Geoscience Reference
In-Depth Information
Remark on misclosures
All misclosures in any acceptable system of heights denoted for the moment
by
h
(not to be confused with ellipsoidal heights) must be zero:
dh
= 0
(4-70)
for any closed path. Height networks consisting of triangles, if computed by
least-squares adjustment, thus must satisfy the condition that the sum of
height differences must be zero for each triangle. Mathematically, this can
be shown to be equivalent to the commutativity of second derivatives:
∂
2
h
∂x ∂y
=
∂
2
h
∂y ∂x
.
(4-71)
4.6
GPS leveling
Spirit leveling (Fig. 4.6) is a very time-consuming operation. GPS has intro-
duced a revolution also here. The basic equation is
H
=
h
−
N.
(4-72)
This equation relates the orthometric height
H
(above the geoid), the el-
lipsoidal height
h
(above the ellipsoid), and the geoidal undulation
N
.If
any two of these quantities are measured, then the third quantity can be
computed.
If
h
is measured by GPS, and if there exists a reliable digital geoid map
of
N
, then the orthometric height
H
can be obtained immediately.
Equation (4-72) can also be used for geoid determination: if
h
is measured
by GPS, and
H
is available from leveling, then the geoid
N
can be determined
as
N
=
h
H
. The same principle can be applied even on the oceans as
satellite altimetry
, as we will see later in Chap. 7, e.g., Eq. (7-47).
GPS leveling implies replacing to some extent the classical leveling by
GPS. Referring to Fig. 4.6 and applying (4-72) to
A
and
B
leads to
−
H
A
=
h
A
−
N
A
,
H
B
=
h
B
− N
B
,
(4-73)
and the height difference
H
B
−
H
A
=
h
B
−
h
A
−
N
B
+
N
A
.
(4-74)
Introducing the notations
δH
AB
=
H
B
−
H
A
,
δh
AB
=
h
B
−
h
A
,and
δN
AB
=
N
B
− N
A
, the relation reduces to
δH
AB
=
δh
AB
−
δN
AB
.
(4-75)
