athanasios dermanis & christopher kotsakis

Athanasios Dermanis & Christopher KotsakisAthanasios Dermanis & Christopher Kotsakis The Aristotle University of Thessaloniki, Department of Geodesy and SurveyingThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying

Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challengesEstimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges

Athanasios Dermanis & Christopher Kotsakis

The Aristotle University of Thessaloniki, Department of Geodesy and Surveying

Estimating crustal deformation parameters from geodetic data:

Review of existing methodologies, open problems and new challenges

International Symposium on Geodetic Deformation Monitoring: From Geophysical to Engineering Roles

17 – 19 March 2005, Jaén (SPAIN)



Athanasios Dermanis & Christopher Kotsakis

The Aristotle University of Thessaloniki Department of Geodesy and Surveying

Estimating crustal deformation parameters from geodetic data:

Review of existing methodologies, open problems and new challenges

International Symposium on Geodetic Deformation Monitoring: From Geophysical to Engineering Roles

17 – 19 March 2005, Jaén (SPAIN)



THE ISSUES:

What should be the end product of geodetic analysis?(Choice of parameters describing deformation)

Which is the role of the chosen reference system(s)?

How should the necessary spatial (and/or temporal) interpolationbe performed?(Trend removal and/or minimum norm interpolation)

2-dimensional or 3-dimensional deformation?(Incorporating height variation information in a reasonable way)

Quality assessment(effect of data errors and interpolation errors on final results)

Data analysis strategy(Data coordinates / displacements deformation parameters)



geometric information(shape alteration)

GEODESY

gravity variationinformation

GEOPHYSICS

acting forces

models forearth behavior

(elasticity, viscocity,...)

equations of motionfor deforming earth -

- constitutional equations

density distribution hypotheses

An interplay between Geodesy and Geophysics:Crustal Deformation as an Inverse Problem y = Ax

yA

xGeodetic product:Free of geophysical hypotheses!



The crustal deformation parametersto be produced by geodetic analysis



The deformation function f

x (λ)x0(λ)O0 O

Shape S0 Shape S

f

x = f ( x0)



x (λ)x0(λ)

O0O

Shape S0 Shape S

Two shapes of the same material curve x (λ) = f (x0(λ))

parametric curvedescriptions with

parameter λ

The deformation gradient F



x (λ)x0(λ)

O0O

Shape S0 Shape S

tangent vectors to the curve shapes


dx0

dλu0 = dxdλu =



ds0

d λ|u0| = ds

d λ|u| =

x (λ)x0(λ)

O0O

Shape S0 Shape S

tangent vector length= rate of length variation





dx0

dλu0 = dxdλu =

x (λ)x0(λ)

O0O

Shape S0 Shape S

F

u = F(u0)



The deformation gradient FRepresentation by a matrix F in chosen coordinate systems

d x0

d λu0 = d xd λu =

x (λ)x0(λ)

O0 O

Shape S0 Shape S

F

u = F u0

d x x du0

d λ x0 d λ=

xx0

F =



time

space(coordinates)

Comparison of shapes at two epochs t0 and t

coordinate linesof material points



time

space(coordinates)


t0 t

x0 = x(P,t0) x = x(P,t)

Observation ofcoordinates

of all material points at 2 epochs:

t0 and t

Spatiallycontinuousinformation



t0 t

x0 x




x0 x

x0

Use initial coordinatesas independent variables




xx

x0


Use coordinates at epoch tas dependent variables



x

x0


Deformation function f :

x = f(x0 )



x

x0

Deformation function f :

x = f(x0 )

Deformation gradient F

at point P =Local slope of

deformation function f :

fx0

F(P) = (P)




x

x0


When only discretespatial information

is available

In order to compute the

deformation gradient F

fx0

F(P) = (P)

We must performspatial interpolation



x

x0


When only discretespatial information

is available

In order to compute the

deformation gradient F

fx0

F(P) = (P)INTERPOLATION

We must performspatial interpolation



Physical interpretation of the deformation gradient

SVDSingular Value Decomposition

F = QT L PFrom diagonalizations:

C = U2 = FTF = PTL2PB = V2 = FFT = QTL2Q

Polar decomposition:

F = QTLP = (QTP)(PTLP) = RU= (QTLQ) QTP = VR

orthogonal diagonal

λ1, λ2, λ3 = singular values

=L 0 λ2 0

λ1 0 0

0 0 λ3



e1(t)

e2(t)


SVD F = QT L P

e1(t0)

e2(t0)

P

R R

Q



e1(t)

e2(t)


SVD F = QT L P

e1(t0)

e2(t0)

This is all we can observe at the two epochsNo relation of coordinate systems possible due to deformation




e1(t0)

e2(t0)

P

R

e1(t)

e2(t) Q

R

The reference systems and cannot be identified in geodesy!We live on the deforming body and not in a rigid laboratory!

e(t0) e(t)

SVD F = QT L P



λ2

Local deformation F = QT L P consists of

λ1

1

elongations L (scaling by λ1, λ2, λ3)

along principal axes

and a rotation R = QTP inaccessible in geodesydue to lack of coordinate system identification

R

Under x0 = S0x0, x = Sx : R = SRS0T ~ ~ ~

principalaxes

S0, S inaccessible but common for all points R(x0) = SR(x0)S0T

~ ~



Local deformation parameters (functions of F = QT L P )

λ1λ2

1

Singular values in L(λ1, λ2, λ3) and functions ψ(λ1,λ2,λ3) - Numerical invariants

R

principalaxes

Angles in P(θ1, θ2, θ3) defining directions of principal axes (physical invariants)

P

Angles in R(ω1, ω2, ω3) defining local rotation (not invariant)

e1(t0)

e2(t0)



Local deformation parameters (functions of F = QT L P )

λ1λ2

1

Singular values in L(λ1, λ2, λ3) and functions ψ(λ1,λ2,λ3) - Numerical invariants

R

principalaxes

e1(t0)

e2(t0)

P

2D(areal) dilatation: Δ = λ1λ2 1

3D(volume) dilatation: Δ = λ1λ2λ3 1

shear: γ = (λ1λ2) (λ1λ2)1/2 shears within principal planes: γik = (λiλk) (λiλk)1/2

ik = 12, 23, 13



The role of thereference system



Coordinates in an “preliminary” system, WGS 84, ITRF, or user defined:

We need a reference system O(t), e1(t), e2(t), e3(t) for every epoch t !(Dynamic or space-time reference system)

epoch t0

epoch t

displacementstoo large !



Definition of a network-intrinsic reference system:

Center of mass preservation:

Vanishing of relative angular momentum:

Mean quadratic scale preservation:

hR = i [xi] vi = 0

L2 = i ||xi – m||2 = const.1n

m = i xi = const. (= 0)1n



Advantages of a network-intrinsic reference system:

Invariant deformation parameters the same – No advantage forcontinuous spatial information

Dermination of motion of the network area as whole: translation and rotation



Advantages of a network-intrinsic reference system:

Invariant deformation parameters the same – No advantage forcontinuous spatial information

Determination of motion of the network area as whole: translation & rotation

Small displacements (trend removal): Essential for proper spatial interpolationof discrete spatial information

Reference systemsat 2 epochs

identified



2-dimensional or3-dimensional deformation?



Crustal deformation is a 3-dimensional physical process

t0 t

F



Usually studied as 2-dimensional by projection of physical surface to a “horizontal” plane

t0 t

F



Proper treatment:Deformation of the 2-dimensional physical surface

as embedded in 3-dimensional space

t0 t

F



Attempts for a 3-dimensional treatment

Extension of the2D finite element method

(triangular elements)to 3D

(quadrilateral elements)



Attempts for a 3-dimensional treatment

deformationof mountain

deformationof air !

quadrilateral elements have much smallervertical extension

We can obtain goodhorizontal information

by interpolation orvirtual densification.

Vertical informationrequires extrapolation(an insecure process)

Derination of 3D crustal deformationfrom 2D deformation surface deformatiom(downward continuation)an improperly posed problem !



Spatial (and/or temporal)interpolation



time

space

The “ideal” situation:

Space continuousTime continuous

To provide deformation parametersat any pointfor any 2 epochs

No interpolation needed !

Geodetic information on crustal deformation - Coordinates x(P,t)



time

space

The “satisfactory” situation:

Space continuousTime discrete

To provide deformation parametersat any pointfor any 2 observation epochs

No interpolation needed !

t1 t2 t3 t4 t5 t6 t7

Geodetic information on crustal deformation - Coordinates x(P,tk)



time

space

Geodetic information on crustal deformation - Coordinates x(P,tk)

The “satisfactory” situation:

Space continuousTime discrete


Temporal interpolation needed !

t1 t2 t3 t4 t5 t6 t7



time

space

The “realistic” situation:

Space discreteTime discrete


Spatial interpolation needed !

t1 t2 t3 t4 t5 t6 t7

P1

P2

P3

P4

P5

P6

Geodetic information on crustal deformation - Coordinates x(Pi,tk)



time

space





t1 t2 t3 t4 t5 t6 t7

P1

P2

P3

P4

P5

P6

Temporal interpolation also needed !




time

space




t1 t2 t3 t4 t5 t6 t7

P1

P2

P3

P4

P5

P6

Not all points observed at each epoch

Temporal interpolation needed !





time

space

t1 t2 t3 t4 t5 t6 t7

P1

P2

P3

P4

P5

P6

Geodetic information on crustal deformation - Coordinates x(Pi,t)

GPS permanent stations

Space discreteTime continuous





time

space

t1 t2 t3 t4 t5 t6 t7

P1

P2

P3

P4

P5

P6

GPS permanent stationsand SAR interferometry

Space discreteTime discrete (SAR) time - continuous (GPS)

To provide deformation parametersat any pointfor any 2 SAR observation epochs

Geodetic information on crustal deformation - Coordinates x(Pi,t), x(Pi,tk)

No spatial interpolationneeded !



time

space

t1 t2 t3 t4 t5 t6 t7

P1

P2

P3

P4

P5

P6

Geodetic information on crustal deformation - Coordinates x(Pi,t), x(Pi,tk)

GPS permanent stationsand SAR interferometry

Space discreteTime discrete (SAR) time - continuous (GPS)





Data analysisstrategies



Alternatives for spatial and/or temporal interpolation

To be inerpolated: displacments u at discrete epochs tk and discrete points Pi

u(Pi, tk) = x(Pi, tk) x(Pi, t0)

u(x0i,tk) = xi(tk) x0i

expressed as

Analytic least squares (smoothing) interpolation

u(x, t)

Minumum norm (exact) interpolation

Sought:

u(x0, t)

Given:

simplified to

for every x0 and t

2 types of interpolation

or

Minimum norm (smoothing) interpolationequivalent to

Minimum mean-square error linear prediction

deterministic

stochastic



Temporal interpolation: Analytic least squares (smoothing) interpolation

deformation evolves slowly with time

Spatial interpolation: Analytic least squares (smoothing) interpolation

Minumum norm (exact) interpolation

or

Minimum norm (smoothing) interpolation

or combination of the two



Alternatives for linear spatial interpolation

Analytic least squares (smoothing) interpolation:

u(x) = g(x, a) = m fm(x) am

a = vector of free parameters ( observations)

least square solution a of u(xi) = g(xi,a) + vi (vTPv = min)

Minumum norm (exact) interpolation:

minimum-norm solution a of u(xi) = g(xi,a) (aTRa = min)

Minumum norm (smoothing) interpolation:

(parameters a > observations, even infinite)

hybrid solution a of u(x0i) = g(x0i, a) + vi (vTPv+aTRa = min)

(parameters a > observations, even infinite)



Deterministic and stochastic interpretation for linear interpolation



aTRa = min

u(x) = fT(x)R1FT(FR1FT)1 b =

= k(x)T K1 b

b = F au(xi) = m fm(xi) am = f(xi)T a

aTRa + vTPv = min

u(P) = kT (P)(K+P1)1 b

Minumum mean square error prediction:

b = F a + v

b = s

b = s + v

e = s(x) - s(x), trE{eeT} = min

s(x) = Cs(x)s Cs1 b

~

~

“Collocation”in geodetic jargon

s(x) = Cs(x)s (Cs + Cv)1 b~

deterministic stochastic



s(x) = Cs(x)s (Cs + Cv)1 b~ =

= f(x)TCaFT (FCaFT)1

b

=

= f(x)TCaFT (FCaFT)1

b

Deterministic and stochastic interpretation for linear interpolation



aTRa = min

u(x) = fT(x)R1FT(FR1FT)1 b =

= k(x)T K1 b

aTRa + vTPv = min

u(P) = kT (P)(K+P1)1 b


b = F a + v b = Fa + v

b = s = F a, s(x) = f(x)Ta

Css= FCaFT, Cs(x)s= f(x)TCaFT

EquivalenceCa = R1

Cv = P1

b = F au(xi) = m fm(xi) am = f(xi)T a

s(x) = Cs(x)s Cs1 b



Temporal interpolation: least-squares analytic

x(x0,t) = x0 + (tt0) v(x0)

u(x0,t) = (tt0) v(x0)

F(x0,t) = I + (tt0) L(x0) L =vx0

+ Spatial interpolation: Combination of least-squares analytic and stochastic prediction

v(x0) = f(x0)T a + z(x0)

s = s1

s2 Csisk(x0,x0)

Spatial interpolation: Combination of least-squares analytic and stochastic prediction

component covariance functionsx(x0) = f(x0)T

a + s(x0)

EXAMPLES OF INTERPOLATION MODELS

Observation epochs t0, t - No temporal interpolation

Czizk(x0,x0)

component covariance functions



Cs2s2(x0,x0) = = Cs2s2

(||x0x0||)

Invariance guaranteed by stationary and isotropic component covariance functions

Invariant Interpolation = independent of used reference systems

Analytic least squares (smoothing) interpolation: u(x) = m fm(x) am = f(x)T a

Not invariant interpolation!Base functions fm(x) depend on coordinate system used

Exception: Finite element method uT(x) = JT x + cT

Different JT, for each triangular element, cT irrelevant (FT = I + JT)

T

1

2

3

u(x2)-u(x1) = J (x2-x1)

u(x3)-u(x1) = J (x3-x1)Solve for J

Same equations for

x = Rx+d u = Ru

Invariant interpolation !

~ ~


Cs1s1(x0,x0) = = Cs1s1

(||x0x0||) Cs1s2(x0,x0) = 0

s(x) = Cs(x)s (Cs + Cv)1 b



Data analysis alternatives

“raw” observations(e.g. GPS baselines)

stationcoordinates

displacementfield

deformationgradient

adjustment

interpolation

“raw” observations(e.g. GPS baselines)

displacementfield

deformationgradient

combinedadjustmentand interpolation

Potential problem:Incorrect separationof observation errorsand displacements



Quality assessmentfor deformation parameters



Problems in quality assessment for deformation parameters

Incorrect statistics for input data (coordinates)

Incorrect assessment of interpolation errors

Incorrect error propagation from deformation gradient to deformation parameters

Typical for GPS coordinates

Improper separation of signal (displacements) from noise

Singular values highly nonlinear functions of deformation gradient

Fit lines to time variationEstimate statistics from residuals

Trial and error

Use propagation with higher derivatives and momentsUse Monte Carlo techniques



FUTURE OUTLOOK:

Permanent GPS stations provide initial coordinates and velocitieswith realistic variance-covariance matrices

Supplementary SAR Interferometry provides spatially interpolated velocities & identifies problematic points

OPEN PROBLEMS:

Optimal merging of GPS with SAR Interferometry data

The missing third dimension in crustal deformation information(Introduce geophysical hypotheses ?)



This presentation will be available at

http://der.topo.auth.gr/

athanasios dermanis & christopher kotsakis

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