on the alternative approaches to itrf formulation. a theoretical comparison
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On the alternative approaches to ITRF formulation. A theoretical comparison. Athanasios Dermanis. Department of Geodesy and Surveying Aristotle University of Thessaloniki. The ITRF Formulation Problem. Given: - PowerPoint PPT PresentationTRANSCRIPT
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):
The ITRF Formulation Problem
, , ,
00 ,
( ) ( ), ( ) ( )
( ) , ( ) ( )
xT i k ITRF i k ITRF T k T i k
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
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):
0 , , ( )i i ITRF ktx v cITRF parameters (initial coordinates, velocities, EOPs):
The ITRF Formulation Problem
The basic stacking model:
, , ,
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
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):
( )ITRF T ktpTransformation parameters from the ITRF reference system
to the reference system of each epoch within each technique:
The ITRF Formulation Problem
The basic stacking model:
, , ,
0 0 ,
( ) ( ), ( ) ( )
(( ) )) , (
xT i k ITRF i k ITRF T k T i k
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
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):
, .( ), ( )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):
The ITRF Formulation Problem
The basic stacking model:
Simplifications of the Problem
4 non-overlapping networks
connected through cross observations
True ITRF formulation problemfor VLBI, SLR, GPS, DORIS:
Simplifications of the Problem
4 non-overlapping networks
connected through cross observations
2 non-overlapping networks
connected through cross observations
Simplifications of the Problem
4 non-overlapping networks
connected through cross observations
2 non-overlapping networks
connected through cross observations
2 overlapping networks
Simplifications of the Problem
4 non-overlapping networks
connected through cross observations
2 non-overlapping networks
connected through cross observations
2 overlapping networks
2 identical networks
Simplifications of the Problem
4 non-overlapping networks
connected through cross observations
2 non-overlapping networks
connected through cross observations
2 overlapping networks
2 identical networks Despite the simplificationsthe fundamental problemcharacteristics are preserved
The simplified cases deservea study in their own
Simplifications of the Problem
4 non-overlapping networks
connected through cross observations
2 non-overlapping networks
connected through cross observations
2 overlapping networks
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
(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 – 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
(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 – 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
same
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 – 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
Difference only in
second steps
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 – 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
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
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
x x Ep p = transformation parameters (rotations, displacement, scale)
Variation of parameters under change of reference system
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
Invariance of observables y = Ax and estimable parameters (functions of y)
x x Ep p = transformation parameters (rotations, displacement, scale)
T AE 0y Ax Ax Ax AEp A P E NEA 0
Variation of parameters under change of reference system
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
T E x 0
Invariance of observables y = Ax and estimable parameters (functions of y)
x x Ep p = transformation parameters (rotations, displacement, scale)
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
Two identical networks
“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 ˆ( )a b a a b b N N x N x N x
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
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 !
Two identical networks
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
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
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 ˆ( )a b a a b b N N x N x N x
Two identical networks
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
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
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 ˆ( )a b a a b b N N x N x N x
Recall that
a a
b b
A E 0 N E 0
A E 0 N E 0
Two identical networks
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
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
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
Normal equations with same weight matricesWeight matrices Na, Nb Weight matrices Na, Nb
ˆ ˆ( )a b a a b b N N x N x N x
Vanishing terms
Two identical networks
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
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 ˆa 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
Normal equations with same weight matricesWeight matrices Na, Nb Weight matrices Na, Nb
ˆ ˆ( )a b a a b b N N x N x N x
Two identical networks
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
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
Normal equations with same weight matricesWeight matrices Na, Nb Weight matrices Na, Nb
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 identical networks
Two overlapping networks
This case would apply to the ITRF formulation problem
if perfect connections were available at collocation sites
Two overlapping networks
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
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
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
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 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
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
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
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 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
Recall that ,a a b b N E 0 N E 0
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
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
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
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
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 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
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
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( )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
Transformation parameters pa, pb undetermined !
Two overlapping networks – Two step solution
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 !
Two overlapping networks – Two step solution
Two overlapping networks
Two step solution(equivalent 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
“wrong” model !Ignores that partial and final solutions
are in different reference systems
correct model !Treats partial and final solutions
in different reference systems
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 !
Transformation parameters pa, pb undetermined !
Two non-overlapping networks connected by observations
This case applies to the ITRF formulation problem
(with error-affected connecting observations at collocation sites)
Two non-overlapping networks connected by observations
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
Two non-overlapping networks connected by observations
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
Two non-overlapping networks connected by observations
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
Two connected non-overlapping networks – One step solution
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
Two connected non-overlapping networks – One step solution
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
Two connected non-overlapping networks – One step solution
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
One step solution = Identical to separate solutions and combination with the 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
Two connected non-overlapping networks – One step solution
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
One step solution = Identical to separate solutions and combination with the model
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
Two connected non-overlapping networks – Two step solution
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
Two connected non-overlapping networks – Two step solution
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
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
Two connected non-overlapping networks – Two step solution
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
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
Two connected non-overlapping networks – Two step solution
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
Combination step (second step)
Two connected non-overlapping networks – Two step solution
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
Combination step (second step)
Two connected non-overlapping networks – Two step solution
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
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 A P A D W E D W E x D W x D W x A P b
E W D E W E 0 p E W x
E W D 0 E W E p E W x
Combination step (second step)
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
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
T T Ta a a a a a a a a aT T Tb b b b b b b b b b
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
E N D E N E 0 p E N x
E N D 0 E N E p E N x
Two connected non-overlapping networks – Two step solution
Combination step (second 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 a a b b b a a a b b b c c c
T T Ta a a a a a a a a aT T Tb b b b b b b b b b
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
E N D E N E 0 p E N x
E N D 0 E N E p E N x
a a N E 0Recall that b b N E 0
Two connected non-overlapping networks – Two step solution
a
b
c
N 0 0
0 N 0
0 0 P
special choice:weight matrix
same as in one step
Combination step (second 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
Two connected non-overlapping networks – Two step solution
a
b
c
N 0 0
0 N 0
0 0 P
special choice:weight matrix
same as in one step
Combination step (second 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
Two connected non-overlapping networks – Two step solution
a
b
c
N 0 0
0 N 0
0 0 P
special choice:weight matrix
same as in one step
Combination step (second 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
Two connected non-overlapping networks – Two step solution
a
b
c
N 0 0
0 N 0
0 0 P
special choice:weight matrix
same as in one step
Combination step (second 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
Two connected non-overlapping networks – Two step solution
a
b
c
N 0 0
0 N 0
0 0 P
special choice:weight matrix
same as in one step
Combination step (second 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
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
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
Two connected non-overlapping networks – Two step solution
a
b
c
N 0 0
0 N 0
0 0 P
special choice:weight matrix
same as in one step
Combination step (second step)
Two-step solution
combination step model
Two non-overlapping networks connected by observations
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
combination step model
Two non-overlapping networks connected by observations
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
“wrong” modelignores dependence of
separate and final solutions on different
reference systems
Two-step solution
combination step model
Two non-overlapping networks connected by observations
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
“wrong” modelignores dependence of
separate and final solutions on different
reference systems
correct modelacknowledges dependence ofseparate and final solutions
on differentreference systems
Two-step solution
combination step model
Two non-overlapping networks connected by observations
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
“wrong” modelignores dependence of
separate and final solutions on different
reference systems
correct modelacknowledges dependence ofseparate and final solutions
on differentreference systems
Neverthelesstransformation
Parameters pa, pbcannot be determined!
Two-step solution
combination step model
Two non-overlapping networks connected by observations
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
combination step model
Two non-overlapping networks connected by observations
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?)
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:
Yes !
Are the weight matrices used in order to obtain equivalent results with theone step approach the correct ones? (Do they lead to BLUUE estimates?)
The choice of weight problem
Are there other correct weight matrices?
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:
Yes !
Yes !
Are the weight matrices used in order to obtain equivalent results with theone step approach the correct ones? (Do they lead to BLUUE estimates?)
The choice of weight problem
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 !
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:
b Ax v
Rao’s Unified Theory for singular covariance matrix and rank deficient design matrix
The choice of weight problem
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
The choice of weight problem
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/