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21-3 Error Propagation in Analysis and Synthesis of a Datum
Problem
Nonlinear error propagation. Dispersion, dispersion transformation, dispersion matrix,
variance-covariance matrix. Stochastic pseudo-observations.
pseudo-observations
by means of a best approximation (
21.46
) subject to (
21.47
), via
nonlinear error propagation
.we
have to consider the
variance-covariance matrix/dispersion matrix
of the datum parameters
x
=
[
t
x
,t
y
,t
z
,α,β,γ,s
] functionally related to the variance-covariance matrix/dispersion matrix of the
pseudo-observations
{l,b}
and
{L, B, H}
. The stochastic pseudo-observations
{l,b}
and
{L, B, H}
enter via the relative data vector (
21.48
) and via the analysis matrix (
21.49
). Accordingly, we
expand
x
(
l,b,L,B,H
) into the dispersion (
21.50
) as outlined in Box
21.9
.Ifa
prior dispersion
matrix
of the form parameters
First, in the
analysis
of the datum parameters from given
{
l,b
}
and
{
L, B, H
}
A
1
,E
2
,a
1
,e
2
is available, it could also be implemented.
Second, the
synthesis
of global ellipsoidal coordinates
{
}
{
L, B
}
from given local ellipsoidal coor-
t
x
,t
y
,t
z
,α,β,γ,s,δa,δe
2
dinates
{
l,b
}
and datum parameters/ellipsoidal form parameters
{
}
,we
t
x
, ..., δe
2
again experience the impact of the dispersion matrix of
{
l,b
}
as well as of
{
}
via nonlinear error propagation. The random character of the pseudo-observations
enters
firstly linearly and secondly nonlinearly via B
0
(
l,b
), while the stochastic
a posteriori
parameters
{
{
l,b
}
enter linearly. Box
21.10
reviews the expansion [
L, B
](
l,b, t
x
,...,δe
2
)towardsthe
dispersion (
21.51
).
t
x
,...,δe
2
}
x
=(A
T
PA)
−
1
A
T
P
y
:= L
y,
(21.46)
L=(A
T
PA)
−
1
A
T
P
,
(21.47)
B
]
T
,
y
:= [
l
−
L, b
−
(21.48)
A=A(
L, B, H
)
,
(21.49)
D(
x
)=M
k
D
kl
(
l,b,L,B,H
)M
l
,
(21.50)
D(
L, B
)=K
μ
D
μν
(
l,b, t
x
,...,δe
2
)K
ν
.
(21.51)
Box 21.9 (Error propagation with respect to analysis
x
(
l,b,L,B,H
)).
d
x
=dL
y
+Ld
y.
(21.52)
dL =
(A
T
A)dA
T
A(A
T
A)
−
1
A
T
+(A
T
A)
−
1
dA
T
(A
T
A)
−
1
A
T
dA(A
T
A)
−
1
A
T
=
=
−
−
(21.53)
NdA
T
AL
LdAL + N
−
1
dA
T
subject to
N:=A
T
A
,
P=I
.
=
−
−
(21.54)
NdA
T
A
x
LdA
x
+N
−
1
dA
T
y
+Ld
y
,
d
x
=
−
−
(A
x
)
T
N] vec dA
T
[(
x
)
T
L] vec dA + [(
y
)
T
N
−
1
]
vec d
x
=[
−
⊗
−
⊗
⊗
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