on the alternative approaches to itrf formulation. a theoretical comparison

On the alternative approaches to ITRF formulation.On the alternative approaches to ITRF formulation.A theoretical comparison.A theoretical comparison.

Department of Geodesy and SurveyingAristotle University of Thessaloniki

Athanasios Dermanis

Given:

Time series of coordinates xT(tk) & EOPS cT(tk) from each space technique T

Find:

The optimal coordinate transformation parameters pT(tk) (rotations, translation, scale)which transform the above time series xT(tk), cT(tk) into new ones xITRF(tk), cITRF(tk)

best fitting the linear-in-time ITRF model for each network station i

with constant initial coordinates x0i and velocities vi

The ITRF Formulation Problem

0 0( ) ( )i k i k it t t x x v

This procedure is called “stacking”

The basic stacking model:

, , ,

0 0 ,

( ) ( ), ( ) ( )

( ) , ( ) ( )

xT i k ITRF i k ITRF T k T i k

xi k i ITRF T k T i k

t t t t

t t t t

x f x p e

f x v p e

.( ) ( ), ( ) ( )cT k ITRF k ITRF T k T i kt t t t c g c p e

Data from a set of 4 non-overlapping networks (VLBI, SLR, GPS, DORIS)

connected through surveying observations between nearby stations

at collocation sites

Coordinates:

Earth Orientation

Parameters (EOPs):


, , ,

00 ,

( ) ( ), ( ) ( )

( ) , ( ) ( )


xk ITRF T k T ki i i

t t t t

t t t t

x f x

v px

p e

f e

.( ) , ( ) ( )( ) cT k ITRFITRF k T k T i kt t tt c g c p e




Coordinates:

Earth Orientation

Parameters (EOPs):

0 , , ( )i i ITRF ktx v cITRF parameters (initial coordinates, velocities, EOPs):



, , ,

0 0 ,

( ) ( ), ( ) ( )

()( ) , )(

xT i k ITR

ITRF T k

F i k ITRF T k T i k

xi k i T i k

t

t

t t t

t t t

x f x

v p

p e

f x e

.( ) ( ( )( )), ITRFc

T k ITRF k T i kT ktt t t c g epc




Coordinates:

Earth Orientation

Parameters (EOPs):

( )ITRF T ktpTransformation parameters from the ITRF reference system

to the reference system of each epoch within each technique:



, , ,

0 0 ,

( ) ( ), ( ) ( )

(( ) )) , (


i k i ITRF T kxT i k

t t t t

t t t t

x f x

f x v p e

p e

.( ) ( ), ( ) ( )T k ITRF k ITRF T kcT i kt tt t c g c ep




Coordinates:

Earth Orientation

Parameters (EOPs):

, .( ), ( )x cT i k T i kt te e

Observation noise - Assumed zero-mean and with known covariance cofactor matrices

(single unknown reference variance 2):



Simplifications of the Problem

4 non-overlapping networks

connected through cross observations

True ITRF formulation problemfor VLBI, SLR, GPS, DORIS:






2 overlapping networks







2 identical networks







2 identical networks Despite the simplificationsthe fundamental problemcharacteristics are preserved

The simplified cases deservea study in their own







2 identical networks

We will restrict to 2 networksin order to keep equationswithin manageable complexity

No loss of generality

TWO STEP APPROACH

(1) Adjustment of data from each technique separately for the estimation of per technique ITRF parameters (stacking per technique)

(2) Combination of the ITRF estimates from each technique into final ITRF estimates

The two alternative approaches

ONE STEP APPROACH

Simultaneous adjustment of data from all techniquesfor the estimation of the ITRF parameters(multi-technique approach – simultaneous stacking)

TWO STEP APPROACH




ONE STEP APPROACH – EQUIVALENT TWO-STEP FORMULATION

(1) Separate solutions

(2) Combination of separate solutions with addition of normal equations Final parameter estimates as weighted mean of separate estimates

TWO STEP APPROACH







same

TWO STEP APPROACH







Difference only in

second steps

TWO STEP APPROACH







Separate solutions produce singular covariance matrices !

Models with rank defect due to lack of reference system definition

b Ax v 2 1( , ) v 0 P rank( )n m

r m n

A

Variation of parameters under change of reference system


b Ax v 2 1( , ) v 0 P rank( )n m

r m n

A

x x Ep p = transformation parameters (rotations, displacement, scale)



b Ax v 2 1( , ) v 0 P rank( )n m

r m n

A

Invariance of observables y = Ax and estimable parameters (functions of y)


T AE 0y Ax Ax Ax AEp A P E NEA 0



b Ax v 2 1( , ) v 0 P rank( )n m

r m n

A

T E x 0

Invariance of observables y = Ax and estimable parameters (functions of y)


T AE 0y Ax Ax Ax AEp A P E NEA 0

(total) inner constraints for reference system choice(usually partial inner constraints or other minimal constraints are employed)

Two identical networks

This case does not apply to the ITRF formulation problem

but has an interest of its own for other network applications

Two identical networks – One step solution

a a a

b b b

b A vx

b A v

12

1,a a

b b

v P 00

v 0 P

( )T T T Ta a a b b b a a a b b b A P A A P A x A P b A P b

( )a b a b N N x u u

ˆ a a x x e

ˆb b bN x uˆa a aN x u

T C x 0 x̂

Identical to separate solutions

and combination using of the model

ˆ b b x x e

2( , )a a e 0 N2( , )a a e 0 N

with weight matrices

,a bN Nˆ ˆ( )a b a a b b N N x N x N x

a aN x u Ta C x 0 2

|ˆ ˆ ( , )aa a aCx x x Q

b bN x u Tb C x 0 2

|ˆ ˆ ( , )bb b bCx x x Q

Step 1: Separate solutions

Normalequations

ˆ a a a x x Ep e

ˆ b b b x x Ep e

2( , )a ae 0 Q2( , )b be 0 Q

ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ ˆ

ˆ

ˆ

a b a b a a b bT T T

a a a a aT T T

b b b b b

W W W E W E x W x W x

E W E W E 0 p E W x

E W 0 E W E p E W x

?a

b

W 0

0 W

Step 2: Combination

Weightmatrix

Minimalconstraints

Two identical networks – Two step solution

Two step solution(equivalent to) one step solution

ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

ˆa a

b b

x eIx

x eI

ˆ ˆ( )a b a a b b N N x N x N x

ˆ ˆ

ˆ

ˆ

a b a bT T

a a aT T

b b b

a a b bT

a aT

b b

W W W E W E x

E W E W E 0 p

E W 0 E W E p

W x W x

E W x

E W x

Normal equationsWeight matrices Na, Nb Weight matrices Wa, Wb


“wrong” model !Ignores that partial and final solutions

are in different reference systems

correct model !Treats partial and final solutions

in different reference systems



ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

ˆa a

b b

x eIx

x eI

Normal equationsWeight matrices Na, Nb Weight matrices Wa, Wb

T Ta b a a a b b b u u A P b A P b

ˆ ˆ

ˆ

ˆ

a b a bT T

a a aT T

b b b

a a b bT

a aT

b b

W W W E W E x

E W E W E 0 p

E W 0 E W E p

W x W x

E W x

E W x

Weight matrices “kill” the dependence of the partial solutions

on different reference systems !







ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

ˆa a

b b

x eIx

x eI

ˆ ˆ

ˆ

ˆ

a b a bT T

a a aT T

b b b

a a b bT

a aT

b b

N N N E N E x

E N E N E 0 p

E N 0 E N E p

N x N x

E N x

E N x

Normal equations with same weight matricesWeight matrices Na, Nb Weight matrices Na, Nb








ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

ˆa a

b b

x eIx

x eI

ˆ ˆ

ˆ

ˆ

a b a bT T

a a aT T

b b b

a a b bT

a aT

b b

N N N E N E x

E N E N E 0 p

E N 0 E N E p

N x N x

E N x

E N x



Recall that

a a

b b

A E 0 N E 0

A E 0 N E 0







ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

ˆa a

b b

x eIx

x eI

ˆ ˆ

ˆ

ˆ

a bT T

a aT T

b b

T

a b

a

b

a a b b

a

b

aT

b

N E N E

E N E N E

E

N N

N

x

0 p

0 E N E

E

p

N x N x

x

x

N

E N



Vanishing terms







ˆ

â a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

â a

b b

x eIx

x eI

ˆ â b a a b b

a

b

N N 0 0 x N x N x

0 0 0 p 0

0 0 0 p 0













ˆ

ˆa a

ab b

b

xx eI E 0

px eI 0 E

p

ˆ

ˆa a

b b

x eIx

x eI


ˆ ˆ( )a b a a b b N N x N x N x ˆ ˆ( )a b a a b b N N x N x N x

0 0

0 0Same results for IERS parameters x !

Transformation parameters pa, pb undetermined !


Two overlapping networks

This case would apply to the ITRF formulation problem

if perfect connections were available at collocation sites


1 31

3a a a a a a a

xb v A x

xA A v 2 1( , )a a v 0 P

2 1( , )b b v 0 P 2 32

3b b b b b b b

xb v A x

xA A v

x3 = parameters of common points

x1 , x2 = parameters of non-common points

Two overlapping networks – Separate solutions

22 232 2 2 3

23 333 2 3 3

T Tb bT b b b b b b

b b b b TT Tb bb b b b b b

N NA P A A P AN A P A

N NA P A A P A

ˆa a aN x u

11 3

3a a a a a a a

xb A A v A x v

x

22 3

3b b b b b b b

xb A A v A x v

x

11 131 1 1 3

13 333 3 3

T Ta aT a a a a a a

a a a a TT Ta aa a a a a a

N NA P A A P AN A P A

N NA P A A P A

11

33

TaT a a a

a a a a Taa a a

uA P bu A P b

uA P b

ˆb b bN x u

22

33

TbT b b b

b b b b Tbb b b

uA P bu A P b

uA P b

normal equations

Network (b) solution:

Network (a) solution:

normal equations

Two overlapping networks – One step solution

11 13 1 11 1 13 3

22 23 2 22 2 23 3

13 23 33 33 3 13 1 33 3 23 2 33 3

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

a a a a a a

b b b b b bT T T Ta b a b a a a a b b b b

N 0 N x N x N x

0 N N x N x N x

N N N N x N x N x N x N x

11 3

22 3

3

a a a a

b b b b

xb A 0 A v

xb 0 A A v

x

a

b

P 0

0 P

11 13

13 33

22 23

23 33

a aTa a

b bTb b

N N 0 0

N N 0 0

0 0 N N

0 0 N N

1 1 1 11

3 3 3 32

2 2 2 23

3 3 3 3

ˆ

ˆ

ˆ

ˆ

a a a

a a a

b b b

b b b

x x e eI 0 0x

x x e e0 0 Ix

x x e e0 I 0x

x x e e0 0 I

11 13 1 1

22 23 2 2

13 23 33 33 3 3 3

ˆ

ˆ

ˆ

a a a

b b bT Ta b a b a b

N 0 N x u

0 N N x u

N N N N x u u

Identical with solution based on separate solutions with model

normalequations

weightmatrix

weightmatrix

Two overlapping networks – Two step solution

ˆ ˆ

ˆ

ˆ

T T T T T Ta a a b b b a a a b b b a a a b b b

T T Ta a a a a a a a a aT T Tb b b b b b b b b b

D W D D W D D W E D W E x D W x D W x

E W D E W E 0 p E W x

E W D 0 E W E p E W x

11 1 1

23 3 3

3

ˆˆ

ˆa a

a a a a a aa a

xx E eI 0 0

x x p D x E p ex E e0 0 I

x

12 2 2

23 3 3

3

ˆˆ

ˆb b

b b b b b bb b

xx E e0 I 0

x x p D x E p ex E e0 0 I

x

ˆ

ˆa a a a

ab b b b

b

xx D E 0 e

px D 0 E e

p

Combination (second) step

1 1 1 1ˆ a a a x x E p e

3 3 3 3ˆ a a a x x E p e

2 2 2 2ˆ b b b x x E p e

3 3 3 3ˆ b b b x x E p e

From network a:

From network b:

Combined a+b:11 13

13 33

22 23

23 33

a aT

a a a

b b bTb b

W W 0 0

W 0 W W 0 0W

0 W 0 0 W W

0 0 W W

normal equations

ˆ ˆ

ˆ

ˆ






normal equations

a a

b b

W 0 N 0W

0 W 0 N

Use of same weights as in the (equivalent to) one step solution

normal equations

ˆ ˆ

ˆ

ˆ



D N D D N D D N E D N E x D N x D N x

E N D E N E 0 p E N x

E N D 0 E N E p E N x


ˆ ˆ

ˆ

ˆ






normal equations

a a

b b

W 0 N 0W

0 W 0 N


normal equations

ˆ ˆ

ˆ

ˆ



D N D D N D D N E D N E x D N x D N x



Recall that ,a a b b N E 0 N E 0


ˆ ˆ

ˆ

ˆ






normal equations

a a

b b

W 0 N 0W

0 W 0 N


normal equations

ˆ ˆ

ˆ

ˆ

a a b bT

T T T T T Ta a a b b b a b a a a b b b

aT T

a a a a a a aT

a aT

b b b b bb bT

b b b

N E N E

E N E

D N D D N

N E E N

E N E N

D D D x D N x D N x

D 0 p x

D 0 E N E p x

Vanishing terms


ˆ ˆ

ˆ

ˆ






normal equations

a a

b b

W 0 N 0W

0 W 0 N


normal equations

ˆ ˆT T T Ta a a b b b a a a b b b

a

b

D N D D N D 0 0 x D N x D N x

0 0 0 p 0

0 0 0 p 0


ˆ ˆ

ˆ

ˆ






normal equations

a a

b b

W 0 N 0W

0 W 0 N


normal equations

ˆ ˆ( )T T T Ta a a b b b a a a b b b D N D D N D x D N x D N x

0 0

0 0



normal equations with same weight matrix as in one-step solution

ˆ ˆ( )T T T Ta a a b b b a a a b b b D N D D N D x D N x D N x

11 13 1 11 1 13 2

22 23 2 22 2 23 3

13 23 33 33 3 13 1 33 2 23 2 33 3

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ

a a a a a a

b b b b b bT T T Ta b a b a a a a b b b b

N 0 N x N x N x

0 N N x N x N x

N N N N x N x N x N x N x

1

2

3 3

a

b

a b

u

u

u u

Same results for parameters x as in the (equivqlent to) one-step solution !




1 1 1

3 3 3

2 2 2

3 3 3

ˆ

ˆ

ˆ

ˆ

a a

a a

b b

b b

x x e

x x e

x x e

x x e

1 1 1 1

3 3 3 3

2 2 2 2

3 3 3 3

ˆ

ˆ

ˆ

ˆ

a a a

a a a

b b b

b b b

x x E p e

x x E p e

x x E p e

x x E p e





Normal equations with same weight matrices

11 13 1 11 1 13 2 1

22 23 2 22 2 23 3 2

13 23 33 33 3 13 1 33 2 23 2 33 3 3 3

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ

a a a a a a a

b b b b b b bT T T Ta b a b a a a a b b b b a b

N 0 N x N x N x u

0 N N x N x N x u

N N N N x N x N x N x N x u u

Same results for IERS parameters x !


Two non-overlapping networks connected by observations

This case applies to the ITRF formulation problem

(with error-affected connecting observations at collocation sites)


Network b

11 2

2a a a a a a a

xb A A v A x v

x

33 4

4b b b b b b b

xb A A v A x v

x

2 2 4 4c c c c b A x A x v Connecting observations

observations of network a

observations of network b

Network a


11 2

2a a a a a a a

xb A A v A x v

x

33 4

4b b b b b b b

xb A A v A x v

x

2 2 4 4c c c c b A x A x v

Network b

observations of network a

observations of network b

Connecting observations

Network a

x3

x2x1x4


11 2

2a a a a a a a

xb A A v A x v

x

33 4

4b b b b b b b

xb A A v A x v

x

2 2 4 4c c c c b A x A x v

Network bNetwork a

x3

x2x1x4

Collocation sites

2 1( , )a a v 0 P

2 1( , )b b v 0 P

2 1( , )c c v 0 P

Two connected non-overlapping networks – Separate solutions

a a a a b A x v

b b b b b A x v

aP

bP

Normal equations & separate solutions

33

44

Tbb b b

b Tbb b b

uA P bu

uA P b

a a aN x u

11

22

Taa a a

a Taa a a

uA P bu

uA P b

33 343 3 3 4

34 444 3 4 4

T Tb bb b b b b b

b TT Tb bb b b b b b

N NA P A A P AN

N NA P A A P A

11 121 1 1 2

12 222 1 2 2

T Ta aa a a a a a

a TT Ta aa a a a a a

N NA P A A P AN

N NA P A A P A

b b bN x u

weightmatrices

Ta a C x 0

Tb b C x 0

1|

2

ˆˆ ˆ

ˆaa C a

xx x

x

3|

4

ˆˆ ˆ

ˆbb C b

xx x

x

+ minimal constraints

+ minimal constraints

Separate solutions = input to: (a) combination step of two step solution(b) 2nd step of equivalent to one step solution

Two connected non-overlapping networks – One step solution

Joint treatment of observations fromnetwork anetwork b

& connecting observations

a

b

c

P 0 0

P 0 P 0

0 0 P

Normal equations


11 2

23 4

32 4

4

a a a a

b b b b

c c c c

xb A A 0 0 v

xb 0 0 A A v

xb 0 A 0 A v

x

11

2 22 2

33

4 44 4

Taa a aTT T

a c c ca a a c c cT

bb b bTT T

b c c cb b b c c c

uA P b

u A P bA P b A P bu

uA P b

u A P bA P b A P b

Weightmatrix

ˆ Nx u

11 12

12 22 2 2 2 4

33 34

4 2 34 44 4 4

a aT T Ta a c c c c c c

b bT T Tc c c b b c c c

N N 0 0

N N A P A 0 A P AN

0 0 N N

0 A P A N N A P A

1

2

3

4

ˆ

ˆˆ

ˆ

ˆ

x

xx

x

x

11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P

One step solution = Identical to separate solutions and combination with the model


1 1 1 11

2 2 2 22

3 3 3 33

4 4 4 44

2 2 4 4 2 4

ˆ

ˆ

ˆ

ˆ

c c c c c c c

x x e I 0 0 0 ex

x x e 0 I 0 0 ex

x x e 0 0 I 0 ex

x x e 0 0 0 I ex

b A x A x v 0 A 0 A v

with weight matrix

This is a two-step equivalent of the one-step solution based on the addition of the partial normal equations

11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P


1 1 1 11

2 2 2 22

3 3 3 33

4 4 4 44

2 2 4 4 2 4

ˆ

ˆ

ˆ

ˆ

c c c c c c c

x x e I 0 0 0 ex

x x e 0 I 0 0 ex

x x e 0 0 I 0 ex

x x e 0 0 0 I ex

b A x A x v 0 A 0 A v

with weight matrix

This model ignores the different reference systems (RS)

in partial and final solution


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P


1 11

2 22

1

2

3 33

4 44

2 4 2 4

2

3

2

3

4

4

1

4

ˆ

ˆ

ˆ

ˆ

c c c c c c c

e I 0 0 0 ex

e 0 I 0 0 ex

e 0 0 I 0 e

x

x

x

x

x

xe 0 0 0 I e

xb A A v 0 Ax 0 A v

x

x

x

x

with weight matrix

RSa

RSb

RSFINAL

This model ignores the different reference systems (RS)

in partial and final solution


Two connected non-overlapping networks – Two step solution

We have already treated the first step (separate solutions)

It remains to examine the second combination step

11 1 1 1 1 1

22 2 2 2 2 2

33 3 3 3 3 3

44 4 4 4 4 4

2 2 4 4 2 4

ˆ

ˆ

ˆ

ˆ

a

a

b

ba

c c c c c c cb

xx x E p e I 0 0 0 E 0 e

xx x E p e 0 I 0 0 E 0 e

xx x E p e 0 0 I 0 0 E e

xx x E p e 0 0 0 I 0 E e

pb A x A x v 0 A 0 A 0 0 v

p


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

W W 0 0 0

W W 0 0 0

0 0 W W 0

0 0 W W 0

0 0 0 0 P

weight matrix

Combination step (second step)

11 1 1 1 1 1

22 2 2 2 2 2

33 3 3 3 3 3

44 4 4 4 4 4

2 2 4 4 2 4

ˆ

ˆ

ˆ

ˆ

a

a

b

ba

c c c c c c cb

xx x E p e I 0 0 0 E 0 e

xx x E p e 0 I 0 0 E 0 e

xx x E p e 0 0 I 0 0 E e

xx x E p e 0 0 0 I 0 E e

pb A x A x v 0 A 0 A 0 0 v

p


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

W W 0 0 0

W W 0 0 0

0 0 W W 0

0 0 W W 0

0 0 0 0 P

weight matrix

This model takes into account the different reference systems (RS)in partial and final solutions by introducing transformation parameters pa, pb


11 1 1 1 1 1

22 2 2 2 2 2

33 3 3 3 3 3

44 4 4 4 4 4

2 2 4 4 2 4

ˆ

ˆ

ˆ

ˆ

a

a

b

ba

c c c c c c cb

xx x E p e I 0 0 0 E 0 e

xx x E p e 0 I 0 0 E 0 e

xx x E p e 0 0 I 0 0 E e

xx x E p e 0 0 0 I 0 E e

pb A x A x v 0 A 0 A 0 0 v

p


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

W W 0 0 0

W W 0 0 0

0 0 W W 0

0 0 W W 0

0 0 0 0 P

weight matrix

partition of matricesfor a more compact

notation


11 1 1 1 1 1

22 2 2 2 2 2

33 3 3 3 3 3

44 4 4 4 4 4

2 2 4 4 2 4

ˆ

ˆ

ˆ

ˆ

a

a

b

ba

c c c c c c cb

xx x E p e I 0 0 0 E 0 e

xx x E p e 0 I 0 0 E 0 e

xx x E p e 0 0 I 0 0 E e

xx x E p e 0 0 0 I 0 E e

pb A x A x v 0 A 0 A 0 0 v

p


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

W W 0 0 0

W W 0 0 0

0 0 W W 0

0 0 W W 0

0 0 0 0 P

weight matrix ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

a

b

c

W 0 0

0 W 0

0 0 P

model in compact notation



weight matrix

ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

a

b

c

W 0 0

0 W 0

0 0 P



weight matrix

ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

a

b

c

W 0 0

0 W 0

0 0 P

Normal equations

ˆ ˆ

ˆ

ˆ

T T T T T T T Ta a a b b b c c c a a a b b b a a a b b b c c c


D W D D W D A P A D W E D W E x D W x D W x A P b




special choice:weight matrix

same as in one step

ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

a

b

c

N 0 0

0 N 0

0 0 P

Normal equations

ˆ ˆ

ˆ

ˆ



D N D D N D A P A D N E D N E x D N x D N x A P b





ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

Normal equations

ˆ ˆ

ˆ

ˆ



D N D D N D A P A D N E D N E x D N x D N x A P b



a a N E 0Recall that b b N E 0


a

b

c

N 0 0

0 N 0

0 0 P


same as in one step


ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

Normal equations

ˆ ˆ

ˆ

ˆ

T T T T T T T Ta a a b b b c c c a b a a a b b b c c c

a

a a b bT T Ta a a a a a aT T Tb

a a

b b bb b b b b b

N E N E

E N

D N D D N D A P A D D x D N x D N x A P b

D 0E N E E N

E N

p x

E N E E ND 0 p x

a a N E 0Recall that b b N E 0

vanishing terms


a

b

c

N 0 0

0 N 0

0 0 P


same as in one step


ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

Normal equations

ˆ ˆT T T T T Ta a a b b b c c c a a a b b b c c c

a

b

D N D D N D A P A 0 0 x D N x D N x A P b

0 0 0 p 0

0 0 0 p 0


a

b

c

N 0 0

0 N 0

0 0 P


same as in one step


ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

Normal equations

ˆ ˆ ˆ( )T T T T T Ta a a b b b c c c a a a b b b c c c D N D D N D A P A x D N x D N x A P b

0 0

0 0


a

b

c

N 0 0

0 N 0

0 0 P


same as in one step


ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

Normal equations


0 0

0 0


a

b

c

N 0 0

0 N 0

0 0 P


same as in one step


No equations containingthe transformationparameters pa, pb

They cannot be determined!

ˆ

ˆa a a a a a a

b b b b b b a b

c c c b c

x x E p D E 0 x e

x x E p D 0 E p e

b b A 0 0 p b

Normal equations


11 12 11 1 12 21

12 22 2 2 2 4 12 1 22 2 22

33 34 33 3 34 43

4 2 34 44 4 4 34 3 44 4 44

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

a a a a a aT T T T Ta a c c c c c c a a a a c c c

b b b b b bT T T T Tc c c b b c c c b b b b c c c

N N 0 0 N x N xx

N N A P A 0 A P A N x N x A P bx

0 0 N N N x N xx

0 A P A N N A P A N x N x A P bx

Identical to those of the one-step solution


a

b

c

N 0 0

0 N 0

0 0 P


same as in one step


Two-step solution

combination step model


One-step solution

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

a

a

b

b

c c c c

x x E p e

x x E p e

x x E p e

x x E p e

b A x A x v

1 1 1

2 2 2

3 3 3

4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

c c c c

x x e

x x e

x x e

x x e

b A x A x v

equivalent model

same weight matrix from separate solutions

11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P

Identical solution for x1, x2, x3, x4

Two-step solution



One-step solution

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

a

a

b

b

c c c c

x x E p e

x x E p e

x x E p e

x x E p e

b A x A x v

1 1 1

2 2 2

3 3 3

4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

c c c c

x x e

x x e

x x e

x x e

b A x A x v

equivalent model


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P


“wrong” modelignores dependence of

separate and final solutions on different

reference systems

Two-step solution



One-step solution

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

a

a

b

b

c c c c

x x E p e

x x E p e

x x E p e

x x E p e

b A x A x v

1 1 1

2 2 2

3 3 3

4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

c c c c

x x e

x x e

x x e

x x e

b A x A x v

equivalent model


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P




reference systems

correct modelacknowledges dependence ofseparate and final solutions

on differentreference systems

Two-step solution



One-step solution

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

a

a

b

b

c c c c

x x E p e

x x E p e

x x E p e

x x E p e

b A x A x v

1 1 1

2 2 2

3 3 3

4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

c c c c

x x e

x x e

x x e

x x e

b A x A x v

equivalent model


11 12

12 22

33 34

34 44

a aTa a

b bTb b

c

N N 0 0 0

N N 0 0 0

0 0 N N 0

0 0 N N 0

0 0 0 0 P




reference systems

correct modelacknowledges dependence ofseparate and final solutions

on differentreference systems

Neverthelesstransformation

Parameters pa, pbcannot be determined!

Two-step solution



One-step solution

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

a

a

b

b

c c c c

x x E p e

x x E p e

x x E p e

x x E p e

b A x A x v

1 1 1

2 2 2

3 3 3

4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

c c c c

x x e

x x e

x x e

x x e

b A x A x v

equivalent model

Why these twodifferent models

lead to equivalentresults ?

Two-step solution



One-step solution

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

a

a

b

b

c c c c

x x E p e

x x E p e

x x E p e

x x E p e

b A x A x v

1 1 1

2 2 2

3 3 3

4 4 4

2 2 4 4

ˆ

ˆ

ˆ

ˆ

c c c c

x x e

x x e

x x e

x x e

b A x A x v

equivalent model

11 12 1

12 22 2

33 34 3

34 4

11 1 12 2

12 1 22 2 2 2

33 3 34 4

34 3 4 44 44 4

ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ

ˆ ˆ

ˆa a a a

T T Ta a a a c c c c c

b b b b

a a aTa a a

T Tb b b b c

b b bTb b bc c

N x N x 0

N x N x A P b A P bu

N x N x

N x N x A P b

N

N N x

N N

N x

x

N N x2

1

444

3

2T

Ta a aTa aTb b

c c c

bTb b

c

T Tc c c c c c

a

b

A P b

A0

A P b

A P b

A P b

P bP b A

P b

A

The particular choice of weight matrix “kills” the dependence on the reference systems !

Normal equations depend only on design (A) and observations (b) !

Why these twodifferent models

lead to equivalentresults ?

The choice of weight problem

In the two step approach, the first step produces singular covariance matrices.Therefore the weight matrices to be used in the second step are not uniquely defined.This fact raises some questions about the choice of weight matrices:

Are the weight matrices used in order to obtain equivalent results with theone step approach the correct ones? (Do they lead to BLUUE estimates?)



Yes !



Are there other correct weight matrices?


Yes !

Yes !



Are there other correct weight matrices?

Do different correct weight matrices lead to the same results?(i.e. to estimates that are connected by a change of the reference system)

Yes !

Yes !

Yes !


b Ax v

Rao’s Unified Theory for singular covariance matrix and rank deficient design matrix


rankn m

r m n

A2( , )v 0 Q rankn n

n

Q

Weight matrix to use: ( )T T W W Σ Q AVA

rank rank( )T Q A Q AVAV = symmetric matrix such that:

( )ΣWΣ Σ

b Ax v

Rao’s Unified Theory for singular covariance matrix and rank deficient design matrix


rankn m

r m n

A2( , )v 0 Q rankn n

n

Q

( )T T W W Σ Q AVA

V = symmetric matrix such that: rank rank( )T Q A Q AVA

a a

b b

c

D E 0

A D 0 E

A 0 0

( )ΣWΣ Σ

1

a b

a a a b

b a b b

T T Ta a a a b c

T Tb b b a

TT Tc c b

x xp xp

xp p p p

xp p p p

V V VQ 0 0 D E 0 D D A

Σ 0 Q 0 D 0 E V V V E 0 0

0 0 P A 0 0 0 E 0V V V

Our case: Combination step

1 1

rank ranka b

a a a b

b a b b

T T Ta a a a a a a b c

T Tb b b b b b a

TT Tc c c c b

x xp xp

xp p p p

xp p p p

V V VD E 0 Q 0 0 Q 0 0 D E 0 D D A

D 0 E 0 Q 0 0 Q 0 D 0 E V V V E 0 0

A 0 0 0 0 P 0 0 P A 0 0 0 E 0V V V

Different weight matrices: from different choices of V and of the generalized inverse

with V satisfying:

Weight matrix to use:

Conclusions

Under the Gauss-Markov assumptions (zero mean noise, single unknown reference variance)

• Both the one-step and the two-step approaches give equivalent results when the used weight matrices are the normal equation matrices from the separate solutions.

• The inclusion of reference frame transformation parameters is meaningless in this case.

• The combination step must be modified to the addition of partial normal equations.

Conclusions

Under the Gauss-Markov assumptions (zero mean noise, single unknown reference variance)

• Both the one-step and the two-step approaches give equivalent results when the used weight matrices are the normal equation matrices from the separate solutions.

• The inclusion of reference frame transformation parameters is meaningless in this case.

• The combination step must be modified to the addition of partial normal equations.

Beyond the Gauss-Markov assumptions (biases, different variance components)

A future study of the effect of biases and the variance component estimationin the two alternative formulations is required.

Different “correct” weight matrices in combination step

A further study of the Rao’s unified theory as applies in our specific problem.Characterization of the whole class weight matrices giving the “same” solution.

Suggestions for further study

Thanks for your attention !

A copy of this presentation can be downloaded from

http://der.topo.auth.gr/

on the alternative approaches to itrf formulation. a theoretical comparison

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