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

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 - 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) - 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 - 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) - New rigorous equations for “horizontal” deformation analysis on the reference ellipsoid Attention: NOT for the physical surface of the earth, NOT 3-dimentional ! 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

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

Deformation as comparison of tw o shapes (at two epochs) x = coordinates at epoch t u = x - x = displacements Mathematical Elasticity: Deformation studied via the deformation gradient local linear approximation to the deformation function Geophysics-Geodesy: Deformation studied via the displacement gradient 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 x 0i and velocities v i at GPS permanent stations P i Displacements: u i = (t – t 0 ) v i

SPATIAL INTERPOLATION for the determination of or Classical horizontal deformation analysis Strain tensor E : description of (quadratic) variation of length element Geodetic data: Discrete initial coordinates x 0i and velocities v i at GPS permanent stations P i Displacements: u i = (t – t 0 ) v i Require: DIFFERENTIATION for the determination of or

Discrete geodetic information at GPS permanent stations Classical horizontal deformation analysis

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 SPATIAL INTERPOLATION

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: diagonalization e max, e min = principal strains  = direction of e max  = small rotation angle Classical horizontal deformation analysis

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

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

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

Horizontal deformational analysis using Singular Value Decomposition from diagonalizations: SVD

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 shear along the 1st axis linear scale factordirection of shear additional rotation not contributing to deformation 2 alternative 4-parametric representations

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

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

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

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

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

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 

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

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

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

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 INTERPOLATION EXTRAPOLATION 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

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

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

HOW IT IS DONE: F q 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: F q refers to local (non orthonormal) coordinate bases: Change to orthonormal bases: Transform F q to orthonormal bases: converting metric matrices to identity 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

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 Separation of rigid motion from deformation SOLUTION: = inertia matrix = relative angular momentum Migrating pole, variable angular velocity versus usual constant rotation (Euler rotation)

Spatial interpolation or prediction

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

TRIANGULAR FINITE ELEMENTSMINIMUM MEAN SQUARE ERROR PREDICTION (COLLOCATION) Spatial interpolation or prediction Deterministic interpolationInterpolation by stochastic prediction true interpolated true interpolated

TRIANGULAR FINITE ELEMENTSMINIMUM MEAN SQUARE ERROR PREDICTION (COLLOCATION) Spatial interpolation or prediction Piecewise linear displacements Piecewise constant deformation parameters Continuous displacements Continuous deformation parameters Deterministic interpolationInterpolation by stochastic prediction true interpolated true interpolated

TRIANGULAR FINITE ELEMENTSMINIMUM MEAN SQUARE ERROR PREDICTION (COLLOCATION) Spatial interpolation or 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 Deterministic interpolationInterpolation by stochastic prediction true interpolated true interpolated

TRIANGULAR FINITE ELEMENTSMINIMUM MEAN SQUARE ERROR PREDICTION (COLLOCATION) Spatial interpolation or 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 Deterministic interpolationInterpolation by stochastic prediction Accuracy estimates of deformation parameters reflect only data uncertainty Accuracy estimates of deformation parameters reflect both data and interpolation uncertainty true interpolated true interpolated

Principal linear elongation factors vs principal strains A comparison

Linear elongations and strains StrainsLinear elongation factors Definitions

Linear elongations and strains StrainsLinear elongation factors Definitions Computation from diagonalizations

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

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

Case study: National permanent GPS network in Central Japan

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 ELEMENTSCOLLOCATION

SNR = Signal to Noise Ration FINITE ELEMENTS COLLOCATION

FINITE ELEMENTS COLLOCATION SNR = Signal to Noise Ration

Linear trends in each sub-region ( max -1)   ( min -1)  10 6    57.1  6.0   10 6       12.1  6.0 ( max -1)   ( min -1)  10 6    89.6  3.9   10 6        3.9 ( max -1)  10 6   ( min -1)  10 6    69.5  5.8   10 6       24.5  5.8 R1 R2 R3 R1 R2 R3

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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