1 establishing global reference frames nonlinar, temporal, geophysical and stochastic aspects...
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Establishing Global Reference FramesEstablishing Global Reference FramesNonlinar, Temporal, Geophysical and Stochastic AspectsNonlinar, Temporal, Geophysical and Stochastic Aspects
Athanasios DermanisAthanasios Dermanis
Department of Geodesy and SurveyingDepartment of Geodesy and SurveyingThe Aristotle University of ThessalonikiThe Aristotle University of Thessaloniki
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ISSUES:ISSUES:
• from space to space-time frame definitionfrom space to space-time frame definition
• alternatives in optimal frame definitions alternatives in optimal frame definitions (Meissl meets Tisserant)(Meissl meets Tisserant)
• discrete networks and continuous earth discrete networks and continuous earth (geodetic and geophysical frames)(geodetic and geophysical frames)
• from deterministic to stochastic frames from deterministic to stochastic frames
(combination of “estimated” networks)(combination of “estimated” networks)
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The (instantaneous) shape manifold The (instantaneous) shape manifold SS
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
bxθRx kk )(λ
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)λ,,,( dθzχx
The geometry of the shape manifoldThe geometry of the shape manifold SS
bzθRx kk )(λ
λ,
b
b
b
,
θ
θ
θ
3
2
1
3
2
1
bθ
Dimension: 7 or 6 (fixed scale) or 3 (geocentric)
Curvilinear coordinates = transformation parameters:
Local Basis:
Local Tangent Space:
ii q
xe E
q
xeee
][ 621 inner constraint
matrix of Meissl !
Exexex ofspacecolumn})(,),({span 61 TS
T321321 ]λbbbθθθ[q
transformationparameters:
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Deformable networks: Deformable networks: the shape-time manifold the shape-time manifold MM
Coordinates:
t
d
θ
p
)(χ
)(),(χ)(
t
ttt
t x
pxx
t
tSM
Optimal Reference Frame: one with minimal length = geodesic !
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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 !
““Geodesic” reference framesGeodesic” reference frames
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Alternative solutions: Meissl and Tisserand reference frames
Meissl Frame:Meissl Frame:
tTSt tt )())(( xx
Generalization of
000 min|| xxxxxx TS
to
)(0
)()(lim)( tt
TSt
tttt xv
xxv
Compare to discrete-time approach:
))()((,0)1(,min|)()1(||)1(| )( kTi tkiiiix xxxExx x
0x
xh
N
kk
kkRF m
t1
][Tisserand Frame:Tisserand Frame:
Vanishing relative angular momentum of network (point masses)
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General ResultsGeneral ResultsAssuming same initial coordinates x0 = x(t0),introducing point masses (weights) mi (special case mi = 1) :
1. Meissl frame = minimal geodesic frameMeissl frame = minimal geodesic frame (Dermanis, 1995)
2. Tisserand frame (Tisserand frame (mmii=1) = Meissl frame=1) = Meissl frame (Dermanis, 1999)
Metric in Network Coordinate Space E3N:
N
kkkEk
N
kkkkkkkk dmzzyyxxm
1
2
1
222 ),(])()()[( xx
N
kkk
Tkkk
T md1
2 )()()()(),( xxxxxxMxxxx
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(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 :
const)()(,)()()()( 011 tt
dt
dtt
dt
dzz bb0
bhCθRθΩ
θ
Realization of solutionRealization of solution
Ti k
RRωωωωΩ )(][,][ 321
dt
dmm k
N
kkkz
N
k
Tkkk
Tkkz
zzhzzIzzC
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][,])[(
)()()()( tttt kk bzθRx
Where:
(matrix of inertia & angular momentum vector of the network)
)(tx
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Network Reference Frame (Geodesy)versus
Earth Reference Frame (Geophysics)
GeophysicsGeophysics: Definition of RF by simplification of Liouville equations -- Reference Frame theoretically imposed
Choices: Axes of inertia (large diurnal variation!)
Tisserant axes (indispensable):
GeodesyGeodesy: Network Meissl-Tisserant axes:
At best (global dense network): a good approximation of
Earth surface ( E) Tisserant axes: Insufficient for geophysical connection !
0][ E
E ddt
dx
xxh
0][, k
kk
kNR mdt
dxxh
0][
E
E dSdt
dxxh with
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Link of geodetic and geophysical Reference FramesLink 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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The statistics of shapesThe statistics of shapes
Given: Network coordinate estimates
Problem: Separate position from shape - estimate optimal shape
),(~ˆ Cxx
fromshape = manifold
toshape = 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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Local - Linear Local - Linear (linearized)(linearized) Approach Approach
mNX 3dim
)( dm
q
xE
dmr
:)( rmG 0]det[ GE
GsEqs
qGEx
xGE
s
q 1
Linearization: )(),( EE RCRM
q “position” (transformation parameters) (d x 1) s “shape” (r x 1)
Do “intrinsic” statistics in R(G) by: ]),ˆ(&[),ˆ(),ˆ( ˆˆˆ qsx CqCsCx
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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.