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⎡
⎤
J
2
:=
∂
(
x, y, z
)
∂
(
l,b,h
)
=
∂x/∂l ∂x/∂b ∂x/∂h
∂y/∂l ∂y/∂b ∂y/∂h
∂z/∂l ∂z/∂b ∂z/∂h
⎣
⎦
,
J
2
:= (J
2
)
−
1
=
∂
(
l,b,h
)
∂
(
x, y, z
)
,
(21.26)
⎡
⎤
−
(
n
+
h
)cos
b
sin
l −
(
m
+
h
)sin
b
cos
l
cos
b
cos
sl
(
n
+
h
)cos
b
cos
l
⎣
⎦
,
J
2
=
−
(
m
+
h
)sin
b
sin
l
cos
b
sin
l
0
(
m
+
h
)cos
b
sin
b
⎡
⎤
sin
l
(
n
+
h
)cos
b
cos
l
(
n
+
h
)cos
b
−
0
⎣
⎦
,
cos
b
m
+
h
cos
b
cos
l
cos
b
sin
l
sin
b
cos
l
sin
b
m
+
h
sin
l
sin
b
m
+
h
J
2
=
−
−
(21.27)
J
23
:=
∂
(
l,b
)
=
−
sin
L
cos
L
(
N
+
H
)cos
B
(
N
+
H
)cos
B
+
0
2
×
3
,
∈
R
cos
L
sin
B
M
+
H
sin
L
sin
B
M
+
H
cos
B
M
+
H
∂
(
x, y, z
)
−
−
taylor
J
1
:=
∂
(
x, y, z
)
∂
(
t
x
,t
y
,t
z
,α,β,γ,s,A
1
,E
2
)
=
(21.28)
taylor
⎡
⎤
M
cos
B
sin
2
B
cos
L
2(1
−E
2
)
N
cos
B
cos
L
A
1
−ZY
X
Y
Z
100
010
001
0
⎣
⎦
∈
R
M
cos
B
sin
2
B
sin
L
2(1
−E
2
)
N
cos
B
sin
L
A
1
3
×
9
.
=
X
−YX
0
Z
0
−
N
sin
B
(1
−
E
2
)
A
1
M
sin
3
B
−
2
N
sin
B
2
Various remarks with respect to the rank and the stability of the Jacobi matrix A have to be
made. First, the Jacobi matrix A indicates that columns seven and eight, namely
a
7
and
a
8
,are
linearly dependent, in particular, A
a
8
=
a
7
. Obviously, the incremental scale
s
and the incre-
mental semi-major axis
δA
cannot be determined independently. Since we cannot discriminate
s
and
δA
, we may consult data files of the global and local reference ellipsoid in order to fix the
values
δA
:=
A
a
as well as
δE
2
:=
E
2
e
2
and remove by
b
(
a
28
δA
+
a
29
δE
2
)the
−
−
−
B
−
δA,δE
2
quantities
{
}
from the analysis. Note that only the differences in latitude are influenced
s, δA, δE
2
by
{
}
, respectively. Second, for a geodetic network for which both
{
l,b
}
and
{
L, B, H
}
are available, the extension in latitude
may lead to an instability within the Jacobi matrix
A as we have experienced. In the analysis of column
a
3
(acting on translation component
t
z
)
and column
a
7
(acting on scale
s
), sin
B
is approximately discriminating the two columns. In
the above-quoted network configuration,
a
3
and
a
7
are nearly linearly dependent. The following
rationale has accordingly been chosen, following the method of partial least squares (
Young 1994
),
for instance.
{
b, B
}
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