Abstract
In this contribution, the regularized Earth’s surface is considered as a graded 2D surface, namely a curved surface, embedded in a Euclidean space \({\mathbb{E}^{3}}\). Thus, the deformation of the surface could be completely specified by the change of the metric and curvature tensors, namely strain tensor and tensor of change of curvature (TCC). The curvature tensor, however, is responsible for the detection of vertical displacements on the surface. Dealing with eigenspace components, e.g., principal components and principal directions of 2D symmetric random tensors of second order is of central importance in this study. Namely, we introduce an eigenspace analysis or a principal component analysis of strain tensor and TCC. However, due to the intricate relations between elements of tensors on one side and eigenspace components on other side, we will convert these relations to simple equations, by simultaneous diagonalization. This will provide simple synthesis equations of eigenspace components (e.g., applicable in stochastic aspects). The last part of this research is devoted to stochastic aspects of deformation analysis. In the presence of errors in measuring a random displacement field (under the normal distribution assumption of displacement field), the stochastic behaviors of eigenspace components of strain tensor and TCC are discussed. It is applied by a numerical example with the crustal deformation field, through the Pacific Northwest Geodetic Array permanent solutions in period January 1999 to January 2004, in Cascadia Subduction Zone. Due to the earthquake which occurred on 28 February 2001 in Puget Sound (M w > 6.8), we performed computations in two steps: the coseismic effect and the postseismic effect of this event. A comparison of patterns of eigenspace components of deformation tensors (corresponding the seismic events) reflects that: among the estimated eigenspace components, near the earthquake region, the eigenvalues have significant variations, but eigendirections have insignificant variations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aoki Y, Scholz C (2003) Vertical deformation of the Japanese Islands, 1996–1999. J Geophys Res 108(B5): ETG 10–1 doi:10.10292002JB002129
Aravind PK (1988) Geometrical interpretation of the simultaneous diagonalization of two quadratic forms. Am J Phys 57(4): 309–311
Beavan J, Matheson D, Denys P, Denham M, Herring T, Hager B, Molnar P (2004) A vertical deformation profile across the southern Alps, New Zealand, from 3.5 years of continuous GPS data. In: Proceedings of the workshop: the state of GPS vertical positioning precision: separation of Earth processes by space geodesy, Cahiers du Centre Européen de Géodynamique et de Séodmologie, vol 24, pp 111–123
Cai J (2004) Statistical inference of the eigenspace components of a symmetric random deformation tensor. Ph.D. thesis, Institute of Geodesy, Universität Stuttgart, Germany
Cai J, Grafarend EW, Schaffrin B (2005) Statistical inference of the eigenspace components of a two-dimensional, symmetric rank-two random tensor. J Geod 78(7–8): 425–436 doi:10.1007/s00190-004-0405-2
Chen X (1998) Continuous GPS monitoring of crustal deformation with the Western Canada Deformation Array, 1992–1995. Master’s thesis, University of New Brunswick, http://www.scientificcommons.org/8897140
Dargert H (1987) The fall (and rise) of central Vancouver Island: 1930–1985. Can J Earth Sci 24: 689–697
Dermanis A, Grafarend EW (1993) The finite element approach to the geodetic computation of two- and three-dimensional deformation parameters: a study of frame invariance and parameter estimability. In: Proceedings of the international conference on cartography, Maracaibo (Venezuela)
Dragert H, Hyndman RD, Rogers GC, Wang K (1994) Current deformation and the width of the seismogenic zone of the northern Cascadia subduction thrust. J Geophys Res 99(B1): 653–668
Dragert H, Chen X, Kouba J (1995) GPS monitoring of crustal strain in southwest British Columbia with the Western Canada Deformation Array. Geomatica 49(3): 301–313
Eisele JA, Mason RM (1970) Applied matrix and tensor analysis. Wiley-Interscience, New York
Ernst LJ (1981) A geometrically nonlinear finite element shell theory. Ph.D. thesis, Department of Mechanical Engineering, TU Delft
Grafarend EW (2006) Linear and nonlinear models: fixed effects random effects and mixed models. Walter de Gruyter, Berlin
Grafarend EW, Engels J (1992) A global representation of ellipsoidal heights—geoidal undulations or topographic heights—in terms of orthonormal functions, part 1: “amplitude-modified” spherical harmonic functions. Manuscripta Geodaetica 17(1): 52–58
Grafarend EW, Krumm FW (2006) Map projections, cartographic information systems. Springer, Berlin
Grafarend EW, Voosoghi B (2003) Intrinsic deformation analysis of the Earth’s surface based on displacement fields derived from space geodetic measurements. Case studies: present-day deformation patterns of Europe and of the Mediterranean area (ITRF data sets). J Geod 77(5–6): 303–326 doi:10.1007/s00190-003-0329-2
Huang J, Véronneau M (2005) Determination of the Canadian Gravimetric Geoid 2005 (CGG05) using GRACE and Terrestrial Gravity Data. AGU Fall Meeting Abstracts
Johansson JM, Davis JL, Scherneck HG, Milne GA, Vermeer M, Mitrovica JX, Bennett RA, Jonsson B, Elgered G, Elosegui P, Koivula H, Poutanen M, Rönnüng BO, Shapiro II. (2002) Continuous GPS measurements of postglacial adjustment in Fennoscandia, 1, Geodetic results. J Geophys Res 107(B8): ETG 3–1 doi:10.10292001JB000400
Kleusberg A, Georgiadou Y, Dragert H (1988) Establishment of crustal deformation networks using GPS: a case study. CISM J ACSGC 42: 341–351
Libai A, Simmonds JG (1976) The Nonlinear theory of elastic shells. Cambridge University Press, Cambridge
Mazzotti S, Dragert H, Henton J, Schmidt M, Hyndman R, James T, Lu Y, Craymer M (2003) Current tectonics of northern Cascadia from a decade of GPS measurements. J Geophys Res 108(B12): ETG 1–1 doi:10.10292003JB002653
Mikhail EM, Ackermann F (1976) Observations and least squares. Univ Pr of Amer, New York
Miller MM, Johnson DJ, Rubin CM, Dragert H, Wang K, Qamar A, Goldfinger C (2001) GPS-determination of along-strike variation in Cascadia margin kinematics: implications for relative plate motion, subduction zone coupling, and permanent deformation. Tectonics 20(2): 161–176
Milne GA, Davis JL, Mitrovica JX, Scherneck HG, Johansson JM, Vermeer M, Koivula H (2001) Space-geodetic constraints on glacial isostatic adjustment in Fennoscandia. Science 291(23): 2381–2385 doi:10.1126/science.1057022
Moghtased K (2007) Surface deformation analysis of dense GPS networks based on intrinsic geometry: deterministc and stochastic aspects. Ph.D thesis, Institute of Geodesy, Universität Stuttgart, Germany
Murray MH, Lisowski M (2000) Strain accumulation along the Cascadia subduction zone. Geophys Res Lett 27: 3631–3634 doi:10.1029/1999GL011127
Mushtari KM, Galimov KZ (1961) The non-linear theory of elastic shells. Israel Program Sci Transl (Translated from Russian), Jerusalem
Naghdi PM (1972) The theory of shells and plates. Handbuch der Physik, VI, A2. Springer, Berlin
Nikolaidis R (2002) Observation of geodetic and seismic deformation with the global positioning system. Ph.D. thesis, University of California, San Diego
Olszak W (1980) The shell theory: new trends and applications. Courses and lectures no. 240. Springer, Berlin
Pietraszkiewicz W (1977) Introduction to the non-linear theory of shells. Technical report 10, Mitteilungen aus dem institut fuer Mechanik, Ruhr-Universität Bochum
Reissner E (1974) Linear and nonlinear theory of shells. In: Thin shell structures. Prentice-Hall, New Jersey, pp 29–44
Rogers GC (1988) An assessment of the megathrust earthquakes potential of the Cascadia subduction zone. Can J Earth Sci 25: 844–852
Stein E (1980) Variational functionals in the geometrical nonlinear theory of thin shells and finite-element-discretizations with applications to stability problems. In: Theory of shells. Proceedings of the 3rd IUTAM symposium on shell theory, Tbilisi 1978, North-Holland, Amsterdam, pp 509–535
Verdonck D (2004) Vertical crustal deformation in Cascadia from historical leveling. In: Denver Annual Meeting
Voosoghi B (2000) Intrinsic deformation analysis of the earth surface based on 3-dimensional displacement fields derived from space geodetic measurements. Ph.D. thesis, Institute of Geodesy, Universität Stuttgart, Germany
Wang K, Dragert H, Melosh HJ (1994) Finite element study of uplift and strain across Vancouver Island. Can J Earth Sci 31: 1510–1522
Xu PL, Grafarend EW (1996) Probability distribution of eigenspectra and eigendirections of a two-dimensional, symmetric rank two random tensor. J Geod 70(7): 419–430 doi:10.1007/BF01090817
Xu PL,Grafarend EW (1996b) Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensor. Geophys J Int 127(3):744–756, doi:10.1111/j.1365-246X.1996.tb04053.x
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Moghtased-Azar, K., Grafarend, E.W. Surface deformation analysis of dense GPS networks based on intrinsic geometry: deterministic and stochastic aspects. J Geod 83, 431–454 (2009). https://doi.org/10.1007/s00190-008-0252-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-008-0252-7