Theoretical Foundations of ITRF Determination

1. The VII Hotine-Marussi Symposium Rome, July 6–10, 2009 Theoretical foundations of ITRF determination The algebraic and the kinematic approach Zuheir Altamimi1 & Athanasios Dermanis2 (1) IGN-LAREG - (2) Aristotle University of Thessaloniki

2. THE ITRF FORMULATION PROBLEM t given a time sequence of sub-network coordinates (one from each technique T = VLBI, SLR, GPS, DORIS) combine them into coordinates for the whole network obeying a time-evolution model Essentially: Determine the model parameters for each network point i

3. THE ITRF FORMULATION PROBLEM t given a time sequence of sub-network coordinates (one from each technique T = VLBI, SLR, GPS, DORIS) combine them into coordinates for the whole network obeying a time-evolution model Essentially: Determine the model parameters for each network point i

4. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t Given a time sequence of sub-network shapes (one from each technique: VLBI, SLR, GPS, DORIS)

5. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t Replace them with a smooth sequence of shapes

6. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t Replace them with a smooth sequence of shapes

7. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t Note that although shape variation is insignificant coordinates may vary significantly due to temporal instability in reference system maintenance

8. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t To remove coordinate variation assign a different reference system at each epoch

9. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t To remove coordinate variation assign a different reference system at each epoch

10. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t To remove coordinate variation assign a different reference system at each epoch such that when networks are viewed in the “same” system

11. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t To remove coordinate variation assign a different reference system at each epoch such that when networks are viewed in the “same” system coordinates vary in a smooth way

12. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t To remove coordinate variation assign a different reference system at each epoch such that when networks are viewed in the “same” system coordinates vary in a smooth way in conformance with a coordinate time-variation model

13. THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM t To remove coordinate variation assign a different reference system at each epoch such that when networks are viewed in the “same” system coordinates vary in a smooth way in conformance with a coordinate time-variation model currently:

14. data: STACKING FOR EACH PARTICULAR TECHNIQUE coordinate transformation parameters: t t coordinate variation model: model parameters: t

15. THE ITRF FORMULATION PROBLEM = SIMULTANEOUS STACKING FOR ALL TECHNIQUES VLBI SLR GPS DORIS ITRF t

16. THE ITRF FORMULATION PROBLEM IN AN OPERATIONALLY CONVENIENT COMPROMISE (1) Separate stackings one for each technique: Provides initial coordinates and velocities for the subnetwork of each technique Separation into 2 steps:

17. THE ITRF FORMULATION PROBLEM IN AN OPERATIONALLY CONVENIENT COMPROMISE (1) Separate stackings one for each technique: Provides initial coordinates and velocities for the subnetwork of each technique Separation into 2 steps: (2) Combination of initial coordinates and velocities: Provides initial coordinates and velocities for the whole ITRF network

18. THE MODEL FOR TIME EVOLUTION OF COORDINATES The (general) model: Point Pi coordinates: ai = point Pi parameters The current model:

19. WHAT THE MODEL DOES SMOOTHING Replaces observes time sequences of sub-network shapes with a single smooth time sequence for the whole ITRF network IMPOSES THE USE OF A REFERENCE SYSTEM so that network shapes are represented by coordinates INTERPOLATION Provides shapes expressed by coordinates for epochs other than observation ones

20. WHAT THE MODEL DOES NOT DO SMOOTHING Replaces observes time sequences of sub-network shapes with a single smooth time sequence for the whole ITRF network IMPOSES THE USE OF A REFERENCE SYSTEM so that network shapes are represented by coordinates It does not resolve the problem of the choice of the reference system INTERPOLATION Provides shapes expressed by coordinates for epochs other than observation ones MAIN ITRF FORMULATION PROBLEM Assign a reference system for each epoch

21. WHAT THE MODEL DOES NOT DO MAIN ITRF FORMULATION PROBLEM Assign a reference system for each epoch It does not resolve the problem of the choice of the reference system

22. WHAT THE MODEL DOES NOT DO MAIN ITRF FORMULATION PROBLEM Assign a reference system for each epoch It does not resolve the problem of the choice of the reference system PROBLEM SOLUTION:Introduce additional minimal constraintsin the Least-Square data analysis problem Minimal constraints: At any epoch t they determine the reference system without affecting the optimal network shape uniquely determined by the least-squares principle for the determination of ITRF parameters

23. WHAT THE MODEL DOES NOT DO MAIN ITRF FORMULATION PROBLEM Assign a reference system for each epoch It does not resolve the problem of the choice of the reference system PROBLEM SOLUTION:Introduce additional minimal constraintsin the Least-Square data analysis problem Minimal constraints: At any epoch t they determine the reference system without affecting the optimal network shape uniquely determined by the least-squares principle for the determination of ITRF parameters How to choose the minimal inner constraints? 2 approaches: (1) The algebraic approach (classical Meissl inner constraints) (2) The kinematic approach (new!)

24. THE ALGEBRAIC APPROACH THE KINEMATIC APPROACH Formulation of Least Squares problem with infinite solutions for different choices of reference system

25. THE ALGEBRAIC APPROACH THE KINEMATIC APPROACH Formulation of Least Squares problem with infinite solutions for different choices of reference system Choice of unique solution by inner constraints: or partial inner constraints:

26. THE ALGEBRAIC APPROACH THE KINEMATIC APPROACH Formulation of Least Squares problem Choice of reference system by minimization of apparent variation of coordinate for network points with infinite solutions for different choices of reference system Choice of unique solution by inner constraints: or partial inner constraints:

27. THE ALGEBRAIC APPROACH THE KINEMATIC APPROACH Formulation of Least Squares problem Choice of reference system by minimization of apparent variation of coordinate for network points with infinite solutions for different choices of reference system Measures of coordinate variation: Choice of unique solution by inner constraints: (1) Minimum relative kinetic energy = = vanishing relative angular momentum Discrete Tisserand Reference System (2) constant network barycenter or partial inner constraints: (3) constant mean quadratic scale

28. THE ALGEBRAIC APPROACH – INNER CONSTRAINTS Inner constraints determined from the linear variation of unknown parameters x when coordinate system changes with small transformation parameters p rotation angles translation vector scale parameter

29. THE ALGEBRAIC APPROACH – INNER CONSTRAINTS Inner constraints determined from the linear variation of unknown parameters x when coordinate system changes with small transformation parameters p rotation angles translation vector scale parameter Determine the parameter variation equations

30. THE ALGEBRAIC APPROACH – INNER CONSTRAINTS Inner constraints determined from the linear variation of unknown parameters x when coordinate system changes with small transformation parameters p rotation angles translation vector scale parameter Determine the parameter variation equations Then the (total) inner constraints are

31. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES Transformation of coordinates in first order approximation

32. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES Transformation of coordinates in first order approximation Model preserving transformations

33. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES Transformation of coordinates in first order approximation Model preserving transformations Transformation of model parameters

34. THE ALGEBRAIC APPROACH – INNER CONSTRAINTS PER STATION For each station Pi determine the parameter variation equations The inner constraints per station are The (total) inner constraints are

35. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES The use of model preserving transformations instead of arbitrary transformations leads to a sub-optimal solution: No matter what the optimality criterion, there exist an arbitrary transformation leading to a better solution which does not conform with the chosen model Strict optimality leads the solution OUTSIDE the adopted model !

36. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES Transformation of model parameters in terms of corrections to approximate values Transformation of corrections to model parameters

37. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES Transformation of model parameters in terms of corrections to approximate values Transformation of corrections to model parameters

38. MODEL PRESERVING APPROXIMATE TRANSFORMATIONS OF INITIAL COORDINATES AND VELOCITIES Transformation of model parameters in terms of corrections to approximate values Transformation of corrections to model parameters inner constraints sub-matrix

39. THE STACKING PROBLEM Original observation model GIVEN SOUGHT Observed coordinates in particular technique at epoch tk ITRF model coordinates at epoch tk NUISANCE Transformation parameters from ITRF system to technique-system at epoch tk

40. THE STACKING PROBLEM Original observation model GIVEN SOUGHT Observed coordinates in particular technique at epoch tk ITRF model coordinates at epoch tk NUISANCE Transformation parameters from ITRF system to technique-system at epoch tk In first order approximation In terms of corrections to approximate values

41. INNER CONSTRAINTS FOR THE STACKING PROBLEM Change of ITRF reference system

42. INNER CONSTRAINTS FOR THE STACKING PROBLEM (Total) inner constraints initial orientation initialtranslation initialscale orientation rate translation rate scale rate

43. INNER CONSTRAINTS FOR THE STACKING PROBLEM Partial inner constraints – Coordinates & velocities initialorientation initialtranslation initialscale orientation rate translation rate scale rate

44. INNER CONSTRAINTS FOR THE STACKING PROBLEM Partial inner constraints – Transformation parameters initialorientation initialtranslation initialscale orientation rate translation rate scale rate

45. THE COMBINATION PROBLEM Observation model Initial coordinates and velocities from each technique T Unknown ITRF initial coordinates and velocities Transformation parameters from ITRF system to technique (stacking) system GIVEN SOUGHT NUISANCE

46. INNER CONSTRAINTS FOR THE COMBINATION PROBLEM Change of ITRF reference system

47. INNER CONSTRAINTS FOR THE COMBINATION PROBLEM (Total) inner constraints initial orientation initialtranslation initialscale orientation rate translation rate scale rate

48. INNER CONSTRAINTS FOR THE COMBINATION PROBLEM Partial inner constraints – Coordinates & velocities initialorientation initialtranslation initialscale orientation rate translation rate scale rate

49. INNER CONSTRAINTS FOR THE COMBINATION PROBLEM Partial inner constraints – Coordinates & velocities initialorientation initialtranslation initialscale orientation rate translation rate scale rate Same as for the stacking problem !

50. INNER CONSTRAINTS FOR THE COMBINATION PROBLEM Partial inner constraints – Transformation parameters initialorientation initialtranslation initialscale orientation rate translation rate scale rate