Aim. Solving the problem associated in using GNSS monitoring data in the ITRS to evaluate deformation of fields of the Earth in terms of elimination of invariance loss errors. Methodology and results. The problem is considered in the context of the geophysical nature of the ITRS in conjunction with the global deformation field of the Earth. The attention is focused on the consequences of deformation of the ITRS system, which manifests itself as invariance loss effects in the interpretation of deformation fields. It is proposed to solve the problem on the basis of the theory of differential presentation of transformations of Riemannian space images in the form of its complicated diffeomorphic manifold such as the tangent Euclidean space, which is parameterized by a rectangular Cartesian coordinate system. As a geometric system, ITRS is a partial case of rectangular Cartesian. On this basis, and with the hypothesis that the transformation of space has a geophysical origin, a method of evaluating deformation fields has been developed. It is foreseen that the direct use of coordinates are obtained by the GNSS-method. Due to this, the evaluation of the topographic surface on which deformation processes are manifested, has been achieved. As a component part of this method, there are working formulas for determining angular and scale distortions of the ITRS at an arbitrary time point relative to the ITRF-realization. Considering the homeomorphism potential of diffeomorphic manifolds, the formulas have taken into account the perspective of the transfer of nonlinear deformation effects. Scientific novelty and practical significance. This used basis has a generalizational character compared to the mathematical theory of elasticity in the framework of its linearly homogeneous model of the infinitely small deformation, which is traditionally used for deformation analysis in geodynamics. Solutions on a generalizational basis have a higher informative resource and provide estimates of deformation fields that are adequate to GNSS data. Given current distortions of the coordinate system, the developed method is able to eliminate the invariance loss effects. Practical recommendations for the formulation and performing deformation analysis tasks by GNSS data in arbitrary observation epochs, which do not coincide with ITRF-realizations of the ITRS, are formulated.
1. Kochin N. E. Vektornoe ischislenie i nachala tenzornogo ischislenija [Vector calculus and beginning of tensor calculus]. Moscow: Science, 1965, 427 p. (in Russian).
2. Rashevskij P. K. Rimanova geometrija i tenzornyj analiz [Riemannian geometry and tensor analysis]. Moscow: Science, 1967, 667 p. (in Russian).
3. Sokol'nikov I. S. Tenzornyj analiz. Teorija i primenenija v geometrii i v mehanike sploshnyh sred. Per. s angl. [Tensor analysis. Theory and applications in geometry and continuum mechanics. Transl. from English]. Moscow: Science, 1971, 376 p. (in Russian).
4. Tadieiev O. A. Otsiniuvannia tryvymirnykh deformatsiinykh poliv Zemli metodamy proektyvno-dyferentsialnoi heometrii. Dylatatsiini polia Zemli [Evaluation of three-dimensional deformation fields of the Earth by methods of the projective differential geometry. Dilatation fields of the Earth]. Suchasni dosiahnennia heodezychnoi nauky ta vyrobnytstva. [Modern Achievements of Geodetic Science and Industry]. 2017, Vol. I (33), pp. 53–60. (in Ukrainian).
5. Altamini Z., Metivier L., Collilieux X. ITRF2008 plate motion model. Journal of Geophysical Research, 2012, Vol. 117 (B7), N. B07402, 14 p. doi: 10.1029/2011JB008930
https://doi.org/10.1029/2011JB008930
6. Altamini Z., Rebischung P., Metivier L., Collilieux X. ITRF2014: A new release of the International Terrestrial Reference Frame modeling nonlinear station motions. Journal of Geophysical Research: Solid Earth, 2016, Vol. 121 (B8), pp. 6109-6131. doi: 10.1002/2016JB013098
https://doi.org/10.1002/2016JB013098
7. Argus D. F., Gordon R. G., DeMets C. Geologically current motion of 56 plates relative to the no-net-rotation reference frame. Geochemistry, Geophysics, Geosystems, 2011, Vol. 12 (11), N. Q11001, 13 p. doi: 10.1029/2011GC003751
https://doi.org/10.1029/2011GC003751
8. Argus D. F., Gordon R. G., Heflin M. B., Ma C., Eanes R., Willis P., Peltier W. R., Owen S. E. The angular velocities of the plates and the velocity of Earths centre from space geodesy. Geophysical Journal International, 2010, Vol. 180 (3), pp. 913-960. doi: 10.1111/j.1365-246X.2009.04463.x
https://doi.org/10.1111/j.1365-246X.2009.04463.x
9. Biagi L., Dermanis A. The treatment of time-continuous GPS observations for the determination of regional deformation parameters. Sanso F., Gil A.J. (Eds.), Geodetic deformation monitoring: from geophysical to geodetic roles. IAG Symposia, March 17-19, 2005, Jaen, Spain. Vol. 131. Berlin: Springer, 2006, pp. 83-94.
https://doi.org/10.1007/978-3-540-38596-7_10
10. DeMets C., Gordon R. G., Argus D. F. Geologically current plate motions. Geophysical Journal International, 2010, Vol. 181 (1), pp. 1-80. doi: 10.1111/j.1365-246X.2009.04491.x
https://doi.org/10.1111/j.1365-246X.2009.04491.x
11. Dermanis A. A study of the invariance of deformation parameters from a geodetic point of view. Kontadakis M. E., Kaltsikis C., Spatalas S., Tokmakidis K., Tziavos I.N. (Eds.), The apple of knowledge. Volume in honor of prof. D. Arabelos. Publication of the school of rural & surveying engineering, Aristotle University of Thessaloniki, 2010, pp. 43-66. http://der.topo.auth.gr/DERMANIS/ENGLISH/Publication_ENG.html
12. Dermanis A., Grafarend E. W. The finite element approach to the geodetic computation of two- and three-dimensional deformation parameters: a study of frame invariance and parameter estimability. Sevilla M. J., Henneberg H. (Eds.), Proceeding Int. Conference "Cartography-Geodesy", 5th Centenary of Americas: 1492-1992, Maracaibo, Venezuela, 24.11-3.12.1992. Madrid: Instituto de astronomia y geodesia, 1993, pp. 66-85.
13. Dermanis A. The evolution of geodetic methods for the determination of strain parameters for earth crust deformation. Arabelos D., Kontadakis M., Kaltsikis Ch., Spatalas S. (Eds.), Terrestrial and stellar environment. Volume in honor of prof. G. Asteriadis. Publication of the school of rural & surveying engineering, Aristotle University of Thessaloniki, 2009, pp. 107–144. http://der.topo.auth.gr/DERMANIS/ENGLISH/ Publication_ENG.html
14. Ferland R., Piraszewski M. The IGS-combined station coordinates, earth rotation parameters and apparent geocenter. Journal of Geodesy, 2009, Vol. 83 (3), pp. 385-392. doi: 10.1007/s00190-008-0295-9
https://doi.org/10.1007/s00190-008-0295-9
15. Hossainali M., Becker M., Groten E., 2011a. Comprehensive approach to the analysis of the 3D kinematics deformation with application to the Kenai Peninsula. Journal of Geodetic Science, 2011, Vol. 1(1), pp. 59-73. doi: 10.2478/v10156-010-0008-1
https://doi.org/10.2478/v10156-010-0008-1
16. Hossainali M., Becker M., Groten E., 2011b. Procrustean statistical inference of deformation. Journal of Geodetic Science, 2011, Vol. 1(2), pp. 170–180. doi: 10.2478/v10156-010-0020-5
https://doi.org/10.2478/v10156-010-0020-5
17. IERS Conventions (2010). Petit G., Luzum B. (Eds.), IERS Technical Note; 36. Frankfurt am Main: Verlag des Bundesamts fur Kartographie und Geodasie, 2010, 179 p. http://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/tn/Tec...
18. International Association of Geodesy. http://iag.dgfi.tum.de/fileadmin/handbook_2012/333_Commission_3.pdf
19. Tadyeyev O. A., 2016a. Evaluation of three-dimensional deformation fields of the Earth by methods of the projective differential geometry. Rigid rotations of the Earth. Geodesy, Cartography and Aerial Photography, 2016, Vol. 84, pp. 25–38.
20. Tadyeyev O. A., 2016b. Evaluation of three-dimensional deformation fields of the Earth by methods of the projective differential geometry. The main linear deformations. Scientific Journal Geodynamics, 2016, No. 2(21), pp. 7–17.
21. Vanicek P., Grafarend E., Berber M. Short note: strain invariants. Journal of Geodesy, 2008, Vol. 82, pp. 263–268. doi: 10.1007/s00190-007-0175-8
https://doi.org/10.1007/s00190-007-0175-8
22. Wu X., Collilieux X., Altamini Z., Vermeersen B. L. A., Gross R. S., Fukumori I. Accuracy of the International Terrestrial Reference Frame origin and Earth expansion. Geophysical Research Letters, 2011, Vol. 38(13), N. L13304, 5 p. doi: 10.1029/2011GL047450
https://doi.org/10.1029/2011GL047450
23. Xu P. L., Shimada S., Fujii Y., Tanaka T. Invariant geodynamical information in geometric geodetic measurement. Geophysical Journal International, 2000, Vol. 142, pp. 586–602.
https://doi.org/10.1046/j.1365-246x.2000.00181.x