Athanasios Dermanis

1
International Symposium on Geodetic Deformation
Monitoring From Geophysical to Engineering
Roles 17 19 March 2005, Jaén (SPAIN)
Estimating 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
2
International Symposium on Geodetic Deformation
Monitoring From Geophysical to Engineering
Roles 17 19 March 2005, Jaén (SPAIN)
Estimating 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
3
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)?
? 2-dimensional or 3-dimensional
deformation? (Incorporating height variation
information in a reasonable way)
? How should the necessary spatial (and/or
temporal) interpolation be performed? (Trend
removal and/or minimum norm interpolation)
? Data analysis strategy (Data ? coordinates /
displacements ? deformation parameters)
? Quality assessment (effect of data errors and
interpolation errors on final results)
4
An interplay between Geodesy and
Geophysics Crustal Deformation as an Inverse
Problem y Ax
GEODESY
GEOPHYSICS
A
acting forces
geometric information (shape alteration)
y
equations of motion for deforming earth - -
constitutional equations
models for earth behavior (elasticity,
viscocity,...)
gravity variation information
x
density distribution hypotheses
Geodetic product Free of geophysical hypotheses!
5
An interplay between Geodesy and
Geophysics Crustal Deformation as an Inverse
Problem y Ax
GEODESY
GEOPHYSICS
A
acting forces
geometric information (shape alteration)
y
equations of motion for deforming earth - -
constitutional equations
models for earth behavior (elasticity,
viscocity,...)
gravity variation information
x
density distribution hypotheses
Geodetic product Free of geophysical hypotheses!
6
The crustal deformation parameters to be produced
by geodetic analysis
7
The deformation function f
f
O0
O
Shape S0
Shape S
8
The deformation gradient F
x (?) f (x0(?))
Two shapes of the same material curve
parametric curve descriptions with parameter ?
O
O0
Shape S0
Shape S
9
The deformation gradient F
tangent vectors to the curve shapes
O
O0
Shape S0
Shape S
10
The deformation gradient F
tangent vector length rate of length variation
ds0
ds
u0
u
d ?
d ?
O
O0
Shape S0
Shape S
11
The deformation gradient F
F
O
O0
Shape S0
Shape S
12
The deformation gradient F Representation by a
matrix F in chosen coordinate systems
d x0
d x
u0
u
d ?
d ?
F
u F u0
x (?)
x0(?)
O0
O
Shape S0
Shape S
13
Comparison of shapes at two epochs t0 and t
space (coordinates)
coordinate lines of material points
time
14
Comparison of shapes at two epochs t0 and t
x0 x(P,t0)
x x(P,t)
space (coordinates)
Observation of coordinates of all material points
at 2 epochs t0 and t
Spatially continuous information
time
t0
t
15
Comparison of shapes at two epochs t0 and t
x0
x
t0
t
16
Comparison of shapes at two epochs t0 and t
x0
x
Use initial coordinates as independent variables
x0
17
Comparison of shapes at two epochs t0 and t
x
x
Use coordinates at epoch t as dependent variables
x0
18
Comparison of shapes at two epochs t0 and t
x
Deformation function f x f(x0 )
x0
19
Comparison of shapes at two epochs t0 and t
x
Deformation function f x f(x0 )
Deformation gradient F at point P Local slope
of deformation function f
x0
20
Comparison of shapes at two epochs t0 and t
x
When only discrete spatial information is
available
We must perform spatial interpolation
In order to compute the deformation gradient F
x0
21
Comparison of shapes at two epochs t0 and t
x
When only discrete spatial information is
available
INTERPOLATION
We must perform spatial interpolation
In order to compute the deformation gradient F
x0
22
Physical interpretation of the deformation
gradient
From diagonalizations C U2 FTF PTL2P B
V2 FFT QTL2Q
diagonal
orthogonal
Polar decomposition F QTLP (QTP)(PTLP)
RU (QTLQ) QTP VR
?1, ?2, ?3 singular values
23
Physical interpretation of the deformation
gradient
SVD
24
Physical interpretation of the deformation
gradient
SVD
This is all we can observe at the two epochs No
relation of coordinate systems possible due to
deformation
25
Physical interpretation of the deformation
gradient
SVD
Q
R
R
P
26
Local deformation F QT L P consists of
principal axes
?2
?1
1
elongations L (scaling by ?1, ?2, ?3) along
principal axes
and a rotation R QTP inaccessible in
geodesy due to lack of coordinate system
identification
27
Local deformation parameters (functions of F
QT L P )
principal axes
R
?2
?1
P
1
Singular values in L(?1, ?2, ?3) and functions
?(?1,?2,?3) - Numerical invariants
Angles in P(?1, ?2, ?3) defining directions of
principal axes (physical invariants)
Angles in R(?1, ?2, ?3) defining local rotation
(not invariant)
28
Local deformation parameters (functions of F
QT L P )
principal axes
R
?2
?1
P
1
Singular values in L(?1, ?2, ?3) and functions
?(?1,?2,?3) - Numerical invariants
2D
3D
(areal) dilatation ? ?1?2 -1
(volume) dilatation ? ?1?2?3 -1
shear ? (?1-?2) (?1-?2)-1/2
shears within principal planes ?ik (?i-?k)
(?i-?k)-1/2 ik 12, 23, 13
29
The role of the reference system
30
Coordinates in an preliminary system, WGS 84,
ITRF, or user defined
epoch t
epoch t0
displacements too large !
31
Definition of a network-intrinsic reference
system
Center of mass preservation
hR ?i xi? vi 0
Vanishing of relative angular momentum
Mean quadratic scale preservation
32
Advantages of a network-intrinsic reference
system
Invariant deformation parameters the same No
advantage for continuous spatial information
Dermination of motion of the network area as
whole
translation
and rotation
33
Advantages of a network-intrinsic reference
system
Invariant deformation parameters the same No
advantage for continuous spatial information
Determination of motion of the network area as
whole
translation
rotation
Small displacements (trend removal)
Essential for proper spatial interpolation of
discrete spatial information
Reference systems at 2 epochs identified
34
2-dimensional or 3-dimensional deformation?
35
Crustal deformation is a 3-dimensional physical
process
F
t0
t
36
Usually studied as 2-dimensional by projection
of physical surface to a horizontal plane
F
t0
t
37
Proper treatment Deformation of the
2-dimensional physical surface as embedded in
3-dimensional space
F
t0
t
38
Attempts for a 3-dimensional treatment
Extension of the 2D finite element
method (triangular elements) to 3D (quadrilateral
elements)
39
Attempts for a 3-dimensional treatment
deformation of air !
We can obtain good horizontal information by
interpolation or virtual densification. Vertical
information requires extrapolation (an insecure
process)
deformation of mountain
Derination of 3D crustal deformation from 2D
deformation surface deformatiom (downward
continuation) an improperly posed problem !
40
Spatial (and/or temporal) interpolation
41
Geodetic information on crustal deformation -
Coordinates x(P,t)
space
The ideal situation Space continuous Time
continuous To provide deformation parameters at
any point for any 2 epochs
No interpolation needed !
time
42
Geodetic information on crustal deformation -
Coordinates x(P,tk)
space
The satisfactory situation Space
continuous Time discrete To provide deformation
parameters at any point for any 2 observation
epochs
No interpolation needed !
t1
t2
t3
t4
t5
t6
t7
time
43
Geodetic information on crustal deformation -
Coordinates x(P,tk)
space
The satisfactory situation Space
continuous Time discrete To provide deformation
parameters at any point for any 2 epochs
Temporal interpolation needed !
t1
t2
t3
t4
t5
t6
t7
time
44
Geodetic information on crustal deformation -
Coordinates x(Pi,tk)
space
The realistic situation Space discrete Time
discrete To provide deformation parameters at
any point for any 2 observation epochs
P6
P5
P4
P3
P2
Spatial interpolation needed !
P1
t1
t2
t3
t4
t5
t6
t7
time
45
Geodetic information on crustal deformation -
Coordinates x(Pi,tk)
space
The realistic situation Space discrete Time
discrete To provide deformation parameters at
any point for any 2 epochs
P6
P5
P4
P3
P2
Spatial interpolation needed !
P1
Temporal interpolation also needed !
t1
t2
t3
t4
t5
t6
t7
time
46
Geodetic information on crustal deformation -
Coordinates x(Pi,tk)
Not all points observed at each epoch
space
The realistic situation Space discrete Time
discrete
P6
P5
To provide deformation parameters at any
point for any 2 observation epochs
P4
P3
P2
Spatial interpolation needed !
P1
Temporal interpolation needed !
t1
t2
t3
t4
t5
t6
t7
time
47
Geodetic information on crustal deformation -
Coordinates x(Pi,t)
space
GPS permanent stations Space discrete Time
continuous To provide deformation parameters at
any point for any 2 epochs
P6
P5
P4
P3
P2
Spatial interpolation needed !
P1
t1
t2
t3
t4
t5
t6
t7
time
48
Geodetic information on crustal deformation -
Coordinates x(Pi,t), x(Pi,tk)
space
GPS permanent stations and SAR interferometry Spa
ce discrete Time discrete (SAR) time -
continuous (GPS) To provide deformation
parameters at any point for any 2 SAR
observation epochs
P6
P5
P4
P3
P2
No spatial interpolation needed !
P1
t1
t2
t3
t4
t5
t6
t7
time
49
Geodetic information on crustal deformation -
Coordinates x(Pi,t), x(Pi,tk)
space
GPS permanent stations and SAR interferometry Spa
ce discrete Time discrete (SAR) time -
continuous (GPS) To provide deformation
parameters at any point for any 2 epochs
P6
P5
P4
P3
P2
Spatial interpolation needed !
P1
t1
t2
t3
t4
t5
t6
t7
time
50
Data analysis strategies
51
Alternatives for spatial and/or temporal
interpolation
To be inerpolated displacments u at discrete
epochs tk and discrete points Pi
Sought
Given
u(Pi, tk) x(Pi, tk) - x(Pi, t0)
u(x0, t)
for every x0 and t
u(x, t)
expressed as
simplified to
u(x0i,tk) xi(tk) - x0i
Analytic least squares (smoothing) interpolation
deterministic
2 types of interpolation
Minumum norm (exact) interpolation
or Minimum norm (smoothing) interpolation
equivalent to Minimum mean-square error linear
prediction
stochastic
52
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
53
Alternatives for linear spatial interpolation
u(x) g(x, a) ?m fm(x) am
Analytic least squares (smoothing) interpolation
a vector of free parameters (? observations)
least square solution a of u(xi) g(xi,a)
vi (vTPv min)
Minumum norm (exact) interpolation
(parameters a gt observations, even infinite)
minimum-norm solution a of u(xi)
g(xi,a) (aTRa min)
Minumum norm (smoothing) interpolation
(parameters a gt observations, even infinite)
hybrid solution a of u(x0i) g(x0i, a) vi
(vTPvaTRa min)
54
Deterministic and stochastic interpretation for
linear interpolation
b F a
u(xi) ?m fm(xi) am f(xi)T a
b s
Minumum norm (exact) interpolation
Minumum mean square error prediction
aTRa min ? u(x) fT(x)R-1FT(FR-1FT)-1 b
k(x)T K-1 b
Minumum norm (smoothing) interpolation
Collocation in geodetic jargon
b F a v
b s v
aTRa vTPv min ? u(P) kT (P)(KP-1)-1 b
deterministic
stochastic
55
Deterministic and stochastic interpretation for
linear interpolation
b s F a, s(x) f(x)Ta ? Css FCaFT,
Cs(x)s f(x)TCaFT
b F a
u(xi) ?m fm(xi) am f(xi)T a
Minumum norm (exact) interpolation
Minumum mean square error prediction
aTRa min ? u(x) fT(x)R-1FT(FR-1FT)-1 b
k(x)T K-1 b
f(x)TCaFT
(FCaFT)-1 b

s(x) Cs(x)s Cs-1 b
Minumum norm (smoothing) interpolation
b Fa v
b F a v
Equivalence Ca R-1 Cv P-1
aTRa vTPv min ? u(P) kT (P)(KP-1)-1 b

f(x)TCaFT (FCaFT)-1 b
56
EXAMPLES OF INTERPOLATION MODELS
Spatial interpolation Combination of
least-squares analytic and stochastic prediction
Observation epochs t0, t - No temporal
interpolation
component covariance functions
x(x0) f(x0)T a s(x0)
Csisk(x0,x?0)
Temporal interpolation least-squares analytic
x(x0,t) x0 (t-t0) v(x0)
F(x0,t) I (t-t0) L(x0)
u(x0,t) (t-t0) v(x0)
Spatial interpolation Combination of
least-squares analytic and stochastic prediction
v(x0) f(x0)T a z(x0)
Czizk(x0,x?0)
component covariance functions
57
Invariant Interpolation independent of used
reference systems
u(x) ?m fm(x) am f(x)T a
Analytic least squares (smoothing) interpolation
Not invariant interpolation! Base functions fm(x)
depend on coordinate system used
uT(x) JT x cT
Exception Finite element method
Different JT, for each triangular element, cT
irrelevant (FT I JT)
s(x) Cs(x)s (Cs Cv)-1 b
Minumum mean square error prediction
Invariance guaranteed by stationary and isotropic
component covariance functions
Cs1s2(x0,x?0) 0
58
Data analysis alternatives
Potential problem Incorrect separation of
observation errors and displacements
59
Quality assessment for deformation parameters
60
Problems in quality assessment for deformation
parameters
? Incorrect statistics for input data
(coordinates)
Typical for GPS coordinates
Fit lines to time variation Estimate statistics
from residuals
? Incorrect assessment of interpolation errors
Improper separation of signal (displacements)
from noise
Trial and error
? Incorrect error propagation from deformation
gradient to deformation parameters
Singular values highly nonlinear functions of
deformation gradient
Use propagation with higher derivatives and
moments Use Monte Carlo techniques
61
FUTURE OUTLOOK
? Permanent GPS stations provide initial
coordinates and velocities with 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 ?)
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This presentation will be available
at http//der.topo.auth.gr/