terrestrial reference system and frame

7/26/2019 Terrestrial Reference System and Frame

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TerrestrialReference System and Frame

Datum transformations

E. Calais

Purdue University - EAS DepartmentCivil 3273 [email protected]


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Need for aTerrestrial Reference GPS geodesy ! estimate position and

displacement (velocity) of points on Earth Positions and velocities:

Are not direct observations, but estimatedquantities

Are not absolute quantities

Need for a Terrestrial Reference in which(or relative to which) positions and velocitiescan be expressed.


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Need for a Terrestrial ReferenceIn practice, GPS data processing can beseen as a 2-step process:

1. From phase and pseudorange observablesto (relative) site positions (and their associated variance-covariance = the wholething is sometimes called quasi-

observations)2. From quasi-observations to positions andvelocities in a well-defined reference frame


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Need for a Terrestial Reference:Datum Defect

Geodetic data are notsufficient by themselvesto calculatecoordinates!

Ex. of triangulation data(angle measurements):origin, orientation, andscale need to be fixed

Ex. of distancemeasurements: originand orientation need tobe fixed, scale is givenby the data


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Datum Defect 3 points (in 3D):

2 independent distance measurements 2 independent angle measurements 2 independent height difference measurements But 9 unknowns

4 points: 12 unknowns, 9 data Datum defect = rank deficiency of the matrix that relates

the observations to the unknowns Solution: define a frame!

Fix or constrain a number of coordinates 3D network: rank deficiency is 7 (or 6) => fix 3 (or 2) points


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Fixing a Frame, but Too many constraints?

Ex.: fixing 2 points in a geodetic solutions means that onefixes:

Scale Orientation Origin twice

A priori constraints used on site positions have an

impact on the final uncertainties: overconstrainingtypically results in artificially small uncertainties The Frame has to be well defined (stable) through

time: time-dependent model


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System vs. Frame Terrestrial Reference System (TRS):

Mathematical definition of the reference in which

positions and velocities will be expressed. Therefore invariable but inaccessible to users in

practice.

Terrestrial Reference Frame (TRF):

Physical materialization of the reference systemby way of geodetic sites.

Therefore accessible but perfectible.


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The ideal TRS Tri-dimensional orthogonal (X,Y,Z). Base vectors have same length. Geocentric: origin close to the Earths center

of mass (including oceans and atmosphere) Equatorial: Z-axis is direction of the Earths

rotation axis Rotating with the Earth.


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3D similarity Under these conditions, the transformation of

cartesian coordinates of any point between 2 TRSs

(1) and (2) is given by a 3D similarity:

Where: X (1) and X (2) are the position vectors in TRS(1) and TRS(2) T 1,2 = translation vector

! 1,2 = scale factor R 1,2 = rotation matrix

X (2) = T 1,2

+ " 1,2

R1,2

X (1)


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3D similarity

Also called a Helmert, or 7-parameter,transformation. If translation (3 parameters), scale (1

parameter) and rotation (3 parameters) areknown, then one can convert between TRSs

If there are common points between 2 TRSs,one can solve for T, " , R: minimum of 3points.

X (2) = T 1,2

+ " 1,2

R1,2

X (1)


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Linearization 3D similarity between TRS1, X 1 and

TRS2, X 2 can also be written as:

With D = scale factor and:

X 2

= X 1

+ T + DX 1

+ RX 1

T =

T 1

T 2

T 3

"

#

$ $ $

%

&

' ' '

R =

0 " R 3 R 2 R

3 0 " R1

" R 2 R1 0

#

$

% % %

&

'

( ( (


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Time dependence X 1, X 2 , T, D, R are generally functions of time

(plate motions, Earths deformation).

Differentiation w.r.t. time gives:

D and R ~ 10 -5 and Xdot ~ 10 cm/yr !

DXdot and RXdot negligible, ~ 0.1 mm/100years, therefore:

X2

= X1

+ T + D X 1

+ D X1

+ R X 1

+ R X1

X2

= X1

+ T + D X 1

+ R X 1


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Estimation The above equations can be written as:

with:

The least-squares solutions are:

Where P x and P v are the weight matrix for station positions andvelocities, respectively

X2

= X1

+ T + D X 1

+ R X 1

" X2

= X1

+ A #

X 2

= X 1

+ T + DX 1

+ RX 1

" X 2

= X 1

+ A #

" = T1, T 2 , T3 , D , R 1, R 2 , R 3[ ]

" = T 1,T 2 ,T 3 , D , R1, R 2 , R 3[ ]

A =

. . . . . . .1 0 0 x 0 z " y0 1 0 y " z 0 x0 0 1 z y " x 0. . . . . . .

#

$

% % % % % %

&

'

( ( ( ( ( (

" = ( A T P x A )# 1 A T P

x ( X 2 # X 1)

" = ( A T P v A )# 1 A T P

v ( X 2 # X 1)


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Summary If we know the position and velocities of a number of geodetic sites

that define a reference frame, we can:1. Use observations at these sites in addition to observations at sites of

unknown position and velocity = reference, or fiducial, sites

2. Equate the position and velocity at these reference sites to theseknown values

3. Solve for 7 or 14 transformation parameters4. Apply the transformation to the position and velocity of the other sites.(3 and 4 can actually be done in one step)

Choice of reference sites: Global distribution Position and velocity precise and accurate Error on their position/velocity and correlations well known

So, now we need to define a reference frame


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The international TerrestrialReference System: ITRS

Definition adopted by the IUGG and IAG Tri-dimensional orthogonal (X,Y,Z), equatorial (Z-

axis coincides with Earths rotation axis) Non-rotating (actually, rotates with the Earth) Geocentric: origin = Earths center of mass, including

oceans and atmosphere. Units = meter and second S.I. Orientation given by BIH at 1984.0. Time evolution of the orientation ensured by

imposing a no-net-rotation condition for horizontalmotions.


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The no-net-rotation (NNR) condition Objective:

Representing velocities without referring to a particular plate. Solve a datum defect problem: ex. of 2 plates ! 1 relative velocity to solve for 2

absolute velocities (what about 3 plates?)

Mean: describe velocities in a frame that minimizes plate velocities (actuallytheir angular momentum), over the whole Earths lithosphere:

Can be discretized as:

Where Q p = inertia tensor of plate p , " p = angular velocity of plate p , P = totalnumber of plates

Q p : depends on plate geometry, but thickness variations are ignored.

X " Vdm = 0 L #

Q p" p = 0 p # P$ (= condition equation)


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The no-net-rotation (NNR) condition

The no-net-rotationcondition states that the

total angular momentum of all tectonic plates shouldbe zero.

See figure for the simple(and theoretical) case of 2plates on a circle

VB/NNR

VB/A P l a

t e 1 P l a t e 2 VA/A=0VA/NNR

A

BEarths radius R

MA = R X V B/NNR MB = R X V A/NNR

# M = 0 ! V B/NNR = V A/NNR


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The no-net-rotation condition The NNR condition has no impact on

relative plate velocities. It is an additional condition used to

define a reference for plate motionsthat is not attached to any particular

plate


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The international TerrestrialReference Frame: ITRF

ITRF is defined by the positions (at a given epoch)and velocities of a set of geodetic sites (+ associated

covariance information) = dynamic datum These positions and velocities are estimated by

combining independent geodetic solutions andtechniques.

Combination: Randomizes systematic errors associated with each

individual solutions Provides a way of detecting blunders in individual solutions Accuracy is equally important as precision


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History of the ITRF 1984: VLBI, SLR, LLR, Transit 1988: TRF activity becomes part of the IERS => first

ITRF = ITRF88 Since then: ITRF89, ITRF90, ITRF92, ITRF93,

ITRF94, ITRF96, ITRF97 Current: ITRF2000 ITRF improves as:

Number of sites with long time series increases New techniques appear Estimation procedures are improved


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ITRF2000

Solutions used: 3 VLBI, 1 LLR, 7 SLR, 6 GPS, 2 DORIS Up to 20 years of data GPS sites defining the ITRF are all IGS sites Wrms on velocities in the combination: 1 mm/yr VLBI, 1-3 mm/yr SLR and GPS


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ITRF and the user ITRF distributed as:

A table (incomplete covariance information)

A Solution Independent Exchange format file =SINEX file (complete covariance information)


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ITRF and the User

Coordinates precise to better than 1 cm and velocities to better than 1 mm/yr for sites with long (10 years and more) observationspans (figure from Altamimi et al., 2002).


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ITRF and the user

Velocities in ITRF2000 a t som e IGS sites


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ITRF and the user Why use it?

Because it provides the most accurate globalreference frame available

Because precise GPS orbits (from IGS) use it If everyone uses it, then, by default, all positions

and velocities will be expressed in the same (and

globally consistent) geodetic frame!

easy tocompare


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ITRF and the user Decide on a number of GPS sites to include in your

data processing

Estimate daily station positions by inverting phaseand pseudorange data = daily solutions Constrain the solutions to ITRF by:

1. First possibility: Fixing, or tightly constraining, the position(and velocity) of reference sites to their ITRF values:

2. Second possibility:1. Estimating a 14 parameter transformation between a loosely

constrained solution and the ITRF, using common sites2. Applying the transformation to the loosely constrained solution


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ITRF and the User Since the ITRF is global, the best procedure is

actually to:1. Include a minimum of reference station in daily GPS

solutions = regional solutions2. Combine (loosely constrained) daily regional solutions with

global solutions3. Estimate 14 parameter transformation using a global set of

reference sites, well distributed4. Apply the transformation.

Also, constraining station positions and/or velocitieswill bias the solution if their coordinates and/or theassociated variance are incorrect.


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ITRF and the User Does ITRF solve the datum defect

problem? Yes, but it is a space-based technique,

therefore: If orbits are fixed => reference frame

already tightly imposed If EOP are fixed => network orientation

imposed


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ITRF and the User How to ensure that the implementation of the

reference frame minimally affects the final solution(estimates and their variance)?

Minimal constraints : for a solution to be in a givenframe, the transformation parameters between thatsolution and the frame must be null.

Application of minimal constraints: Remove constraints from solution if they were implemented

=> loose solution Decide on a number of sits that will be used to estimate the

transformation parameters (= to implement the referenceframe)

Add condition equation that reflect the null transformation


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Minimal Constraints


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Link with Inertial System Conventional Inertial System :

Orthogonal system, center = Earth center of mass, defined at standard epochJ2000 (January 1 st , 2000, 12:00 UT)

Z = position of the Earths angular momentum axis at standard epoch J2000

X = points to the vernal equinox It is materialized by precise equatorial coordinates of extragalactic radio

sources observed in Very Long Baseline Interferometry (VLBI) = InertialReference Frame .

First realization of the International Celestial Reference Frame (ICRF) in 1995.

Systems are defined and frames are realized in the framework of aninternational service: the International Earth Rotation Service, IERS(http://www.iers.org/ )

! Mission of the IERS: To provide to the worldwide scientific and technical community reference values for Earth orientation parameters and referencerealizations of internationally accepted celestial and terrestrial referencesystems


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Link with Inertial System The Earth's orientation is defined as the rotation

between : A rotating geocentric set of axes linked to the Earth (the

terrestrial system materialized by the coordinates of observingstations) = Terrestrial System

A non-rotating geocentric set of axes linked to inertialspace (the celestial system materialized by coordinates of stars, quasars, objects of the solar system) = Inertial System

The transformation from Inertial to Terrestrial can bemade using a standard matrix rotation:

precession nutation spin Polar motion terrestrialinertial

t i xW S N P x =


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Link with Inertial Frame Rotation parameters are given in astronomical tables

provided by the IERS: for the most precisepositioning, they are used as a priori values andsometimes adjusted in the data inversion.

In practice, the IERS provides five EarthOrientation Parameters (EOP) : Celestial pole offsets (dPsi, dEps): corrections to a given

precession and nutation model

Universal time (UT1) = UT corrected for polar motion,provided as UT1-TAI 2 coordinates of the pole (x,-y) of the CEP with respect to

Earths geographic pole axis = polar motion


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Questions? How to transform ITRF velocities to

plate-fixed velocities? Are velocities at reference sites really

constant? Most legal reference frames are static

terrestrial reference system and frame

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