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

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

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**On the alternative approaches to ITRF formulation.**A theoretical comparison. Athanasios Dermanis Department of Geodesy and Surveying Aristotle University of Thessaloniki**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”**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**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**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**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**Simplifications of the Problem**True ITRF formulation problem for VLBI, SLR, GPS, DORIS: 4 non-overlapping networks connected through cross observations**Simplifications of the Problem**4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations**Simplifications of the Problem**2 overlapping networks 4 non-overlapping networks connected through cross observations 2 non-overlapping networks connected through cross observations**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**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**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**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**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**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**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**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 !**Models with rank defect due to lack of reference system**definition**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)**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)**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)**Two identical networks**This case does not apply to the ITRF formulation problem but has an interest of its own for other network applications**Two identical networks – One step solution**Identical to separate solutions and combination using of the model with weight matrices**Two identical networks – Two step solution**Step 1: Separate solutions Minimal constraints Step 2: Combination Weight matrix Normal equations**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**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 !**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**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**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**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**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 !**Two overlapping networks**This case would apply to the ITRF formulation problem if perfect connections were available at collocation sites**Two overlapping networks**x1,x2= parameters of non-common points x3= parameters of common points**Two overlapping networks – Separate solutions**Network (a) solution: normal equations Network (b) solution: normal equations**Two overlapping networks – One step solution**weight matrix normal equations Identical with solution based on separate solutions with model weight matrix**Two overlapping networks – Two step solution**Combination (second) step From network a: From network b: Combined a+b: normal equations**Two overlapping networks – Two step solution**normal equations Use of same weights as in the (equivalent to) one step solution normal equations**Two overlapping networks – Two step solution**normal equations Use of same weights as in the (equivalent to) one step solution normal equations Recall that**Two overlapping networks – Two step solution**normal equations Use of same weights as in the (equivalent to) one step solution normal equations Vanishing terms**Two overlapping networks – Two step solution**normal equations Use of same weights as in the (equivalent to) one step solution normal equations**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 !**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 !**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 !**Two non-overlapping networks connected by observations**This case applies to the ITRF formulation problem (with error-affected connecting observations at collocation sites)**Two non-overlapping networks connected by observations**observations of network a observations of network b Connecting observations Network a Network b**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**Two non-overlapping networks connected by observations**Collocation sites x2 x4 x1 x3 Network a Network b**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**Two connectednon-overlapping networks – One step solution**Joint treatment of observations from network a network b & connecting observations**Two connectednon-overlapping networks – One step solution**Weight matrix Normal equations