1. Crustal Deformation Analysis from Permanent GPS Networks
European Geophysical Union General Assembly - EGU2009
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 motion of (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 motion of (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. x = coordinates at epoch t x = coordinates at epoch t
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. x = coordinates at epoch t x = coordinates at epoch t
Deformation as comparison of two shapes (at two epochs)
x = coordinates at epoch t
x = coordinates at epoch t
u = x - x = displacements
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 gradient J = 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. SVD 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)

51. Horizontal deformation on ellipsoidal surface
Our approach to horizontal deformation:
Project surface points on reference ellipsoid,
Study the deformation of the derived (abstract) ellipsoidal surface

52. Horizontal deformation on ellipsoidal surface
HOW IT IS DONE:
Use curvilinear coordinates on the surface (geodetic coordinates)
Formulate coordinate gradient

53. Horizontal deformation on ellipsoidal surface
HOW IT IS DONE:
Fq refers to local (non orthonormal) coordinate bases:

54. Horizontal deformation on ellipsoidal surface
HOW IT IS DONE:
Fq refers to local (non orthonormal) coordinate bases:
Change to orthonormal bases:
converting metric matrices to identity

55. Horizontal deformation on ellipsoidal surface
HOW IT IS DONE:
Fq refers to local (non orthonormal) coordinate bases:
Change to orthonormal bases:
converting metric matrices to identity
Transform Fq to orthonormal bases:

56. Horizontal deformation on ellipsoidal surface
HOW IT IS DONE:
THEN PROCEED AS IN THE PLANAR CASE

57. Separation of rigid motion from deformation
The concept of the
discrete Tisserant reference system
best adapted to a particular region

58. Separation of rigid motion from deformation
SPATIAL INTERPOLATION

59. Separation of rigid motion from deformation

60. Separation of rigid motion from deformation
BAD SPATIAL INTERPOLATION

61. Separation of rigid motion from deformation
GOOD SPATIAL INTERPOLATION
PIECEWISE INTERPOLATION
INVOLVES DISCONTINUITIES
= FAULTS !

62. Separation of rigid motion from deformation
Horizontal Displacements

63. Separation of rigid motion from deformation
Horizontal Displacements

64. Separation of rigid motion from deformation
Different displacements behavior in 3 regions
Apart from internal deformation regions are in relative motion

65. Separation of rigid motion from deformation
HOW TO REPRESENT THE MOTION OF A DEFORMING REGION AS A WHOLE ?
BY THE MOTION OF A REGIONAL OPTIMAL REFERENCE SYSTEM !
OPTIMAL = SUCH THAT THE CORRESPONDING DISPLACEMENTS
(OR VELOVITIES) BECOME AS SMALL AS POSSIBLE

66. Separation of rigid motion from deformation
ORIGINAL
REFERENCE
SYSTEM
OPTIMAL
REFERENCE
SYSTEM
Motion as whole ( = motion of reference system)
+ internal deformation ( = motion with respect to the reference system)

67. Separation of rigid motion from deformation
Horizontal motion on earth ellipsoid (  sphere): Rotation around an axis with angular velocity 

68. Separation of rigid motion from deformation
Horizontal motion on earth ellipsoid (  sphere): Rotation around an axis with angular velocity 
DEFINITION OF OPTIMAL FRAME:
Minimization of relative kinetic energy
of regional network
Discrete Tisserant reference system

69. Separation of rigid motion from deformation
Horizontal motion on earth ellipsoid (  sphere): Rotation around an axis with angular velocity 
DEFINITION OF OPTIMAL FRAME:
Minimization of relative kinetic energy
of regional network
Discrete Tisserant reference system
SOLUTION:
Migrating pole, variable angular velocity
versus usual
constant rotation (Euler rotation)
= inertia matrix
= relative angular momentum

70. Spatial interpolation or prediction

71. Spatial interpolation or prediction
TRIANGULAR FINITE ELEMENTS
MINIMUM MEAN SQUARE ERROR
PREDICTION (COLLOCATION)
interpolated
interpolated
true
true

72. Spatial interpolation or prediction
TRIANGULAR FINITE ELEMENTS
MINIMUM MEAN SQUARE ERROR
PREDICTION (COLLOCATION)
interpolated
interpolated
true
true
Deterministic interpolation
Interpolation by stochastic prediction

73. Spatial interpolation or prediction
TRIANGULAR FINITE ELEMENTS
MINIMUM MEAN SQUARE ERROR
PREDICTION (COLLOCATION)
interpolated
interpolated
true
true
Deterministic interpolation
Interpolation by stochastic prediction
Piecewise linear displacements
Piecewise constant deformation parameters
Continuous displacements
Continuous deformation parameters

74. Spatial interpolation or prediction
TRIANGULAR FINITE ELEMENTS
MINIMUM MEAN SQUARE ERROR
PREDICTION (COLLOCATION)
interpolated
interpolated
true
true
Deterministic interpolation
Interpolation by stochastic prediction
Piecewise linear displacements
Piecewise constant deformation parameters
Continuous displacements
Continuous deformation parameters
Observation errors completely absorbed
in deformation parameters estimates
Observation errors partially removed
by interpolation smoothing

75. Spatial interpolation or prediction
TRIANGULAR FINITE ELEMENTS
MINIMUM MEAN SQUARE ERROR
PREDICTION (COLLOCATION)
interpolated
interpolated
true
true
Deterministic interpolation
Interpolation by stochastic prediction
Piecewise linear displacements
Piecewise constant deformation parameters
Continuous displacements
Continuous deformation parameters
Observation errors completely absorbed
in deformation parameters estimates
Observation errors partially removed
by interpolation smoothing
Accuracy estimates of deformation
parameters reflect only data uncertainty
Accuracy estimates of deformation
parameters reflect both data and
interpolation uncertainty

76. Principal linear elongation factorsvs
principal strains
A comparison

77. Linear elongations and strains
Linear elongation factors
Definitions

78. Linear elongations and strains
Linear elongation factors
Definitions
Computation from diagonalizations

79. Linear elongations and strains
Linear elongation factors
Definitions
Computation from diagonalizations
Interpretation
clear
meaning !
Meaning ?

80. Linear elongations and strains
Linear elongation factors
Definitions
Computation from diagonalizations
Interpretation
clear
meaning !
Meaning ?
Relation

81. National permanent GPS network
Case study:
National permanent GPS network
in Central Japan

83. Case study: Central Japan
Original velocities

84. Case study: Central Japan
Original velocities
Reduced velocities (removal of rotation)

85. Case study: Central Japan
Reduced velocities

86. Case study: Central Japan
Division in 3 regions. Relative velocities w.r. region R2 after removal of rigid rotations

87. Linear elongation factors max = 1, min = 2
FINITE ELEMENTS
SEPARATE COLLOCATIONS
IN EACH REGION

88. Dilatation  and shear 
FINITE ELEMENTS
COLLOCATION

89. SNR = Signal to Noise Ration
FINITE
ELEMENTS
COLLOCATION

90. SNR = Signal to Noise Ration
FINITE
ELEMENTS
COLLOCATION

91. Linear trends in each sub-region
(max-1)  106
0.001  0.013
(min-1)  106
0.116  0.021 
69.5  5.8
  106
0.117  0.024
  106
0.116  0.026 
24.5  5.8 R3 R3
(max-1)  106
0.004  0.007
(min-1)  106
0.072  0.008 
89.6  3.9
  106
0.068  0.011
  106
0.076  0.011 
134.6  3.9 R2 R2 R1
(max-1)  106
0.002  0.006
(min-1)  106
0.044  0.008 
57.1  6.0
  106
0.043  0.010
  106
0.046  0.010 
12.1  6.0 R1

92. Conclusions
Minimum Mean Square Error Prediction (collocation)
has the following advantages:
- Produces continuous results
for any desired point in the region of application
-Provides smooth results
where the effect of the data errors
is partially removed
- Provides more realistic variances-covariances
which in addition to the data uncertainty
reflect also the interpolation uncertainty

93. http://der.topo.auth.gr/ THANKS FOR YOUR ATTENTION
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