm R R 2 δ = dγ 2 [sin 2 z + (4 5sin z) sin z]cos( A a). (13) 3 2gr r 13

The correction evaluating the effect of the Moon and Sun is calculated following the formula: The coefficients ( ) δ = dk + δ = d K + K + δ, (14) MS MS z M S z K M and KMS= KM+ KS. K S are calculated by the following equation: 3Gm1 R R 2 K = γ 2 [sin 2 z + (4 5sin z) sin z]cos( A a), (15) 3 2gr r where δ z is the correction of the transition to the zero geoid: δ z γ = 45.6 dx sin 2Bv, (16) g 2 1 where dx is the increase of the plane coordinatex of the benchmarks, between which the height difference is determined and B v is the average latitude. When executing the assessment of the effect of the Moon and Sun in the measurements of the vertical first class network, the assumption is made that g= 981507 mgal. To determine the coordinates and the distances to the celestial bodies, the digital version of the astronomic chronicle was used. In case of the assumption that the levelling between the two points of the vertical network has to bear the unvaried azimuth and has to be proceeded equally, the correction evaluating the effect δhms of the Moon and Sun was calculated according to the following formulas: δ h n = d K + δ, (17) MS s MSs z s= 1 KMSpa KMSpr KMSs = KMSpr + s n 1 ( 1), (18) where n is the number of the levelling stations in the levelling run; d s is the distance between the levelling rods in the station; K is the value of coefficient K MS at the beginning of the run; MSpr K MSpa is the value of coefficient 14

K MS at the end of the run. The methodology proposed was applied while assessing the effect of the Moon and the Sun in the first class geodetic vertical network measurements. The corrections are obtained by applying the proposed methodology and compared with the corrections received by making use of the previously used methods where only the first member of the tide potential was assessed and was not evaluated the change of the geocentric distance to the celestial body. The differences of the Lunisolar corrections are submitted in figure 2. Thus, the differences of the corrections made could measure to 0.04 mm. Moreover, it is possible to observe that the greater part of the correction differences are negative within the direct levelling, while in the reverse levelling the received data are opposite, namely, they appeared to be positive. Thus is related to the transition towards the zero geoid. δh mm 0.04 0.03 0.02 0.01 0-0.01-0.02-0.03-0.04 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 Nr. Fig. 2. Difference in Lunisolar corrections ( d grey, t d a black), mm The calculations derived in accordance with the upgraded methodology were also compared with the methodology applied in the Finish Geodesy Institute (FGI). After the corrections were calculated for the 84th sections of the levelling area, there were registered 0.01 mm differences within the 9 the 15

segments (10.7%). However, the corrections coincided in the rest of the segments. For the vertical network lines (Fig. 3) in the 4 th polygon of the first class vertical network of Lithuania, according to the misalignments of the differences of heights measured in the direct and reverse directions, there were calculated the root mean square error values of the height differences for one km length line when there was taken into consideration the effect of the celestial bodies and when the effect of the celestial bodies was not assessed. Fig. 3. Results of the evaluation of celestial bodies effect While carrying out the analysis of the errors there appeared that the error for the lines of the experimental polygon after the evaluation of the Lunisolar effect tends to be reduced. The maximum reduced value of the error appeared to 16

be 0.004 mm/km. Besides that, the closing error of the differences of the heights proved to be reduced. On the basis of the attained theoretical investigations presented in chapter 2 as well as the research carried on the gravity field elements submitted in chapter 3 there were worked out several methodologies for a detailed assessment to be executed in connection with the celestial bodies effect for the gravimetric, vertical network, geodetic astronomy measurements. There was investigated the influence of the Moon and Sun on to the GNSS measurements. Based on the analysis of the zonal waves of the tide potential there were investigated two possibilities available for the selection of the equipotential surface characterizing the geoid. The research was carried concerning the possibilities for the recovery of the zonal waves of the tide potential and the alternative choice for the gravity system. General conclusions 1. There have been compiled the new series of the tide potential, determined by the effect of the celestial bodies, which have been expressed by the spherical functions, where the arguments of which are considered to be the celestial body coordinates. When these series are used, then their members are expressed by the finite equations and then it is available in more precise and simpler way to assess the impact of the celestial bodies on the geodetic measurements if compared with the usage of the available known series in which the elements of the orbits are used. 2. By applying the derived equations for the zonal waves of the tide potential there has been worked out the methodology enabling to specify the geoid surface and gravity system, estimating the permanent part of the celestial body impact. 3. By applying the horizontal and equatorial coordinates instead of the orbit elements there has been investigated the impact of the celestial bodies on the elements of the gravity field related to the geodetic measurements, namely to gravity, to the vertical and the equipotential surface. It has been determined that on the whole Earth s surface, due to the Lunisolar impact, the maximum range of the gravity changes is correspondingly the following 247.27 µgal and 92.11 µgal, the declination of the vertical in terms of the Earth s surface is equal to 0.0541" and 0.0204", for the equipotential surface they are the following 0.4824 m and 0.1861 m. In the territory of Lithuania the maximum range of the gravity changes is considered to be the 17

following 199.50 µgal and 68.42 µgal, the range of the declination of the vertical is correspondingly equal to 0.0642" and 0.0204" that of the equipotential surface is 0.3885 m and 0.1338 m. 4. The received results and findings of the carried out scientific research in applying the modified assessment methodology of Lunisolar impact on the geodetic measurements when compared with the findings of the methodologies applied in the Institute of Geodesy in Finland and the Institute of Geodesy and Cartography in Poland as well as International Earth s Rotation Service Centre indicate that the differences of the errors of the gravity acceleration do not exceed 0.76 µgal. Thus proves the proposed modified methodology to be considered for the approval in practice and to be widely implemented. 5. There has been analysed the impact of the celestial bodies on the first class geodetic vertical network measurements of Lithuania. It has been determined that the errors could amount up to 0.102 mm/km. By applying the developed perfected assessment methodology regarding the Lunisolar impact on the geodetic measurements, in the experimental polygon of the vertical network, the mean square error of the height differences for one kilometre has decreased on all the levelling lines of the polygon. The closing error of the height differences has decreased by 1.3 time. Because of that it is necessary to assess the impact of the celestial bodies by executing the measurements of the vertical network. It is recommended to assess the impact of the celestial bodies at each levelling station. 6. The research work presents the perfected methodology of the assessment of the impact of the celestial bodies on geodetic measurements, which allows the reduction of the data of various geodetic measurements to be introduced into the uniform assessment system dealing with the impact of the celestial bodies. List of Published Works on the Topic of the Dissertation In the reviewed scientific periodical publications Popovas, D. 2011. Analysis of celestial bodies impact to measured height difference, Geodesy and Cartography, 37(3): 101 104 (in Lithuanian). ISSN 2029-6991 (Compendex). Petroškevičius, P.; Popovas, D. 2010. Analysis of zonal waves of the tide generating potential caused by celestial bodies, Geodesy and Cartography, 36(4): 133 138 (in Lithuanian). ISSN 1392-1541 (Compendex). 18

Petroškevičius, P.; Popovas, D. 2010. Evaluation of the tide generating potential to gravity field equipotential surface, Geodesy and Cartography, 36(3): 91 96 (in Lithuanian). ISSN 1392 1541 (Compendex). Petroškevičius, P.; Popovas, D. 2008. Evaluation of the celestial bodies efect to gravity field, Geodesy and Cartography, 34(1): 19 22 (in Lithuanian). ISSN 1392-1541 (Compendex). In the other editions Popovas, D. 2011. Estimation of Lunisolar correction in precise levelling. In Proceedings of the 8th International Conference "Environmental Engineering" Selected papers, Vol. III, 19 20May [8-osios tarptautinės konferencijos Aplinkos inžinerija, įvykusios Vilniuje 2011 m. gegužės 19 20 d., medžiaga], 1432 1435. ISBN 978-9955- 28-827-5. Petroškevičius, P.; Popovas, D.; Krikštaponis, B.; Putrimas, R.; Obuchovski, R.; Būga, A. 2008. Estimation of gravity field non-homogeneity and variation for the vertical network observations, in Proceedings of the 7th International Conference "Environmental Engineering" Selected papers, Vol. III, 22 23May [7-osios tarptautinės konferencijos Aplinkos inžinerija, įvykusios Vilniuje 2008 m. gegužės 22 23 d., medžiaga], Vilnius: Technika, 1439 1445. ISBN 978-9955-28-265-5 (Thomson ISI Proceedings). Popovas, D. 2007. Readjustment and analysis of Lithuanian GPS network, in Republic scientific-practical conference Measurement Engineering and GIS, Mastaičiai, 68 70. ISBN 978-9955-27-033-1. Popovas, D. 2010. Earth tide efects to the coordinates of geodetic bechmarks, in Republic scientific-practical conference Measurement Engineering and GIS, Mastaičiai, 126 129. ISSN 2029-5790. About the author Darius Popovas was awarded the Bachelor degree in the field of Geodesy at the Vilnius Gediminas Technical University in 1998. In 1999 he was granted the professional degree that of Engineer of Geodesy. In 2001 he was awarded the Master of Science degree in the field of GPS Technology at the Aalborg University (Denmark). He continued his postgraduate studies in 2006 2011 as Vilnius Gediminas Technical University as PhD student. In 2007 Darius Popovas had an internship course at the ETH Zurich (Swiss Federal Institute of Technology). At present the author is engaged as an assistant at the Department of Geodesy and Cadastre at the Vilnius Gediminas Technical University. 19

Letter of thanks The author expresses sincere thanks for the original scientific guidance, remarks and suggestions on the dissertation to the supervisor of the thesis Prof Dr Habil Petras Pertoškevičius. The author is grateful for the possibility to have the internship at Zürich Federal Technology University to Prof Dr Habil Vytautas Giniotis. The author expresses credit and appreciation to the staff of the Department of Geodesy and Cadastre and Institute of Geodesy at Vilnius Gediminas Technical University for the support and conditions available for the research work to be finalized. DANGAUS KŪNŲ ĮTAKOS GEODEZINIAMS MATAVIMAMS VERTINIMAS Problemos formulavimas. Geodeziniai matavimai susieti su gravitacijos lauku, kuris laiko bėgyje keičiasi. Tam didelės įtakos turi dangaus kūnų, ypač Mėnulio ir Saulės, poveikis. Judančių dangaus kūnų trauka keičia Žemės sunkio lauką bei jos formą. Mėnulio ir Saulės poveikis sukelia periodines bei pastovias Žemės deformacijas. Minėtos deformacijos įtakoja vertikalės kryptį, sunkį, sunkio lauko ekvipotencialinius paviršius ir kt., o tai savo ruožtu įtakoja ir geodezinius matavimus. Potvynio pataisų skaičiavimui naudojama potvynio potencialo sferinių funkcijų eilutė. Šiuolaikiniai tikslūs potvynio potencialo skaičiavimai, naudojant Saulės ir Mėnulio koordinates, leidžia tiksliai įvertinti sunkio lauko pokytį absoliučiai kietai Žemei. Tačiau Žemė nėra kietas ir homogeniškas kūnas ir reaguoja į dangaus kūnų poveikį daug sudėtingiau, todėl esami potvynių modeliai ir algoritmai daugiau ar mažiau skiriasi tarpusavyje. Klasikinės geodezijos metoduose (niveliacijoje, gravimetrijoje) tebetaikomos paprastesnės ir nevienodos potvynių pataisų taikymo metodikos, kurios didėjant matavimų tikslumui turėtų būti peržiūrėtos ir patikslintos. Darbo aktualumas. Dangaus kūnų poveikį būtina įvertinti atliekant geodezinius matavimus, nustatant punktų koordinates ir Žemės gravitacijos lauko parametrus. Šie klausimai aktualūs sprendžiant geodezinius uždavinius ne tik klasikiniais metodais, bet taip pat ir kosminės geodezijos metodais. Kosminės geodezijos metodais gaunami duomenys dažnai naudojami geofizikos ir geodinamikos uždaviniams spręsti, globaliems ir regioniniams geodeziniams tinklams sudaryti, todėl šių duomenų pataisoms naudojami potvynių modeliai turi atitikti reikalaujamą tokiems duomenims tikslumą. Didėjantis geodezinių ir gravimetrinių matavimų tikslumas reikalauja detalesnių 20

dangaus kūno poveikio tyrimų ir tikslesnio jo įvertinimo. Tai aktualu ir Lietuvoje atliekamiems geodeziniams darbams, kurie būtini kuriant šiuolaikinį geodezinį pagrindą. Tyrimų objektas. Dangaus kūnų įtaka sunkio lauko elementams bei geodeziniams matavimams, sunkio lauko kitimo įvertinimo geodeziniuose matavimuose metodika. Darbo tikslas. Darbo tikslas ištirti ir įvertinti dangaus kūnų poveikį sunkio lauko elementams ir patobulinti dangaus kūnų poveikio geodeziniams matavimams įvertinimo metodiką. Darbo uždaviniai 1. Atlikti dangaus kūnų lemiamo potvynio potencialo tyrimus. 2. Išnagrinėti dangaus kūnų poveikį sunkio lauko elementams. 3. Įvertinti ir ištirti dangaus kūnų poveikį geodeziniams matavimams. 4. Patobulinti dangaus kūnų poveikio geodeziniams matavimams vertinimo metodiką. Tyrimų metodika. Teoriniai ir eksperimentiniai Žemės sunkio lauko kitimo tyrimai taikant potvynio potencialo teoriją, sferinių funkcijų eilutes, dangaus horizontinę ir pusiaujinę koordinačių sistemas ir įvertinant Žemės tamprumą. Darbo mokslinis naujumas Disertaciniame darbe buvo gauti šie matavimų inžinerijos mokslui nauji rezultatai: 1. Išnagrinėtas, įvertintas ir ištirtas dangaus kūnų poveikis sunkio lauko elementams vietoje orbitos elementų naudojant dangaus kūnų koordinates. 2. Detaliai išnagrinėtos potvynio potencialo zoninės bangos, įvertintas jų poveikis sunkio lauko elementams. 3. Patobulinta dangaus kūnų poveikio geodeziniams matavimams įvertinimo metodika. 4. Geoido paviršiaus ir sunkio sistemos patikslinimo metodika, įvertinant dangaus kūnų poveikio sunkiui pastoviąją dalį. 21

5. Dangaus kūnų poveikio sunkio lauko elementams Lietuvos teritorijoje vertinimo rezultatai ir jų sklaida. Darbo rezultatų praktinė reikšmė. Atlikto darbo rezultatus galima panaudoti vertinant dangaus kūnų poveikį geodeziniams, gravimetriniams ir geodezinės astronomijos matavimams taip pat pasirenkant Žemės ir geoido paviršius bei sunkio sistemą dangaus kūnų poveikio įvertinimo atžvilgiu. Mėnulio ir Saulės įtakos aukščių skirtumui metodika taikyta Lietuvos geodezinio vertikaliojo pirmosios klasės tinklo matavimuose. Ginamieji teiginiai 1. Dangaus kūnų lemiamo potvynių potencialo naujos eilutės, išreikštos sferinėmis funkcijomis, kurių argumentai yra dangaus kūnų koordinatės, leidžia vertinti dangaus kūnų poveikį naudojant baigtines formules kiekvienam eilutės nariui. 2. Dangaus pusiaujinių koordinačių panaudojimas leidžia išskirti ilgo periodo potvynio potencialo zonines bangas. Tai sudaro galimybę pasirinkti skirtingas sunkio sistemas bei ekvipotencialinius paviršius geoidui apibrėžti. 3. Taikant patobulintą dangaus kūnų poveikio geodeziniams matavimams vertinimo metodiką galima įvertinti pastovią ir kintamą dangaus kūnų poveikio dalį. 4. Siūloma metodika leidžia redukuoti įvairių geodezinių matavimų duomenis į vieningą dangaus kūnų įtakos vertinimo sistemą. Darbo apimtis Disertaciją sudaro įvadas, 4 skyriai ir rezultatų apibendrinimas. Taip pat yra 2 priedai. Darbo apimtis yra 108 puslapiai, neskaitant priedų, tekste panaudotos 138 numeruotos formulės, 45 paveikslai ir 1 lentelė. Rašant disertaciją buvo panaudoti 153 literatūros šaltiniai. Pirmasis skyrius skirtas mokslinės literatūros analizei. Jame nagrinėjami esami dangaus kūnų sukeliamų potvynių įvertinimo metodai ir jų tikslumas. Išryškintos problemos dangaus kūnų poveikio vertinime geodezijoje ir suformuluoti uždaviniai, kuriuos disertaciniame darbe tikslinga išspręsti. 22

Antrajame disertacijos skyriuje pateiktas teorinis tyrimas. Nagrinėjamas dangaus kūno lemiamo potvynio potencialo eilutės narių poveikis sunkio lauko elementams: sunkiui, vertikalei ir ekvipotencialiniam paviršiui. Tyrimai atlikti naudojant horizontinę ir pusiaujinę dangaus koordinačių sistemas. Trečiajame disertacijos skyriuje pateikti Mėnulio ir Saulės poveikio sunkio lauko elementams tyrimai, įvertinamas zoninių bangų poveikis. Įvertinti sunkio, vertikalės nuokrypio bei ekvipotencialinio paviršiaus kitimo diapazonai, ištirtas pastovios potvynio dalies poveikis. Ketvirtajame disertacijos skyriuje pateikta patobulinta dangaus kūnų poveikio įvertinimo geodeziniuose matavimuose metodika. Bendrosios išvados Remiantis disertaciniame darbe atliktais teoriniais ir eksperimentiniais tyrimais, gautos šios apibendrintos išvados: 1. Sudarytos naujos dangaus kūnų lemiamo potvynio potencialo eilutės, išreikštos sferinėmis funkcijomis, kurių argumentai yra dangaus kūnų koordinatės. Taikant šias eilutes, jų nariai išreiškiami baigtinėmis formulėmis ir dėl to paprasčiau galima įvertinti dangaus kūnų poveikį geodeziniams matavimams, lyginant su žinomomis eilutėmis, kuriose taikomi orbitos elementai. 2. Taikant gautas potvynio potencialo zoninių bangų formules parengta geoido paviršiaus ir sunkio sistemos patikslinimo metodika, įvertinanti dangaus kūnų poveikio pastoviąją dalį. 3. Taikant dangaus kūnų horizontines ir pusiaujines koordinates vietoje orbitos elementų, ištirtas dangaus kūnų poveikis sunkio lauko elementams, susietiems su geodeziniais matavimais: sunkiui, vertikalei ir ekvipotencialiniam paviršiui. Nustatyta, kad visame Žemės paviršiuje dėl Mėnulio ir Saulės poveikio maksimalus sunkio pokyčių diapazonas atitinkamai 247,27 µgal ir 92,11 µgal, vertikalės nuokrypio Žemės paviršiaus atžvilgiu 0,0541" ir 0,0204", bei ekvipotencialinio paviršiaus 0,4824 m ir 0,1861 m. Lietuvos teritorijoje maksimalus sunkio pokyčių diapazonas 199,50 µgal ir 68,42 µgal, vertikalės nuokrypio 0,0642" ir 0,0204", bei ekvipotencialinio paviršiaus 0,3885 m ir 0,1338 m. 4. Gautus tyrimo rezultatus, taikant patobulintą Mėnulio ir Saulės poveikio geodeziniams matavimams vertinimo metodiką, palyginus su Suomijos geodezijos instituto, Lenkijos geodezijos ir kartografijos instituto ir Tarptautinės Žemės Sukimosi tarnybos taikomų metodikų rezultatais, sunkio pagreičio pataisų skirtumai neviršija 0,76 µgal. Tas 23

rodo, kad siūloma patobulinta metodika tinkama praktiniam naudojimui. 5. Ištirtas dangaus kūnų poveikis Lietuvos geodezinio vertikaliojo pirmosios klasės tinklo matavimams. Nustatyta, kad paklaidos gali siekti iki 0,102 mm/km. Taikant patobulintą Mėnulio ir Saulės poveikio geodeziniams matavimams vertinimo metodiką, tirtame vertikaliojo tinklo poligone, aukščių skirtumo vidutinė kvadratinė paklaida vienam kilometrui, sumažėjo visose poligono linijose. Aukščių skirtumo nesąryšis sumažėjo 1,3 karto. Todėl atliekant vertikaliojo tinklo matavimus būtina įvertinti dangaus kūnų poveikį. Rekomenduojama dangaus kūnų poveikį vertinti kiekvienoje matavimų stotyje. 6. Darbe pateikta patobulinta dangaus kūnų poveikio geodeziniams matavimams vertinimo metodika leidžia redukuoti įvairių geodezinių matavimų duomenis į vieningą dangaus kūnų įtakos vertinimo sistemą. Trumpos žinios apie autorių Darius Popovas 1998 metais Vilniaus Gedimino technikos universitete įgijo geodezijos bakalauro kvalifikacinį laipsnį, o 1999 metais diplomuoto geodezijos inžinieriaus laipsnį. 2001 metais įgijo GPS technologijų magistro laipsnį Alborgo universitete Danijoje. 2006 2011 metais Vilniaus Gedimino technikos universiteto doktorantas. Darius Popovas 2007 m. stažavosi Ciuricho Federaliniame technologijos universitete. Šiuo metu dirba asistentu Vilniaus Gedimino technikos universiteto Geodezijos ir kadastro katedroje. Padėkos Autorius nuoširdžiai dėkoja darbo vadovui prof. habil. dr. Petrui Pertoškevičiui už originalius mokslinius patarimus, pagalbą bei pastabas ir pasiūlymus rengiant disertaciją. Autorius dėkoja prof. habil. dr. Vytautui Giniočiui už pagalbą organizuojant stažuotę Ciuricho Federaliniame technologijos universitete. Padėka visam Geodezijos instituto ir Geodezijos ir kadastro katedros kolektyvui už sudarytas sąlygas ir palaikymą. 24