Geoscience Reference
In-Depth Information
B
±n
±
H
B
A
H
B
H
A
geoid
Fig. 4.2. Leveling and orthometric height
where
g
is the gravity at the leveling station and
g
is the gravity on the
plumb line of
B
at
δH
B
. Hence,
g
g
δH
B
=
δn
=
δn .
(4-3)
There is, thus, no direct geometrical relation between the result of lev-
eling and the orthometric height, since (4-3) expresses a physical relation.
What, then, if not height, is directly obtained by leveling? If gravity
g
is also
measured, then
δW
=
−
gδn
(4-4)
is determined, so that we obtain
B
W
B
−
W
A
=
−
gδn.
(4-5)
A
Thus, leveling combined with gravity measurements furnishes
potential dif-
ferences
, that is, physical quantities.
It is somewhat more rigorous theoretically to replace the sum in (4-5)
by an integral, obtaining
B
W
B
−
W
A
=
−
gdn.
(4-6)
A
Note that this integral is independent of the path of integration; that is,
different leveling lines connecting the points
A
and
B
(Fig. 4.3) should give
the same result. This is evident because
W
is a function of position only;
therefore, to every point there corresponds a unique value
W
. If the leveling
line returns to
A
, then the total integral must be zero:
gdn
=
−W
A
+
W
A
=0
.
(4-7)
