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

THE ITRF FORMULATION PROBLEM 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 t

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

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

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

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

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

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

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

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

THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM 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 t

THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM 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 t

THE COORDINATE-FREE APPROACH (ALMOST) TO THE ITRF FORMULATION PROBLEM currently: 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 t

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

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

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

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

The (general) model: Point P i coordinates: a i = point P i parameters The current model: THE MODEL FOR TIME EVOLUTION OF COORDINATES

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

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 MAIN ITRF FORMULATION PROBLEM Assign a reference system for each epoch INTERPOLATION Provides shapes expressed by coordinates for epochs other than observation ones 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

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

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 constraints in 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

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 constraints in 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!)

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

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

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

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

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

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

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

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

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

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

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

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 !

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

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

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

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

INNER CONSTRAINTS FOR THE STACKING PROBLEM Change of ITRF reference system

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

INNER CONSTRAINTS FOR THE STACKING PROBLEM Partial inner constraints – Coordinates & velocities initial orientation initial translation initial scale orientation rate translation rate scale rate

INNER CONSTRAINTS FOR THE STACKING PROBLEM Partial inner constraints – Transformation parameters initial orientation initial translation initial scale orientation rate translation rate scale rate

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

INNER CONSTRAINTS FOR THE COMBINATION PROBLEM Change of ITRF reference system

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

INNER CONSTRAINTS FOR THE COMBINATION PROBLEM initial orientation initial translation initial scale orientation rate translation rate scale rate Partial inner constraints – Coordinates & velocities

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

INNER CONSTRAINTS FOR THE COMBINATION PROBLEM initial orientation initial translation initial scale orientation rate translation rate scale rate Partial inner constraints – Transformation parameters

THE KINEMATIC APPROACH Translation Orientation Scale Establish a reference system in such a way that the apparent motion of network points (variation of their coordinates) is minimized with respect to:

THE KINEMATIC APPROACH Establish a reference system in such a way that the apparent motion of network points (variation of their coordinates) is minimized with respect to: Translation: The network barycenter does not move Scale Orientation

THE KINEMATIC APPROACH Establish a reference system in such a way that the apparent motion of network points (variation of their coordinates) is minimized with respect to: Translation: The network barycenter does not move Orientation:The relative kinematic energy is minimized = = the relative angular momentum vanishes Scale

THE KINEMATIC APPROACH Establish a reference system in such a way that the apparent motion of network points (variation of their coordinates) is minimized with respect to: Translation: The network barycenter does not move Orientation:The relative kinematic energy is minimized = = the relative angular momentum vanishes Scale:The network mean quadratic scale remains constant

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Initial translation: Initial orientation: Initial scale: Translation rate: Orientation rate: Scale rate: NOT available (to be borrowed from the algebraic approach)

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: Under the choice Initial translation: Initial orientation: Initial scale: NOT available (to be borrowed from the algebraic approach)

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: Under the choice Initial translation: Initial orientation: Initial scale: NOT available (to be borrowed from the algebraic approach)

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: Under the choice Initial translation: Initial orientation: Initial scale: NOT available (to be borrowed from the algebraic approach)

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: Under the choice Initial translation: Initial orientation: Initial scale: NOT available (to be borrowed from the algebraic approach)

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: NOT available (to be borrowed from the algebraic approach) Under the choice Initial translation: Initial orientation: Initial scale:

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: Under the choice Initial translation: Initial orientation: Initial scale: NOT available (to be borrowed from the algebraic approach)

MINIMAL CONSTRAINTS IN THE KINEMATIC APPROACH Translation rate: Orientation rate: Scale rate: Under the choice Same as the partial inner constraints of the algebraic approach ! Initial translation: Initial orientation: Initial scale: NOT available (to be borrowed from the algebraic approach)

SUMMARY AND CONCLUSIONS MODEL for smooth shape variation (removal of data noise) OPTIMALITY CRITERION Best reference system among all equivalent ones connected by arbitrary transformations INCONCISTENT

SUMMARY AND CONCLUSIONS MODEL for smooth shape variation (removal of data noise) OPTIMALITY CRITERION Best reference system among all equivalent ones connected by arbitrary transformations INCONCISTENT

SUMMARY AND CONCLUSIONS MODEL for smooth shape variation (removal of data noise) OPTIMALITY CRITERION Best reference system among all equivalent ones connected by approximate transformations INCONCISTENT

SUMMARY AND CONCLUSIONS MODEL for smooth shape variation (removal of data noise) OPTIMALITY CRITERION Best reference system among all equivalent ones connected by approximate transformations CONCISTENT which preserve the model

SUMMARY AND CONCLUSIONS MODEL for smooth shape variation (removal of data noise) OPTIMALITY CRITERION Best reference system among all equivalent ones connected by approximate transformations CONCISTENT which preserve the model SUB-OPTIMALITY

SUMMARY AND CONCLUSIONS SUB-OPTIMAL REFERENCE SYSTEM (close to the identity, model preserving transformations) BY USING MINIMAL CONSTRAINTS

SUMMARY AND CONCLUSIONS SUB-OPTIMAL REFERENCE SYSTEM (close to the identity, model preserving transformations) BY USING MINIMAL CONSTRAINTS ALGEBRAIC APPROACH Minimization of parameter sum of squares  PARTIAL INNER CONSTRAINTS KINEMATIC APPROACH Minimization of apparent coordinate variation  MINIMAL CONSTRAINTS

SUMMARY AND CONCLUSIONS SUB-OPTIMAL REFERENCE SYSTEM (close to the identity, model preserving transformations) BY USING MINIMAL CONSTRAINTS ALGEBRAIC APPROACH Minimization of parameter sum of squares  PARTIAL INNER CONSTRAINTS IDENTICAL RESULTS under proper choice of approximate values KINEMATIC APPROACH Minimization of apparent coordinate variation  MINIMAL CONSTRAINTS

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