The definition of a geophysically meaningful International Terrestrial Reference System Problems and prospects The definition of a geophysically meaningful

The definition of a geophysically meaningful International Terrestrial Reference System

Problems and prospects

The definition of a geophysically meaningful International Terrestrial Reference System

Problems and prospects

Athanasios Dermanis

Department of Geodesy and SurveyingAristotle University of Thessaloniki X

Y

Z

EGU General Assembly 2006April 2-7, Vienna

Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS)


Objective (practice): Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions xi(t), yi(t), zi(t) of

points Pi in a global geodetic network









Available data: Coordinates xi(tk), yi(tk), zi(tk) of points in subnetworks at

discrete epochs tk (e.g. daily or weekly solutions) with respect to a different reference systems for each subnetwork













Problem (theory): Introduce a principle leading to an optimal choice of the reference frame

Problem (theory): Introduce a principle leading to an optimal choice of the reference frame

Problem (practice): Combine the known shapes of the overlapping subnetworks into the shape of a global network expressed by coordinates with respect to the already defined optimal reference system

Problem (practice): Combine the known shapes of the overlapping subnetworks into the shape of a global network expressed by coordinates with respect to the already defined optimal reference system

The optimal choice of reference systemGeneral characteristic: Apparent motion (coordinate variation) of mass points should

be minimal


be minimal

For the whole earthFor the whole earth

dmx

R [ ]d

dmdt

x

h x 0Earth

T

R mind d

T dmdt dt

x x

Earth

Tisserand principle:

Vanishing relative angular momentum

Minimal relative kinetic energy


be minimal


be minimal

For the whole earthFor the whole earth For a network of discrete pointsFor a network of discrete points

dmx

R [ ]d

dmdt

x

h x 0Earth

T

R mind d

T dmdt dt

x x

Earth




R [ ] ii i

i

dm

dt x

h x 0

T

R mini ii

i

d dT m

dt dt

x x

Discrete Tisserand principle:



xi

i


be minimal


be minimal

For the whole earthFor the whole earth For a network of discrete pointsFor a network of discrete points

dmx

Earth

M dm


1const.i i

i

mM

m x

Discrete Tisserand principle:

Define orientation of the reference system.For the position:

xi

i

Earth

1const.dm

M m x

ii

M m

Preservation of the center of mass:

Trivial choice:mi =1

The “linear” model The “linear” model

Is there a linear model for deformation? No !It is not possible to have all shape related parameterswhich are coordinate invariantsas linear functions of time!


Is there a linear model for deformation? No !

A planar counterexample:

s1 s2

s4 s5

s3 si = ai t + bi , i = 1,2,3,4,5

The 5 shape defining sides are linear functions of time:

It is not possible to have all shape related parameterswhich are coordinate invariantsas linear functions of time!


Is there a linear model for deformation? No !

A planar counterexample:

s1 s2

s4 s5

s3

d

si = ai t + bi , i = 1,2,3,4,5

d = nonlinear in t

The 5 shape defining sides are linear functions of time:

The 2nd diagonal of the quadrilateral is not a linear function of time!

It is not possible to have all shape related parameterswhich are coordinate invariantsas linear functions of time!


What is the meaning of linear model for coordinate functions?

Linear coordinates in one reference system

x(t) = x0 t + vx ,

y(t) = y0 t + vy ,

z(t) = z0 t + vz ,

x(t) = x0 t + v x(t) = R(t) x0 t + R(t) v + c(t)

x0(t) t + vx´(t) = R(t) x(t) + c(t)

Non-linear coordinates in another

CHANGE OF REFERENCE SYSTEM


x(t)

x´(t) = R(t) x(t) + c(t)are both Tisserand

x(t)


R(t) = R = const.

c(t) = c = const.

R(t) = R = const.

c(t) = c = const.

What is the meaning of linear model for coordinate functions? x(t) = x0 t + vx ,

y(t) = y0 t + vy ,

z(t) = z0 t + vz ,


Ad hoc definition of “linear deformation” model:

Coordinates in one (and hence all) Tisserand systemare linear functions of time

x(t)


x(t)


Note: Tisserand reference system here means:Orientation by Tisserand principle (zero relative angular momentum)Position by center of mass preservation principle

R(t) = R = const.

c(t) = c = const.

R(t) = R = const.

c(t) = c = const.

What is the meaning of linear model for coordinate functions? x(t) = x0 + t vx ,

y(t) = y0 + t vy ,

z(t) = z0 + t vz ,

The Tisserand reference system conditions for the linear modelThe Tisserand reference system conditions for the linear model

0( )i i it t x x v 1,2, ,i N

0( ) [ ]R i i ii

t m h x v 0

0

1 1( ) const.i i i i

i i

t m t mM M

m x v 0

Vanishing relative angular momentum:

Constant (zero) center of mass coordinates:

Tisserand conditions:

0[ ]i i ii

m x v 0

0i ii

m x 0

i ii

m v 0

Usual choice:

1im


0[ ]i ii

x v 0

0ii

x 0

ii

v 0

The Tisserand reference system conditions for the linear modelThe Tisserand reference system conditions for the linear model

0( )i i it t x x v 1,2, ,i N

0( ) [ ]R i i ii

t m h x v 0

0

1 1( ) const.i i i i

i i

t m t mM M

m x v 0

Vanishing relative angular momentum:

Constant (zero) center of mass coordinates:


0[ ]i i ii

m x v 0

0i ii

m x 0

i ii

m v 0

Usual choice:

1im


0[ ]i ii

x v 0

0ii

x 0

ii

v 0

Used at the stage of data analysisas additional constraints

+Usual inner constraints for the

definition of reference frame at epoch t0

=Selection of a single Tisserand frameout of a set of dynamically equivalent

ones

Used at the stage of data analysisas additional constraints

+Usual inner constraints for the

definition of reference frame at epoch t0

=Selection of a single Tisserand frameout of a set of dynamically equivalent

ones

Accessing a geophysically meaningful reference systemAccessing a geophysically meaningful reference system

x(t)(earth) Tisserand


Earth

[ ]R

ddm

dt

xh x 0

[ ] iR i i

i

dm

dt x

h x 0

~x(t) = R(t) x(t)~x(t)

(discrete) Tisserand

x(t)(discrete) Tisserand

dm

To access R(t) transforming hR(Earth) to hR(Earth) = 0 we need an approximation of hR(Earth)~

Accessing a geophysically meaningful reference systemAccessing a geophysically meaningful reference system



Earth

[ ]R

ddm

dt

xh x 0

[ ] iR i i

i

dm

dt x

h x 0

~x(t) = R(t) x(t)~x(t)

(discrete) Tisserand

x(t)(discrete) Tisserand

dm

To access R(t) transforming hR(Earth) to hR(Earth) = 0 we need an approximation of hR(Earth)~

Earth E = P1 + … + PK + … + PL + P0

Network D = D1 + … + DK + … + DL

Plates Interior

Subnetwork on plate PK

To access R(t) transforming hR(Earth) to hR(Earth) = 0 we need an approximation of hR(Earth)

0KR P PK

h h h 0

0

Use DK to access hPK(t) !

Estimation of the rigid motion of a plateEstimation of the rigid motion of a plate

Plate PK

Subnetwork DK

x = RK x Transformation such that R K(D ) h 0

original velocities

X

Y

Z

X

Y

Z


Plate PK

Subnetwork DK

RK

K


original velocities

transformed velocities

determine rotation RK

to reference systembest fitted to DK

X

Y

Z

X

Y

Z


Plate PK

Subnetwork DK

RK

K


original velocities


velocities due to rigidplate rotation only



assume rigid motion of plate PK

due only to rotation RK

X

Y

Z

X

Y

Z


Plate PK

Subnetwork DK

RK

K


original velocities


velocities due to rigidplate rotation only



assume rigid motion of plate PK

due only to rotation RK

compute relative angular momentum of plate PK due only to rotation RK

( ) [ ]K

KR

P

P dm h x v

X

Y

Z

X

Y

Z

Final transformation to an approximate Earth-Tisserand reference systemFinal transformation to an approximate Earth-Tisserand reference system

( )RK

R KPh h R 0hx = R x~

Previously estimated


( )RK



Approximate (numerically sufficient) solution for the linear model: xi(t) = x0i + t vi


( )RK




0[ ]K

K i ii D

h x v

0 0 0[ ][ ]K

K i ii D

C x x

relative angular momentum

inertia matrix

Subnetwork DK1


( )RK




0[ ]K

K i ii D

h x v

0 0 0[ ][ ]K

K i ii D

C x x

10 0K K K

ω C h

0 0[ ][ ]K

K

P

P

dm C x x


inertia matrix

rotation vector

inertia matrix

Subnetwork DK

Plate PK

1

2


( )RK




0[ ]K

K i ii D

h x v

0 0 0[ ][ ]K

K i ii D

C x x

10 0K K K

ω C h

0 0[ ][ ]K

K

P

P

dm C x x

00

` 1

K KP P KK K

C ωω C


inertia matrix

rotation vector

inertia matrix

Subnetwork DK

Plate PK

total rotation vector1

2

3


( )RK




0[ ]K

K i ii D

h x v

0 0 0[ ][ ]K

K i ii D

C x x

10 0K K K

ω C h

0 0[ ][ ]K

K

P

P

dm C x x

00

` 1

K KP P KK K

C ωω C

0( )i i it t x x v

0 0ii x x

0 0[ ]i i i ωv xv


inertia matrix

rotation vector

inertia matrix

Subnetwork DK

Plate PK

total rotation vector1

2

3

4 new linear model

only velocities updated !

A copy of this presentation can be downloaded from:

http://der.topo.auth.gr

A copy of this presentation can be downloaded from:

http://der.topo.auth.gr

Thanks for your attention !

VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy:

Challenge and Role of Modern GeodesyMay 29 - June 2, 2006, Wuhan, China

VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy:

Challenge and Role of Modern GeodesyMay 29 - June 2, 2006, Wuhan, China

the Intercommission Committee on Theory (ICCT) of the International Association of Geodesy (IAG)

Documents

The definition of a geophysically meaningful International Terrestrial Reference System Problems and prospects The definition of a geophysically meaningful