Crustal Deformation Analysis from Permanent GPS Networks 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

Our approach - Departure from classical horizontal deformation analysis:

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

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 !

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)

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

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

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

Classical horizontal deformation analysis A short review

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

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

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

Classical horizontal deformation analysis Discrete geodetic information at GPS permanent stations

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

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

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

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

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

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 

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 

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

Horizontal deformational analysis using Singular Value Decomposition from diagonalizations: SVD

Horizontal deformational analysis using Singular Value Decomposition

Horizontal deformational analysis using Singular Value Decomposition

Horizontal deformational analysis using Singular Value Decomposition

Horizontal deformational analysis using Singular Value Decomposition

Horizontal deformational analysis using Singular Value Decomposition

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

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

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

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

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

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

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

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

Rigorous derivation of invariant deformation parameters shear along direction 

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

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

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

Rigorous derivation of invariant deformation parameters Derivation of dilatation 

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

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

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

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

Horizontal deformation on the surface of the reference ellipsoid

Horizontal deformation on ellipsoidal surface Actual deformation is 3-dimensional

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

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

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

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)

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

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

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

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

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:

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

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

Separation of rigid motion from deformation SPATIAL INTERPOLATION

Separation of rigid motion from deformation

Separation of rigid motion from deformation BAD SPATIAL INTERPOLATION

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

Separation of rigid motion from deformation Horizontal Displacements

Separation of rigid motion from deformation Horizontal Displacements

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

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

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)

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

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

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

Spatial interpolation or prediction

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

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

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

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

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

Principal linear elongation factors vs principal strains A comparison

Linear elongations and strains Linear elongation factors Definitions

Linear elongations and strains Linear elongation factors Definitions Computation from diagonalizations

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

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

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

Case study: Central Japan Original velocities

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

Case study: Central Japan Reduced velocities

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

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

Dilatation  and shear  FINITE ELEMENTS COLLOCATION

SNR = Signal to Noise Ration FINITE ELEMENTS COLLOCATION

SNR = Signal to Noise Ration FINITE ELEMENTS COLLOCATION

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

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

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