Reference Systems: Definition and Realization

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Reference Systems Definition and Realization

• Zuheir ALTAMIMI
• Laboratoire de Recherche en Géodésie
• Institut Géographique National
• France
• E-mail altamimi_at_ensg.ign.fr

AFREF Technical Workshop, University of Cape
Town, July 09-13 , 2006
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OUTLINE

• Concept and Definition Terrestrial Reference
  Systems (TRS)
• TRS Realization by a Frame (TRF)
• Transformation between TRS/TRF
• Coordinate Systems
• Map Projections
• Implementation of TRF in Space geodesy
• Datum Definition

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Geodesy

• Etymologically comes from Greek  geôdaisia 
   dividing the Earth 
• Study of the form, dimensions, rotation and
  gravity field of the Earth
• Main geodesy activity determination of
  point/object positions over the Earth surface or
  near-by space
• There is a need for a Terrestrial Reference
  System and a Coordinate System

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Reference Systems Terminology

• Ideal Reference System theoretical definition
  (not accessible)
• Conventional Reference System set of
  conventions, algorithms, constants used to
  determine object positions in an IRS
• Conventional Reference Frame
• - Set of physical objects with their coordinates
• - Realization of an Ideal Reference System
• Coordinate System cartesian (X,Y,Z), geographic
  (l, f, h),...

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Ideal Terrestrial Reference System
A tridimensional reference frame (mathematical
sense) Defined in an Euclidian affine space of
dimension 3 Affine Frame (O,E) where O point
in space (Origin) E vector base orthogonal with
the same length - unit vectors co-linear to the
base (Orientation) - unit of length (Scale)
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Affine Frame

• Origin
• Barycentric (Center of Mass of the solar system)
• Geocentric CoM of the Earth
• Orientation
• Ecliptic
• Equatorial
• Unit of length (Scale) Same norm for the 3
  vectors

Z
P
k
j
i
o
Y
X
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Ideal Terrestrial Reference System in the Context
of Space Geodesy

• Origin Geocentric Earth Center of Mass
• Scale SI Unit
• Orientation Equatorial (Z axis is the direction
  of the Earth pole)

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Coordinate Systems

• 3D
• Cartesian X, Y, Z
• Ellipsoidal l, j, h
• Mapping E, N, h
• Spherical R, q, l
• Cylindrical l, l, Z
• 2D
• Geographic l, j
• Mapping E, N
• 1D Height system H

Z
P
R
z
q
o
Y
l
l
X
Rcosq cosl OP Rcosq sinl Rsinq
l cosl OP l sinl z
Spherical
Cylindrical
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General Transformation Diagram
TRS-1
TRS-2
Cartesian X, Y, Z
Cartesian X, Y, Z
7-Parameter Similarity
Ellipsoidal l, j , h
Ellipsoidal l, j , h
Molodensky Transformation
H
H
Mapping E, N, h
Mapping E, N, h
H
H
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Ellipsoidal and Cartesian CoordinatesEllipsoid
definition
a semi major axis b semi minor axis f
flattening e eccentricity
b
a
(a,b), (a,f ), or (a,e2) define entirely
and geometrically the ellipsoid
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Ellipsoidal and Cartesian Coordinates
Z
Zero meridian
Local meridian
h
b
j
o
a
Y
l
equator
X
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(X, Y, Z) (l,j,h)
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Map Projection
Mathematical function from an ellipsoid to a
plane (map)
E f(l,j) N g(l,j)
Z
P
h
N
f g
p
P0
E
j
Y
o
l
Mapping coordinates (E,N,h)
X
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Space Geodesy Techniques

• Very Long Baseline Interferometry (VLBI)
• Lunar Laser Ranging (LLR)
• Satellite Laser Ranging (SLR)
• DORIS
• Global Positioning System (GPS)
• Others (PRARE, GLONASS, GALILEO)

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Very Long Baseline Interferometry VLBI
Quasar direction
Wave front
Geometric Delay
dg
Baseline
Radiotelescope 1
Radiotelescope 2
Earth surface
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Lunar Satellite
LLR SLR
Laser Ranging
Moon
Measuring Time Propagation
Passive Satellite
LLR Telescope
Earth
SLR Telescope
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Global Positioning System GPS
Satellite
Satellite Orbit
GPS Antenna
Navigation Message sent by each satellite -
Orbit parameters - Clock corrections GPS
Measurements - Pseudorange - Phase
Earth
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DORISDoppler Orbitography and Radiopositioning
Integrated by Satellite

• French Technique developed by CNES, IGN and GRGS
• Uplink System on-board receiver measures the
  doppler shift on the signal emitted by the ground
  beacon

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Relation between Terrestrial and Celestial
reference Systems
Celestial System XC
Terrestrial System XT
P Precession N Nutation S Earth
rotation
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Polar Motion
Y(mas)
X (mas)
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Precession- Nutation
Q
Precession
Nutation
P
Q
9.21
6.87
e
2327
Equator
e
Ecliptic
Q
Displacement in 26000 yrs of the pole axis along
a cone of semi-angle e 2327
Elliptic Motion along the precession cone
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Classical versus Space Geodesy

• Space Geodesy
• Provide 3-D positions expressed in a Global
  Terrestrial Reference System

• Classical Geodesy
• Directions and Distances
• Levelling
• Leads to
• Horizontal coordinates
• (l, j)
• Vertical coordinates (H)
• Need a geoid to establish
• 3-D coordinate system

Topographic surface
H
h
Geoid
N
Ellipsoid
h H N
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Transformation between TRS (1/2)
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Transformation between TRS (2/2)
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Crust-based TRF
The instantaneous position of a point on Earth
Crust at epoch t could be written as
X0 point position at a reference epoch t0
point linear velocity high frequency
time variations - solid Earth tide - ocean
loading - atmosphere loading - geocenter
motion
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Comparison of Two TRFs
Estimation of the Transformation parameters
between the Two
or
q is solved for using Least Squares adjustment
And in case of velocities
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Combination of TRFs
Based on the Transformation Formula of 7
parameters For each individual TRF s, we have

• The unknowns are
• Xc station positions ( velocities)
• transformation parameters ( rates) from TRF c
  to TRF s
• Solved for using least Squares adjustment

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Implementation of a TRF

• Definition at a chosen epoch, by selecting
• 7 transformation parameters, tending to satisfy
  the theoretical definition of the corresponding
  TRS
• A law of time evolution, by selecting 7 rates of
  the 7 transformation parameters, assuming linear
  station motion!
• 14 parameters are needed to define a TRF

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How to define the 14 parameters ? Datum
definition 

• Origin rate CoM (Dynamical Techniques)
• Scale rate depends on physical parameters
• Orientation conventional
• Orient. Rate conventional Geophysical meaning
  (Tectonic Plate Motion)
• Lack of information for some parameters
• Orientation rate (all techniques)
• Origin rate in case of VLBI
• Rank Deficiency in terms of Normal Eq. System

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Datum Definition
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TRF implementation in practice
The initial NEQ system of space geodesy
observations could be written as
Where
are the linearized unknowns

• Normal matrix is singular having a rank
  deficiency
• Equal to the number of TRF parameters not reduced
• by the observations. Some constraints are needed
• Tight constraints ( ? ? 10-10 ) m
• Removable constraints ( ? ? 10-5 ) m
• Loose constraints ( ? ? 1) m
• Minimum constraints (applied over the TRF
• parameters and not over station coordinates)

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Combination in the era of times series

• Daily/Weekly/Monthly solutions of Station
  positions allow to detect
• station non-linear and seasonal motions,
  discontinuities and other problems
• geocenter motion
• loading effects (common mode)
• Ensure TRF EOP consistency in the combination
• But how to ensure the TRF long-term stability
  (well defined time evolution) in presence of
  non-linear variations ?
• Basic question real non-linear variations vs
  real geophysical motions ?

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Time series combination (Rigourosly stacking)

• Input
• Weekly Station Positions X(t)
• Daily Polar motion ( rates), UT1, LOD
• Output Long-Term Solution (LTS)
• Station positions at a reference epoch t0
• Station Velocities
• Daily EOPs
• Time series of the transformation parameters
  between each week and the LTS

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Stacking
time series
Datum Definition with Minimum Constraints Over a
Reference Set of stations
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Datum definition current principlesfor time
series stacking

• (1) Define the frame at a given epoch t0
• 7 degrees of freedom to be selected/fixed
• (2) Define a linear (secular) time evolution
• 7 degrees of freedom to be selected/fixed
• Assume linear station motion
• Add break-wise approach for discontinuities
• Investigate the non-linear part in the time
  series of the residuals

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Ways of implementation

• (1) Select an external frame as a "reference"
• and apply minimum constraints approach
• or
• (2) Considering that for any Transf. Param.
• apply "inner" conditions

and