Geoscience Reference
In-Depth Information
5.7.3
Three-dimensional transformation between WGS 84
and a local system
In the case of a datum transformation between WGS 84 and a local sys-
tem, some simplifications will arise. Referring to the necessary approximate
values, the approximation
µ
approx
= 1 is appropriate and the relation
µ
=
µ
approx
+
δµ
=1+
δµ
(5-45)
is obtained. Furthermore, the rotation angles
ε
i
in (5-44) are small and may
be treated as differential quantities. Introducing these quantities into (5-44),
setting cos
ε
i
=1andsin
ε
i
=
ε
i
, and considering only first-order terms gives
⎡
⎤
1
ε
3
−
ε
2
⎣
⎦
=
I
+
δ
R
,
R
=
−ε
3
1
ε
1
(5-46)
ε
2
−
ε
1
1
where
I
is the unit matrix and
δ
R
is a (skewsymmetric) differential rotation
matrix. Thus, the approximation
R
approx
=
I
is appropriate. Finally, the
shift vector is split up in the form
x
0
=
x
0approx
+
δ
x
0
,
(5-47)
where the approximate shift vector
x
0approx
=
X
T
−
X
(5-48)
follows by substituting the approximations for the scale factor and the rota-
tion matrix into Eq. (5-41).
Introducing Eqs. (5-45), (5-46), (5-47) into (5-41) and skipping de-
tails which can be found, for example, in Hofmann-Wellenhof et al. (1994:
Sect. 3.3) gives the linearized model for a single point
i
. This model can be
writtenintheform
X
T
i
−
X
i
−
x
0approx
=
A
i
δ
p
,
(5-49)
where the left side of the equation is known and may formally be considered
as an observation. The design matrix
A
i
and the vector
δ
p
, containing the
unknown parameters, are given by
⎡
⎤
100
X
i
0
−Z
i
Y
i
⎣
⎦
,
A
i
=
010
Y
i
Z
i
0
−
X
i
(5-50)
001
Z
i
−
Y
i
X
i
0
δ
p
=[
δx
0
δy
0
δz
0
δµ
ε
1
ε
2
ε
3
]
.
