1. The definition of a geophysically meaningful
International Terrestrial Reference System
Problems and prospects Z
EGU General Assembly 2006
April 2-7, Vienna Y
Athanasios Dermanis
Department of Geodesy and Surveying
Aristotle University of Thessaloniki X

2. Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS)
Objective (practice): Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions xi(t), yi(t), zi(t) of points Pi in a global geodetic network

3. Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS)
Objective (practice): Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions xi(t), yi(t), zi(t) of points Pi in a global geodetic network
Available data: Coordinates xi(tk), yi(tk), zi(tk) of points in subnetworks at discrete epochs tk (e.g. daily or weekly solutions) with respect to a different reference systems for each subnetwork

4. Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS)
Objective (practice): Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions xi(t), yi(t), zi(t) of points Pi in a global geodetic network
Available data: Coordinates xi(tk), yi(tk), zi(tk) of points in subnetworks at discrete epochs tk (e.g. daily or weekly solutions) with respect to a different reference systems for each subnetwork
Problem (theory): Introduce a principle leading to an optimal choice of the reference frame
Problem (practice): Combine the known shapes of the overlapping subnetworks into the shape of a global network expressed by coordinates with respect to the already defined optimal reference system

5. The optimal choice of reference system
General characteristic: Apparent motion (coordinate variation) of mass points should be minimal
For the whole earth dm x
Tisserand principle:
Vanishing relative angular momentum 
Minimal relative kinetic energy

6. i dm xi x The optimal choice of reference system  
General characteristic: Apparent motion (coordinate variation) of mass points should be minimal
For the whole earth
For a network of discrete points dm x i xi
Tisserand principle:
Discrete Tisserand principle:
Vanishing relative angular momentum
Vanishing relative angular momentum  
Minimal relative kinetic energy
Minimal relative kinetic energy

7. i dm xi x mi =1 The optimal choice of reference system
General characteristic: Apparent motion (coordinate variation) of mass points should be minimal
For the whole earth
For a network of discrete points dm x i xi
Tisserand principle:
Discrete Tisserand principle:
Define orientation of the reference system.
For the position:
Preservation of the center of mass:
Trivial
choice:
mi =1

8. No ! The “linear” model It is not possible to have all
shape related parameters
which are coordinate invariants
as linear functions of time!
Is there a linear model for deformation?
No !

9. No ! s1 s2 s3 si = ai t + bi , i = 1,2,3,4,5 s4 s5 The “linear” model
It is not possible to have all
shape related parameters
which are coordinate invariants
as linear functions of time!
Is there a linear model for deformation?
No !
A planar counterexample:
The 5 shape defining sides
are linear functions of time: s1 s2 s3
si = ai t + bi , i = 1,2,3,4,5 s4 s5

10. The “linear” model
It is not possible to have all
shape related parameters
which are coordinate invariants
as linear functions of time!
Is there a linear model for deformation?
No !
A planar counterexample:
The 5 shape defining sides
are linear functions of time: s1 s2 s3
si = ai t + bi , i = 1,2,3,4,5 d s4 s5
 d = nonlinear in t
The 2nd diagonal of the quadrilateral
is not a linear function of time!

11. x(t) = R(t) x0 t + R(t) v + c(t)  x0(t) t + v
The “linear” model
x(t) = x0 t + vx ,
y(t) = y0 t + vy ,
z(t) = z0 t + vz ,
What is the meaning of linear model for coordinate functions?
Linear coordinates in one reference system
Non-linear coordinates in another
CHANGE OF REFERENCE SYSTEM
x(t) = x0 t + v
x(t) = R(t) x0 t + R(t) v + c(t)
 x0(t) t + v
x´(t) = R(t) x(t) + c(t)

12. x(t) = x0 t + vx , y(t) = y0 t + vy , z(t) = z0 t + vz , x(t)
The “linear” model
x(t) = x0 t + vx ,
y(t) = y0 t + vy ,
z(t) = z0 t + vz ,
What is the meaning of linear model for coordinate functions?
x(t)
x´(t) = R(t) x(t) + c(t)
are both Tisserand
R(t) = R = const.
c(t) = c = const.

13. x(t) = x0 + t vx , y(t) = y0 + t vy , z(t) = z0 + t vz , x(t)
The “linear” model
x(t) = x0 + t vx ,
y(t) = y0 + t vy ,
z(t) = z0 + t vz ,
What is the meaning of linear model for coordinate functions?
x(t)
x´(t) = R(t) x(t) + c(t)
are both Tisserand
R(t) = R = const.
c(t) = c = const.
Ad hoc definition of “linear deformation” model:
Coordinates in one (and hence all) Tisserand system
are linear functions of time
Note: Tisserand reference system here means:
Orientation by Tisserand principle (zero relative angular momentum)
Position by center of mass preservation principle

14. The Tisserand reference system conditions for the linear model
Vanishing relative angular momentum:
Constant (zero) center of mass coordinates:
Tisserand conditions:
Tisserand conditions:
Usual choice:

15. + = The Tisserand reference system conditions for the linear model
Used at the stage of data analysis
as additional constraints +
Usual inner constraints for the definition
of reference frame at epoch t0 =
Selection of a single Tisserand frame
out of a set of dynamically equivalent ones
Vanishing relative angular momentum:
Constant (zero) center of mass coordinates:
Tisserand conditions:
Tisserand conditions:
Usual choice:

16. Accessing a geophysically meaningful reference system
x(t) = R(t) x(t) ~ ~
x(t)
(discrete) Tisserand
x(t)
(earth) Tisserand dm ~
To access R(t) transforming hR(Earth) to hR(Earth) = 0 we need an approximation of hR(Earth)

17. Accessing a geophysically meaningful reference system
x(t) = R(t) x(t) ~ ~
x(t)
(discrete) Tisserand
x(t)
(earth) Tisserand dm ~
To access R(t) transforming hR(Earth) to hR(Earth) = 0 we need an approximation of hR(Earth)
To access R(t) transforming hR(Earth) to hR(Earth) = 0 we need an approximation of hR(Earth)
Plates
Interior
Earth  E = P1 + … + PK + … + PL + P0
Subnetwork on plate PK
Network  D = D1 + … + DK + … + DL
Use DK to access hPK(t) !

18. Estimation of the rigid motion of a plate
x = RK x
Transformation such that
original velocities Z Z
Subnetwork DK Y Y
Plate PK X X

19. Estimation of the rigid motion of a plate
x = RK x
Transformation such that K 
original velocities Z Z
determine rotation RK
to reference system
best fitted to DK RK
Subnetwork DK
transformed velocities Y Y
Plate PK X X

20. Estimation of the rigid motion of a plate
x = RK x
Transformation such that K 
original velocities Z Z
determine rotation RK
to reference system
best fitted to DK RK
Subnetwork DK
transformed velocities Y
assume rigid motion of plate PK
due only to rotation RK Y
Plate PK X
velocities due to rigid
plate rotation only X

21. Estimation of the rigid motion of a plate
x = RK x
Transformation such that K 
original velocities Z Z
determine rotation RK
to reference system
best fitted to DK RK
Subnetwork DK
transformed velocities Y
assume rigid motion of plate PK
due only to rotation RK Y
Plate PK X
velocities due to rigid
plate rotation only X
compute relative angular momentum
of plate PK due only to rotation RK

22. Final transformation to an approximate Earth-Tisserand reference system
x = R x ~
Previously estimated

23. Final transformation to an approximate Earth-Tisserand reference system
x = R x ~
Previously estimated
Approximate (numerically sufficient) solution for the linear model: xi(t) = x0i + t vi

24. Final transformation to an approximate Earth-Tisserand reference system
x = R x ~
Previously estimated
Approximate (numerically sufficient) solution for the linear model: xi(t) = x0i + t vi 1
Subnetwork DK
relative angular momentum
inertia matrix

25. Final transformation to an approximate Earth-Tisserand reference system
x = R x ~
Previously estimated
Approximate (numerically sufficient) solution for the linear model: xi(t) = x0i + t vi 1
Subnetwork DK
relative angular momentum
inertia matrix 2
Plate PK
rotation vector
inertia matrix

26. Final transformation to an approximate Earth-Tisserand reference system
x = R x ~
Previously estimated
Approximate (numerically sufficient) solution for the linear model: xi(t) = x0i + t vi 1 3
Subnetwork DK
total rotation vector
relative angular momentum
inertia matrix 2
Plate PK
rotation vector
inertia matrix

27. Final transformation to an approximate Earth-Tisserand reference system
x = R x ~
Previously estimated
Approximate (numerically sufficient) solution for the linear model: xi(t) = x0i + t vi 1 3
Subnetwork DK
total rotation vector
relative angular momentum
inertia matrix 4
new linear model 2
Plate PK
rotation vector
inertia matrix
only velocities updated !

28. http://der.topo.auth.gr Thanks for your attention !
A copy of this presentation can be downloaded from:

29. hope to see you there ! Challenge and Role of Modern Geodesy
the Intercommission Committee on Theory (ICCT)
of the International Association of Geodesy (IAG)
invites you to the
VI Hotine-Marussi Symposium
of Theoretical and Computational Geodesy:
Challenge and Role of Modern Geodesy
May 29 - June 2, 2006, Wuhan, China
hope to see you there !