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1 Establishing Global Reference Frames Nonlinar, Temporal, Geophysical and Stochastic Aspects Athanasios Dermanis Department of Geodesy and Surveying The Aristotle University of Thessaloniki

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2 ISSUES: from space to space-time frame definitionfrom space to space-time frame definition alternatives in optimal frame definitions (Meissl meets Tisserant)alternatives in optimal frame definitions (Meissl meets Tisserant) discrete networks and continuous earth (geodetic and geophysical frames)discrete networks and continuous earth (geodetic and geophysical frames) from deterministic to stochastic frames (combination of “estimated” networks)from deterministic to stochastic frames (combination of “estimated” networks)

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3 The (instantaneous) shape manifold S S = all networks with the same shape = same network in different placements w.r. to reference frame = different placements of reference frame w.r. to the network

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4 The geometry of the shape manifold S Dimension: 7 or 6 (fixed scale) or 3 (geocentric) Curvilinear coordinates = transformation parameters : Local Basis: Local Tangent Space: inner constraint matrix of Meissl ! transformation parameters:

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5 Deformable networks: the shape-time manifold M Coordinates: Optimal Reference Frame: one with minimal length = geodesic !

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6 Geodesic of minimum length from S 0 to S F : perpendicular to both. Problem: all minimal geodesics are “parallel” (p(t) =const.) = have same length Solution: Must fix x 0 arbitrarily ! “Geodesic” reference frames

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7 Alternative solutions: Meissl and Tisserand reference frames Meissl Frame: Generalization of to Compare to discrete-time approach: Tisserand Frame: Vanishing relative angular momentum of network (point masses)

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8 General Results Assuming same initial coordinates x 0 = x(t 0 ), introducing point masses (weights) m i (special case m i = 1 ) : Meissl frame = minimal geodesic frame 1. Meissl frame = minimal geodesic frame (Dermanis, 1995) Tisserand frame (m i =1) = Meissl frame 2. Tisserand frame (m i =1) = Meissl frame (Dermanis, 1999) Metric in Network Coordinate Space E 3N :

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9 (a)Compute any (minimal) “reference” solution z(t): discrete (but dense) arbitrary solution, smoothing interpolation. (b)Find transformation parameters (t), b(t) by solving: (c)Transform to optimal (Meissl-Tisserant) solution : Realization of solution Where: (matrix of inertia & angular momentum vector of the network)

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10 Network Reference Frame (Geodesy) versus Earth Reference Frame (Geophysics) Geophysics Geophysics : Definition of RF by simplification of Liouville equations - - Reference Frame theoretically imposed Choices:Axes of inertia(large diurnal variation!) Tisserant axes(indispensable): Geodesy Geodesy:Network Meissl-Tisserant axes: At best (global dense network): a good approximation of Earth surface ( E) Tisserant axes: Insufficient for geophysical connection ! with

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11 Link of geodetic and geophysical Reference Frames Need:For comparison of theory with observation. Solution:Introduce geophysical hypotheses in the geodetic RF. Example:Plate tectonics Establish a common global network frame Establish a separate frame for each plate Detect “outlier” stations (local deformations) and remove Compute angular momentum change due to each plate motion Determine transformation so that total angular momentum change vanishes Transform to new global frame (approximation to Earth Tisserant Frame) Requirement: density knowledge Improvement: Introduce model for earth core contribution to angular momentum

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12 The statistics of shapes Given: Network coordinate estimates Problem: Separate position from shape - estimate optimal shape from shape = manifold to shape = point Get marginal distribution from X = R 3N to section C Find coordinates system for C Do statistics intrinsically in C (non-linear !)

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13 Local - Linear (linearized) Approach Linearization: q “position” (transformation parameters) (d x 1) s “shape” (r x 1) Do “intrinsic” statistics in R(G) by:

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14 CONCLUSIONS - We need: (a) Global geodetic network (ITRF) - for “positioning” Few fundamental stations (collocated various observations techniques). Frame choice principle for continuous coordinate functions x(t). A discrete realization of the principle. Removal of periodic variations. Specific techniques for optimal combination of shape estimates. Separate estimation of geocenter and rotation axis position. (a) Modified earth network - link with geophysical theories Large number of well-distributed stations (mainly GPS). Implementation of geophysical hypotheses for choice of optimal frame. (Plate tectonic motions, Tisserant frame). Inclusion of periodic variations present in theory of rotation deformation.

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