Ludovico Biagi & Athanasios Dermanis Politecnico di Milano, DIIAR

1. European Geophysical Union General Assembly - EGU2009 19 -24 April 2009, Vienna, Austria Crustal Deformation Analysis from Permanent GPS Networks Ludovico Biagi & Athanasios Dermanis Politecnico di Milano, DIIAR Aristotle University of Thessaloniki, Department of Geodesy and Surveying

2. Our approach - Departure from classical horizontal deformation analysis:

3. Our approach - Departure from classical horizontal deformation analysis: - New rigorous equations for deformation invariant parameters: Principal (max-min) linear elongation factors & their directions Dilatation, (maximum) shear strain & its direction

4. Our approach - Departure from classical horizontal deformation analysis: - New rigorous equations for deformation invariant parameters: Principal (max-min) linear elongation factors & their directions Dilatation, (maximum) shear strain & its direction - New rigorous equations for “horizontal” deformation analysis on the reference ellipsoid Attention: NOT for the physical surface of the earth, NOT 3-dimentional !

5. Our approach - Departure from classical horizontal deformation analysis: - New rigorous equations for deformation invariant parameters: Principal (max-min) linear elongation factors & their directions Dilatation, (maximum) shear strain & its direction - New rigorous equations for “horizontal” deformation analysis on the reference ellipsoid Attention: NOT for the physical surface of the earth, NOT 3-dimentional ! - Separation of relative rigid motionof (sub)regions from actual deformation: Identification of regions with different kinematic behavior (clustering) Use of best fitting reference system for each region (Concept of regional discrete Tisserant reference system)

6. Our approach - Departure from classical horizontal deformation analysis: - New rigorous equations for deformation invariant parameters: Principal (max-min) linear elongation factors & their directions Dilatation, (maximum) shear strain & its direction - New rigorous equations for “horizontal” deformation analysis on the reference ellipsoid Attention: NOT for the physical surface of the earth, NOT 3-dimentional ! - Separation of relative rigid motionof (sub)regions from actual deformation: Identification of regions with different kinematic behavior (clustering) Use of best fitting reference system for each region (Concept of regional discrete Tisserant reference system) PLUS Study of signal-to-noise ratio (significance) of deformation parameters from spatially interpolated GPS velocity estimates using: - Finite element method (triangular elements) - Minimum Mean Square Error Prediction (collocation) CASE STUDY: Central Japan

7. Deformation as comparison of two shapes (at two epochs) x = coordinates at epoch t x = coordinates at epoch t Mathematical Elasticity: Deformation studied via the deformation gradient local linear approximation to the deformation function

8. Deformation as comparison of two shapes (at two epochs) x = coordinates at epoch t u = x-x = displacements x = coordinates at epoch t Mathematical Elasticity: Deformation studied via the deformation gradient Geophysics-Geodesy: Deformation studied via the displacement gradient local linear approximation to the deformation function and approximation to strain tensor

9. Classical horizontal deformation analysis A short review

10. Classical horizontal deformation analysis Strain tensor E : description of (quadratic) variation of length element

11. Classical horizontal deformation analysis Strain tensor E : description of (quadratic) variation of length element Geodetic data: Discrete initial coordinates x0i and velocities vi at GPS permanent stations Pi Displacements: ui = (t – t0) vi

12. Classical horizontal deformation analysis Strain tensor E : description of (quadratic) variation of length element Geodetic data: Discrete initial coordinates x0i and velocities vi at GPS permanent stations Pi Displacements: ui = (t – t0) vi Require: SPATIAL INTERPOLATION for the determination of or DIFFERENTIATION for the determination of or

13. Classical horizontal deformation analysis Discrete geodetic information at GPS permanent stations

14. Classical horizontal deformation analysis SPATIAL INTERPOLATION Discrete geodetic information at GPS permanent stations Interpolation to obtain continuous information, e.g. displacements at every point

15. Classical horizontal deformation analysis SPATIAL INTERPOLATION Discrete geodetic information at GPS permanent stations Interpolation to obtain continuous information, e.g. displacements at every point Differentiation to obtain the deformation gradient F or displacement gradientJ = F - I

16. Classical horizontal deformation analysis Analysis of the displacement gradient J into symmetric and antisymmetric part:

17. Classical horizontal deformation analysis Analysis of the displacement gradient J into symmetric and antisymmetric part:  = small rotation angle

18. Classical horizontal deformation analysis Analysis of the displacement gradient J into symmetric and antisymmetric part:  = small rotation angle emax, emin = principal strains  = direction of emax diagonalization

19. Classical horizontal deformation analysis Analysis of the displacement gradient J into symmetric and antisymmetric part:  = small rotation angle emax, emin = principal strains  = direction of emax diagonalization  = dilataton = maximum shear strain = direction of 

20. Classical horizontal deformation analysis Analysis of the displacement gradient J into symmetric and antisymmetric part:  = small rotation angle emax, emin = principal strains  = direction of emax diagonalization  = dilataton = maximum shear strain = direction of 

21. Horizontal deformational analysis using the Singular Value Decomposition (SVD) A new approach SVD

22. Horizontal deformational analysis using Singular Value Decomposition from diagonalizations: SVD

23. Horizontal deformational analysis using Singular Value Decomposition

24. Horizontal deformational analysis using Singular Value Decomposition

25. Horizontal deformational analysis using Singular Value Decomposition

26. Horizontal deformational analysis using Singular Value Decomposition

27. Horizontal deformational analysis using Singular Value Decomposition

28. Rigorous derivation of invariant deformation parameters without the approximations based on the infinitesimal strain tensor

29. Rigorous derivation of invariant deformation parameters 2 alternative 4-parametric representations shear along the 1st axis linear scale factor direction of shear additional rotation not contributing to deformation

30. Rigorous derivation of invariant deformation parameters 2 alternative 4-parametric representations shear along the 1st axis linear scale factor direction of shear additional rotation not contributing to deformation

31. Rigorous derivation of invariant deformation parameters 2 alternative 4-parametric representations shear along the 1st axis linear scale factor direction of shear additional rotation not contributing to deformation

32. Rigorous derivation of invariant deformation parameters 2 alternative 4-parametric representations shear along the 1st axis linear scale factor direction of shear additional rotation not contributing to deformation

33. Rigorous derivation of invariant deformation parameters 2 alternative 4-parametric representations shear along the 1st axis linear scale factor direction of shear additional rotation not contributing to deformation

34. Rigorous derivation of invariant deformation parameters 2 alternative 4-parametric representations shear along the 1st axis linear scale factor direction of shear additional rotation not contributing to deformation

35. Rigorous derivation of invariant deformation parameters shear along the 1st axis

36. Rigorous derivation of invariant deformation parameters shear along direction 

37. Rigorous derivation of invariant deformation parameters additional rotation  (no deformation)

38. Rigorous derivation of invariant deformation parameters additional scaling (scale factor s)

39. Rigorous derivation of invariant deformation parameters Compare the two representations and express s, , ,  as functions of 1, 2, , 

40. Rigorous derivation of invariant deformation parameters Derivation of dilatation 

41. Rigorous derivation of invariant deformation parameters Derivation of shear , and its direction  Use Singular Value Decomposition and replace

42. Rigorous derivation of invariant deformation parameters Derivation of shear , and its direction  Compare

43. Rigorous derivation of invariant deformation parameters Derivation of shear , and its direction 

44. Rigorous derivation of invariant deformation parameters Derivation of shear , and its direction 

45. Horizontal deformation on the surface of the reference ellipsoid

46. Horizontal deformation on ellipsoidal surface Actual deformation is 3-dimensional

47. Horizontal deformation on ellipsoidal surface But we can observe only on 2-dimensional earth surface !

48. Horizontal deformation on ellipsoidal surface INTERPOLATION EXTRAPOLATION Why not 3D deformation? 3D deformation requires not only interpolation but also an extrapolation outside the surface Extrapolation from surface geodetic data is not reliable – requires additional geophysical hypothesis

49. Horizontal deformation on ellipsoidal surface Standard horizontal deformation: Project surface points on horizontal plane, Study the deformation of the derived (abstract) planar surface

50. Horizontal deformation on ellipsoidal surface Why not study deformation of actual earth surface? Local surface deformation is a view of actual 3D deformation through a section along the tangent plane to the surface. For variable terrain: we look on 3D deformation from different directions ! Horizontal and vertical deformation caused by different geophysical processes (e.g. plate motion vs postglacial uplift)