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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.