VI Hotine-Marussi Symposium
of Theoretical and Computational Geodesy:
Challenge and Role of Modern Geodesy
May 29 - June 2, 2006, Wuhan, China
FELIX QUI POTUIT RERUM
COGNOSCERE CAUSAS
The ITRF Beyond the “Linear” Model
Choices and Challenges
Athanasius Dermanis
Department of Geodesy and Surveying - Aristotle University of Thessaloniki
A reference system for a deformable body or point network:
A time-wise smooth choice at every epoch t of
(a) a point - the origin, O(t)
(b) three directed straight lines



and a unit of length - the vectorial basis e1(t), e2(t), e3(t)
such that the apparent motion of the body masses (or network points),
as seen with respect to the reference system, is minimized
The reference system separates the total motion
with respect to the inertial background into:
(a) the translational motion and the rotation of the body/network
as represented by the selected reference system
(b) the remaining “deformation”
The optimal choice of the reference system requires the introduction of
an optimality criterion = a measure of the “deformation” to be minimized
The optimality criterion should be applied on the realization
of the reference system by a set of point coordinates expressed as
functions of time for a selected global terrestrial network:
The International Terrestrial Reference Frame (ITRF):
xi(ai, t), i = 1, 2, …, N
making use of model parameters ai [e.g. xi(t0), vi - presently]
Therefore the optimality criterion must be realized by means of
a set of mathematical conditions on the coordinate model parameters:
Fk(a1, a2, …, aN) = 0,
k = 1, 2, …, L
Plus:
arbitrary choice among one of dynamically equivalent reference systems
( xi ~ xi'

xi' = R xi + d,
R, d = constant )
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
origin = geocenter

x(t ) dmx ( t )  0 ,
t
Earth
axes = Tisserand axes

Earth
[x(t )]
dx
(t ) dmx ( t )  0 ,
dt
t
vanishing relative angular momentum =
= minimal relative kinetic energy
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
origin: constant network barycenter
origin = geocenter
1 N
xi (t )  m  constant ,

N i 1

t
dxi
[xi (t )]
(t )  0 ,

dt
i 1
t
Earth
axes = discrete Tisserand principle
N
x(t ) dmx ( t )  0 ,
t
vanishing discrete relative angular momentum
network points = treated as (unit) mass points
axes = Tisserand axes

Earth
[x(t )]
dx
(t ) dmx ( t )  0 ,
dt
t
vanishing relative angular momentum =
= minimal relative kinetic energy
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
origin: constant network barycenter
origin = geocenter
1 N
xi (t )  m  constant ,

N i 1

t
dxi
[xi (t )]
(t )  0 ,

dt
i 1
t
Earth
axes = discrete Tisserand principle
N
x(t ) dmx ( t )  0 ,
t
vanishing discrete relative angular momentum
orasa(unit)
combination
network points = treated
mass points
axes = Tisserand axes

Earth
[x(t )]
dx
(t ) dmx ( t )  0 ,
dt
t
vanishing relative angular momentum =
of the two approaches:
= minimal relative kinetic energy
one for the geocenter, the other for the axes
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
ADVANTAGES
Physical meaning!
Compatibility with theories of
- orbit of the earth
(translational motion of geocenter)
&
- earth rotation
(rotation of the Tisserand axes)
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
DISADVANTAGES
ADVANTAGES
No physical meaning!
Physical meaning!
Lack of compatibility with
reference systems
implicitly defined in theories
of earth orbital motion
and earth rotation
Compatibility with theories of
(additional discrepancies between
theory and observations due to
different reference system definitions)
- orbit of the earth
(translational motion of geocenter)
&
- earth rotation
(rotation of the Tisserand axes)
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
ADVANTAGES
Coordinates suffer only from errors
in estimating the network shape!
No additional errors arising
from uncertainty in the position
of the origin and axes
with respect to the network!
Pure geodetic-positional approach
free from geophysical hypotheses!
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
ADVANTAGES
DISADVANTAGES
Coordinates suffer only from errors
in estimating the network shape!
No additional errors arising
from uncertainty in the position
of the origin and axes
with respect to the network!
Coordinates suffer also from errors
in estimating the position
of the origin and axes
with respect to the network!
Pure geodetic-positional approach
free from geophysical hypotheses!
Geocenter = mean position of centers
of oscillating ellipses of satellite orbits.
Estimated position of Tisserand axes
heavily depends on geophysical
assumptions about density and motion
of internal earth masses
Two possible approaches to the reference system choice for the ITRF
Mathematical approach
Physical approach
Introduce origin and axes in a
purely mathematical way by minimizing
the apparent motion
of the ITRF network points
with respect to the reference system
Introduce origin and axes in a
physical way by accessing
a point and lines depending on the
physical constitution of the earth
(though still mathematically defined)
ADVANTAGES
DISADVANTAGES
Coordinates suffer only from errors
in estimating the network shape!
No additional errors arising
from uncertainty in the position
of the origin and axes
Dermanis (2006)
with respect to theSee:
network!
EGU Vienna
Pure geodetic-positional approach
free from geophysical hypotheses!
Coordinates suffer also from errors
in estimating the position
of the origin and axes
with respect to the network!
Geocenter = mean position of centers
of oscillating ellipses of satellite orbits.
Estimated position of Tisserand axes
heavily depends on geophysical
assumptions about density and motion
of internal earth masses
Which approach to use ?
My suggestion: Both!
FIRST:
Establish an ITRF by purely geodetic-“positional” means
with origin and axes defined mathematically by optimality conditions
(constant position of network barycenter – vanishing discrete relative angular momentum of “unit mass” stations)
ITRF coordinates (functions of time) reflect only network shape and its
temporal variation (deformation), free from additional positional uncertainties
Which approach to use ?
My suggestion: Both!
FIRST:
Establish an ITRF by purely geodetic-“positional” means
with origin and axes defined mathematically by optimality conditions
(constant position of network barycenter –
– vanishing discrete relative angular momentum of “unit mass” stations)
ITRF coordinates (functions of time) reflect only network shape and its
temporal variation (deformation), free from additional positional uncertainties
SECOND:
Use geodetic satellite observations to estimate the time-varying geocenter
position with respect to the ITRF.
Use best available geophysical hypotheses about unobservable internal
(subsurface) earth composition and motions to
estimate the time-varying position of the Tisserand axes.
Transform the ITRF into an “earth reference frame” for comparison with
earth rotation and other geophysical processes,
while understanding the influence of additional estimation errors
and accepted geophysical hypotheses.
About the ITRF linear model
Coordinates linear functions of time:
xi (t )  xi (t0 )  (t  t0 ) vi
Is there a linear network deformation model? No!
Transformation to another equally legitimate reference system destroys linearity!
xi (t )  xi (t0 )  (t  t0 ) vi
xi (t )  R(t )xi (t )  d(t )
xi (t )  R(t )xi (t0 )  (t  t0 )R(t ) vi  d(t )
About the ITRF linear model
xi (t )  xi (t0 )  (t  t0 ) vi
Coordinates linear functions of time:
Is there a linear network deformation model? No!
Transformation to another equally legitimate reference system destroys linearity!
xi (t )  xi (t0 )  (t  t0 ) vi
xi (t )  R(t )xi (t )  d(t )
xi (t )  R(t )xi (t0 )  (t  t0 )R(t ) vi  d(t )
Same holds for spectral analysis (Fourier series model):
xi (t )  R(t )xi (t )  d(t )
xi (t )  a    aik cos(kt )  bik sin(kt ) 
M
0
i
k 1
xi (t )  R (t )a    R (t )aik cos(kt )  R (t )bik sin(kt )   d(t )
M
0
i
k 1
There exists no Fourier analysis independent from the choice of reference system!
Frequency components appearing in one coordinate system are different from those
in another one, where also the contributions of frequencies in R(t) and d(t) are present!
However:
If x(t) and x'(t) both satisfy the discrete Tisserand conditions
(constant barycenter, zero discrete relative angular momentum)
xi (t )  R(t )x(t )  d(t )
xi (t )  xi (t0 )  (t  t0 ) vi

R (t )  R  constant
&
d(t )  d  constant
xi (t )  Rxi (t )  d
xi (t )  [Rxi (t0 )  d]  (t  t0 )[Rvi ]  xi (t0 )  (t  t0 ) vi
Linearity is preserved!
Ad hoc definition of a “linear deformation model”:
We say that a point network or body deform in a linear way if the coordinates of any point
with respect to any Tisserand reference system are linear functions of time
However:
If x(t) and x'(t) both satisfy the discrete Tisserand conditions
(constant barycenter, zero discrete relative angular momentum)
xi (t )  R(t )x(t )  d(t )
xi (t )  xi (t0 )  (t  t0 ) vi
R (t )  R  constant

&
d(t )  d  constant
xi (t )  Rxi (t )  d
xi (t )  [Rxi (t0 )  d]  (t  t0 )[Rvi ]  xi (t0 )  (t  t0 ) vi
Linearity is preserved!
Ad hoc definition of a “linear deformation model”:
We say that a point network or body deform in a linear way if the coordinates of any point
with respect to any Tisserand reference system are linear functions of time
In a similar way we may speak about the spectral analysis of the deformation of a network
or body, meaning the spectral analysis of the coordinate functions of any point with respect
to any Tisserand reference system
xi  Rxi (t )  d
xi (t )  a    a cos(kt )  b sin(kt ) 
M
0
i
k 1
k
i
k
i
xi (t )  [Ra  d]   [Raik ]cos(kt )  [ Rbik ]sin( kt )  
M
0
i
k 1
 ai 0    ai k cos(kt )  bi k sin(kt ) 
M
k 1
Beyond the “Linear” Model
Extended time period of ITRF relevance
and particular station behavior
necessitate richer time-evolution models
First choices
Polynomials:
xi (t )  ci0  c1i t  ci2t 2  ...  cint n
M
Fourier series:
xi (t )  ai0   aik cos(kt )  bik sin(kt )
k 1
A general class of models
M
Linear combinations of base functions:
xi (t )   cikk (t )
k 1
{k} = a class of functions:
 closed under differentiation:
dm M n
  Amn
dt
n 1
L
 closed under multiplication:
M L
L
dm M n
n
j
k
  Amkn   Am Bk ,n j   Ckj,m j
dt
n 1
n 1 j 1
j 1
kn   Bkj,n j
j 1
The discrete Tisserand conditions
M
xi   c 
k 1
k
i k
dxi M k dk
  ci
dt k 1
dt
L
dm
k
  Ckj,m j
dt
j 1
(a) Vanishing (discrete) “relative angular momentum”:
network points treated
as (unit) mass points
The discrete Tisserand conditions
M
xi   c 
k 1
k
i k
dxi M k dk
  ci
dt k 1
dt
L
dm
k
  Ckj,m j
dt
j 1
network points treated
as (unit) mass points
(a) Vanishing (discrete) “relative angular momentum”:
N
hR  
i 1
dxi N
[xi ]

dt
i 1
M
M

k 1
m 1
dm N
[c ]c 

dt
i 1
k
i
m
i k
M
M

k 1
m 1
[c ]c
k
i
L
m
i
C
j 1
j  0
j
k ,m
The discrete Tisserand conditions
M
xi   c 
k 1
k
i k
dxi M k dk
  ci
dt k 1
dt
L
dm
k
  Ckj,m j
dt
j 1
network points treated
as (unit) mass points
(a) Vanishing (discrete) “relative angular momentum”:
N
hR  
i 1
dxi N
[xi ]

dt
i 1
M
M

k 1
m 1
dm N
[c ]c 

dt
i 1
k
i
m
i k
L
h R (t )  
j 1
 N

 i 1
M
M

[c ]c
k
i
L
m
i
C
k 1
m 1
M
M
k 1
m 1

Ckj,m [cik ]cim   j (t )  0


j 1
j  0
j
k ,m
t
The discrete Tisserand conditions
M
xi   c 
k 1
k
i k
dxi M k dk
  ci
dt k 1
dt
L
dm
k
  Ckj,m j
dt
j 1
network points treated
as (unit) mass points
(a) Vanishing (discrete) “relative angular momentum”:
N
hR  
i 1
dxi N
[xi ]

dt
i 1
M
M

k 1
m 1
dm N
[c ]c 

dt
i 1
k
i
m
i k
L
h R (t )  
j 1
N
axes orientation
conditions
M
 N

 i 1
M
 C
i 1 k 1 m 1
j
k ,m
M
M

[c ]c
k
i
L
m
i
C
k 1
m 1
M
M
k 1
m 1

Ckj,m [cik ]cim   j (t )  0


j 1
j  0
j
k ,m
t
[cik ]cim   (Ckj,m  Cmj ,k )[cik ]cim  0
i
k m
j  1,2,
,L
The discrete Tisserand conditions
M
xi   c 
m 1
m
i m
dm M j
  Am j
dt
j 1
(b) Constant (discrete) barycenter:
1 N
m(t )   xi  const.
N i 1
dxi M m dm M m M j
  ci
  ci  Am j
dt m1
dt
m 1
j 1
The discrete Tisserand conditions
M
xi   c 
m 1
m
i m
dxi M m dm M m M j
  ci
  ci  Am j
dt m1
dt
m 1
j 1
dm M j
  Am j
dt
j 1
(b) Constant (discrete) barycenter:
1 N
m(t )   xi  const.
N i 1

dm 1 N dx i
1 N
 
 
dt
N i 1 dt
N i 1
M
M
 c A 
m 1
m
i
j 1
j
m
j
0
The discrete Tisserand conditions
M
xi   c 
m 1
m
i m
dxi M m dm M m M j
  ci
  ci  Am j
dt m1
dt
m 1
j 1
dm M j
  Am j
dt
j 1
(b) Constant (discrete) barycenter:
1 N
m(t )   xi  const.
N i 1

dm 1 N dx i
1 N
 
 
dt
N i 1 dt
N i 1
M

j 1
 N

 i 1
M

m 1

Amj cim  j (t )  0

t
M
M
 c A 
m 1
m
i
j 1
j
m
j
0
The discrete Tisserand conditions
M
xi   c 
m 1
m
i m
dxi M m dm M m M j
  ci
  ci  Am j
dt m1
dt
m 1
j 1
dm M j
  Am j
dt
j 1
(b) Constant (discrete) barycenter:
1 N
m(t )   xi  const.
N i 1

dm 1 N dx i
1 N
 
 
dt
N i 1 dt
N i 1
M

j 1


 i 1
N
M

m 1

Amj cim  j (t )  0

t
origin
conditions
M
M
 c A 
m 1
m
i
j 1
N
M
i 1
m 1

j
m
j
0
Amj cim  0
j  1,2,
,M
k (t )  t k 1
Example 1: Polynomials
N
Origin conditions:
dx i
0

i 1 dt
dxi
 c1i  2ci2t  3ci3t 2
dt
xi  ci0  c1i t  ci2t 2  ci3t 3
N
c
1
i
i1
0
N
c
i1
2
i
0
N
c
i1
3
i
0
k (t )  t k 1
Example 1: Polynomials
N
dx i
0

i 1 dt
Origin conditions:
N
Orientation conditions:
[xi ]
i 1
[xi ]
dxi
 c1i  2ci2t  3ci3t 2
dt
xi  ci0  c1i t  ci2t 2  ci3t 3
N
c
1
i
N
c
0
i1
2
i
0
i1
N
c
3
i
0
i1
dx i
0
dt
dxi
 [ci0 ]  [c1i ]t  [ci2 ]t 2  [ci3]t 3  c1i  2ci2t  3ci3t 2  
dt
 [ci0 ]c1i    2[ci0 ]ci2  [c1i ]c1i  t   3[ci0 ]ci3  2[c1i ]ci2  [ci2 ]c1i  t 2 
  3[c1i ]c3i  2[ci2 ]ci2  [c3i ]c1i  t 3   3[ci2 ]c3i  2[c3i ]ci2  t 4   3[c3i ]c3i  t 5
N
[c ]c
0
i
1
i
0
i1
[a]b  [b]a 

[a]a  0 
[c ]c
0
i
2
i
0
i1
1
i
3
i
0
3
i
0
N
[c ]c
2
i
i1
  3[c ]c
N
0
i
i1
[c ]c
i1
N

N
3
i
 [c1i ]ci2   0
Example 2: Fourier series
kC  cos(kt )
kS  sin(kt )
d
dt
kC  kkS
d
dt
dxi
  (a1i1S  b1i1C  2ai22S  2bi22C )
dt
kS  kkC
N
Origin conditions:
xi  ai0  a1i1C  b1i1S  ai22C  bi22S
dx i
0

dt
i 1
N
a
1
i
i1
0
N
b
1
i
i1
0
N
a
i1
2
i
0
N
b
i1
2
i
0
Example 2: Fourier series
kC  cos(kt )
d
dt
kS  sin(kt )
xi  ai0  a1i1C  b1i1S  ai22C  bi22S
kC  kkS
d
dt
dxi
  (a1i1S  b1i1C  2ai22S  2bi22C )
dt
kS  kkC
N
Origin conditions:
dx i
0

dt
i 1
N
a
1
i
i1
N
Orientation conditions:
i 1
2

[xi ]
dx i
[x ] dt
i
0
N
b
1
i
i1
0
N
a
i1
2
i
0
N
b
2
i
0
i1
0
dxi
 2 [ai0 ]  [a1i ]1C  [b1i ]1S  [ai2 ]2C  [bi2 ]2S  a1i1S  b1i1C  2ai22S  2b i22C  
dt
 2[ai0 ]a1i1S  2[ai0 ]b1i1C  4[ai0 ]ai22S  4[ai0 ]bi22C
[a1i ]a1i (21C1S )  [a1i ]b1i (21C1C )  2[a1i ]ai2 (21C2S )  2[a1i ]bi2 (21C2C )
[b1i ]a1i (21S1S )  [b1i ]b1i (21S1C )  2[b1i ]ai2 (21S2S )  2[b1i ]bi2 (21S2C )
[ai2 ]a1i (22C1S )  [ai2 ]b1i (22C1C )  2[ai2 ]ai2 (22C2S )  2[ai2 ]bi2 (22C2C )
[bi2 ]a1i (22S1S )  [bi2 ]b1i (22S1C )  2[bi2 ]ai2 (22S2S )  2[bi2 ]bi2 (22S2C )
Use of trigonometric
identities:
2kCmC  |kCm|  kC m
2kSmS  |kC m|  kC m
2kCmS  kSm  |kk mm| |kSm|
2kSmC  kSm  |kk mm| |kSm|
2

dxi
  [a1i ]b1i  [b1i ]a1i  2[ai2 ]bi2  2[bi2 ]ai2 0C
dt
  2[ai0 ]b1i  2[a1i ]bi2  2[b1i ]ai2  [ai2 ]b1i  [b i2 ]a1i 1C
[xi ]
  2[ai0 ]a1i  2[a1i ]ai2  2[b1i ]bi2  [ai2 ]a1i  [bi2 ]b1i 1S
  4[ai0 ]bi2  [a1i ]b1i  [b1i ]a1i 2C   4[ai0 ]ai2  [a1i ]a1i  [b1i ]b1i 2S
  2[a1i ]bi2  2[b1i ]ai2  [ai2 ]b1i  [bi2 ]a1i 3C   2[a1i ]ai2  2[b1i ]b i2  [ai2 ]a1i  [b i2 ]b1i 3S
  2[ai2 ]bi2  2[bi2 ]ai2 4C   4[ai2 ]ai2  2[bi2 ]bi2 4S
Orientation conditions:
C
0
  2[a ]b
 :
  2[a ]b
 :
C
1
 :
S
1
N
1
i
i 1
1
i
 4[ai2 ]bi2   0
 3[a1i ]bi2  3[ai2 ]b1i   0
N
0
i
i 1
1
i
  2[a ]a
N
0
i
i 1
1
i
 3[a1i ]ai2  3[b1i ]b i2   0
C
2
  4[a ]b   0
 :
  4[a ]a   0
 :
S
2
 :
C
3
N
0
i
i 1
2
i
N
0
i
i 1
 [a ]b
2
i
N
i 1
1
i
2
i
 [ai2 ]b1i   0
S
3
  [a ]a
 :
  2[a ]b
 :
C
4
 :
S
4
N
i 1
1
i
2
i
N
i 1
2
i
2
i
  4[a ]a
 [b1i ]bi2   0
 2[ai2 ]bi2   0
N
i 1
2
i
2
i
 2[bi2 ]bi2   0
No condition produced!
No condition produced!
How to implement the optimality conditions?
A-priori at the level of data analysis
For ITRF construction
Requires single epoch data xi(tk)
(daily or weekly solutions).
The data enter free of reference system
as shapes of sub-networks. They are rotated
& translated to the optimal reference system
by applying the optimality constraints.
A-posteriori by transformation from
an arbitrary to the optimal system
How to implement the optimality conditions?
A-priori at the level of data analysis
For ITRF construction
A-posteriori by transformation from
an arbitrary to the optimal system
Requires single epoch data xi(tk)
(daily or weekly solutions).
The data enter free of reference system
as shapes of sub-networks. They are rotated
& translated to the optimal reference system
by applying the optimality constraints.
Analysis is performed in any convenient
reference system and coordinates are then
converted to the optimal one by solving for
appropriate rotation R(t) and translation d(t),
such that the optimality conditions
are satisfied
How to implement the optimality conditions?
A-priori at the level of data analysis
For ITRF construction
A-posteriori by transformation from
an arbitrary to the optimal system
Requires single epoch data xi(tk)
(daily or weekly solutions).
The data enter free of reference system
as shapes of sub-networks. They are rotated
& translated to the optimal reference system
by applying the optimality constraints.
Analysis is performed in any convenient
reference system and coordinates are then
converted to the optimal one by solving for
appropriate rotation R(t) and translation d(t),
such that the optimality conditions
are satisfied
DISADVANTAGES - Advantages
How to implement the optimality conditions?
A-priori at the level of data analysis
For ITRF construction
A-posteriori by transformation from
an arbitrary to the optimal system
Requires single epoch data xi(tk)
(daily or weekly solutions).
The data enter free of reference system
as shapes of sub-networks. They are rotated
& translated to the optimal reference system
by applying the optimality constraints.
Analysis is performed in any convenient
reference system and coordinates are then
converted to the optimal one by solving for
appropriate rotation R(t) and translation d(t),
such that the optimality conditions
are satisfied
DISADVANTAGES - Advantages
 Only single-epoch data acceptable.
 It is not possible to introduce time-evolution
models at the preprocessing stage
(at the data analysis centers dealing with
regional sub-networks).
 Significant sub-network overlap is required
to secure successful “patchwork” into a
single global shape, before introducing the
optimal reference system.
How to implement the optimality conditions?
A-priori at the level of data analysis
For ITRF construction
A-posteriori by transformation from
an arbitrary to the optimal system
Requires single epoch data xi(tk)
(daily or weekly solutions).
The data enter free of reference system
as shapes of sub-networks. They are rotated
& translated to the optimal reference system
by applying the optimality constraints.
Analysis is performed in any convenient
reference system and coordinates are then
converted to the optimal one by solving for
appropriate rotation R(t) and translation d(t),
such that the optimality conditions
are satisfied
DISADVANTAGES - Advantages
 Only single-epoch data acceptable.
 It is not possible to introduce time-evolution
models at the preprocessing stage
(at the data analysis centers dealing with
regional sub-networks).
 Significant sub-network overlap is required
to secure successful “patchwork” into a
single global shape, before introducing the
optimal reference system.
 Time-evolution models can be introduced
during the preprocessing level at the
analysis centers of regional sub-networks.
 Optimality can be only approximately
achieved, because preservation of the
time evolution model requires restriction
of unknown rotation R(t) and translation
d(t) to the same model class
(linear combinations of the same
base functions)
The ultimate problem:
Use observed deformation or a “smoothed” version?
Spectral analysis is typically used for isolating estimation errors (noise) from signal:
 High frequencies are attributed to noise and are removed.
 Low frequencies are retained as signal.
 Middle frequencies?
Are middle frequencies in network deformation real ?
The ultimate problem:
Use observed deformation or a “smoothed” version?
Spectral analysis is typically used for isolating estimation errors (noise) from signal:
 High frequencies are attributed to noise and are removed.
 Low frequencies are retained as signal.
 Middle frequencies?
Are middle frequencies in network deformation real ?
Difficult to answer !
Systematic errors in space-techniques have atmospheric
origins with spectral characteristics related to the
annual cycle.
Real middle frequency deformations should have
origins with the similar spectral characteristics.
Hard or even impossible to distinguish!
Future hope: Better monitoring of the atmosphere and
effective removal of systematic errors.
The ultimate problem:
Use observed deformation or a “smoothed” version?
Spectral analysis is typically used for isolating estimation errors (noise) from signal:
 High frequencies are attributed to noise and are removed.
 Low frequencies are retained as signal.
 Middle frequencies?
Are middle frequencies in network deformation real ?
Even if middle frequencies are real should they be retained in the ITRF model ?
The ultimate problem:
Use observed deformation or a “smoothed” version?
Spectral analysis is typically used for isolating estimation errors (noise) from signal:
 High frequencies are attributed to noise and are removed.
 Low frequencies are retained as signal.
 Middle frequencies?
Are middle frequencies in network deformation real ?
Even if middle frequencies are real should they be retained in the ITRF model ?
Answer depends on the specific use of the ITRF.
Note: Earth-tide deformations are already removed
at the data preprocessing level.
Other periodic components may be removed to produce
an ITRF version reflecting only secular deformation
(compare: UT2 versus UT1).
All removed frequencies must be restored before any
comparison with actual geophysical observations.
Discontinuous episodic deformations must be modeled
by step functions and removed from the final solution.
Thanks for your attention!
A copy of this presentation can be found
at my personal web page:
http://der.topo.auth.gr