Determination of the horizontal strain rates tensor in Western Ukraine
Received: July 18, 2019
Revised: November 18, 2019
Accepted: December 05, 2019
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University; Carpathian branch of S. I. Subbotin name Institute of geophysics of NAS of Ukraine
Lviv Polytechnic National University, Institute of Geodesy
General Commission for Survey, Saudi Arabia

Doppler Orbitography and Radio-positioning Integrated by Satellite (CORS) observations from 37 Global Navigation Satellite System (GNSS) stations located in the Western Ukraine area were processed using Bernese Processing Engine module (BPE) of Bernese GNSS Software version 5.2 for a time span of about 2.5 years. To get a better agreement for constrains, the IGS stations closest to the surrounding area of study were chosen with fixed coordinates of ITRF2008 at epoch 2005.0. Eastern and Northern components of velocities of GNSS observations from these 37 permanent stations, calculated from GNSS measurements, were used to construct a 2D model of horizontal strain rates field for the area. This study is presented in three parts. Firstly, two exact solutions for the components of the 2D strain rate tensor derived on the geosphere based on solving the eigenvalues – eigenvectors problem were analyzed, including skew symmetric rotational rate tensor. Secondly, based on the most simple and useful formulas from the first stage, a rigorous estimation of the accuracy of components of the 2D strain rate tensor were obtained based on the covariance propagation rule. Finally, the components of the 2D strain rate tensor, dilatation rate and components of the sheer rate tensor in the region were computed. A model of the rotation rate tensor was constructed for the described area, which led to the conclusion that the region of study should be interpreted as a deformed territory. Based on the computations from the GNSS-data model of components of horizontal deformations, the rates of principal values and rates of principal axes of the Earth’s crust deformation were found. To be consistent, the main tectonic formations are shown as the background intensity of different components of velocities, the rotation rate and strain rate tensors. Topographic features of the region were based on the SRTM-3 model (Shuttle Radar Topography Mission) with resolution 3²´3². At the first sight, the maximum sheer rates have greatest values in the areas located around the Ukrainian Carpathians. The dilatation rate has also a similar distribution.  Nevertheless, because in the paper only eigenvalue – eigenvector problem without accuracy estimation has been considered, which possibly leads to doubtful conclusions regarding interpretation and requires an additional solution of a purely mathematical problem. The full covariance matrix of the strain rate tensor should be found based on given full covariance matrix of the velocity components obtained by Bernese software. As a matter of fact, the study region is very complex in terms of crustal movements, which, according to the results obtained, require further densification of permanent GNSS stations.

  1. 1. Bird, P. (2003). An updated digital model of plate boundaries. Geochemistry, Geophysics, Geosystems, 4(3). art. no. 1027, doi:10.1029/2001GC000252, 1-52
    2. Crespi, M., Pietrantonio, G., & Riguzzi, F. (2000) Strain tensor estimation by GPS observations: Software and applications. Bollettino di geodesia e scienze affini, 59(3), 261-280.
    3. DeMets, C., Gordon, R. G., Argus, D. F., & Stein, S. (1990). Current plate motions. Geophysical journal international, 101(2), 425-478.
    4. DeMets, C., Gordon, R. G., Argus, D. F., & Stein, S. (1994). Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophysical research letters, 21(20), 2191-2194
    5. England, P., & Molnar, P. (1997). The field of crustal velocity in Asia calculated from Quaternary rates of slip on faults. Geophysical Journal International, 130(3), 551-582.
    6. Haines, A. J., & Holt, W. E. (1993). A procedure for obtaining the complete horizontal motions within zones of distributed deformation from the inversion of strain rate data. Journal of Geophysical Research: Solid Earth, 98(B7), 12057-12082.
    7. Julliette, L., Altamimi, Z., & Olivier, J. (2006). Interpolation of the European velocity field using least squeares collocation method. Paper presented at the EUREF Symposium 2006. Riga, Latvia, 14-17 June, 2006
    8. Kreemer, C., Haines, J., Holt, W. E., Blewitt, G., & Lavallee, D. (2000). On the determination of a global strain rate model. Earth, Planets and Space, 52(10), 765-770.
    9. Marchenko, A. N. (2003). A note on the eigenvalue - eigenvector problem. In: Kühtreiber N. (Ed.), Festschrift dedicated to Helmut Moritz on the occasion of his 70th birthday. Graz University of Technology, pp. 143-152.
    10. Marchenko, A. N., & Schwintzer, P. (2003). Estimation of the Earth's tensor of inertia from recent global gravity field solutions. Journal of geodesy, 76(9-10), 495-509.
    11. Marchenko, A. N., Tretyak, K. R., & Serant, O. (2010). On the accuracy estimation of components of the strain tensor. In: Modern Achievements of Geodetic Science and Industry. 2(20), 41-43 (in Ukrainian)
    12. Marchenko, A. N., Marchenko, D.A. & Lopushansky, A.N. (2016). Gravity field models derived from the second degree radial derivatives of the GOCE mission: a case study. Annals of Geophysics, 59(6), 0649-0659.
    13. Minster, J. B., & Jordan, T. H. (1978). Present‐day plate motions. Journal of Geophysical Research: Solid Earth, 83(B11), 5331-5354.
    14. Moritz, H., & Mulle,r I. I. (1987). Earth's Rotation. Theory and estimations, New York, Ungar
    15. Petit, G. & Luzum, B. (2010). IERS Conventions (2010), IERS Technical Note, No.36, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main.
    16. Vaníček, P., Grafarend, E. W., & Berber, M. (2008). Short note: Strain invariants. Journal of Geodesy, 82(4-5), 263-268.
    17. Ward, S. N. (1998). On the consistency of earthquake moment rates, geological fault data, and space geodetic strain: the United States. Geophysical Journal International, 134(1), 172-186.