Compatibility of the IERS earth rotation representation and its relation to the NRO conditions
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Transcript of Compatibility of the IERS earth rotation representation and its relation to the NRO conditions
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Compatibility of the IERS earth rotation representationand its relation to the NRO conditions
Athanasios Dermanis
Department of Geodesy and SurveyingThe Aristotle University of Thessaloniki
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
C C C C1 2 3[ ]e e ee
T T T T1 2 3[ ]e e ee
C C C1 2 3( , )O e e e
T T T1 2 3( , )O e e e
Earth Rotation:Relation of Terrestrial to Celestial Reference System
Celestial Reference System:
Terrestrial Reference System:
Mathematical model: C Te e R
( ) ( )t tR R a = orthogonal rotation matrix
T1 2 m( ) [ ( ) ( ) ( )]t a t a t a ta = earth rotation parameters
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
To every orthogonal rotation matrix R(t) corresponds a unique rotation vector:
TT e ω
defined by TT[ ]
d
dt
Rω R
R
Notation:3 2
3 1
2 1
0
[ ] 0
0
a a
a a
a a
a
1
2
3
a
a
a
a
[a] is the antisymmetric matrix with axial vector a
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
R = QDW: Separation of earth rotation in 3 parts:
3( )GASTR
3( )R
3( )R
3( )R
D = Diurnal rotation
Number of independent parameters needed: 3 (geometric description)6 (dynamic description – state vector)
9 parameters
NRO conditions:
3 2 3 1 3 1( ) ( ) ( ) ( ) ( ) ( )z R R R R R RR
3 2 3( ) ( ) ( )E d E s R R RR
0 3( , ) ( )X Y s RR Q
1 2( ) ( ) R R R
Q = Precession-Nutation
s = s (g,F) = s (xP,yP) s = s(d,E) = s(X,Y)
Classical model:
IERS model (IAU 2000):
2 1( ) ( )P Px yR R
3 2 3( ) ( ) ( )s F g F R R R
OSU Report Nr. 245, 1977:
2 1( ) ( )P Px yR R
3 2 1( ) ( ) ( )P Ps x yR R R
W = Polar motion
5 parameters
7 parameters reduced to 5by 2 NRO conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Characteristics of the IERS earth rotation representation
C C C C1 2 3[ ]e e ee
T T T T1 2 3[ ]e e ee
IC IC IC IC1 2 3[ ]e e ee
IT IT IT IT1 2 3[ ]e e ee
Q
D
W
PrecessionNutation
Diurnal Rotationaround
PolarMotion
fromtheory
fromobservations
R
high frequency termsremoved from
precession-nutation
CIP
Consequences on model-compatible
rotation vector
TT e ω
T[ ] T d
dt
Rω R
IC IT3 3
1
| |e e
Rotation vector not aligned to common 3rd axis of intermediate systems
Magnitude not equal torate of diurnal rotation angle
| |d
dt
3( ) D R
IC IT3 3p e e
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Objective: Construct a compatible representation with a 3 part separation Objective: Construct a compatible representation with a 3 part separation
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Objective: Construct a compatible representation with a 3 part separation Objective: Construct a compatible representation with a 3 part separation
3 2 3 3 3 2 3{ ( ) ( ) ( )} ( ) { ( ) ( ) ( )}E d E s s F g F R QDW R R R R R R R
Involving 2 intermediate reference systems:
Find a representation of the same separated form a the IERS representation
IC IC IC IC C T1 2 3[ ]e e e e e Q
IT IT IT IT C T T T1 2 3[ ]e e e e e Q D e W
Intermediate Celestial:
Intermediate Terrestrial:
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Objective: Construct a compatible representation with a 3 part separation Objective: Construct a compatible representation with a 3 part separation
3 2 3 3 3 2 3{ ( ) ( ) ( )} ( ) { ( ) ( ) ( )}E d E s s F g F R QDW R R R R R R R
Involving 2 intermediate reference systems:
Find a representation of the same separated form a the IERS representation
Subject to the following (natural) compatibility conditions:
3 3
1
| |IC ITp e e n
| |d
dt
Intermediate Celestial:
Intermediate Terrestrial:
2 directional conditions:
1 magnitude condition:
IC IC IC IC C T1 2 3[ ]e e e e e Q
IT IT IT IT C T T T1 2 3[ ]e e e e e Q D e W
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Decomposition of the rotation vector in 3 relative rotation vectors
3( ) R QDW QR W T[ ] T d
dt
Rω RT
T e ω
Ce Te
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Decomposition of the rotation vector in 3 relative rotation vectors
3( ) R QDW QR W
Relative rotation vectors:
T[ ] T d
dt
Rω RT
T e ω
Ce ICe
ITe Te
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Decomposition of the rotation vector in 3 relative rotation vectors
3( ) R QDW QR W
Q IC[( ) ] T d
dt
Qω Q
ICQ IC e ω
of Intermediate Celestial with respect to Celestial
Relative rotation vectors:
Defined by:
T[ ] T d
dt
Rω RT
T e ω
Ce ICe
ITe Te
Q
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Decomposition of the rotation vector in 3 relative rotation vectors
3( ) R QDW QR W
Q IC[( ) ] T d
dt
Qω Q D IT 3[( ) ] [ ]T d d
dt dt
Dω D i
ICQ IC e ω
of Intermediate Celestial with respect to Celestial
Relative rotation vectors:
ITD IT e ω
of Intermediate Terrestrial with respect to Intermediate Celestial
Defined by:
T[ ] T d
dt
Rω RT
T e ω
Ce ICe
ITe Te
Q
D
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Decomposition of the rotation vector in 3 relative rotation vectors
3( ) R QDW QR W
Q IC[( ) ] T d
dt
Qω Q D IT 3[( ) ] [ ]T d d
dt dt
Dω D i W[ ] T d
dt
Wω W
ICQ IC e ω
of Intermediate Celestial with respect to Celestial
Relative rotation vectors:
ITD IT e ω
of Intermediate Terrestrial with respect to Intermediate Celestial
TW T e ω
of Terrestrial with respect to Intermediate Terrestrial
Defined by:
T[ ] T d
dt
Rω RT
T e ω
Ce ICe
ITe Te
Q
D W
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Decomposition of the rotation vector in 3 relative rotation vectors
3( ) R QDW QR W
Q D W
Q D W
Q IC[( ) ] T d
dt
Qω Q D IT 3[( ) ] [ ]T d d
dt dt
Dω D i W[ ] T d
dt
Wω W
ICQ IC e ω
of Intermediate Celestial with respect to Celestial
Relative rotation vectors:
ITD IT e ω
of Intermediate Terrestrial with respect to Intermediate Celestial
TW T e ω
of Terrestrial with respect to Intermediate Terrestrial
Defined by:
T[ ] T d
dt
Rω RT
T e ω
Ce ICe
ITe Te
Q
D W
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
In the Intermediate Celestial reference system ICIC e ω
1 1 1IC IC IC2 2 2
IC IC IC IC3 3 3 3 3 3IC Q IC D IC W IC Q IC W IC( ) ( ) ( ) ( ) ( )
ω
2 2 T IC 2 IC 2 IC 2IC IC 1 2 3| | [ ] [ ] [ ] ω ω
IC 2 IC 2 IC IC 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
IC IT3 3p e e
In the Intermediate Celestial reference system
= Celestial Pole (direction of diurnal rotation), e.g. CEP, CIP
ICIC e ω
1 1 1IC IC IC2 2 2
IC IC IC IC3 3 3 3 3 3IC Q IC D IC W IC Q IC W IC( ) ( ) ( ) ( ) ( )
ω
2 2 T IC 2 IC 2 IC 2IC IC 1 2 3| | [ ] [ ] [ ] ω ω
IC 2 IC 2 IC IC 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
1n
= Compatible Celestial Pole (CCP)
= Compatible rotation vector (derived from rotation matrix R)
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
0 3 3 3 0( , ) ( ) ( ) ( ) ( , )E d s s F g R QDW Q R R R W
7 parameters instead of 3 minimum required = 4 conditions needed !
The compatibility conditions
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Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
1IC2
IC IC3 3
Q IC W IC( ) ( )
ω
0 3 3 3 0( , ) ( ) ( ) ( ) ( , )E d s s F g R QDW Q R R R W
7 parameters instead of 3 minimum required = 4 conditions needed !
IC
0
0
1
p
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
p n
1IC2
IC IC3 3
Q IC W IC( ) ( )
ω
2 direction conditions:IC1 0 IC
2 0
0 3 3 3 0( , ) ( ) ( ) ( ) ( , )E d s s F g R QDW Q R R R W
7 parameters instead of 3 minimum required = 4 conditions needed !
IC
0
0
1
p
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
p n
1IC2
IC IC3 3
Q IC W IC( ) ( )
ω
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:IC1 0 IC
2 0 1 magnitude condition:
0 3 3 3 0( , ) ( ) ( ) ( ) ( , )E d s s F g R QDW Q R R R W
7 parameters instead of 3 minimum required = 4 conditions needed !
IC
0
0
1
p
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
p n
1IC2
IC IC3 3
Q IC W IC( ) ( )
ω
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:IC1 0 IC
2 0 1 magnitude condition:
0 3 3 3 0( , ) ( ) ( ) ( ) ( , )E d s s F g R QDW Q R R R W
Missing 4th condition:
7 parameters instead of 3 minimum required = 4 conditions needed !
0 3 3 3 0 0 3 3 3 0( ) ( ) ( ) ( ) ( ) ( )ss s ss s R Q R R R W Q R R R W
IC
0
0
1
p
s = arbitrary !
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
p n
1IC2
IC IC3 3
Q IC W IC( ) ( )
ω
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:IC1 0 IC
2 0 1 magnitude condition:
0 3 3 3 0( , ) ( ) ( ) ( ) ( , )E d s s F g R QDW Q R R R W
Missing 4th condition:
7 parameters instead of 3 minimum required = 4 conditions needed !
0 3 3 3 0 0 3 3 3 0( ) ( ) ( ) ( ) ( ) ( )ss s ss s R Q R R R W Q R R R W
IC
0
0
1
p
s = arbitrary !
4th condition = arbitrary definition of origin of diurnal rotation angle
The compatibility conditions
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
2 direction conditions:IC1 0 IC
2 0
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:
1 magnitude condition:
IC1 0 IC
2 0
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
IC1 0 IC
2 0
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
The 4 independent compatibility conditions
2 direction conditions:
2 NRO conditions:IC
Q 3( ) 0 ICW 3( ) 0
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:
1 magnitude condition:
IC1 0 IC
2 0
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
IC1 0 IC
2 0
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
The 4 independent compatibility conditions
2 direction conditions:
2 NRO conditions:IC
Q 3( ) 0 ICW 3( ) 0
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:
1 magnitude condition:
IC1 0 IC
2 0
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
IC1 0 IC
2 0
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
The 4 independent compatibility conditions
2 direction conditions:
2 NRO conditions:IC
Q 3( ) 0 ICW 3( ) 0
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:
1 magnitude condition:
IC1 0 IC
2 0
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
IC1 0 IC
2 0
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
The 4 independent compatibility conditions
2 direction conditions:
2 NRO conditions:IC
Q 3( ) 0 ICW 3( ) 0
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:
1 magnitude condition:
IC1 0 IC
2 0
2 2 2 2 20 0 [0 0]
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
The NRO conditions in relation to the compatibility conditions
IC1 0 IC
2 0
The 2 NRO (Non Rotating Origin) conditions:
Q p
CEO (Celestial Ephemeris Origin) :
W p
TEO (Terrestrial Ephemeris Origin) :
ICQ 3( ) 0
ICW 3( ) 0
The 4 independent compatibility conditions
2 direction conditions:
2 NRO conditions:IC
Q 3( ) 0 ICW 3( ) 0
2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]
2 direction conditions:
1 magnitude condition:
IC1 0 IC
2 0
2 2 2 2 20 0 [0 0]
magnitude condition satisfied !
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The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Explicit form of the 4 compatibility conditions
1
2
sin sin( ) ( )IC
IC
E d F gS S
d g
R R
3 3 3( ) ( ) ( cos ) ( cos )IC Q IC W IC E d S F g S
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S s E
S s F
Explicit form of the 4 compatibility conditions
1 cos sin sin cos( )sin sin( ) 0IC S d E S d S g F S g
1
2
sin sin( ) ( )IC
IC
E d F gS S
d g
R R
2 sin sin cos sin( )sin cos( ) 0IC S d E S d S g F S g
Direction conditions:
3 3 3( ) ( ) ( cos ) ( cos )IC Q IC W IC E d S F g S
![Page 33: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/33.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
S s E
S s F
Explicit form of the 4 compatibility conditions
1 cos sin sin cos( )sin sin( ) 0IC S d E S d S g F S g
1
2
sin sin( ) ( )IC
IC
E d F gS S
d g
R R
2 sin sin cos sin( )sin cos( ) 0IC S d E S d S g F S g
3( ) cos 0Q IC E d S NRO conditions:
Direction conditions:
3 3 3( ) ( ) ( cos ) ( cos )IC Q IC W IC E d S F g S
3( ) cos 0W IC S F g
( , )S S E d
( , )S S F g
( , )s S E s E d
( , )s S F s F g
![Page 34: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/34.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
S s E
S s F
Explicit form of the 4 compatibility conditions
1 cos sin sin cos( )sin sin( ) 0IC S d E S d S g F S g
1
2
sin sin( ) ( )IC
IC
E d F gS S
d g
R R
2 sin sin cos sin( )sin cos( ) 0IC S d E S d S g F S g
3( ) cos 0Q IC E d S NRO conditions:
Direction conditions:
3 3 3( ) ( ) ( cos ) ( cos )IC Q IC W IC E d S F g S
3( ) cos 0W IC S F g
cos sin sin cos( )sin sin( )S g F S g S d E S d
Direction conditions + NRO conditions :
( , )S S E d
When , E, d [and s(E,d)] are known
then F, g [and s(F,g)]are uniquely determined !
( , )S S F g
( , )s S E s E d
( , )s S F s F g
sin sin cos sin( )sin cos( )S g F S g S d E S d
![Page 35: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/35.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Construct a compatible separated model from observations only
( ) ( ) ( )k k m m n n R R R RAnalyze observations using a 3 parameter model:
![Page 36: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/36.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Construct a compatible separated model from observations only
( ) ( ) ( )k k m m n n R R R RAnalyze observations using a 3 parameter model:
( ) ( ) ( )T k k m k k m n k k m m n ω i R i R R i
T TT T C C ω ω ω ω
1[ ]TT TX Y Z n ω
( ) ( ) ( )C k n n m m k m n n m n n ω R R i R i i 1[ ]TC C n ω
Compute rotation vector components, magnitude & directions (CCP components):
![Page 37: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/37.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Construct a compatible separated model from observations only
( ) ( ) ( )k k m m n n R R R RAnalyze observations using a 3 parameter model:
( ) ( ) ( )T k k m k k m n k k m m n ω i R i R R i
T TT T C C ω ω ω ω
1[ ]TT TX Y Z n ω
( ) ( ) ( )C k n n m m k m n n m n n ω R R i R i i 1[ ]TC C n ω
Compute rotation vector components, magnitude & directions (CCP components):
Compute precession-nutation and polar motion angles:
arctanY
EX
2 2
arctanX Y
dZ
arctanF
2 2
arctang
![Page 38: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/38.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Construct a compatible separated model from observations only
( ) ( ) ( )k k m m n n R R R RAnalyze observations using a 3 parameter model:
( ) ( ) ( )T k k m k k m n k k m m n ω i R i R R i
T TT T C C ω ω ω ω
1[ ]TT TX Y Z n ω
( ) ( ) ( )C k n n m m k m n n m n n ω R R i R i i 1[ ]TC C n ω
Compute rotation vector components, magnitude & directions (CCP components):
Compute precession-nutation and polar motion angles:
arctanY
EX
2 2
arctanX Y
dZ
arctanF
2 2
arctang
Determine s and s from NRO conditions: (cos 1)s E d (cos 1)s F g
![Page 39: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions](https://reader035.fdocuments.net/reader035/viewer/2022062800/56814122550346895dacf8df/html5/thumbnails/39.jpg)
The Aristotle University of Thessaloniki Department of Geodesy and Surveying
Athanasios Dermanis Journées des Systèmes de Référence Spatio-Temporels – Warsaw 2005
Construct a compatible separated model from observations only
( ) ( ) ( )k k m m n n R R R RAnalyze observations using a 3 parameter model:
( ) ( ) ( )T k k m k k m n k k m m n ω i R i R R i
T TT T C C ω ω ω ω
1[ ]TT TX Y Z n ω
( ) ( ) ( )C k n n m m k m n n m n n ω R R i R i i 1[ ]TC C n ω
Compute rotation vector components, magnitude & directions (CCP components):
Compute precession-nutation and polar motion angles:
arctanY
EX
2 2
arctanX Y
dZ
arctanF
2 2
arctang
Compute diurnal rotation angle: 3 ( ) ( , , ) ( , , ) ( , , )T Tk m nE d s F g s R Q R W
Determine s and s from NRO conditions: (cos 1)s E d (cos 1)s F g