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

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 x i (t), y i (t), z i (t) of points P i 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 x i (t), y i (t), z i (t) of points P i in a global geodetic network Available data:Coordinates x i (t k ), y i (t k ), z i (t k ) of points in subnetworks at discrete epochs t k (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 x i (t), y i (t), z i (t) of points P i in a global geodetic network Available data:Coordinates x i (t k ), y i (t k ), z i (t k ) of points in subnetworks at discrete epochs t k (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 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. The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal 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 Tisserand principle: Vanishing relative angular momentum Minimal relative kinetic energy  Discrete Tisserand principle: Vanishing relative angular momentum Minimal relative kinetic energy  xixi i

7. The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal 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 Tisserand principle:Discrete Tisserand principle: Define orientation of the reference system. For the position: xixi i Preservation of the center of mass: Trivial choice: m i =1

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

9. The “linear” model Is there a linear model for deformation? No ! A planar counterexample: s1s1 s2s2 s4s4 s5s5 s3s3 s i = a i t + b i,i = 1,2,3,4,5 The 5 shape defining sides are linear functions of time: It is not possible to have all shape related parameters which are coordinate invariants as linear functions of time!

10. The “linear” model Is there a linear model for deformation? No ! A planar counterexample: s1s1 s2s2 s4s4 s5s5 s3s3 d s i = a i t + b i,i = 1,2,3,4,5  d = nonlinear in t The 5 shape defining sides are linear functions of time: The 2nd diagonal of the quadrilateral is not a linear function of time! It is not possible to have all shape related parameters which are coordinate invariants as linear functions of time!

11. The “linear” model What is the meaning of linear model for coordinate functions? Linear coordinates in one reference system x(t) = x 0 t + v x, y(t) = y 0 t + v y, z(t) = z 0 t + v z, x(t) = x 0 t + vx(t) = R(t) x 0 t + R(t) v + c(t)  x 0 (t) t + v x´(t) = R(t) x(t) + c(t) Non-linear coordinates in another CHANGE OF REFERENCE SYSTEM

12. The “linear” model x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand R(t) = R = const. c(t) = c = const. R(t) = R = const. c(t) = c = const. What is the meaning of linear model for coordinate functions? x(t) = x 0 t + v x, y(t) = y 0 t + v y, z(t) = z 0 t + v z,

13. The “linear” model Ad hoc definition of “linear deformation” model: Coordinates in one (and hence all) Tisserand system are linear functions of time x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand Note:Tisserand reference system here means: Orientation by Tisserand principle (zero relative angular momentum) Position by center of mass preservation principle R(t) = R = const. c(t) = c = const. R(t) = R = const. c(t) = c = const. What is the meaning of linear model for coordinate functions? x(t) = x 0 + t v x, y(t) = y 0 + t v y, z(t) = z 0 + t v z,

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

15. The Tisserand reference system conditions for the linear model Vanishing relative angular momentum: Constant (zero) center of mass coordinates: Tisserand conditions: Usual choice: Tisserand conditions: Used at the stage of data analysis as additional constraints + Usual inner constraints for the definition of reference frame at epoch t 0 = Selection of a single Tisserand frame out of a set of dynamically equivalent ones Used at the stage of data analysis as additional constraints + Usual inner constraints for the definition of reference frame at epoch t 0 = Selection of a single Tisserand frame out of a set of dynamically equivalent ones

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

17. Accessing a geophysically meaningful reference system x(t) (earth) Tisserand x(t) (earth) Tisserand ~ x(t) = R(t) x(t) ~ x(t) (discrete) Tisserand x(t) (discrete) Tisserand dm To access R(t) transforming h R (Earth) to h R (Earth) = 0 we need an approximation of h R (Earth) ~ Earth  E = P 1 + … + P K + … + P L + P 0 Network  D = D 1 + … + D K + … + D L PlatesInterior Subnetwork on plate P K To access R(t) transforming h R (Earth) to h R (Earth) = 0 we need an approximation of h R (Earth) 0 Use D K to access h P K (t) !

18. Estimation of the rigid motion of a plate Plate P K Subnetwork D K x = RK xx = RK x Transformation such that original velocities X Y Z X Y Z

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

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

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

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: x i (t) = x 0i + t v i

24. Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix Subnetwork D K 1

25. Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix rotation vector inertia matrix Subnetwork D K Plate P K 1 2

26. Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix rotation vector inertia matrix Subnetwork D K Plate P K total rotation vector 1 2 3

27. Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix rotation vector inertia matrix Subnetwork D K Plate P K total rotation vector 1 2 3 4 new linear model only velocities updated !

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

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