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Establishing Global Reference Frames Nonlinar, Temporal, Geophysical and Stochastic Aspects

Establishing Global Reference Frames Nonlinar, Temporal, Geophysical and Stochastic Aspects. Athanasios Dermanis Department of Geodesy and Surveying The Aristotle University of Thessaloniki. ISSUES:. from space to space-time frame definition

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Establishing Global Reference Frames Nonlinar, Temporal, Geophysical and Stochastic Aspects

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

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

  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

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

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

  6. “Geodesic” reference frames Geodesic of minimum length from S0 to SF: perpendicular to both. Problem: all minimal geodesics are “parallel” (p(t) =const.) = have same length Solution: Must fix x0 arbitrarily !

  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)

  8. General Results Assuming same initial coordinates x0 = x(t0), introducing point masses (weights)mi(special casemi = 1) : 1.Meissl frame = minimal geodesic frame(Dermanis, 1995) 2.Tisserand frame (mi=1) = Meissl frame(Dermanis, 1999) Metric in Network Coordinate Space E3N:

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

  10. Network Reference Frame (Geodesy) versus Earth Reference Frame (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: Network Meissl-Tisserant axes: At best (global dense network): a good approximation of with Earth surface ( E) Tisserant axes: Insufficient for geophysical connection !

  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

  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 fromX = R 3Nto sectionC Find coordinates system forC Do statistics intrinsically inC(non-linear !)

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

  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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