compatibility of the iers earth rotation representation and its relation to the nro conditions...

Compatibility of the IERS earth rotation representationand its relation to the NRO conditions

Athanasios Dermanis

Department of Geodesy and SurveyingThe Aristotle University of Thessaloniki

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



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



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



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



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:




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



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




3( ) R QDW QR W

Relative rotation vectors:

T[ ] T d

dt

Rω RT

T e ω

Ce ICe

ITe Te




3( ) R QDW QR W

Q IC[( ) ] T d

dt

Qω Q

ICQ IC e ω

of Intermediate Celestial with respect to Celestial


Defined by:

T[ ] T d

dt

Rω RT

T e ω

Ce ICe

ITe Te

Q




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



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




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



ITD IT e ω


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




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



ITD IT e ω


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



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



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)




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 !




1IC2

IC IC3 3

Q IC W IC( ) ( )

ω



IC

0

0

1

p




p n

1IC2

IC IC3 3

Q IC W IC( ) ( )

ω

2 direction conditions:IC1 0 IC

2 0



IC

0

0

1

p




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 0 1 magnitude condition:



IC

0

0

1

p




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[ ] [ ] [( ) ( ) ]




Missing 4th condition:


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 !




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[ ] [ ] [( ) ( ) ]




Missing 4th condition:


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 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





Q p


W p


ICQ 3( ) 0

ICW 3( ) 0


2 0





Q p


W p


ICQ 3( ) 0

ICW 3( ) 0

2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]

2 direction conditions:


IC1 0 IC

2 0




IC1 0 IC

2 0


Q p


W p


ICQ 3( ) 0

ICW 3( ) 0

The 4 independent compatibility 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[ ] [ ] [( ) ( ) ]



IC1 0 IC

2 0




IC1 0 IC

2 0


Q p


W p


ICQ 3( ) 0

ICW 3( ) 0



2 NRO conditions:IC

Q 3( ) 0 ICW 3( ) 0

2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]



IC1 0 IC

2 0

2 2 2 2 20 0 [0 0]




IC1 0 IC

2 0


Q p


W p


ICQ 3( ) 0

ICW 3( ) 0



2 NRO conditions:IC

Q 3( ) 0 ICW 3( ) 0

2 IC 2 IC 2 IC IC 2 21 2 Q 3 W 3[ ] [ ] [( ) ( ) ]



IC1 0 IC

2 0

2 2 2 2 20 0 [0 0]

magnitude condition satisfied !



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



S s E

S s F


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:




S s E

S s F



1

2

sin sin( ) ( )IC

IC

E d F gS S

d g

R R


3( ) cos 0Q IC E d S NRO conditions:



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



S s E

S s F



1

2

sin sin( ) ( )IC

IC

E d F gS S

d g

R R


3( ) cos 0Q IC E d S NRO conditions:



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



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):







1[ ]TT TX Y Z n ω



Compute precession-nutation and polar motion angles:

arctanY

EX

2 2

arctanX Y

dZ

arctanF

2 2

arctang







1[ ]TT TX Y Z n ω




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







1[ ]TT TX Y Z n ω




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

compatibility of the iers earth rotation representation and its relation to the nro conditions...

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