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The ITRF Beyond the “Linear” Model Choices and Challenges Athanasius Dermanis Department of Geodesy and Surveying - Aristotle University of Thessaloniki FELIX QUI POTUIT RERUM COGNOSCERE CAUSAS VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy May 29 - June 2, 2006, Wuhan, China

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such that the apparent motion of the body masses (or network points), as seen with respect to the reference system, is minimized A reference system for a deformable body or point network: A time-wise smooth choice at every epoch t of (a)a point - the origin, O(t) (b)three directed straight lines and a unit of length - the vectorial basis e 1 (t), e 2 (t), e 3 (t) The reference system separates the total motion with respect to the inertial background into: (a)the translational motion and the rotation of the body/network as represented by the selected reference system (b)the remaining “deformation”

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Therefore the optimality criterion must be realized by means of a set of mathematical conditions on the coordinate model parameters: F k (a 1, a 2, …, a N ) = 0,k = 1, 2, …, L The optimal choice of the reference system requires the introduction of an optimality criterion = a measure of the “deformation” to be minimized The optimality criterion should be applied on the realization of the reference system by a set of point coordinates expressed as functions of time for a selected global terrestrial network: The International Terrestrial Reference Frame (ITRF): x i (a i, t), i = 1, 2, …, N making use of model parameters a i [e.g. x i (t 0 ), v i - presently] Plus: arbitrary choice among one of dynamically equivalent reference systems ( x i ~ x i ' x i ' = R x i + d,R, d = constant )

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach origin = geocenter axes = Tisserand axes vanishing relative angular momentum = = minimal relative kinetic energy Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach origin = geocenter axes = Tisserand axes origin: constant network barycenter axes = discrete Tisserand principle vanishing relative angular momentum = = minimal relative kinetic energy vanishing discrete relative angular momentum network points = treated as (unit) mass points Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach origin = geocenter axes = Tisserand axes origin: constant network barycenter axes = discrete Tisserand principle vanishing relative angular momentum = = minimal relative kinetic energy vanishing discrete relative angular momentum network points = treated as (unit) mass points or a combination of the two approaches: one for the geocenter, the other for the axes Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach ADVANTAGES Physical meaning! Compatibility with theories of - orbit of the earth (translational motion of geocenter) & - earth rotation (rotation of the Tisserand axes) Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach ADVANTAGES DISADVANTAGES Physical meaning! Compatibility with theories of - orbit of the earth (translational motion of geocenter) & - earth rotation (rotation of the Tisserand axes) No physical meaning! Lack of compatibility with reference systems implicitly defined in theories of earth orbital motion and earth rotation (additional discrepancies between theory and observations due to different reference system definitions) Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach ADVANTAGES Coordinates suffer only from errors in estimating the network shape! No additional errors arising from uncertainty in the position of the origin and axes with respect to the network! Pure geodetic-positional approach free from geophysical hypotheses! Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach DISADVANTAGES ADVANTAGES Coordinates suffer also from errors in estimating the position of the origin and axes with respect to the network! Geocenter = mean position of centers of oscillating ellipses of satellite orbits. Estimated position of Tisserand axes heavily depends on geophysical assumptions about density and motion of internal earth masses Coordinates suffer only from errors in estimating the network shape! No additional errors arising from uncertainty in the position of the origin and axes with respect to the network! Pure geodetic-positional approach free from geophysical hypotheses! Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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DISADVANTAGES ADVANTAGES Coordinates suffer also from errors in estimating the position of the origin and axes with respect to the network! Geocenter = mean position of centers of oscillating ellipses of satellite orbits. Estimated position of Tisserand axes heavily depends on geophysical assumptions about density and motion of internal earth masses Coordinates suffer only from errors in estimating the network shape! No additional errors arising from uncertainty in the position of the origin and axes with respect to the network! Pure geodetic-positional approach free from geophysical hypotheses! See: Dermanis (2006) EGU Vienna Two possible approaches to the reference system choice for the ITRF Mathematical approach Physical approach Introduce origin and axes in a purely mathematical way by minimizing the apparent motion of the ITRF network points with respect to the reference system Introduce origin and axes in a physical way by accessing a point and lines depending on the physical constitution of the earth (though still mathematically defined)

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Which approach to use ?My suggestion: Both! FIRST: Establish an ITRF by purely geodetic-“positional” means with origin and axes defined mathematically by optimality conditions (constant position of network barycenter - – vanishing discrete relative angular momentum of “unit mass” stations) FIRST: Establish an ITRF by purely geodetic-“positional” means with origin and axes defined mathematically by optimality conditions (constant position of network barycenter - – vanishing discrete relative angular momentum of “unit mass” stations) ITRF coordinates (functions of time) reflect only network shape and its temporal variation (deformation), free from additional positional uncertainties ITRF coordinates (functions of time) reflect only network shape and its temporal variation (deformation), free from additional positional uncertainties

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Which approach to use ?My suggestion: Both! FIRST: Establish an ITRF by purely geodetic-“positional” means with origin and axes defined mathematically by optimality conditions (constant position of network barycenter – – vanishing discrete relative angular momentum of “unit mass” stations) FIRST: Establish an ITRF by purely geodetic-“positional” means with origin and axes defined mathematically by optimality conditions (constant position of network barycenter – – vanishing discrete relative angular momentum of “unit mass” stations) ITRF coordinates (functions of time) reflect only network shape and its temporal variation (deformation), free from additional positional uncertainties ITRF coordinates (functions of time) reflect only network shape and its temporal variation (deformation), free from additional positional uncertainties Transform the ITRF into an “earth reference frame” for comparison with earth rotation and other geophysical processes, while understanding the influence of additional estimation errors and accepted geophysical hypotheses. Transform the ITRF into an “earth reference frame” for comparison with earth rotation and other geophysical processes, while understanding the influence of additional estimation errors and accepted geophysical hypotheses. SECOND: Use geodetic satellite observations to estimate the time-varying geocenter position with respect to the ITRF. Use best available geophysical hypotheses about unobservable internal (subsurface) earth composition and motions to estimate the time-varying position of the Tisserand axes. SECOND: Use geodetic satellite observations to estimate the time-varying geocenter position with respect to the ITRF. Use best available geophysical hypotheses about unobservable internal (subsurface) earth composition and motions to estimate the time-varying position of the Tisserand axes.

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Coordinates linear functions of time: About the ITRF linear model Is there a linear network deformation model? No! Transformation to another equally legitimate reference system destroys linearity!

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Coordinates linear functions of time: About the ITRF linear model Is there a linear network deformation model? No! Transformation to another equally legitimate reference system destroys linearity! Same holds for spectral analysis (Fourier series model): There exists no Fourier analysis independent from the choice of reference system! Frequency components appearing in one coordinate system are different from those in another one, where also the contributions of frequencies in R(t) and d(t) are present!

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However: If x(t) and x'(t) both satisfy the discrete Tisserand conditions (constant barycenter, zero discrete relative angular momentum) Ad hoc definition of a “linear deformation model”: We say that a point network or body deform in a linear way if the coordinates of any point with respect to any Tisserand reference system are linear functions of time Ad hoc definition of a “linear deformation model”: We say that a point network or body deform in a linear way if the coordinates of any point with respect to any Tisserand reference system are linear functions of time Linearity is preserved!

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However: If x(t) and x'(t) both satisfy the discrete Tisserand conditions (constant barycenter, zero discrete relative angular momentum) Ad hoc definition of a “linear deformation model”: We say that a point network or body deform in a linear way if the coordinates of any point with respect to any Tisserand reference system are linear functions of time Ad hoc definition of a “linear deformation model”: We say that a point network or body deform in a linear way if the coordinates of any point with respect to any Tisserand reference system are linear functions of time In a similar way we may speak about the spectral analysis of the deformation of a network or body, meaning the spectral analysis of the coordinate functions of any point with respect to any Tisserand reference system Linearity is preserved!

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Beyond the “Linear” Model Extended time period of ITRF relevance and particular station behavior necessitate richer time-evolution models First choices Polynomials: Fourier series:

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A general class of models Linear combinations of base functions: { k } = a class of functions: closed under differentiation: closed under multiplication:

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The discrete Tisserand conditions (a) Vanishing (discrete) “relative angular momentum”: network points treated as (unit) mass points

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The discrete Tisserand conditions (a) Vanishing (discrete) “relative angular momentum”: network points treated as (unit) mass points

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The discrete Tisserand conditions (a) Vanishing (discrete) “relative angular momentum”: network points treated as (unit) mass points

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The discrete Tisserand conditions (a) Vanishing (discrete) “relative angular momentum”: network points treated as (unit) mass points axes orientation conditions

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The discrete Tisserand conditions (b) Constant (discrete) barycenter:

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The discrete Tisserand conditions (b) Constant (discrete) barycenter:

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The discrete Tisserand conditions (b) Constant (discrete) barycenter:

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The discrete Tisserand conditions (b) Constant (discrete) barycenter: origin conditions

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Example 1: Polynomials Origin conditions:

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Example 1: Polynomials Orientation conditions: Origin conditions:

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Example 2: Fourier series Origin conditions:

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Example 2: Fourier series Orientation conditions: Origin conditions:

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Use of trigonometric identities:

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Orientation conditions: No condition produced!

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How to implement the optimality conditions? A-priori at the level of data analysis For ITRF construction A-priori at the level of data analysis For ITRF construction A-posteriori by transformation from an arbitrary to the optimal system A-posteriori by transformation from an arbitrary to the optimal system Requires single epoch data x i (t k ) (daily or weekly solutions). The data enter free of reference system as shapes of sub-networks. They are rotated & translated to the optimal reference system by applying the optimality constraints.

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Analysis is performed in any convenient reference system and coordinates are then converted to the optimal one by solving for appropriate rotation R(t) and translation d(t), such that the optimality conditions are satisfied How to implement the optimality conditions? A-priori at the level of data analysis For ITRF construction A-priori at the level of data analysis For ITRF construction A-posteriori by transformation from an arbitrary to the optimal system A-posteriori by transformation from an arbitrary to the optimal system Requires single epoch data x i (t k ) (daily or weekly solutions). The data enter free of reference system as shapes of sub-networks. They are rotated & translated to the optimal reference system by applying the optimality constraints.

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Analysis is performed in any convenient reference system and coordinates are then converted to the optimal one by solving for appropriate rotation R(t) and translation d(t), such that the optimality conditions are satisfied How to implement the optimality conditions? A-priori at the level of data analysis For ITRF construction A-priori at the level of data analysis For ITRF construction A-posteriori by transformation from an arbitrary to the optimal system A-posteriori by transformation from an arbitrary to the optimal system Requires single epoch data x i (t k ) (daily or weekly solutions). The data enter free of reference system as shapes of sub-networks. They are rotated & translated to the optimal reference system by applying the optimality constraints. DISADVANTAGES - Advantages

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Analysis is performed in any convenient reference system and coordinates are then converted to the optimal one by solving for appropriate rotation R(t) and translation d(t), such that the optimality conditions are satisfied How to implement the optimality conditions? A-priori at the level of data analysis For ITRF construction A-priori at the level of data analysis For ITRF construction A-posteriori by transformation from an arbitrary to the optimal system A-posteriori by transformation from an arbitrary to the optimal system Only single-epoch data acceptable. It is not possible to introduce time-evolution models at the preprocessing stage (at the data analysis centers dealing with regional sub-networks). Significant sub-network overlap is required to secure successful “patchwork” into a single global shape, before introducing the optimal reference system. Requires single epoch data x i (t k ) (daily or weekly solutions). The data enter free of reference system as shapes of sub-networks. They are rotated & translated to the optimal reference system by applying the optimality constraints. DISADVANTAGES - Advantages

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Time-evolution models can be introduced during the preprocessing level at the analysis centers of regional sub-networks. Optimality can be only approximately achieved, because preservation of the time evolution model requires restriction of unknown rotation R(t) and translation d(t) to the same model class (linear combinations of the same base functions) Analysis is performed in any convenient reference system and coordinates are then converted to the optimal one by solving for appropriate rotation R(t) and translation d(t), such that the optimality conditions are satisfied How to implement the optimality conditions? A-priori at the level of data analysis For ITRF construction A-priori at the level of data analysis For ITRF construction A-posteriori by transformation from an arbitrary to the optimal system A-posteriori by transformation from an arbitrary to the optimal system Only single-epoch data acceptable. It is not possible to introduce time-evolution models at the preprocessing stage (at the data analysis centers dealing with regional sub-networks). Significant sub-network overlap is required to secure successful “patchwork” into a single global shape, before introducing the optimal reference system. Requires single epoch data x i (t k ) (daily or weekly solutions). The data enter free of reference system as shapes of sub-networks. They are rotated & translated to the optimal reference system by applying the optimality constraints. DISADVANTAGES - Advantages

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The ultimate problem: Use observed deformation or a “smoothed” version? The ultimate problem: Use observed deformation or a “smoothed” version? Spectral analysis is typically used for isolating estimation errors (noise) from signal: High frequencies are attributed to noise and are removed. Low frequencies are retained as signal. Middle frequencies? Are middle frequencies in network deformation real ?

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The ultimate problem: Use observed deformation or a “smoothed” version? The ultimate problem: Use observed deformation or a “smoothed” version? Spectral analysis is typically used for isolating estimation errors (noise) from signal: High frequencies are attributed to noise and are removed. Low frequencies are retained as signal. Middle frequencies? Are middle frequencies in network deformation real ? Difficult to answer ! Systematic errors in space-techniques have atmospheric origins with spectral characteristics related to the annual cycle. Real middle frequency deformations should have origins with the similar spectral characteristics. Hard or even impossible to distinguish! Future hope: Better monitoring of the atmosphere and effective removal of systematic errors.

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The ultimate problem: Use observed deformation or a “smoothed” version? The ultimate problem: Use observed deformation or a “smoothed” version? Spectral analysis is typically used for isolating estimation errors (noise) from signal: High frequencies are attributed to noise and are removed. Low frequencies are retained as signal. Middle frequencies? Are middle frequencies in network deformation real ? Even if middle frequencies are real should they be retained in the ITRF model ?

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The ultimate problem: Use observed deformation or a “smoothed” version? The ultimate problem: Use observed deformation or a “smoothed” version? Spectral analysis is typically used for isolating estimation errors (noise) from signal: High frequencies are attributed to noise and are removed. Low frequencies are retained as signal. Middle frequencies? Are middle frequencies in network deformation real ? Even if middle frequencies are real should they be retained in the ITRF model ? Answer depends on the specific use of the ITRF. Note:Earth-tide deformations are already removed at the data preprocessing level. Other periodic components may be removed to produce an ITRF version reflecting only secular deformation (compare: UT2 versus UT1). All removed frequencies must be restored before any comparison with actual geophysical observations. Discontinuous episodic deformations must be modeled by step functions and removed from the final solution.

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