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

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Given: Time series of coordinates x T (t k ) & EOPS c T (t k ) from each space technique T Find: The optimal coordinate transformation parameters p T (t k ) (rotations, translation, scale) which transform the above time series x T (t k ), c T (t k ) into new ones x ITRF (t k ), c ITRF (t k ) best fitting the linear-in-time ITRF model for each network station i with constant initial coordinates x 0i and velocities v i The ITRF Formulation Problem This procedure is called “stacking”

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

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

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

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

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

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

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

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

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

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

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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 Simultaneous adjustment of data from all techniques for the estimation of the ITRF parameters (multi-technique approach – simultaneous stacking)

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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

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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

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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

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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 Separate solutions produce singular covariance matrices !

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Models with rank defect due to lack of reference system definition

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

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

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

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

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

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

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Two step solution(equivalent to) one step solution Normal equations Weight matrices N a, N b Weight matrices W a, W b Two identical networks “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems

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Two step solution(equivalent to) one step solution Normal equations Weight matrices N a, N b Weight matrices W a, W b Weight matrices “kill” the dependence of the partial solutions on different reference systems ! Two identical networks “wrong” model ! Ignores that partial and final solutions are in different reference systems correct model ! Treats partial and final solutions in different reference systems

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

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

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

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

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

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

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Two overlapping networks x 3 = parameters of common points x 1, x 2 = parameters of non-common points

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

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

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

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

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

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

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

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normal equations Use of same weights as in the (equivalent to) one step solution normal equations Transformation parameters p a, p b undetermined ! Two overlapping networks – Two step solution

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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 – Two step solution

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Two overlapping networks Two step solution(equivalent to) one 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 p a, p b undetermined !

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

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

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Two non-overlapping networks connected by observations Network b observations of network a observations of network b Connecting observations Network a x3x3 x2x2 x1x1 x4x4

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Two non-overlapping networks connected by observations Network bNetwork a x3x3 x2x2 x1x1 x4x4 Collocation sites

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

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Two connected non-overlapping networks – One step solution Joint treatment of observations from network a network b & connecting observations

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Normal equations Two connected non-overlapping networks – One step solution Weight matrix

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One step solution = Identical to separate solutions and combination with the model Two connected non-overlapping networks – One step solution with weight matrix This is a two-step equivalent of the one-step solution based on the addition of the partial normal equations

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Two connected non-overlapping networks – One step solution with weight matrix This model ignores the different reference systems (RS) in partial and final solution One step solution = Identical to separate solutions and combination with the model

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Two connected non-overlapping networks – One step solution with weight matrix RS a RS b RS FINAL This model ignores the different reference systems (RS) in partial and final solution One step solution = Identical to separate solutions and combination with the model

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Two connected non-overlapping networks – Two step solution We have already treated the first step (separate solutions) It remains to examine the second combination step

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Two connected non-overlapping networks – Two step solution weight matrix Combination step (second step)

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Two connected non-overlapping networks – Two step solution weight matrix This model takes into account the different reference systems (RS) in partial and final solutions by introducing transformation parameters p a, p b Combination step (second step)

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Two connected non-overlapping networks – Two step solution weight matrix partition of matrices for a more compact notation Combination step (second step)

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Two connected non-overlapping networks – Two step solution weight matrix model in compact notation Combination step (second step)

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Two connected non-overlapping networks – Two step solution weight matrix Combination step (second step)

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Two connected non-overlapping networks – Two step solution weight matrix Normal equations Combination step (second step)

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special choice: weight matrix same as in one step Normal equations Two connected non-overlapping networks – Two step solution Combination step (second step)

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Normal equations Recall that Two connected non-overlapping networks – Two step solution special choice: weight matrix same as in one step Combination step (second step)

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Normal equations Recall that vanishing terms Two connected non-overlapping networks – Two step solution special choice: weight matrix same as in one step Combination step (second step)

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Normal equations Two connected non-overlapping networks – Two step solution special choice: weight matrix same as in one step Combination step (second step)

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Normal equations Two connected non-overlapping networks – Two step solution special choice: weight matrix same as in one step Combination step (second step)

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Normal equations Two connected non-overlapping networks – Two step solution special choice: weight matrix same as in one step Combination step (second step) No equations containing the transformation parameters p a, p b They cannot be determined!

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Normal equations Identical to those of the one-step solution Two connected non-overlapping networks – Two step solution special choice: weight matrix same as in one step Combination step (second step)

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Two-step solution combination step model Two non-overlapping networks connected by observations One-step solution equivalent model same weight matrix from separate solutions Identical solution for x 1, x 2, x 3, x 4

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Two-step solution combination step model Two non-overlapping networks connected by observations One-step solution equivalent model same weight matrix from separate solutions Identical solution for x 1, x 2, x 3, x 4 “wrong” model ignores dependence of separate and final solutions on different reference systems

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Two-step solution combination step model Two non-overlapping networks connected by observations One-step solution equivalent model same weight matrix from separate solutions Identical solution for x 1, x 2, x 3, x 4 “wrong” model ignores dependence of separate and final solutions on different reference systems correct model acknowledges dependence of separate and final solutions on different reference systems

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Two-step solution combination step model Two non-overlapping networks connected by observations One-step solution equivalent model same weight matrix from separate solutions Identical solution for x 1, x 2, x 3, x 4 “wrong” model ignores dependence of separate and final solutions on different reference systems correct model acknowledges dependence of separate and final solutions on different reference systems Nevertheless transformation Parameters p a, p b cannot be determined!

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Two-step solution combination step model Two non-overlapping networks connected by observations One-step solution equivalent model Why these two different models lead to equivalent results ?

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Two-step solution combination step model Two non-overlapping networks connected by observations One-step solution equivalent model The particular choice of weight matrix “kills” the dependence on the reference systems ! Normal equations depend only on design ( A ) and observations ( b ) ! Why these two different models lead to equivalent results ?

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The choice of weight problem In the two step approach, the first step produces singular covariance matrices. Therefore the weight matrices to be used in the second step are not uniquely defined. This fact raises some questions about the choice of weight matrices:

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Are the weight matrices used in order to obtain equivalent results with the one step approach the correct ones? (Do they lead to BLUUE estimates?) The choice of weight problem In the two step approach, the first step produces singular covariance matrices. Therefore the weight matrices to be used in the second step are not uniquely defined. This fact raises some questions about the choice of weight matrices: Yes !

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Are the weight matrices used in order to obtain equivalent results with the one step approach the correct ones? (Do they lead to BLUUE estimates?) The choice of weight problem Are there other correct weight matrices? In the two step approach, the first step produces singular covariance matrices. Therefore the weight matrices to be used in the second step are not uniquely defined. This fact raises some questions about the choice of weight matrices: Yes !

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Are the weight matrices used in order to obtain equivalent results with the one step approach the correct ones? (Do they lead to BLUUE estimates?) The choice of weight problem Are there other correct weight matrices? Do different correct weight matrices lead to the same results? (i.e. to estimates that are connected by a change of the reference system) Yes ! In the two step approach, the first step produces singular covariance matrices. Therefore the weight matrices to be used in the second step are not uniquely defined. This fact raises some questions about the choice of weight matrices:

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Rao’s Unified Theory for singular covariance matrix and rank deficient design matrix The choice of weight problem Weight matrix to use: V = symmetric matrix such that:

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Rao’s Unified Theory for singular covariance matrix and rank deficient design matrix The choice of weight problem V = symmetric matrix such that: Our case: Combination step Different weight matrices: from different choices of V and of the generalized inverse with V satisfying: Weight matrix to use:

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Conclusions Under the Gauss-Markov assumptions (zero mean noise, single unknown reference variance) Both the one-step and the two-step approaches give equivalent results when the used weight matrices are the normal equation matrices from the separate solutions. The inclusion of reference frame transformation parameters is meaningless in this case. The combination step must be modified to the addition of partial normal equations.

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Conclusions Under the Gauss-Markov assumptions (zero mean noise, single unknown reference variance) Both the one-step and the two-step approaches give equivalent results when the used weight matrices are the normal equation matrices from the separate solutions. The inclusion of reference frame transformation parameters is meaningless in this case. The combination step must be modified to the addition of partial normal equations. Beyond the Gauss-Markov assumptions (biases, different variance components) A future study of the effect of biases and the variance component estimation in the two alternative formulations is required. Different “correct” weight matrices in combination step A further study of the Rao’s unified theory as applies in our specific problem. Characterization of the whole class weight matrices giving the “same” solution. Suggestions for further study

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