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

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 manifold S
transformation parameters
Curvilinear coordinates transformation
parameters
Dimension 7 or 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 case mi 1)
1. Meissl frame minimal geodesic frame
(Dermanis, 1995)
2. Tisserand frame (mi1) Meissl frame
(Dermanis, 1999)
Metric in Network Coordinate Space E3N
9
Realization of solution
(a) Compute any (minimal) reference solution
z(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 from X R 3N to
section C Find coordinates system for C Do
statistics intrinsically in C (non-linear !)
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
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.