Geoscience Reference
In-Depth Information
for the baseline. Similar to (5-49), the linearized form is
X
ij
(LS)
=
X
ij
(GPS)
+
A
ij
δ
p
,
(5-87)
where now the vector
δ
p
and the design matrix
A
ij
are given by
ε
3
]
T
,
δ
p
=
δµ
ε
1
ε
2
⎡
⎣
⎤
⎦
X
ij
0
−
Z
ij
Y
ij
(5-88)
A
ij
=
Y
ij
Z
ij
0
−
X
ij
.
Z
ij
−
Y
ij
X
ij
0
(GPS)
Note that the rotations
ε
i
refer to the axes of the system used in GPS. If
they should refer to the local system, then the signs of the rotations must be
changed, i.e., the signs of the elements of the last three columns of matrix
A
ij
must be reversed.
The vector
X
ij
(LS)
on the left side of (5-87) contains the points
X
i
(LS)
and
X
j
(LS)
in the local system. If these points are unknown, then they are
replaced by known approximate values and unknown increments
X
i
(LS)
=
X
i
0
(LS)
+
δ
X
i
(LS)
,
X
j
(LS)
=
X
j
0
(LS)
+
δ
X
j
(LS)
,
(5-89)
where the coecients of these unknown increments (+1 or
−
1) together with
matrix
A
ij
form the design matrix.
The vector
X
ij
(GPS)
in (5-87) is regarded as measurement quantity. Thus,
finally,
X
ij
(GPS)
=
δ
X
j
(LS)
− δ
X
i
(LS)
−
A
ij
δ
p
+
X
j
0
(LS)
−
X
i
0
(LS)
(5-90)
is the linearized observation equation.
In principle, any type of geodetic measurement can be employed if the
integrated geodesy adjustment model is used. The basic concept is that any
geodetic measurement can be expressed as a function of one or more posi-
tion vectors
X
and of the gravity field
W
of the earth. The usually non-
linear function must be linearized where the gravity field
W
is split into
the normal potential
U
of an ellipsoid and the disturbing potential
T
,thus,
W
=
U
+
T
. Applying a minimum principle leads to the collocation formulas
(Moritz 1980 a: Chap. 11).
Many examples integrating GPS and other data can be found in technical
publications. For example, there are attempts to detect earth deformations
from GPS and terrestrial data.
