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# On the alternative approaches to ITRF formulation. A theoretical comparison.

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1. On the alternative approaches to ITRF formulation. A theoretical comparison. Athanasios Dermanis Department of Geodesy and Surveying Aristotle University of Thessaloniki

2. The ITRF Formulation Problem Given: Time series of coordinates xT(tk) & EOPS cT(tk) from each space technique T Find: The optimal coordinate transformation parameters pT(tk) (rotations, translation, scale) which transform the above time series xT(tk), cT(tk) into new ones xITRF(tk), cITRF(tk) best fitting the linear-in-time ITRF model for each network station i with constant initial coordinates x0i and velocities vi This procedure is called “stacking”

3. The ITRF Formulation Problem The basic stacking model: Coordinates: Earth Orientation Parameters (EOPs): Data from a set of 4 non-overlapping networks (VLBI, SLR, GPS, DORIS) connected through surveying observations between nearby stations at collocation sites

4. The ITRF Formulation Problem The basic stacking model: Coordinates: Earth Orientation Parameters (EOPs): ITRF parameters (initial coordinates, velocities, EOPs): Data from a set of 4 non-overlapping networks (VLBI, SLR, GPS, DORIS) connected through surveying observations between nearby stations at collocation sites

5. The ITRF Formulation Problem The basic stacking model: Coordinates: Earth Orientation Parameters (EOPs): Transformation parameters from the ITRF reference system to the reference system of each epoch within each technique: Data from a set of 4 non-overlapping networks (VLBI, SLR, GPS, DORIS) connected through surveying observations between nearby stations at collocation sites

6. The ITRF Formulation Problem The basic stacking model: Coordinates: Earth Orientation Parameters (EOPs): Observation noise - Assumed zero-mean and with known covariance cofactor matrices (single unknown reference variance 2): Data from a set of 4 non-overlapping networks (VLBI, SLR, GPS, DORIS) connected through surveying observations between nearby stations at collocation sites

7. Simplifications of the Problem True ITRF formulation problem for VLBI, SLR, GPS, DORIS: 4 non-overlapping networks connected through cross observations

8. Simplifications of the Problem 4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations

9. Simplifications of the Problem 2 overlapping networks 4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations

10. Simplifications of the Problem 2 identical networks 2 overlapping networks 4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations

11. Simplifications of the Problem 2 identical networks Despite the simplifications the fundamental problem characteristics are preserved The simplified cases deserve a study in their own 2 overlapping networks 4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations

12. Simplifications of the Problem 2 identical networks We will restrict to 2 networks in order to keep equations within manageable complexity No loss of generality 2 overlapping networks 4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations

13. The two alternative approaches ONE STEP APPROACH Simultaneous adjustment of data from all techniques for the estimation of the ITRF parameters (multi-technique approach – simultaneous stacking) TWO STEP APPROACH (1) Adjustment of data from each technique separately for the estimation of per technique ITRF parameters (stacking per technique) (2) Combination of the ITRF estimates from each technique into final ITRF estimates

14. The two alternative approaches ONE STEP APPROACH – EQUIVALENT TWO-STEP FORMULATION (1) Separate solutions (2) Combination of separate solutions with addition of normal equations Final parameter estimates as weighted mean of separate estimates TWO STEP APPROACH (1) Adjustment of data from each technique separately for the estimation of per technique ITRF parameters (stacking per technique) (2) Combination of the ITRF estimates from each technique into final ITRF estimates

15. The two alternative approaches ONE STEP APPROACH – EQUIVALENT TWO-STEP FORMULATION (1) Separate solutions (2) Combination of separate solutions with addition of normal equations Final parameter estimates as weighted mean of separate estimates same TWO STEP APPROACH (1) Adjustment of data from each technique separately for the estimation of per technique ITRF parameters (stacking per technique) (2) Combination of the ITRF estimates from each technique into final ITRF estimates

16. The two alternative approaches ONE STEP APPROACH – EQUIVALENT TWO-STEP FORMULATION (1) Separate solutions (2) Combination of separate solutions with addition of normal equations Final parameter estimates as weighted mean of separate estimates Difference only in second steps TWO STEP APPROACH (1) Adjustment of data from each technique separately for the estimation of per technique ITRF parameters (stacking per technique) (2) Combination of the ITRF estimates from each technique into final ITRF estimates

17. The two alternative approaches ONE STEP APPROACH – EQUIVALENT TWO-STEP FORMULATION (1) Separate solutions (2) Combination of separate solutions with addition of normal equations Final parameter estimates as weighted mean of separate estimates TWO STEP APPROACH (1) Adjustment of data from each technique separately for the estimation of per technique ITRF parameters (stacking per technique) (2) Combination of the ITRF estimates from each technique into final ITRF estimates Separate solutions produce singular covariance matrices !

18. Models with rank defect due to lack of reference system definition Variation of parameters under change of reference system p= transformation parameters (rotations, displacement, scale)

19. Models with rank defect due to lack of reference system definition Variation of parameters under change of reference system p= transformation parameters (rotations, displacement, scale) Invariance of observables y = Ax and estimable parameters (functions of y)

20. Models with rank defect due to lack of reference system definition Variation of parameters under change of reference system p= transformation parameters (rotations, displacement, scale) Invariance of observables y = Ax and estimable parameters (functions of y) (total) inner constraints for reference system choice (usually partial inner constraints or other minimal constraints are employed)

21. Two identical networks This case does not apply to the ITRF formulation problem but has an interest of its own for other network applications

22. Two identical networks – One step solution Identical to separate solutions and combination using of the model with weight matrices

23. Two identical networks – Two step solution Step 1: Separate solutions Minimal constraints Step 2: Combination Weight matrix Normal equations

24. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations Weight matrices Na, Nb Weight matrices Wa, Wb

25. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations Weight matrices Na, Nb Weight matrices Wa, Wb Weight matrices “kill” the dependence of the partial solutions on different reference systems !

26. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations with same weight matrices Weight matrices Na, Nb Weight matrices Na, Nb

27. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations with same weight matrices Weight matrices Na, Nb Weight matrices Na, Nb Recall that

28. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations with same weight matrices Weight matrices Na, Nb Weight matrices Na, Nb Vanishing terms

29. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations with same weight matrices Weight matrices Na, Nb Weight matrices Na, Nb

30. Two identical networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations with same weight matrices Weight matrices Na, Nb Weight matrices Na, Nb Same results for IERS parameters x ! Transformation parameters pa, pb undetermined !

31. Two overlapping networks This case would apply to the ITRF formulation problem if perfect connections were available at collocation sites

32. Two overlapping networks x1,x2= parameters of non-common points x3= parameters of common points

33. Two overlapping networks – Separate solutions Network (a) solution: normal equations Network (b) solution: normal equations

34. Two overlapping networks – One step solution weight matrix normal equations Identical with solution based on separate solutions with model weight matrix

35. Two overlapping networks – Two step solution Combination (second) step From network a: From network b: Combined a+b: normal equations

36. Two overlapping networks – Two step solution normal equations Use of same weights as in the (equivalent to) one step solution normal equations

37. Two overlapping networks – Two step solution normal equations Use of same weights as in the (equivalent to) one step solution normal equations Recall that

38. Two overlapping networks – Two step solution normal equations Use of same weights as in the (equivalent to) one step solution normal equations Vanishing terms

39. Two overlapping networks – Two step solution normal equations Use of same weights as in the (equivalent to) one step solution normal equations

40. Two overlapping networks – Two step solution normal equations Use of same weights as in the (equivalent to) one step solution normal equations Transformation parameters pa, pb undetermined !

41. Two overlapping networks – Two step solution normal equations with same weight matrix as in one-step solution Same results for parameters x as in the (equivqlent to) one-step solution !

42. Two overlapping networks (equivalent to) one step solution Two step solution “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems Normal equations with same weight matrices Same results for IERS parameters x ! Transformation parameters pa, pb undetermined !

43. Two non-overlapping networks connected by observations This case applies to the ITRF formulation problem (with error-affected connecting observations at collocation sites)

44. Two non-overlapping networks connected by observations observations of network a observations of network b Connecting observations Network a Network b

45. Two non-overlapping networks connected by observations observations of network a observations of network b Connecting observations x2 x4 x1 x3 Network a Network b

46. Two non-overlapping networks connected by observations Collocation sites x2 x4 x1 x3 Network a Network b

47. Two connectednon-overlapping networks – Separate solutions weight matrices Normal equations & separate solutions + minimal constraints + minimal constraints Separate solutions = input to: (a) combination step of two step solution (b) 2nd step of equivalent to one step solution

48. Two connectednon-overlapping networks – One step solution Joint treatment of observations from network a network b & connecting observations

49. Two connectednon-overlapping networks – One step solution Weight matrix Normal equations