Geoscience Reference
In-Depth Information
Ellipsoidal height differences
The “measured” ellipsoidal height difference is represented by
h
ij
=
h
j
−
h
i
.
(5-81)
The heights involved are obtained by transforming the Cartesian coordinates
into ellipsoidal coordinates according to (5-36) or by using the iterative pro-
cedure given in Sect. 5.6.1. The height difference is approximately (neglecting
the curvature of the earth) given by the third component of
x
ij
in the local
level system. Hence,
h
ij
=
u
i
·
X
ij
(5-82)
or, by substituting
u
i
according to (5-64), the relation
h
ij
=cosΦ
i
cos Λ
i
X
ij
+cosΦ
i
sin Λ
i
Y
ij
+sinΦ
i
Z
ij
(5-83)
is obtained. This equation may be differentiated with respect to the Carte-
sian coordinates. If the differentials are replaced by the corresponding dif-
ferences,
δh
ij
=cosΦ
j
cos Λ
j
δX
j
+cosΦ
j
sin Λ
j
δY
j
+sinΦ
j
δZ
j
(5-84)
−
cos Φ
i
cos Λ
i
δX
i
−
cos Φ
i
sin Λ
i
δY
i
−
sin Φ
i
δZ
i
is obtained, where the coordinate differences were decomposed into their
individual coordinates.
Baselines
From relative GPS measurements, baselines
X
ij
(GPS)
=
X
j
(GPS)
−
X
i
(GPS)
in
the WGS 84 are obtained. The position vectors
X
i
(GPS)
and
X
j
(GPS)
may be
transformed by a three-dimensional (7-parameter) similarity transformation
to a local system indicated by LS. According to Eq. (5-41), the transforma-
tion formula reads
X
LS
=
x
0
+
µ
RX
GPS
,
(5-85)
where the meaning of the individual quantities is the following:
X
LS
...
position vector in the local system
,
X
GPS
...
position vector in the WGS 84
,
x
0
...
shift vector
,
R
...
rotation matrix
,
µ
...
scale factor
.
Forming the difference of two position vectors, i.e., the baseline
X
ij
,the
shift vector
x
0
is eliminated. Using (5-85), there results
X
ij
(LS)
=
µ
RX
ij
(GPS)
(5-86)
